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Gas-phase ion-molecule chemistry of chromium nitrosyl complex CpCr(NO)₂CH₃ and coulomb interaction between… Chen, Shu-Ping 1992

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GAS-PHASE ION-MOLECULE CHEMISTRY OF H 3 C 2 CpCr(NO) CHROMIUM NITROSYL COMPLEX , AND COULOMB INTERACTION BETWEEN IONS IN FOURIER TRANSFORM ION CYCLOTRON RESONANCE MASS SPECTROMETRY by  SHU-PING CHEN B. Sc., Xiamen University, 1977 M. Sc., Xiamen University, 1982 A THESIS SUBMflTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemistry We accept this thesis as conforming to  the reonired standard  THE UNiVERSITY OF BRiTISH COLUMBIA February, 1992 © Shu-Ping Chen, 1992  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives, It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  Chemistry  The University of British Columbia Vancouver, Canada Date  DE-6 (2)88)  April, 1992  —“  ABSTRACT This work is devoted to application and performance modifications of Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR): (1) gas-phase chemistry of chromium nitrosyl; (2) Coulomb interactions between ions in ion cyclotron motion. Chromium nitrosyl CpCr(NO) 3 C 2 H (Cp  =  ) produces a series of ions which H 5 ?7-C  has been observed to fifth kinetic order. The ions of 3 CH show many products 2 CpCr(NO) in which the oxygen of the NO ligand is retained and tl nitrogen is lost as part of a neutral product. An empirical method was proposed for calibrating nominal pressures of transition metal complexes to determine real rate constants. The Cr ions in an excited state can be quenched in collision with molecules N ,H 2 0, 3 2 NH and CH , . The ground state Cr 4 ,H 2 ions prefer charge transfer reactions which result in different products from those of the , and 4 3 CH also can condensation reactions of the excited state Cr ions. H ,H 2 O, NH 2 CH to produce the ammine ligand. 2 CpCr(NO) react with the nitrosyl ligand in 3 A point model and a line model, which correspond more closely to physical reality than some prior models, are proposed to account for the Coulomb-induced frequency shifts observed in FT-ICR. The first model consists of two point charges which undergo cyclotron orbits with the same orbit centers at their respective frequencies. The model predicts that each excited cyclotron motion should induce a negative frequency shift in the other’s cyclotron motion. The line model, created by extension of the point model, gives rise to a position-dependent frequency shift which is synonymous with inhomogeneous Coulomb broadening. A disk model for the Coulomb shifting, unlike the point model, has a finite average radial Coulomb force. It consists of a uniformly charged disk, whose excited cyclotron motion is perturbed by a second excited, uniformly charged disk. The average radial force is found to be a function of ratio of the cyclotron radius to the disk radius. This allows characterization in terms of an “apparent Coulomb distance”. This distance, when applied  -ifi  in a charged-cylinder model, accounts for Coulomb-induced line broadening and frequency shifting. The charged-cylinder model agrees with experiments. Absolute mass calibration of FT-ICR spectra is enhanced by the Coulomb correction. The charged-point and charged-disk models are valid when the Coulomb interaction is much smaller than the Lorentz force. When the Coulomb force is comparable to the Lorentz force, a strong coupling interaction arises. A strong coupling Coulomb interaction  for small spatial separations between two ion species is developed using a Taylor’s expansion method based on two tetragonal ion clouds. Under strong coupling, the twoion mass peaks will merge.  —iv—  TABLE OF CONTENTS  ii  Abstract List of Tables  xii  List of Figures  xiv xviii  List of Abbreviations Acknowledgements Chapter 1 1.1  General  XX  Introduction  A brief history of ion cyclotron resonance mass spectrometry  1 2  1.2 The background of Fourier transform ion cyclotron resonance mass spectrometry  1.3  5  1.2.1 Basic operating principle of FT-ICR  5  1.2.2 Fourier transform of a time signal  8  Ion-molecule reactions and FT-ICR multiple resonance  12  1.4 Performance of FT-ICR  13  References  16  Chapter 2  Ion-Molecule Condensation Chemistry of -2,4-Cyclopentadien-1-yl)methyldinitrosylchromium 5 (7 CH 2 CpCr(NO) 3  2.1  Introduction  2.2 Experimental section  2.3  21 22 25  2.2.1 Experimental hardware  25  2.2.2 Multiple resonance technique  26  2.2.3 Pressure calibration  27  Condensation chemistry of the positive ions 2.3.1 Cr reactions  32 37  —v—  2.3.1.1 Charge transfer and electrophilic additions of Cr  37  O reactions 2 2.3.1.2 CpCr  39  O reactions 2 2.3.1.3 CpCr  41  2.3.2 CpCr reactions  42  2.3.2.1 Charge transfer and electrophilic additions of CpCr  42  D r C 2 Cp ,O Cr 2 D 4 H 5 CpC ,O Cr 2 Cp 2.3.2.2 O reactions O r reactions C 2 Cp 2.3.2.3 N 2.3.3 CpCrNO reactions  43 43 43  D CpCr(NO) C 2.3.4 CpCrNOCD, CpCr(NO), and 2 reactions D r reactions 2 C 4 H 5 C 2.3.5 C  45  2.3.6 CpCrO reactions  45  2.4 Ion-molecule chemistry of the negative ions 2.5  44  Rate constants  46 49  2.5.1 Rate constants of the positive ions  49  2.5.2 Rate constants of the negative ions  52  2.6 Proposed ion structures 2.6.1  56  Cluster ions from the Cr-molecule condensation reactions  56  2.6.2 Cluster ions from the CpCr-mo1ecule condensation reactions  58  2.6.3 Other bimetallic cluster ions  58  2.6.4 The negative ions  61  2.7 Discussion  62  —  vi  —  65  References Chapter 3  Reactions of the Fragment Ions of 3 CH 2 CpCr(NO) , and 3 with Small Molecules N ,H 2 ,H 2 0, D 2 0, NH 2 4 in the Gas-Phase CH  3.1  Introduction  3.2 Experimental section 3.3  Calibrations of the nominal pressures  2 3.4 Ion-molecule chemistry of 3 CH with hydrogen, H 2 CpCr(NO) 3.4.1  Collisional quenching and charge-transfer reactions  3.4.2 Ligand substitutions  3.5  72 73 74 75 76 76 82  Ion-molecule chemistry of 3 CH with water and 2 CpCr(NO) 0 2 heavy water, H 0 and D 2  84  CH with ammonia, NH 2 CpCr(NO) 3 3.6 Ion-molecule chemistry of 3  85  3.7  CH with methane, CH 2 CpCr(NO) 4 Ion-molecule chemistry of 3  86  3.8  CH with nitrogen, N 2 CpCr(NO) Ion-molecule chemistry of 3 2  86  3.9 Discussion  87  3.9.1 The electron transfer reaction  87  3.9,2 The condensation reactions  89  3.9.3 Further work  91  References Chapter 4  93  Simple Physical Point and Line Charge Models for Coulomb-Induced Frequency Shift and Inhomogeneous Broadening in FT-ICR Mass Spectrometry  4.1  Space charge effects in FT-ICR mass spectrometry 4.1.1  Space charge effects and mass measurement  4.1.2 Prior research on the space charge effects  95 96 96 97  -vi’-  4.2 The point charge model  —  Coulomb shifting of unlike ions  99  4.3 The line charge model—Coulomb shifting and broadening of unlike ions  104  4.4 Discussion  107  4.4.1 Choice of Model  107  4.4.2 Inhomogeneous broadening in FT-ICR  113  4.4.3 Coulomb shifting and broadening of like ions  114  4.4.4 ICR cell design and Coulomb effects  114  References  Chapter 5  116  Simple Charge-Disk Model and Simple Charged-Cylinder  Model for Coulomb Shifting and Coulomb Broadening in FT-ICR Mass Spectrometry 5.1  Introduction  119 120  5.2 The charge-disk model 5.2.1 Two charged disks in a rotating laboratory-frame  122 122  5.2.2 Radial Coulomb force on Disk 1 for non-overlapping disks (‘I’  > touch)  125  5.2.3 Radial Coulomb force on Disk 1 overlapping disks (‘  129  <  5.2.4 Instantaneous radial Coulomb force  135  5.2.5 Average radial Coulomb force  140  5.2.6 Apparent Coulomb distance, D  140  5.2.7 Validity of the charged-disk model  143  5.2.7.1 Assumption of small like-ion interactions  143  5.2.7.2 Assumption of small unlike-ion interactions  147  —  viii—  5.2.7.3 Assumptionofhighfrequency perturbations  5.3 The charged-cylinder model  148 150  5.3.1 Apparent Coulomb distance and charged-cylinder 150  model  5.3.2 Validity of the charged-cylinder model 5.2.7.1 Assumption of small like-ion interactions  152 152  5.3.2.2 Assumption of small unlike-ion interactions  155  5.3.2.3 Assumption of high frequency perturbations  156  5.4 The Coulomb-induced frequency shift and inhomogeneous broadening of the charged-cylinder model  157  5.5 Experimental tests of the charged-cylinder model  159  5.5.1 The unlike-ion Coulomb-induced frequency shifting from experiment of Francl et al  159  5.5.2 Unlike-ion Coulomb-induced frequency shifting in FT-ICR  162  5.5.3 Experimental inhomogeneous broadening in FT-ICR  168  5.6 Discussion  170  References  174  Chapter 6  Taylor’s Expansion Approximation of Ion-Ion  Strong Coupling Coulomb Interaction in FT-ICR Mass Spectrometry 6.1  Introduction  176 177  6.2 Potential energy of a single tetragonal ion cloud  181  6.3 Potential energy of two tetragonal ion clouds  184  —  ix  —  6.3.1 Taylor’s expansion approximation of the Coulomb potential energy  184  6.3.2 Integration of the Coulomb potential energy  187  6.4 Motion equations and solutions of two tetragonal ion clouds  191  6.5  193  Strong coupling Coulomb interaction between two ion clouds  6.5.1 Strong coupling condition  193  6.5.2 Two oscillations under the strong coupling condition  194  6.5.3 Strong coupling critical curve  195  6.5.4 Strong coupling condition for N 1  N  6.6 Weak coupling Coulomb interaction  197 200  6.6.1 Weak coupling condition  200  6.6.2 Validity of the weak coupling condition  201  6.7  Electric trapping potential and frequency shifts  203  6.8  Discussion  208  6.8.1 The charge-cylinder model and Taylor’s expansion approximation  208  6.8.2 The electric quadrupole trap and analyzer cell of FT-ICR  209  6.8.3 Intermediate coupling condition  210  6.8.4 Ion distribution functions  212  6.8.5 Further work  214  References  216  Appendixes  218  Al  219  Appendixes of Chapter 2 A1.1  The “twenty-second” ion-molecule reaction mass spectra  219  —x—  A 1.2  A1.3 A2  Coordination modes of metal carbonyls and metal nitrosyls  220  Known oxo chromium complexes  224  Appendixes of Chapter 3 Kinetic behavior of the Cr produced from 3 CH 2 CpCr(NO)  A3  in H 0, NH 2 , CH 3 , and N 4 2 media  228  Appendixes of Chapter 5  230  A3.1  The FORTRAN program for calculating ion-ion Coulomb interaction between two charged disks based on double Gaussian numerical computation  A3.2  230  The FORTRAN programs for calculating ion-ion Coulomb interaction between two charged disks based on double Romberg numerical computation A3.2.1  Program “DISK.ROMB” for two charged disks without overlapping  A3.2. 1  234  Program “OVER.ROMB” for two charged disks with overlapping  A3.3  234  236  The results from the double Romberg numerical computation A3.3. 1  240  Average radial Coulomb force for =  1 mm and r  =  1 cm  240  A3.3.2 Average radial Coulomb force for =  A3.3.3  =  1 cm  241  Average radial Coulomb force for r’  A3.4  0.5 mm and r  =  0.1 mm and r  =  1 cm  244  Curve fitting for “apparent Coulomb distance” vs. r/r’  246  —  A4  xi  —  248  Appendixes of Chapter 6 A4. 1  A4.2  Four particular integrals applied in the Taylor’s expansion approximation of ion-ion Coulomb interaction  248  Integrations of the Coulomb interaction coefficient  252  A4.2. 1  Integration for  A4.2.2 Integration for z 1 A4.2.3 Integration fory’ 1 A4.2.4  =  =  2 z  y  253 256  Value of the Coulomb interaction coefficient, C  References  252  =  266 269  -xl’-  LIST OF TABLES  Table 1.1  Summary of FT-ICR line shape formulas of a time signal  Table 1.2  Comparison of typical mass spectrometer performance  11  parameters  15  Table 2.1  The important electric and Fourier transform parameters  26  Table 2.2  A comparison between literary data of average a values and theoretical calculations of a for some transition metal complexes  33  Table 2.3  The primary positive ions from 3 CD 2 CpCr(NO)  34  Table 2.4  CH 2 CpCr(NO) The identification for the ions from 3  38  Table 2.5  CD 2 CpCr(NO) The primary negative ions from 3  46  Table 2.6  The experimental rate constants of the primary positive ions from 3 CD 2 CpCr(NO)  Table 2.7  The experimental rate constants of the positive condensation CH 2 CpCr(NO) ions from 3  Table 2.8  55  A summary of carbonyls and nitrosyls in ion-molecule condensation chemistry of transition metal complexes  Table 3.1  55  The experimental rate constants of the primary negative ions from 3 CH 2 CpCr(NO)  Table 2.11  54  The experimental rate constants of the primary negative ions from 3 CD 2 CpCr(NO)  Table 2.10  53  The experimental rate constants of the positive condensation ions from 3 CD 2 CpCr(NO)  Table 2.9  51  64  0, NH 2 , 3 Calibrations of the sampling pressures of N , 1I2, H 2 CH 2 CpCr(NO) , and 3 4 CH  76  —  Table 3.2  xlii—  , 2 Rate constants of CpCrNOCH and CpCr(NO) in H , CH 3 , and N 4 0, NH 2 H 2 media  Table 5.1  89  Apparent Coulomb distance for Coulomb shifting and Coulomb broadening in FT-ICR  Table 5.2  The validities of Eqs. (5.70b), (5.71b), and (5.73b) for 1 Tesla, r  B  =  =  1 electron charge, m 1  and D Table 5.3  142  =  =  1 cm, 1  =  =  2.4 cm, N 1  =  2 N  =  iO, q 1  =  2 q  250 Daltons, m 2 = 251 Daltons,  1.327 r and 1.215 r  The effective cyclotron frequency,fff, of CpMn with the change of electron emission current, EMC  Table 5.4  159  168  The peak width at 50% height of CpCr with the change of electron emission current, EMC  169  Table A1.1  Coordination modes of carbon monoxide  220  Table A1.2  Coordination modes of nitric oxide  222  Table A1.3  Known oxo chromium complexes  224  Table A3.1  Comparison of the values of D’ calculated from Eqs.(A3.12) —  (A3.15) with theoretical values of D  247  —  xiv  —  LIST OF FIGURES  Figure 1.1  The first ICR spectrometer—the Omegatron mass spectrometer  4  Figure 1.2  A conventional cubic trapped-ion cell of FT-ICR  6  Figure 1.3  (a) Ion excitation;  Figure 1.4  The elementary experimental sequence of FT-ICR  Figure 1.5  The absorption, the dispersion, and the magnitude Fourier  (b) Ion detection  7 9  transform spectra of a time domain signal K 0 cos  11  Figure 2.1  The vacuum system of the FT-ICR mass spectrometer  25  Figure 2.2  A positive ion mass spectrum of 3 CD 2 CpCr(NO)  26  Figure 2.3  The triple resonance procedure of FT-ICR mass spectrometry  26  Figure 2.4  (a) A pure Cr÷ spectrum by using the triple resonance technique; (b) The spectrum under the same conditions after a reaction delay time of 100 ms  Figure 2.5  The temporal behaviors of products from the condensation CH 2 CpCr(NO) reactions of Cr with the parent molecule 3  Figure 2.6  41  The temporal variations of the negative ions from 48  CD 2 CpCr(NO) 3 Figure 2.10  40  Cr and 3 Cp O The temporal behaviors (solid lines) of 4 C 3 Cp C O 4 D r  Figure 2.9  36  Two proposed mechanisms for the bent nitrosyl ligand in the Cr-molecule condensation reaction  Figure 2.8  35  The temporal behaviors of products from the condensation CD 2 CpCr(NO) reactions of Cr with the parent molecule 3  Figure 2.7  30  Plot of in rate constants vs. electron deficiency for the primary ions from  CH 2 CpCr(NO) 3  52  —  Figure 2.11  xv  —  Proposed structures of the cluster ions from the Cr-molecule condensation reactions  Figure 2.12  57  Proposed structures of the cluster ions from the CpCr-molecule condensation reactions  Figure 2.13  59  Proposed structures of all the other positive bimetallic cluster ions  60  Figure 2.14  Proposed structures of CpCrO and 2 C Cp ( C NO) H r  61  Figure 3.1  (a) Spectrum of pure Cr obtained by the triple resonance (b) After a 100 ms delay time, large amounts of product ions from charge transfer of the Cr to 3 CH appeared 2 CpCr(NO)  Figure 3.2  Temporal behaviors of Cr, CpCrNOCH, CpCr(NO), O, and CpCr 2 CpCr NO in H 2 2 medium  Figure 3.3  79  Curve fitting for ion intensity ([Cr+*]+[cr+J) vs. time t. Experimental data were obtained in H 2 medium  Figure 3.4  85  Temporal variations of the chromium ammine complexes produced from the ion-molecule reactions of 3 CH with CH 2 CpCr(NO) 4  Figure 3.6  83  Temporal variations of the chromium animine complexes produced from the ion-molecule reactions of 3 CH with H 2 CpCr(NO) 2  Figure 3.5  77  87  Temporal variations of CpCr and the secondary ions Cr Cp O , 3 C Cp ( NO)O r in Crt 2 2 Cp  H2  medium  90  Figure 4.1  Point model for Coulomb effect  100  Figure 4.2  The average distance approach  102  Figure 4.3  Line model for Coulomb effect  105  Figure 4.4  Coulomb shifting and broadening for the line model of Fig. 4.3  109  Figure 4.5  Same graph as that in Fig. 4.4 for 1= 8 cm and  2 N  =  6 ions 3.33x10  110  —  Figure 4.6  xvi  —  Same graph as that in Fig. 4.4 for 1= 15 cm and  2 N  =  6.25x 106 ions  110  Figure 5.1  Uniformly charge-disk model for FT-ICR  Figure 5.2  The averaging of tangential components of the Coulomb interaction between two ions m 1 and m 2  Figure 5.3  123  124  Differential elements of the charge-disk model for Coulomb shifting in FT-ICR  126  Figure 5.4  Overlapping charged disks  130  Figure 5.5  Sections of overlapping disks used for piecewise evaluation of the total radial Coulomb force from Disk 2 to Disk 1  Figure 5.6  132  Radial Coulomb force on Disk 1, a uniformly charged disk of m 1 ions, due to a uniformly charged disk m ions, as a 2 function of the position of Disk 2  Figure 5.7  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for r/r’  Figure 5.8  138  =  10  138  =  20  139  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for r/r’  Figure 5.11  5  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for r/r’  Figure 5.10  =  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for r/r’  Figure 5.9  137  =  100  139  Apparent Coulomb distance, D, as a function of nT’, the ratio of the cyclotron radius to the disk radius, for the charge-disk model  144  Figure 5.12  The charge-cylinder model  151  Figure 5.13  Coulomb shifting and broadening for the charge-cylinder model of Fig. 5.12  158  —  Figure 5.14  xvii  —  Drift in the 1.9 Tesla magnetic field used in the experiment after preheating 30 mm  Figure 5.15  165  The effective cyclotron frequency,fff, of CpMn vs. number of ions with and without unlike-ion Coulomb interactions  167  Figure 6.1  A cylinder in cylindrical coordinates (p, q, z)  178  Figure 6.2  The charge-tetragon model  180  Figure 6.3  The coordinate expression of singular points for 1 y =y, z  =  =  188  2 z  Figure 6.4  Strong coupling critical curve  Figure 6.5  The cubic ion-trapped cell configuration employed in the FT-ICR mass spectrometer  Figure A1.1  198  204  The “twenty-second” ion-molecule reaction mass spectra (a) Parent molecule 3 CH (b) Parent molecule 2 CpCr(NO) ; CD 2 CpCr(N0) 3  Figure A2  219  Kinetic behavior of Cr produced from 3 CH in 2 CpCr(NO) , CH 3 , and N 4 2 media 1120, NH  Figure A3.1  (a) Temporal behavior of Cr in H 0 2  228  (b) Temporal behavior of Cr in NH 3  228  (c) Temporal behavior of Cr in CH 4  229  (d) Temporal behavior of Cr in N 2  229  Radial Coulomb force on Disk 1, a unifonxily charged disk of 1 ions, due to a uniformly charged disk of m m 2 ions, as a  function of the position of Disk 2 Figure A3.2  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for rir’  Figure A3.3  242  =  20  243  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for r/r’  =  100  245  —  xviii  —  LIST OF ABBREVIATIONS  A  Angstrom  arnu  atomic mass unit  Bu  (-CH C 2 ) H Butyl 3  C  Coulomb  CU)  Collision-Induced Dissociation  Cp  Cyclopentadienyl 5 -C ( ) H  Eq.  Equation  Et  CH 2 (-CH ) Ethyl 3  eV  electronVolt  Fig.  Figure  FT-ICR  Fourier Transform Ion Cyclotron Resonance mass spectrometry  FT-JR  Fourier Transform InfraRed spectroscopy  FT-MS  Fourier Transform ion cyclotron resonance Mass Spectrometry  FT-NMR  Fourier Transform Nuclear Magnetic Resonance spectroscopy  Hz  Hertz  ICR  Ion Cyclotron Resonance mass spectrometry  3  Joule  k  kilo  kHz  kiloHertz  K  Kelvin  Me  Methyl (-CH ) 3  MHz  MegaHertz  m  meter  mm  minute  m/q, m/z  mass-to-charge ratio of ion  -xix  ms  millisecond  MS-MS  Mass Spectrometry-Mass Spectrometry  pA  microampere  N  Newton  nA  nanoAmpere  Pa  Pascal  Ph  Phenyl (-C ) 5 H 6  ppm  parts per million  if  radiofrequency  see  second  SI units  Standard International units  V  Volt  —  xx  —  ACKNOWLEDEGMENTS My sincere gratitude goes to my research supervisor, Professor Melvin B. Comisarow, for being a constant source of support to me (both academically and personally) over years. Without question, whatever I have achieved academically over the last five years has been largely due to his great inspiration and energetic enthusiasm for hard work. I also want to thank the occupants of E61/63, Sandra M. Taylor, and Ziyi Kan, for providing the friendly research environment, and special thanks to Ms. Sandra M. Taylor for proofreading the manuscript of this thesis. My gratitude goes to Professor Peter Legzdins and Dr. George Bannerman Richter Addo for their kindness in providing chromium nitrosyl samples and the information on their new development in metal nitrosyls; to Dr. Xingru Zhang (Department of Mathematics, UBC), Mr. Peter Yueqiang Zhu (Department of Physics, UBC), and Dr. Wenzhu Zhang for their helpful discussions in mathematics, physics, and computer science, respectively. I would also like to thank the people working in the Electronic shop and Mechanical shop of UBC Chemistry department for the maintenance of the FT-ICR mass spectrometer. Finally, my wife and my parents have to be thanked for their comfort and encouragement.  —  xxi  —  This dissertation is dedicated to my youngest brother, Shu-Huan Chen  —1—  CHAPTER 1  GENERAL INTRODUCTION  —2—  The first English book on mass spectrometry which I read in China in 1976 was McDowell’s “Mass Spectrometry”.(’) At that time I did not expect that 10 years later I would study mass spectrometry in the Department of Chemistry formerly headed by this very professor. In 1980, when my Master’s Degree supervisor, Professor Ou Ji (at that time an associate professor), introduced Fourier Transform Ion Cyclotron Resonance ) to me, I was very surprised and had many 2 mass spectrometry (FT-ICR or FT-MS)( questions. How could a mathematical method enhance the resolution of ion cyclotron resonance mass spectrometry? Was there another transform method better than the Fourier transform? What were the elementary differences between Ion Cyclotron Resonance (ICR) mass spectrometry and FT-ICR mass spectrometry? How was mass spectrometry mass spectrometry run in a single FT-ICR? I even argued with my former supervisor. I had thought that the high resolution capability of FT-ICR was due to the long relaxation time of ions and Ou Ji thought that the Fourier transform itself produced the high resolution. In a word, because I was attracted by its beautiful “first name”  —  Fourier  transform, I travelled to the birthplace of FT-ICR. This work is devoted to the application of FT-ICR and its performance modifications. There are two parts: (1) gas-phase chemistry of chromium nitrosyls; (2) Coulomb interaction between ions in ion cyclotron motion, including treatment for averaging the Coulomb interaction and treatment for the Coulomb interaction within small separations.  1.1  A Brief History of Ion Cyclotron Resonance Mass Spectrometry The fundamental basis for ICR mass spectrometry was established in the early 1930’s  by Ernest 0. Lawrence. He was the first to demonstrate that charged particles could be accelerated to high energies by an oscillatory electric field having the same frequency as ) In a uniform static magnetic field B, an excited charged 3 their cyclotron frequency.( particle is constrained to move in a circular orbit with a frequency given by  —3—  %= qB/m  (1.1)  where Co 0 is termed the natural cyclotron frequency,( ) q and in are the charge and mass of 4 this particle. The cyclotron particle accelerator, in which a charged particle can be excited to very high translational energy, is a powerful tool for high-energy nuclear physics, but it is not very useful in chemistry. The first mass spectrometer based on the cyclotron resonance principle was developed in 1949 by Hipple and co-workers.( ) At that time the device shown in Figure 5 1.1 was called the Omegatron mass spectrometer. After the gas in the Omegatron is ionized by an electron beam (Figure 1.1), an electric radiofrequency (ii) field is applied between a pair of planar electrodes to excite the newly-formed ions, and the field is homogenized by feeding a stack of guard rings from a voltage divider. If the frequency of the rf field is equal to the cyclotron frequency, w , of an ion, the ion formed in the center of the device 0 moves in a continuously expanding spiral normal to the direction of the magnetic field B. When the ion eventually reaches the collector, it is detected as an ion current signal. The plot of this ion signal against ion resonance frequency (or ion mass according to Eq.(1. 1)) is the ion mass spectrum. Because ion-molecule coffisions and the space charge effects from the electron beam severely degrade the performance of the Omegatron, its resolution was very poor above m/q 50. In 1960’s Wobschall et a!. detected ICR ion signals by measuring the ion cyclotron resonance power absorption from the rf field rather than from ) and Varian Associates produced the first commercial ICR mass 6 the direct ion current,( spectrometer, in which a three section ICR drift cell was developed to separate the ionization, resonance, and detection procedures.( ) The three section ICR drift cell 7 eliminated the ion cyclotron frequency shift from space charge effects of the electron beam. Mclver made important innovations in 1970 for ICR mass spectrometry: the pulsing technique and a one-region trapped-ion analyzer cell.( ) The pulsing technique allowed 8  —4—  B  Path of ions at resonance Guard rings  ‘—  Filament  = =  7’ h,  /“ ‘,‘  =  Ion collector Trapping voltage  SIDE VIEW  EN]) VIEW  Figure 1.1  The first ICR spectrometer  —  the Omegatron mass spectrometer. After  ions are formed in the Omegatron, a rf field is applied across the top and bottom plates, and the field is homogenized by a stack of guard rings. When the cyclotron frequency of an ion is equal to the the frequency of the rf field, it will move in a spiral orbit until it strikes the collector plate.  one to control the ICR experiments electronically. Ionization, resonance, and detection of ICR experimental events were separated in a time sequence rather than in three spatial regions. This innovation combined the advances of power absorption detection and the three section ICR drift cell. However, in all ICR mass spectrometries mentioned above, the ion species were excited and detected one by one by scanning the magnetic field or the frequency of the if field. Thus, the m/q of an ion had to be calculated from the frequency of the if field. A dramatic development in 1974 was Fr-ICR, invented by Comisarow and  —5—  Marshall in our Department of Chemistry.( ) In FT-ICR, all ions are excited and detected 2 simultaneously. The ion image signals on the receiver plates are detected all at once, both for ion intensities and ion cyclotron frequencies.( ) The interest in FT-ICR is illustrated by 9 many reviews on the topic. So far, there have been 51 reviews (in four languages: English, German, Chinese, and Japanese) on the theory, instrumentation, and applications of FT-ICR.( ) FT-ICR has been shown to be an active area in spectrometry.( 1060 ) 61  1.2  The Background of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry  1.2.1 Basic operating principle of FT-ICR A cubic trapped-ion cell( ’ 62) is shown in Figure 1.2. When a sample to be 17 analyzed is introduced into this cubic cell, it can be first ionized by various ionization techniques, such as electron ionization, photo-ionization, or chemical ionization. These ions are confined in the cubic cell by a static trapping voltage. In order to measure the cyclotron frequencies and the intensities of an ensemble of ions, the experiment is divided into two steps (see Figure 1.3). At first, a broadband rf rapid sweep is applied across the cubic cell to excite all the ions successively. When the radiofrequency is equal to the natural cyclotron resonance frequency radius of cyclotron motion.  of an ion, this ion will be excited into its orbital  This process is the ion excitation step of FT-ICR.  Secondarily, after the if sweep is turned off, the excited cyclotron motion goes into a circular orbit by maintaining its resonance frequency. The orbiting ion induces an alternating image cunent on two opposing parallel receiver plates and the voltage induced across this circuit is the FT-ICR signal.( ) This process is the ion detection step of FT 9 ICR. The ion cyclotron motion is detected as a time domain signal. Then, the signal is digitally sampled and Fourier transformed to give a frequency spectrum in a computer.  —6—  C y  Figure 1.2  A conventional cubic trapped-ion cell of FT-ICR. The cell is composed of six square plates. Two T’s are trapping plates to which is applied a direct voltage, typically,  +  1 V for positive ions and  1 V for negative ions. N  and S are receiver plates for detection of ion cyclotron motion. E and W are transmitter plates for rf excitation.  C is an electron collector for  attracting and monitoring the electron current. G is a grid for shielding the strong ionization voltage. F is a thermionic filament. P is a preamplifier for ion detection. A is a transmitter amplifier for the if field. B is a  magnetic field applied across the cubic cell.  —7—  GB  I  I.  I  I  I  I  Figure 1.3  (a)  (b)  The radiofrequency is on  The radiofrequency is off  (a) Ion excitation. When a radiofrequency field is applied across the cubic trapped-ion cell, the ion is excited and its cyclotron radius increases in a spiral orbit. (b) Ion detection. When the rf field is off, the ion cyclotron radius is constant, and the ion cyclotron motion induces an image current 1(t) that is converted to an alternating voltage signal VQ) and amplified by the preamplifier.  —8—  From the relationship between the cyclotron frequency and the mass of an ion in Eq. (1.1), the frequency spectrum is a mass spectrum, too.  In either the ion excitation or the ion  detection, broadband circuitry is used, which is different from the narrow-band inductivecapacitive circuitry used in Fourier Transform Nuclear Magnetic Resonance spectroscopy ) 9 (FT-NMR).( All the experimental procedures of FT-ICR consist of a sequence of pulses controlled by a computer. The FT-ICR experimental pulse sequence is indicated in Figure 1.4. 1.2.2 Fourier transform of a time signal Any periodic well-behaved function can be expressed as a sum of Fourier series, the familiar relationship,  f (t)  cos (n cot) +  =  b sin (n co t)  (1.2)  where a and b are series coefficients, and co = 2itfT, T being the period of the function, a) 63 f(t). For an aperiodic function, the summation becomes an integration(  00  f(t)  I  =  Jo  00  A(co) cos (cot) dw +  J  B(o)) sin (cot) dco.  (1.3)  JO  A(w) is called the pure cosine frequency content of f(t), and B(w) is called the pure sine frequency content off(t). These two frequency spectra are given by  00  A(w)  =  -  )f(t)cos(wt)dt  (1.4)  00  B(co)  =  -  JfQ)sin(wt)dt.  (1.5)  —9—  Figure 1.4  The elementary experimental sequence of FT-ICR. Neutral atoms or molecules are ionized in the first pulse, ion formation. These ions can be temporally retained in the cubic cell and then excited in the second pulse, or immediately excited without any delay. The ion cyclotron motion is monitored in the third pulse after the rf is shut down. In the fourth pulse, all ions are cleared out, and then the above procedure is repeated in order to  enhance the ion signal.  —10—  When the lower limits of the integrals in Eqs. (1.4) and (1.5) are zero (i.e.,  t>  0), then  A(w) and B(a) are called the Fourier cosine transform and Fourier sine transform, respectively. Excited cyclotron motion produces a time domain signal (Figure 1.3 (b))  fQ)  2it 0 K cos w t, 0  =  0  <t <  (1.6)  T  where K 0 is a constant proportional to the number of excited ions and T is the signal acquisition time or the duration of the detection period. The basic Fourier transform ) 16 formulas of this time domain signal are:(  A()  7t  B(o.)  =  sin[(w—w K ) 0 TJ  f(t)cos tdt=  =  1 —  0  {1—cos[(o.—w K ) 0 TJ}  .  I f(t)srnwtdr=  =  2  {[A(co)1 +[B(w)]  2 112  }  (1.8)  0 (0—CL)  ltJO  C(co)  (1.7)  (OW  =  0 2K  sin[(a— w,.) T/2]  (1.9)  CL) —  A(w), B(w), and C(w) are called the absorption, the dispersion, and the magnitude (absolute-value) spectra. Usually, magnitude mode is used in FT-ICR. These three kinds of frequency domain spectra are shown in Figure 1.5. The theoretical spectral linewidth, &o, is defmed as the full linewidth at half of the maximum peak height, &05%, Aco From the line shapes of Eqs. (1.7)  —  =  (1.10)  (1.9), the full linewidths of the absorption, the  dispersion, and the magnitude spectra are solved and summarized in Table 1.1. The theoretical spectral linewidths are all inversely proportional to the the acquisition time T at  —  11—  öiö (w —  Time, in sec  0  D  20  -20 -10  0  0  The absorption, the dispersion, and the magnitude Fourier transform spectra  Figure 1.5  0 cos %t. of a time domain signal 2t K  Table 1.1 Summary of FT-ICR line shape formulas of a time signalf(t)  =  0 cos 2itK  0< t < T, and their extrema and full linewidths Zero-pressure line shape’ 2  Spectral display mode Absorption  Dispersion  Magnitude (Absolute value)  )T] 0 sin[(a — ci. co.-o  A(w)=  B(oi)  C(a)  urn (ct—w ) 0  1 =  2  —  0.7246 b  )TJ 0 cos[(w — w w—  sin[(0  —  )T/2] 0 o,  I  for simplicity.  b These are the extrema of B(w) at w—  ± 2.331/T.  =  —*  T  a The scaling coefficient K 0 has been omitted  C  20  in see-’  T  This value is the distance based on the extrema on either side of w.  0  50 Full hnewidth w  3.791 T 4.662 c T 7.582 T  —  12—  the low-pressure limit. Thus, if there is no damping force, the relaxation time of the excited ion cyclotron motion will become arbitrarily long and the resolution will be  arbitrarily high. The longer relaxation times (at a lower sample pressure) of ions are, the more precisely the cyclotron frequencies of these ions can be measured, which is the uncertainty principle in FT-ICR.( ) Table 1.1 shows that the absorption mode has the 16 narrowest linewidth. Fourier analysis (the use of Fourier series and Fourier transform in analysis) is also b) The use of Fourier transform in FT-ICR actually is a 63 called frequency analysis.( technique for measuring the frequency content of a time signal, and frequency can be measured very accurately.( ) Therefore, high resolving power of FT-ICR comes not from 17 the mathematical method, but rather from precise electronic techniques for the measurement of frequency.  1.3  Ion-Molecule Reactions and FT-ICR Multiple Resonance An important application of ICR mass spectrometry is gas phase ion-molecule  chemistry. The trapping direct voltage applied on the cubic ion-trapped cell has the capability to confine ions. After the ions are formed, they can be maintained within the cubic cell for a while (from microseconds to tens of seconds) to allow the ions to react with the neutral molecules whose pressure is typically iO  —  iO Torr. Then, the products of  these ion-molecule reactions are detected by FT-ICR. The ICR multiple resonance technique that was invented for conventional ICR by Anders et at. in l966,(M) and then transplanted to FT-ICR by Comisarow et at. in 1978(65) can help to monitor the pathways of these ions. If the amplitude of a specific if sweep of FT-ICR is strong or the duration of the if sweep is long, the cyclotron resonance motion of the corresponding ions in the cubic cell will be excited enough that these ions hit the walls of the cubic cell and their charges neutralized. This process is called an ion ejection in ICR mass spectrometry. Because the  —13—  experimental sequence of FT-ICR is controlled by computer, further experimental events, such as ion ejections, can be added into the sequence shown in Figure 1.4. If only one ion ejection is used to eject a desired mass range of ions, it is called double resonance because there are two resonance events in this sequence: ion ejection and ion excitation. If two ion ejections are used to eject two desired mass ranges of ions, it is called triple resonance, and so on. Double resonance can be used to eject only one ion mass, leaving all other ions, and triple resonance can be used to eject all the ions except ions of one mass. Triple resonance experiments for identification of the ion-molecule reaction pathways are easily interpreted. In triple resonance, the FI’-ICR spectrum is simplified to be that of only a single mass of ions. These ions are then trapped in the cubic trapped-ion cell for ion-molecule reactions. Consequently, all products can be assigned to be the products of the single ion with the neutral molecules. These processes will be discussed in Chapter 2 in more detail. The ICR multiple resonance is a Mass Spectrometry-Mass Spectrometry (MS-MS) technique in which the first stage of mass separation is designed to select parent ions of a given m/q, and the second stage is designed to detect all of the daughter ions resulting from fragmentation or ion-molecule reaction of those initially selected parent ions. Multiple resonance of FT-ICR is executed in only a single analyzer cell rather than in multiple mass analyzers as is the MS-MS experiment in other tandem mass spectrometers.  1.4  Performance of FT-ICR Fr-ICR has the same advantages as other Fourier transform spectroscopies, such as  Fourier Transform InfraRed spectroscopy (FT-IR) and FT-NMR. These advantages are the Fellgett (multiplex) advantage for great speed and great signal-to-noise ratio. The whole spectrum can be detected simultaneously rather than one mass at a time by scanning as in a conventional ICR mass spectrometer, and a low signal-to-noise ratio can be enhanced by accumulation. FT-ICR has other advantages, such as ultrahigh mass  —  14—  resolution, wide mass range, trapped-ion ability, and multiple resonance capability. Its transmission and mass resolution are better than other commercial mass spectrometers. A comparison of Fr-ICR features with those of other conventional mass spectrometries( ) is 66 given in Table 1.2. The performance of FT-ICR has undergone much development in the last three Ar ions in a 1 8 for ‘ years. The resolution of FT-ICR has increased amazingly to over 4x10  ) 67 7 Tesla magnetic field.(  Furthermore, by using high-precision techniques of trap  construction, operation, and analysis, very high accuracy in mass measurement has been ° has been obtained 1 demonstrated as being possible. A mass ratio accuracy of up to 4x10— He with ) for the light ions (CO compared with N ,1 2 H, 2 H, and 3 C).( 2 ‘ 6 68 9 In order to 69 without regard to 68 one,( obtain greater accuracy, these light ions were detected one by ) the multichannel advantage of conventional FT-ICR (all ions are detected simultaneously). These methods are not appropriate to chemistry as yet. An ion of m/q to 31, 000 Daltons has been measured,( ) but this is far from the theoretical prediction that a 2.5 cm cubic cell 70 at a trapping voltage 1 V in a magnetic field of 13 Tesla is capable of storing an ion having an m/q as large as 950, 000 Daltons.( ) It is well known that the relative peak heights in 28 FT-ICR do not always reflect the ion abundances accurately. The most important error sources are that ion signal intensity of FT-ICR not only depends on the ion abundance, but ). Because FT-ICR is 71 ) and the ion z-mode ejection( 9 also the cyclotron radius of an ion( not usually applied to quantitative chemical analysis, this problem is not of much concern yet. Recently, Koning et al.( ) have used a segmented Fourier transform method to 72 enhance the accuracy of ion abundance measurements of FT-ICR. In their method, the time domain signal is divided into two parts for t from 0 to T/2 and t from T12 to T, where T is the acquisition period. The individual components of the signal are separated by  Fourier transform, and then the amplitudes of ion intensities are calibrated against the decay of the ion cyclotron motion. For a narrow mass range, such as among xenon isotopes, the measurement of ion abundances with accuracies of about 0.1% was demonstrated.  —15— Table 1.2  Comparison of Typical Mass Spectrometer Performance Parameters  Instrument Type Quadrupole  Transmission  (%)  Minimum #Ions Needed to be Fumd  Mass Resolving Power 102_103  0.01  Mass Range (1)alton) 3 i0  *  Magnetic Sector —low resolution —high resolution  10  10  0.001  5 i0  Time of Flight  10—50  FT-ICR  (— 100)  102  1—10  b0_102  Wien ExB  *  10  2—10  10  104_b?  **  —io 4 io 5  _to? 4 10  “  104_to?  Actually, the mass range of a mass spectrometer may depend on the ionization approach. A new  ionization technique: electrospray ionization, began a revolution in high mass measuring. For example, after an electrospray ion source was attached to a quadrupole analyzer, as high as 5 million Daltons of poly (ethylene glycols) with thousands of positive charge on each molecule was measured, although the mass range of the quadrupole was only b0 (T. Nohmi and J. B. Fenn, Proceedings of the 38th ASMS Conference on Mass Spectrometry and Allied Topics, 1990, 10-11). **  The highest mass/charge ratio is 274,800 Daltons obtained by an matrix-assisted laser desorption ion  source (F. Hillenkarnp, M. Karas, Proceedings of the 37th ASMS Conference on Mass Spectrometry and Allied Topics, 1989, 1168-1169). The resolution of Time of Flight mass spectrometry has reached 35,000 for Cs atoms and 22,000 at 111,000 Daltons (T. Bergmann, T. P. Martin, and H. Schaber, Rev. Sci. Instrum., 60, 1989, 792-793 and T. Bergmann, H. Goehlich, T. P. Martin, and H. Schaber, Rev. Sd.  Instrum., 61, 1990, 2585-2591). ***  Because the ions are formed and detected in the same region, in the cubic trapped-ion cell, transmission  of FT-ICR can be over 80%. Detection limits of FT-ICR for minimum ion number are 10 ions (M. B. Comisarow, Analytica Chimica Acta, 178, 1985, 1-15.) and after a special technique was added, a single ion has been detected CE. A. Cornell, R. M. Weisskoff, K. R. Boyce, Robert W. Flanagan, Jr., G. P. Lafyatis, and D. B. Pritchard, Phys. Rev. Leu., 63, 1989, 1674-1677). The resolution of FT-ICR is up to 8 for 40 4x10 Ar (K. P. Wanczek, Proceedings of the 7th International Dynamic Mass Spectrometry Symposium, Salford, 1989). The current highest mass/charge ratio is 31830 Daltons with an accuracy of 2.26% (C. B. Lebrilla, D. T.-S. Wang, R. L. Hunter, and R. T. Mclver, Jr., Anal. Chem., 62, 1990, 878880).  —16—  References  1.  McDowell, C. A. “Mass Spectrometry”; McGraw-Hill: New York, 1963.  2.  Comisarow, M. B.; Marshall, A. G. Chem. Phys. Lett. 1974,25, 282-283.  3.  (a) Lawrence, E. 0. and Edlefsen, N. E. Science 1930, 72, 376-377. (b) Lawrence, E. 0.; Livingston, M. S. Phys. Rev. 1932,40, 19-35.  4.  Henis, 3. M. S. in “Ion-Molecule Reactions”; Franklin, 3. L. Ed.; Plenum, New York, 1972; p 398.  5.  (a) Hipple, 3. A.; Sommer, H.; Thomas, H. A. Phys. Rev. 1949, 76, 1977-1878. (b) Sommer, H.; Thomas, H. A.; Hipple, 3. A. Phys. Rev. 1951, 82, 697-702.  6.  Wobschall, D.; Graham, J. R. Jr.; Malone, D. P. Phys. Rev. 1963, 131, 15651571.  7.  The first commercial ICR mass spectrometer was designed and constructed under the direction of Dr. P. Llewellyn at Varian Associates for Professor 3. D. Baldeschwieler at Stanford University in 1965.  8.  Mclver, R. T. Jr. Rev. Sci. Instrum. 1970, 41, 555-558.  9.  Comisarow, M. B. J. Chem. Phys. 1978, 69, 4097-4104.  10.  Asamoto, B. Spectroscopy 1988, 3, 38-46.  11.  Baykut, G.; Eyler, J. R. Trends Anal. Chem. 1986,5, 44-49.  12.  Buchanan, M. V. “Fourier Transform Mass Spectrometry: Evolution, Innovation, and Applications”; American Chemical Society: Washington, DC, 1987.  13.  Chiarelli, M. P.; Gross, M. L. in “Analytical Applications of Spectroscopy”; Creaser, C. S.; Davies, A. M. C., Ed.; Royal Society of Chemistry: London, 1988; pp 263273.  14.  Cody, R. B. Jr. Analysis 1988, 16, 30-36.  —17—  15.  Comisarow, M. B. Adv. Mass Spectrom. 1978, 7, 1042-1046.  16.  Comisarow, M. B. in “Transform Techniques in Chemistry”; Griffiths, P. R., Ed.; Plenum Press: New York, 1978; pp 257-284.  17.  Comisarow, M. B. Adv. Mass. Spec. 1980, 8, 1698—1706.  18.  Comisarow, M. B. in “Fourier, Hadamard and Hubert Transform in Chemistry”; Marshall, A. 0., Ed.; Plenum Press: New York, 1982; pp 125-146.  19.  Comisarow, M. B. Anal. Chim. Acta 1985, 178, 1-15.  20.  Comisarow, M. B.; Nibbering, N. M. M. “Special Issue, Fourier Transform Ion Cyclotron Resonance Mass Spectrometry”;  mt. J. Mass Spectrom. Ion Proc. 1986,  72, (No. 1-2). 21.  Connes, P. Mikrochim. Acta 1987, III, 337-352.  22.  Eller, K.; Schwarz, H. Chem. Rev. 1991,91, 1121-1177.  23.  Fang, Y. Huaxue Tongbao (Chinese) 1988, 14-18.  24.  Freiser, B. S. Talanta 1985,32, 697-708.  25.  Freiser, B. S. in “Techniques for the Study of Ion Molecule Reactions”; Farrar, J. M.; Saunders, W. H., Ed.; John Wiley & Sons: New York, 1988; pp 61-118.  26.  Freiser, B. S. Chemtracts-Analyt. Phys. Chem. 1989, 1, 65-109.  27.  Gord, 3. R.; Freiser, B. S. Anal. Chim. Acta 1989,225, 11-24.  28.  Gross, M. L.; Rempel, D. L. Science 1984,226, 261-268.  29.  Gross, M. L. Mass Spectrom. Rev. 1989, 8, 165-197.  30.  Hanson, C. D.; Kerley, E. L.; Russell, D. H. in “Treatise on Analytical Chemistry”; Winefordner, J. D., Bursey, M. M.; Koithoff, I. M., Ed.; John Wiley & Sons: New York, 1989; Vol. 11, pp 117-187.  31.  Inoue, M. Bunseki (Japanese) 1988, 490-496.  —18—  32.  Johiman, C. L.; White, R. L.; Wilkins, C. L. Mass Spectrom. Rev. 1983,2, 389415.  33.  Land, D. P.; Pettiette-Hall, C. L.; Hemminger, J. C.; Mclver, R. T. Jr. Acc. Chem. Res. 1991,24, 42-47.  34.  Laude, D. A.; Johlman, C. L.; Brown, R. S.; Weil, D. A.; Willcins, C. L. Mass Spectrom. Rev. 1986, 5, 107-166.  35.  Ledford, E. B. 3.; Ghaderi, S.; Wilkins, C. L.; Gross, M. L. Adv. Mass Spectrom. 1980, 8, 1707-1724.  36.  Lubman, D. M. “Lasers in Mass Spectrometry”; Oxford U. Press: New York, 1990, Chapters 7, 11, 12, 13, 14, and 15.  37.  Marshall, A. G. Ace. Chem. Res. 1985, 18, 3 16-322.  38.  Marshall, A. G. in “New Dir. Chem. Anal. Proc. Symp. md. Univ. Coop. Chem. Program Dep. Chem., Tex. A & M Univ.”; Shapiro, B. L., Ed.; Tex. A & M Univ. -  Press: College Station, Tex., 1985; pp 111-134. 39.  Marshall, A. G. in “Mass Spectrometry in the Health and Life Sciences”; Burlingame, A. L.; Castagnoli, N. J., Ed.; Elsevier: Amsterdam, 1985; pp 265-286.  40.  Marshall, A. G. Adv. Mass Spectrom. 1989, 11, 65 1-669.  41.  Marshall, A. G.; Verdun, F. R. “Fourier Transforms in NMR, Optical, and Mass Spectrometry”; Elsevier: Amsterdam, 1990; Chapter 7.  42.  Marshall, A. G.; Grosshans P. B. Anal. Chem. 1991, 63, 215A-229A.  43.  McCrery, D. A.; Sack, T. M.; Gross, M. L. Spectrosc.  44.  Mclver, R. T. Jr.; Bowers, W. D. in “Tandem Mass Spectrometry”; McLafferty, F. W., Ed.; John Wiley &Sons: New York, 1983; pp 287-301.  45.  Nibbering, N. M. M. Nachr. Chem., Tech. Lab. (German) 1984,32, 1044-1050.  46.  Nibbering, N. M. M. Kem-Kemi 1984, 11, 11-12.  47.  Nibbering, N. M. M. Adv. Mass Spectrom. 1985, 10, 417-435.  mt. J.  1984, 57-71.  —19—  48.  Nibbering, N. M. M. Mass Spectrom. 1985,8, 141-160.  49.  Nibbering, N. M. M. Comments At. Mo!. Phys. 1986, 18, 223-234.  50.  Nibbering, N. M. M. Adv. Phys. Org. Chem. 1988,24, 1-55.  51.  Nibbering, N. M. M. Adv. Mass Spectrom. 1989, 11, 101-125.  52.  Roth, L. M.; Freiser, B. S. Mass Spectrom. Rev. 1991, 10, 303-328.  53.  Russell, D. H. Mass Spectrom. Rev. 1986,5, 167-189.  54.  Sharpe, P.; Richardson, D. E. Coord. Chem. Rev. 1989, 93, 59-85.  55.  Wanczek, K.-P.  mt. J. Mass Spectrom. Ion Proc.  1984, 60, 11-60.  56.  Wanczek, K.-P.  mt. J. Mass Spectrom. Ion Proc.  1989, 95, 1-38.  57.  Wilkins, C. L. Anal. Chem. 1978,50, 493A-500A.  58.  Wilkins, C. L.; Gross, M. L. Anal. Chem. 1981,53, 166 1A-1676A.  59.  Wilkins, C. L.; Brown, R. S. in “Mass Spectrometry in the Analysis of Large Molecules”; McNeal, C. 3., Ed.; John Wiley & Sons: Chichester, 1986; pp 191-198.  60.  Wilkins, C. L.; Chowdhury, A. K.; Nuwaysir, L. M.; Coates, M. L. Mass Spectrom. Rev. 1989, 8, 67-92.  61.  Braun, T.; Schubert, A. Trends Anal. Chem. 1991, 10, 1-3.  62.  Comisarow, M. B.  63.  (a) Weaver, H. 3. “Applications of Discrete and Continuous Fourier Analysis”; John Wiley & Sons: New York, 1983; pp 57-58. (b) Ibid, p 1.  64.  Anders, L. R.; Beauchamp, 3. L.; Dunbar, R. C.; Baldeschwieler, 3. D. J. Chem. Phys. 1966, 45, 1062-1063.  65.  Comisarow, M. B.; Grassi, V.; Parisod, G. Chem. Phys. Lett. 1978,57, 413-416.  mt. J. Mass Spectrom. Ion Phys. 1981, 37, 251-257.  —20—  66.  Colton, R. 3.; Kidwell, D. A.; Ross, M. M. in “Mass Spectrometry in the Analysis of Large Molecules”; McNeal, C. 3., Ed.; John Wiley & Sons: Chichester, 1986; p 39.  67. Wanczek, K.-P. Proceedings of the 7th International Dynamic Mass Spectrometry Symposiwn, 1989 (unpublished). 68.  Cornell, E. A.; Weisskoff, R. M.; Boyce, K. R.; Flanagan, R. W. Jr.; Lafyatis, G. P.; Pritchard, D. E. Phys. Rev. Lett. 1989,63, 1674-1677.  69.  Van Dyck, R. S. Jr.; Moore, F. L.; Farnham, D. L.; Schwinberg, P. B. Phys. Rev. A 1989, 40, 6308-6313.  70.  Lebrilla, C. B.; Wang, D. T.-S.; Hunter, R. L.; Mclver, R. T. Jr. Anal. Chem. 1990, 62, 878-8 80.  71.  Hofstadler, S. A.; Laude, D. A. Jr. 65-78 and many references therein.  72.  Koning, L. 3. de; Kort, C. W. F.; Pinkse, F. A.; Nibbering, N. M. M. Spectrom and Ion Proc. 1989,95,71-92.  mt. J. Mass Spectrom and Ion Proc. 1990, 101, mt. J. Mass  —21—  CHAPTER 2  ION-MOLECULE CONDENSATION CHEMISTRY OF -2,4-CYCLOPENTADIEN-1-YL) 5 ( METHYLDINITROSYLCHROMIUM 3 CH 2 CpCr(NO)  —22—  2.1  Introduction Gas-phase ion-molecule reactions in hydrogen were discovered by Dempster as early  as 1916.() After almost fifty years, the gas-phase ion-molecule condensation reactions of the transition metal complexes ferrocene and nickelocene were first reported by Schmacher  and Taubenest in 1964.(2) These early studies were performed on magnetic sector mass spectrometers and the reaction yields were very limited. Only in the past decade, the development of gas-phase ion chemistry has been promoted intensely by new instrumentation and new experimental methods.( ) Just twenty years ago (in 1971), 317 Foster and Beauchamp first used ICR mass spectrometry to investigate the ion-molecule chemistry of iron pentacarbonyl.( ) FT-ICR,( 18 ) with its greater mass range, higher 19 resolution, and more efficient double resonance technique, was a significant improvement for studying ion-organometallic molecule chemistry. FT-ICR mass spectrometry was first ) Studies in 20 used in 1980 by Parisod and Comisarow to study organometaffic chemistry.( ion-molecule chemistry of transition metal complexes by ICR and FT-ICR can be classified into three types: (1) Bond energetics and thermochemistry of transition metal complexes. (2) The chemistry of the ions (especially bare metal ions) generated from transition metal complexes with small inorganic and organic gas molecules. This will be discussed in Chapter 3. (3) Condensation reactions of transition metal complexes. This will be dealt with in the present chapter. ) The 211 There are several recent reviews for these three applications of Fr-ICR.( ,( 25.-29) 5 Fe(CO) ’ studies of the condensation chemistry (type 3) by ICR are clustering of 18 ) 31 ,( 25 6 Cr(CO) ) 25 ,( 30) Cp 4 Ni(CO) ’ Fe (Cp is the abbreviation of Cyclopentadienyl),( 2 Ni,( 33 2 Cp ) 32 ,( 34 2 CpCo(CO) ) ,( Ti, V÷, Cr, Mn, Fe, Cot, Ni with 3 CoNO(CO) ) ,( and 10 4 Ni(CO) ) 6 and 35 Cr(CO) (CO) Most investigations involving transition 2 Re ) 36 .(  —23—  metal condensation chemistry by Fr-ICR have been concerned with transition metal carbonyls, such as CpMn(CO) , CpCrNO(CO) 3 , and 20 2 ,( Cr with 2 CpCrNS(CO) ) ,( Fe, Cot, FeCH, and FeOH with 38 6 Cr(CO) ) 37 ,( Fe with 39 5 Fe(CO) ) (CO) 2 Co , 8 ) ( Co with Fe(CO) , CoC 5 H with 40 4 Fe,( V with 41 2 Cp ) ,( V with V(CO) 5 Fe(CO) ) , Cr 6 with Cr(CO) , Fe— with Fe(CO) 6 , Co with 8 5 (CO) Mo with 42 2 Co , ,( 6 Mo(CO) ) (CO) 10 2 Re ) 43 ,( 10 (CO) and ReMn(CO) 2 Mn ) Cr(CO) 44 ,( 10 6 and 45 ,( Ni(CO) 5 Fe(CO) ) 4 and ,( and their hetemuclear ionic clusters,( 3 CoNO(CO) ) 46 ) Fe and 47  with Fe(CO) 5 and  (CO) 2 Co , 8 ) 48 ( La and Rli with Fe(CO) , Rh with 49 5 ,( Sc, Ti, V÷, Cr, 3 CoNO(CO) ) Fe, Cot, Nit, Cu, Nb, and Ta with 50 ,( Cu with 51 5 Fe(CO) ) ,( 5 Fe(CO) ) 0 2 H ( 3 ) 52 ,( 10 CO) s La 2 with 53 ,( large cluster ions containing up to 40 metal 5 Fe(CO) ) atoms from Fe(CO) 5 and Re (CO) 2 (CO) 2 Re  bare metal clusters from Fe(CO) 5 and  Mn with 56 ,( 57 6 Cr(CO) ) ,( Fe( 4 CpV(CO) ) 13 Fe(CO) ,(58) CO) with 5  -arene)(CO) 6 Cr( , 3 ) 59 ( and 6 Fe(r O H . 3 O )(CO) C ( ) Gas-phase ion-molecule condensation chemistry of transition metal nitrosyls is a largely undeveloped field, although organometallic nitrosyl chemistry already has become a special subject in inorganic chemistry.( ) Cleavage of the N-O bond was not observed in 61 ion-molecule condensation studies of 34 ,( 46,49) as there is a strong Co-NO 3 CoNO(CO) ’ bond in CoNO(CO) 3 (which is expected since the Co electronic structure is 2 4s and the 7 3d , linear NO ligand is a three electron donor). The elimination of nitrogen from the NO ligand  and oxo chromium clusters containing up to four metal atoms have been observed in clustering of 20 ,( but no further study was attempted. To make a 2 CpCrNO(CO) ) comparison with the clustering of CpCrNO(CO) , some extensions were made in this 2 work. We used a chromium mtrosyl compound containing no carbonyl, 3 CH 2 CpCr(NO) , and because the mass of NO is almost equal to the mass sum of two CH 3 groups, the deuterated compound 3 CD was used to confirm our results. It should be noted 2 CpCr(NO) that we have to study a compound containing nitrosyls with other ligands, since only two thermally stable binary metal nitrosyls, Cr(NO) 4 and Co(NO) , are known at present.( 3 ) 62  —24—  Reaction pathways were determined by the multiple resonance technique of FT-ICR. The kinetics were studied by monitoring temporal variations of the reactions for up to twenty seconds, and by calibration of nominal pressure. Cluster structural information was proposed, even though the structure-reactivity rule( ) for gas-phase metal carbonyl ions, 29 which was based on the 18-electron rule, is not suitable to metal nitrosyls. CH was first synthesized by Piper and Wilkinson in 1955.(63) Their 2 CpCr(NO) 3 method involved reaction of a Grignard reagent, CH MgI, with the iodo-complex 3 1. 2 CpCr(NO)  Another method for preparation of 3 CH in which an 2 CpCr(NO) ,  organoaluminum reagent, (CH A1, was used to react with the chioro-complex ) 3 C1, was reported by Hoyato, Legzdins, and Malito in 1975.(64) 2 CpCr(NO) b) The 63 CH possesses a “three-legged piano stool” molecular structure.( 2 CpCr(NO) 3 nitrosyls in 3 CH are linear so that the complex has an 18-electron structure. 2 CpCr(NO)  C? 3 H  c’ 4  ‘N  D N 3 0  0  (?-2,4-Cyclopentadien- l-yl)  (i-2,4-Cyclopentadien- 1-yl)  methyldinitrosylchromium  -dinitrosylchromium 3 methyl-d  The bond energy of N—0 in metal complexes is generally weaker than that of C) This is consistent with their respective vibration frequencies (vco( 61 C_O.( ): 1850 65 2125 cm 1 and VNO( ): 1520 66  —  —  ) in metal complexes. The N—O bond can be 1 1950 cm  broken in ion-molecule reactions to form oxygenated complexes, especially for the CH is 2 CpCr(NO) oxophilic early transition metals. Thus, the condensation chemistry of 3 much richer than that of metal carbonyl complexes.  —25—  2.2  Experimental Section  2.2.1 Experimental Hardware All experiments were performed in a home-built FT-ICR spectrometer equipped with a Nicolet FT-MS 1000 console. The size of the cubic ion-trapped cell is 2.54 cm x 2.54 cm x 2.54 cm. Spectra were obtained by electron ionization using a 1.9 Tesla magnetic field. Background pressures in the cubic ion-trapped cell and sampling manifold were maintained below 5 x i0 Torr and 5 x 10 Torr, respectively. A schematic diagram of the vacuum system of the FT-ICR mass spectrometer is shown in Figure 2.1. The basic operating  Roughing Valves  Cubic Cell Magnet  Figure 2.1  The vacuum system of the FT-ICR mass spectrometer. The sample is introduced from a sampling tube through a roughing valve, through a leak valve to the high vacuum region containing the cubic cell. Sample pressure is held constant by continuous pumping by the ion pump.  —26—  principle of FT-ICR has been described in Chapter 1. Compounds CpCr(NO) 3 C 2 H and 3 C 2 CpCr(NO) D are dark green crystals, which were kindly provided by the research group of Professor P. Legzdins. No impurity was detected in the mass spectra of these two compounds. The samples were evaporated into the FT-ICR spectrometer at ambient temperature through variable leak valves. Pressure was controlled in the nominal pressure range 6.2  —  6.8 x i0- Torr as measured by a VARIAN Dual Range Ionization Gauge  Model 971-1008. Other important parameters employed in the FT-ICR experiment are listed in Table 2.1. The rate constants were measured on different days, using different emission currents, and different excitation levels in order to increase the reliability. The measured kinetic data were plotted with Cricket Graph on a Macintosh microcomputer. Reaction rate constants and error data were calculated using Cricket Graph.  Table 2.1  The important electhc and Fourier transform parameters  Parameter name  Parameter value  Ionization voltage  —25.0 V for positive ions, —1.2 V for negative ions  Trapping voltage  +1 V for positive ions, —1 V for negative ions  Number of data points  64 k  for FT  Remark  for a good resolution, o%>30O. 5 (m/Em)  Zero filling for FT  1  Ionization time  20 ms  for good ion signals.  2.2.2 Multiple Resonance Technique Through the triple resonance experiment described in Chapter 1, all ion masses except one can be ejected. Figure 2.2 is a positive ion mass spectrum of 3 CD under 2 CpCr(NO)  —27—  25 eV electron ionization, which corresponds to a normal mass spectrum obtained from conventional mass spectrometers without ion-molecule reactions. If only  (mlz 52) is to  be studied, the multiple resonance procedure is run as follows. The ions whose masses are below m/z 51 are ejected by a if sweep within their resonance frequencies and this process is called ejection 1 in Figure 2.3. The ions whose masses are over mlz 53 are ejected by a second if sweep within their resonance frequencies and this process is called ejection 2 in Figure 2.3. A third if sweep whose amplitude is weaker than those of the prior two if sweeps is used to excite Cr ions for detection. A spectrum of pure Cr (see Figure 2.4 (a)) is obtained after three if sweeps  —  a triple resonance procedure. Then, the same  procedure is run again except that a delay time, e. g., 100 ms, is added to permit reactions  between the Cr ions and the neutral molecules. All mass peaks containing more than two chromium atoms shown in Figure 2.4 (b) can be assigned to be ion-molecule condensation CD Comparing with Fig. 2.2, many “primary” ions are 2 CpCr(NO) . products of Cr with 3 also found in Fig. 2.4 (b), especially CpCrNOCD and CpCr(NO). These ions are formed from the reactions of Cr ions with the parent molecules.by charge transfer followed by fragmentation.  2.2.3 Pressure Calibration The reactivity of a metal (or metal cluster) ion is a reflection of its coordinative ) Therefore, determination of the rate constants of gas-phase metal 29 unsaturation.( ion-molecule reactions is very important. Ion-molecule reactions in mass spectrometry usually are pseudo-first order, since [molecule] >> [ions]. For example, the ion density in Fr-ICR is generally 106 ions/cm . According to the ideal-gas law, number of molecules N 3 in a volume V and at a pressure P is N=k’  ,  (2.1)  —28—  100  -  —CpCrNOCD CpCr —  1  50-  CpCr(NO)  +  /  7  i  ion  /r_moIUuIa  _..tI1i  0—  100  Figure 2.2  11111  200  300 MASS In  400 A.M.U.  500  II  600  A positive ion mass spectrum of 3 CD The operating parameters 2 CpCr(NO) . are listed in Table 2.1. Because of a 20 ms experimental beam time, some ion-molecule reactions are observed.  Ion excitation (rf sweep)  Ion formation  Normal procedure formation  Ion ejection 1 (if sweep)  Ion quench  [j [j  _J]aable delay time  Ion  Ion detection  Repeat  Ion  Ion  Ion  Ion  ejection 2 (rf sweep)  excitation (rf sweep)  detection  quench  UULfU1ZL  Triple resonance  Figure 2.3  Variable delay time  The triple resonance procedure of FT-ICR mass spectrometry. Two ion ejections are simply added between “Ion fonnation” and “Ion excitation”.  —29—  where the Boltzmann’s constant kB  =  , and T is temperature in 1 1.380658x 10-23 3 K-  5 Pa) and a temperature of 300 K, the Kelvin. At a pressure of 1 x i0 Torr (1.3 x iO— 3 is number of molecules in a volume of 1 cm N  =  —5’  (  —6’  (  =  ) x 300 23 (1.380658x10—  3x10 (molecules).  That is, the density of molecules is generally 3 x iO times the ion density in FT-ICR. If the true pressure of the neutral gas is known, the second order rate constants of these reactions then can be calculated. The true pressure is not the nominal value from the vacuum meter, because the chemical sensitivity of an ion gauge varies from gas to gas. Bartmess and Georgiadis have measured the relative sensitivity of the ion gauge for various ) The true pressure of a sample x is given by the 67 small inorganic and organic molecules.( formula —  x  nominal pressure of x R  22  where R is relative sensitivity of the ion gauge with respect to the sample x with. They compared several empirical methods for calculating the sensitivities and gave the best general equation R = Sx/SN  =  a + 0.30 (0.36/A ) 3  (2.3)  where S, is the chemical sensitivity of a sample x, relative to the chemical sensitivity of nitrogen SN  =  1.00, 1. e., the nominal pressure of nitrogen is defined as a standard, and a  . The molecular polarizability of a molecule is not a 3 is the molecular polarizability of x in A simple sum of its atomic polarizabilities.  Nevertheless, Aroney, Le Fèvre, and  Somasundaram in 1960 found that the polarizability and molar Kerr constant of ferrocene is ) Since only a 68 equivalent to that of a krypton atom sandwiched between two Cp planes.( few data on polarizabiities of metal complexes are available, we assumed:  —30—  100  •1  100  200  400 300 MASS IN A.M.U.  500  600  500  600  (a) 1 I C,,  z  LU I  z  LU ‘I  I-  4 -J Ui  100  200  400 300 MASS in A.M.U. (b)  Figure  2.4  (a) A pure Cr spectrum by using the triple resonance technique; (b) The spectrum under the same conditions after a reaction delay time of 100 ms (nominal pressure  =  7 Torr). CpCrNOCD and CpCr(NO) came 6.2x10  CD 2 CpCr(NO) . from collision induced dissociation of Cr with 3  —31—  (1) The average a of first, second, and third-row transition metal complexes, following the 1 8-electron rule, is a sum of the polarizabifities of krypton and the ligands. (2) Polarizabifity of the Cp ligand and the methyl group could be taken from their corresponding molecules benzene( ) and ethane, respectively. Because terminal nitrosyl 69 and carbonyl both contain it-back-bonding to a metal and there is only one electron difference in their numbers of electrons donated, the a of nitrosyl is assumed to be approximately the same as that of carbonyl. (3) Eq. (2.2) is suitable for organometallic compounds. On the basis of these three hypotheses, the relative sensitivity of 3 CH is 2 CpCr(NO) calculated as follows. The polarizabilities of benzene and ethane are known: a(C ) H 6 ) 6 H 2 a(C  =  4.45  .C70) 3 A  =  10.32  A,  and  The a of carbonyl in Cr(CO) 6 and 6 3 ( ) H Cr(CO) C can be  derived to be 2.82 A .(7’) Thus, according to hypothesis (2) 3 ) H 5 a(C  =  10.32  ) 3 a(.CH  =  4.45  a(nitrosyl)  3x A  3x A  =  =  8.60  2.23  , 3 A  , 3 A  3 2.82A  where a(CH ) is quite close to the theoretical value 2.22 3 method suggested by Miller and Savchik.( ) a(Kr) 72  =  3 obtained from an empirical A  2.4844 A3.(70) The polarizabiity of  3 C 2 CpCr(NO) H is calculated according to hypothesis (1) to be 3 C 2 a(CpCr(NO) ) H  =  ) +2 x a(nitrosyl) H 5 cz(C  =  8.60+2x2.82+2.23+2.4844 18.95  ). 3 (A  +  ) 3 a(•CH  +  a(Kr)  —32—  The chemical sensitivity of 3 CH is then, according to hypothesis (3), 2 CpCr(NO) RCpcr(NO)CH  =  0.36 x 18.95 + 0.3  =  7.1.  CH the calibrated 2 CpCr(NO) , Hence, for a nominal pressure of 6.3 x i0 Torr of 3 pressure is given by CH 2 “CpCr(NO) 3  =  6.3 x i0 Torr I 7.1  =  8.9 x 10 (Torr).  CH as indicated on the ion 2 CpCr(NO) The calculation shows that the nominal pressure of 3 gauge may be very different from the true pressure. For sixteen transition metal complexes whose average molecular polarizabilities have been measured experimentally,( ’ 70-7 1, 73) their theoretical molecular polarizabilities are 68 4 and calculated on the basis of the three hypotheses and given in Table 2.2. Except TiC1 , whose structures do not obey the 1 8-electron rule, the theoretical polarizabilities of 4 0s0 all the other carbonyl complexes agree with the literature polarizabilities within an error range<5%.  2.3  Condensation Chemistry of the Positive Ions 3 group and those of the Cp To avoid confusion between the hydrogens of the CH  D rather than 3 C 2 group, the condensation chemistry is presented using CpCr(NO) H Under electron ionization of 25 eV, the important primary ions (relative 3 C 2 CpCr(NO) . D are Cr, CpCr, CpCrNO, 3 C 2 intensities all over 10%, see Table 2.3) from CpCr(NO) CD. Of 2 Cr, CpCrO, and CpCr(NO) 2 D 4 H 6 NCD, C H 5 CpCrNOCD, CpCr(NO), C NCD, is very unreactive in gas-phase H 5 these primary ions, the methylpyridinium ion, C due to its aromaticity. Temporal behavior in ion intensity from zero to one second of some D are shown in Figures 2.5 and 2.6, 3 C 2 H and CpCr(NO) 3 C 2 ions from CpCr(NO)  —  Table 2.2  33—  A comparison between literature data of average a values and theoretical calculations of a values for some transition metal complexes  Complex  EXPenmefltal cx (As)  error  Theoretical a (A) t  (%)  Remark  Ref.  4 TiC1  16.4  17.6  + 7.3  Fe 2 Cp  19.0  19.7  +3.7  (68)  19.4  The Os has 16 valence (70) electrons.  4 0s0  8.17  9.76  6 Cr(CO)  19.4  )Cr(CO) H 6 (C 3  21.7  21.3  3 (toluene)Cr(CO)  23.6  23.2  3 (p-xylene)Cr(CO)  25.7  25.0  3 (mesitylene)Cr(CO)  27.6  26.9  3 (mesitylene)Mo(CO)  29.8  28.4  3 (mesitylene)W(CO)  30.1  29.7  3 (durene)Cr(CO)  29.0  28.7  3 Cr(CO)  30.5  )Cr(CO) C 6 (Me 3 3 (t-butylbenzene)Cr(CO)  +  —  —  The Ti has 8 valence  electrons.  CO in it as a standard.  (70)  (71)  1.8  (71)  1.7  (71)  2.7  (71)  2.5  (71)  4.7  (73)  1.3  (73)  1.0  (71)  30.6  + 0.3  (71)  31.8  32.4  + 1.9  (71)  28.8  28.7  0.3  (71)  35.9  36.1  + 0.6  (71)  42.4  43.5  +  2.6  (71)  —  —  —  —  —  —  —  (pentamethylbenzene)  —  (p-di-t-butylbenzene)  3 Cr(CO) (1 ,3,5-tri-t-butylbenzene)  3 Cr(CO)  ¶ The polarizabffities of the ligands are directly taken from reference 70 or calculated from the method suggested by reference 72.  —34—  where relative ion intensity is plotted for each ion intensity I relative to total ion current intensity El 1 Relative ion intensity Reaction times as long as  =  (2.4)  If E I.  twenty seconds have been utilized and no significant change in  final products was observed (see A1.1 of Appendix Al). Elemental composition assignments of the ions can be determined by accurate mass measurements.  Our FT-ICR mass spectrometer is capable of mass measurements of good  accuracy to permit elemental composition assignments. Mass measurement errors are generally less than 40 ppm as shown in Table 2.4 with every mass peak calibrated by its left and right neighboring peaks. CpCrNOCH had a maximum ion intensity, and hence  Table 2.3  The primary positive ions from 3 CD (25 eV electron 2 CpCr(NO) ionization and sample nominal pressure  Primary positive ion  mlz  =  7 Torr) 6.5x10-  Relative intensity  (%)  Electron count for Cr  Fragment ions: Cr  52  CpCr  117  16 49  CpCrNO CpCrNOCD  147 165  17 100  CpCr(NO)  177  Rearrangement ions: NCD H 5 C  9 g  5 10 13 14 16  97  11  Cr 2 D 4 H 6 C  132  26  9orll  CpCrO  133  14  12  Molecular ion CD 2 CpCr(NO)  195  13  17  ¶ After a short delay time, this ion intensity increases to over 10%.  —35  —  9  c;1 C 1  0.0  Figure 2.5  0.2  0.4 0.6 Time (second)  0.8  1.0  The temporal behaviors of products from the condensation reactions of Cr with the parent molecule 3 CH Operating parameters are those 2 CpCr(NO) . in Table 2.1 and the sample nominal pressure  =  7 Torr. Cr is the 6.4x10  primary ion; CpCr O is the secondary ion; 3 2 C 2 Cp ( NO)O r is the tertiary  ion; 4 Cr and 2 3 Cp O C 3 Cp ( 4 0 NO) r + are the quarternary ions; C 4 Cp ( 5 3 NO)0 r is the quinary ion.  —36  —  7 6 5 c;1  C  4 3 2 1 0 Time (sec)  Figure 2.6  The temporal behaviors of products from the condensation reactions of Cr with the parent molecule 3 CD Operating parameters are those 2 CpCr(NO) . in Table 2.1 and the sample nominal pressure  =  7 Torr. Cr is the 6.5x10  primary ion; CpCr 4 is the secondary ion; 3 O 2 C 2 Cp ( NO)O r is the tertiary ion; 4 Cr and 2 3 Cp O C 3 Cp ( 4 O NO) r are the quarternary ions; C 4 Cp ( 5 NO)O r is the quinary ion. The relative ion intensity of CpCrO (m/z 185) has been calibrated for the isotopic peak H 10 C C 3 D 7 r (m/z 185).  —  37  —  there was a non-identical unlike-ion interaction for CpCrNOCH ions relative to the other ions. This interaction will be discussed in Chapters 4—6. The resolution of the FT-ICR mass spectrometry was not very good for ion mass range over 400 Daltons. Therefore, the large errors (over 40 ppm) in mass measurements of ions CpCrNOCH, 3 C Cp ( NO)O, r Cr 3 Cp O 4 ,4 C 3 Cp ( NO)O, r and 5 C 4 Cp ( NO)0 r can be expected. The ion-molecule condensation products are deduced from precision mass measurements for the compositions of the product ions; the strong Cp-Cr bond (Cp-metal bond is generally inert to both nucleophilic and electrophilic reagents.( )). The neutral 74 products in the following reaction equations are suggested because of their stabilities, and their simple compositions. 2.3.1. CrReactions 2.3.1.1 Charger transfer and electrophiic additions of Cr Charge transfer between ion and molecule followed by subsequent dissociation of the molecule is very common in MS-MS experiments.( ) Using triple resonance, charge 75 transfer reactions between Cr and the parent molecule are confirmed as in Eq. (2.4). NCD + Cr + Cr0 + NO H 5 (C CP  +  CD 2 CpCr(NO) 3  CpCrNOCD  +  Cr + NO  CpCr(NO) + Cr  +  (2.4)  3 CD  177  The mass-to-charge ratio (mlz) of every ion is indicated just below its formula. The Cr in CpCrNOCD has a 14-electron structure and the Cr in CpCr(NO) a 16-electron structure. These two ions are probable ion fragments, as they are even-electron ion species and the metals in them are close to the 18-electron structure. Their subsequent reactions are discussed in the individual sections for each of these two primary ions. Charge transfer Cr, and CpCr0 with the parent 2 D 4 H 6 was found in the reactions of CpCr, CpCrN0, C  —38—  Table 2.4  CH 2 CpCr(NO) The identification for the ions from 3  Error (ppm)  Calculated miz  Measured mlz  94.065126  94.064138  CpCr  116.979086  116.981384  +  Cr H 6 C  129.986911  129.990027  +23.98  CpCiO  132.974001  132.971305  —20.27  CpCrNO  146.977075  146.982775  +  CpCrNOCH  162.000550  161.987292  —81.84  CpCr(NO)  176.975064  176.979096  +  H CpCr(NO) C 2  191.998539  191.996018  Cr 2 Cp  182.018212  182.014131  —22.42  O 2 CPCr  184.914511  184.919162  +25.15  NO 2 CpCr  198.917585  198.910078  —37.74  Cr Cp O 2  249.953636  249.959373  +  r C 2 Cp O H  250.961461  250.954970  —25.86  Cr Cp O 2  265.948551  265.941574  —26.23  r C 2 Cp N OCH  278.980185  278.970193  —35.81  r 3 C Cp N 2 ) O(CH  294.003660  294.002970  H r C Cp ( 2 C NO)  308.978174  308.981741  r CH 3 C Cp ( ( 2 ) NO)  324.001649  323.998438  r 3 C 2 Cp ( NO)O  331.892134  331.891092  r C 3 Cp ( NO)O  396.93 1259  396.901698  Cr 3 Cp O 4  450.863610  450.836560  r 2 C 3 Cp ( 4 NO)0  464.866684  464.824477  r 3 C 4 Cp ( 5 NO)0  597.841233  597.894157  Stoichiometry of ion  N 8 H 6 C  ¶ Calculated ion mass has been calibrated for electron mass.  10.50  —  19.65  38.78  22.78 13.13  —  22.95  —  2.35  11.55  +  —  9.91  —3.14 —  74.47  —59.99 —  +  90.79 88.53  —39—  molecules as well. In this chapter, we are concerned with the condensation chemistry such as electrophilic additions by  shown in Eq. (2.5), where the dominant reaction is marked  byastar*. O 2 CPCr +  +  CD 2 CpCr(NO) 3 —  +  0 2 N  +  *  .033  (2.5)  3 NO + NO + •Q3 2 cpCr 199  The generation of CpCr O and CpCr 2 NO showed that a linear NO ligand in 2 CD became bent under the attack of Cr. The reason that the yield of the 2 CpCr(NO) 3 O(oxygen)-bridge product, CpCr O, predominates over the yield of the N(nitrogen) 2 bridge product, CpCr NO, could be attributed to the extra lone pair of electrons on the 2 oxygen of bent NO ligand (see Figure 2.7). This process is comparable to the reaction of Fe with Fe(CO) 5 in which the CO served as a 4-electron bridge ligand.( ) 45  CpCr Reactions O 2.3.1.2 2 O 2 CpCr  CD 2 CpCr(NO) 3  +  —*  C 2 Cp ( 3 NO)O r  185  +  NO  +  . 3 •CD  3 + NO + N 2+ f CpCrO 451  I C 2 Cp ( 3 NO) rO  (2.6)  332  +  CD 2 CpCr(NO) 3  Cr4NO) O + NO 3 Cp  +  3 CD 3 CD  465  C 3 Cp O 4 + 2 DCN+D +N r  332  (27)  467  C 3 Cp ( 4 O 2 NO) r  +  479  Cr + 4 3 Cp O 4 CD -, 5 2 CpCr(NO) C 4 Cp ( NO)O r 451  C 3 Cp ( 4 2 NO)0 r 465  +  NO  NO + +  . 3 CD  (2.8)  598 +  CD 2 CpCr(NO) 3  —,  C 4 Cp ( 5 NO)O r  +  O 2 N  +  . 3 CD  (2.9)  598  Cr (m/z 467, close to mlz 465) cannot be determined by the 3 Cp O Subsequent reaction of 4  -40--  Cr  (a) 0-bridging  (b) N-bridging  Figure 2.7  Two proposed mechanisms for the bent nitrosyl ligand in the Cr-molecule condensation reaction. (a) 0-bridging to Cr; (b) N-bridging to Cr.  Cr 3 Cp O multiple resonance at present due to insufficient mass resolution. Because 4 r (mlz 465) and there C 3 Cp ( N0)0 Cr (mlz 451) and 4 Cp 3 O shows a lower reactivity than 4 r (m/z 483) whose intensity grows more slowly than that of C 3 Cp C 0 D is an ion species 4  —41—  r C 3 Cp O -molecule r (m/z 479) as shown in Figure 2.8, the product of 4 NO) 2 C 3 Cp ( 4 O D r 2 C 3 Cp C O 4 reaction could be .  3.0 2.5  b  1.0 0.5 0.0  0.0  0.2  0.4  0.6  0.8  1.0  Time (second) Figure 2.8  D. r C 3 Cp C O r and 4 4 C 3 Cp The temporal behaviors (solid lines) of O Operating parameters are those in Table 2.1 and the nominal sample pressure  =  7 Torr. The chemical reactivity of 4 6.5x10 Cr is less than 3 Cp O  D r C 3 Cp C O r and 4 r Ion intensity of 4 4 C 3 Cp those of O C 3 Cp ( NO)O. . r NO) C 3 Cp ( 4 O grows more slowly than that of 2  2.3.1.3 CpCr NO Reactions 2  0 CpCr N 2 199  +  D 3 C 2 CpCr(NO)  -  r NO)O 3 C 2 Cp ( 332  Eq. (2.7)  +  O 2 N  +  3 •CD  (2.10)  —42—  The condensation of CpCr NO includes addition of an oxygen ligand. Conversely, the 2 condensation of CpCr O includes addition of a nitrosyl ligand. 2  Cr and 3 Cp O 4  C 3 Cp ( 4 NO)O r have the same tendency. The products in Eqs. (2.6)—(2.1O) indicate that only when the positive charge moves to the Cr unconnected to Cp, can the ion-molecule condensation proceed. This is apparently due to spatial freedom of the Cr unconnected to Cp. Without exception, the largest cluster always comes from the reaction of a bare metal ion with molecules because of its largest electron deficiency and unsaturated coordination number. The largest cluster here is the quinary ion, 5 C 4 Cp ( NO)O. r Typically, the clusters from the reactions of Cr with the parent molecule, 3 CD are of the form CpCr(NO)O, where 2 CpCr(NO) , m=x+y,andn=m+ 1. 2.3.2 CpCr Reactions 2.3.2.1 Charger transfer and electrophilic additions of CpCr Charge transfer from CpCr to the parent molecule produces CpCrNOCD and CpCr(NO). The condensation processes of CpCr with the parent molecule are quite  similar to those of the Cr-molecule reactions in previous section: Cr + N (Cp 0 2 0 + cDD 2 3 250 1 DCr + N 4 H 5 CpC O 2 0 + •CHD 2 I2  CpCr  +  CD 2 CpCr(NO) 3  117  —  251  C Cp O 2 D r+N 2  +  O 2 CD  *  (2 11)  *  252  C Cp N 2 O r + NO 264  +  3 cD  In the condensations of CpCr with 3 CD 0-bridging is still favored over 2 CpCr(NO) , N-bridging from 2 Cr [Cp O ]  +  DCr 4 H 5 [CpC O 2 9  +  C [Cp O 2 D] r  >>  C [Cp N 2 O]. r  —43—  DCr (m/z 251) was 4 H 5 CpC O Moreover, a-hydrogen transfer from the methyl to form 2 observed in the CpCr-molecule reaction. 2.3.2.2 2 Cr Cp O ,2 DCr 4 H 5 CpC O , and 2 C Cp O D r Reactions Resolution of our multiple resonance experiments is not accurate enough to distinguish these three secondary ions because their masses are very close. Their reactions are discussed together. DCr * 4 H 5 Crt CpC 2 Cp 183 182 Cp Cr O C 3 Cp ( NO) rO 250 + D 4 H 5 CpC O C 2 — 3 +CpCr(NO) Cr D D 4 H 5 C 2 Cp ( 3 NO) Cr 0 251  (2.12)  398  +  C Cp ( 3 N0)O rD  252  399  The tertiary ion Cp Cr is known to be very stable.( 2 ) 76 2.3.2.3 2 C Cp N O r Reactions O+M 2 (CpCr N H 5 +C C Cp N 2 O r+ 3 CD 2 CpCr(NO) 264  —  I  185  C 2 Cp ( 3 NO)OC rD  +  N H 5 C  +  NO  (2.13)  350  where M represents the parent molecule. Reactions of CPCr C) have been given in Section 2 2.3.1.2. The intensity of 3 C 2 Cp ( NO)OCD r is weak and its subsequent reaction pathways are uncertain. 2.3.3 CpCrNO Reactions CD during coffision with 2 CpCr(NO) CpCrNO preferred to transfer its charge to 3 CD to produce CpCr (which may come from a dissociation of CpCrNO), 2 CpCr(NO) 3  CpCrNOCD and CpCr(NO). It did not show obvious condensation chemistry.  -44-  2.3.4. CpCrNOCD, CpCr(NO), and CpCr(NO) CD Reactions 2 These primary ions have similar reaction pathways and are discussed together.  CpCrNOCD is the most abundant primary ion. Consequently, bimetal complexes are major products in the condensation chemistry of 3 CD 2 CpCr(NO) .  Cr + NO + N C O 6 D 2 2+C fI 2 266 Cr + NO Cp D O 2  I  CpCrNOCD + 3 CD 2 CpCr(NO) 165  I  +  2 N  +  5 D 2 •C  268  C Cp N 2 0CD r + 2N0  +  3 •cD  *  282  C Cp N 2 ) 3 O(CD r + 2N0  *  (2.14)  300  C Cp ( C 2 NO) D r + NO + D 3 312  C ( ( 2 ) 3 NO) CD r + NO 330  These product ions in Eq. (2.14), except 2 C Cp N OCD r (mlz 282), reacted slowly with the parent molecules ( three second reaction time under the pressure range of 6.2  —  6.8x iO- Torr) to produce more stable 3 C Cp ( O 2 ) CD r (mlz 318) and a deuterated ion of m/z 286 (more precisely, m/z  =  285.97) whose composition remains to be proved yet.  -C ( C ) 4 H 5 D N C r (m/z Several possible compositions exist for the latter ion, such as 2 C Cp N 3 C D r (mlz 285.976) etc. -C 4 ( C ) 6 H 5 D 3 O r (m/z 285.973), and 2 285.964), 2 The molecular ion CpCr(NO) CD can fragment to CpCrNOCD and CpCr(NO) 2 through collisions with the parent molecule. Condensation of the molecular ion with the parent molecule occurs in the same way as does CpCrNOCD, except there is one more CD reaction pathways 2 NO in the right-hand of Eq. (2.14). CpCrNOCD and CpCr(NO) were also confirmed at ionization threshold (nominal 9.7 eV). At the ionization threshold, CD, and 2 the mass spectrum showed only two mass peaks, CpCrNOCD and CpCr(NO) their condensation products did not change with higher electron ionization energy.  —45—  Since, there is no methyl ligand in CpCr(N0),  the condensation reactions of  CpCr(N0) with the parent molecule are deduced as: 2 + ‘CD 3 Cr + 2N0 + N (CP 0 2 266 I 0D + N 2 CpCr 0+N 2 2 + CD 20 268k  CD 2 CpCr(N0)+CpCr(N0) 3  (2,15)  282  C Cp ( C 2 N0) D r + 2N0 312  CD 4 H 5 C C r Reactions 2.3.5 2 CD charge 2 CpCr(N0) , During collision of 2 CD 4 H 5 C C r with the parent molecule, 3 transfer occurred to produce Cr (which may come from a dissociation of 2 Crj, 4 H 5 C C D CpCr, CpCrNOCD, and CpCr(N0).  Another more complicated reaction of  CD 4 H 5 C C 2 r with the molecule was:  CD 4 H 5 C C 2 r  +  CD 2 CpCr(NO) 3  132  —  CrC 2 4 C D 5 H 6  +  Cr  +  2N0  +  . 2 D  (2.16)  211  Similar reactions (both the fragmentation and condensation) were also observed in the ionmolecule chemistry of (r -benzene)Cr with 77 6 -benzene)Cr(C0) 6 (r . 3 ) ( Therefore, the composition of this primary ion seems to be (i7 -benzene)Cr. 6  2.3.6 CpCr0 Reactions Because ion mass of 2 CD 4 H 5 C C r (m/z 132) is very close to that of CpCr0 (m/z 133), the reactions of CpCr0 with the protium molecule 3 CH were monitored 2 CpCr(NO) using triple resonance. Products from charge transfer between CpCr0 and the parent molecule, 3 CH are CpCrNOCH and CpCr(N0). Condensation reactions of 2 CpCr(N0) , CpCiO with the molecule are:  —46— C (Cp O 2 CH r + 2N0 265  +3 CH 2 CpCr(NO) 133  Cr + N Cp H O 2 2  +  O 2 CH  (2.17)  267  C Cp C O 2 H r+ N 2  +  OH  280  2.4  Ion-Molecule Chemistry of the Negative Ions Negative ion chemistry of 3 CH is simpler than its positive ion chemistry. 2 CpCr(NO)  There are only four primary negative ions from 3 CD CpCrO, CpCrNOCD, 2 CpCr(NO) : CpCr(NO), and CpCr(NO) CD (Table 2.5) formed with 1.2 eV electron ionization. The 2 nitrosyls in these primary ions are considered to be linear. The negative charge in either CpCr(NO) or CpCr(NO) CD may delocalize onto the two NO ligands.( 2 ) 78  c7 30 H  Table 2.5  The primary negative ions from 3 CD (1.2 eV 2 CpCr(NO) electron ionization and nominal pressure  Primary negative ion CpCrO (superoxide) CpCrNOCD CpCr(NO) CD 2 CpCr(NO)  0  mlz  =  7 Torr) 6.5x10  Relative intensity  (%)  Electron count for Cr  149  2  13  165 177  100 5 24  16  195  ¶ If the negative charge is delocalized onto the dinitrosyl.  17 18  ‘  —47  —  CpCrO and CpCr(NO) underwent electron transfer reactions to the parent molecule, 3 C 2 CpCr(NO) . D The reactions of CpCrNOCD with the molecule gave rise to a mononitrosyl chromium complex and a bimetallic condensation product. CpCrO + CpCr(NO) 3 C 2 D  CpCr(NO) C 2 D  —  149  +  ] 2 [CpCrO  (2.18)  +  [CpCr(NO) ] 2 .  (2.19)  195  CpCr(NO) + CpCr(NO) 3 C 2 D  CpCr(NO) C 2 D  -*  177  195  ) + [CpCr(NO )2]• 3 (cpcrNO(cD CpCrNOCD  +  183  3 C 2 CpCr(NO) D  (2.20)  C Cp ( 2 C NO) D r  165  +  3 DD  *  342  17-electron radical CpCrNO(CD ) should possess a “three-legged piano stool” molecular 3 structure, like those of the 17-electron mononitrosyl chromium complexes, CpCrNO(L)I (L =  , P(OPh) 3 PPh , P(OEt) 3 ) and ) 3 79 CpCrNO(PP ) S 2 . 3 CH iMe ( h Triple resonance experiment showed that some 2 CpCr(NO) C D ions decomposed to  produce CpCrO and CpCrNOCD.  The temporal variation of the 2 CpCr(NO) C D  intensity displayed in Figure 2.9 shows that two kinds of 2 CpCr(NO) C D ion species CpCr(NO) C D* and ground state 2 probably exist: excited state 2 CpCr(NO) C D (The unusual temporal behavior (decrease and then increase) was reproducible and measured four times (Table 2.9)).  We believe that the excited state negative molecular ion is a  CpCrO ion source (Eqs. (2.21)). There are three possible decay pathways for the excited CpCr(NO) C D*ion: radiation decay, unimolecular decomposition, and bimolecular state 2 ) Further studies are needed. 80 reaction.( CpCr(NO) C 2 D* 195  —  CpCrO  +  149  .1. Eq. (2.18)  2 N  +  . 3 CD  (2.21a)  —50— d  = —  k’ [A]  (2.24)  where  k’  k [MJ.  (2.25)  Integration of Eq. (2.24) gives in [Ak]  =  —  , 0 k’ t + in [A]  (2.26)  where [Aj 0 is the initial intensity of A. In order to correct for fluctuations in ion intensity measurement of FT-ICR, the normalized ion intensities [Aj 0 and [Ak] are used to plot against time. If [M] is calibrated by the method suggested in Section 2.2.3 TT Pressure Calibration’, the rate constant of an ion-molecule reaction can then be calculated,  k  =  k’ / [M].  (2.27)  Here we follow the method given by Meckstroth, Ridge, and Reents( ) to measure the rate 43 constants for each ion from the decay portion of each temporal intensity curve. The decay portions of the curves were found to give satisfactorily linear logarithmic plots. The rate constants, averaged from six measurements, of the primary positive ions from 3 C 2 CpCr(NO) , H are given in Table 2.6. Since gas-phase reactions can be conveniently studied by measuring partial pressures of the reactants and products, the units of the rate constants given here are 1 •sec 1 Torr’sec Torr . 1  =  17 . 3.1x10 •molecules 3 cm s 1 ec  In Table 2.6, the uncertainty of the rate constant measurements for highly reactive ions is less than ± 10- 15%. The big errors in the rate constants of CpCrNO and CpCr(NO) are due to their low ion intensity. It is interesting to note that the reaction rates of the primary positive ions in which Cr has an even electron structure are not proportional to their electron deficiencies: CpCr(NO) (16-electron) is not more active than 2 CpCr(NO) C H (17-electron), CpCrO (12-electron) is not more active than CpCrNO (13-electron), and  —51—  the reactivity of CpCr (10-electron) is almost equal to that of 6 C C H r (11-electron). Although a smooth relationship curve between the rate constants of these primary ions and their electron deficiencies, plotted in Figure 2.10, also can be found, as proposed by Wronka and Ridge,( ) there is at least a 1 electron error in the electron deficiency model. 29 The rate constants of all positive condensation ions from CpCr(NO) 3 C 2 H shown in Eq. (2.5)  —  Eq. (2.17) are listed in Table 2.7. The measurement errors are calculated at  95% probability. The precision of rate constant measurements is less when the reactivities of the ions listed in Table 2.7 are less or when there is poor mass resolution in the high ion mass region. For comparison, the rate constants of all positive ions from CpCr(NO) 3 C 2 D are listed in Table 2.8. There is little difference between the rate constants of ion 3 C 2 CpCr(NO) H reactions and those of 3 ion-CpCr(NO C 2 D) reactions. The relative rate constants in Table 2.6 are independent of the pressure calibration accuracy.  Table 2.6 The experimental rate constants of the primary positive ions from 3 C 2 CpCr(NO) H (25 eV electron ionization) k x i0 Torr•sec  Positive primary ions  Relative rate (no unit) Average Error  1  2  3  4  5  6  C?  23.4  23.7  23.8  22.7  24.0  21.6  23.2  ±0.9  1.000  CpC?  19.3  20.3  16.8  19.5  21.2  19.1  19.4  ±1.5  0.836  C C H 6 r  19.9  20.1  16.3  18.6  19.0  20.1  19.0  ±1.5  0.819  CpCrO  16.7  18.4  14.8  18.6  16.0  15.4  16.6  ±1.6  0.716  CpCrNO  16.9  20.8  15.8  19.0  22.9  15.9  18.6  ±3.0  0.802  CpCrNOCH  11.2  10.2  9.45  10.2  ±0.9  0.440  9.26  9.59  11.3  CpCr(NO)  5.00  7.47  6.56  8.32  9.06  4.97  6.9  ±1.8  0.30  CpCr(NO) C 2 H  9.71  8.94  7.21  7.43  9.95  8.05  8.5  ±1.2  0.37  —52—  19.5  19.2  18.9  Ink 18.6  18.3  18.0  17.7 0  2  4  6  8  10  12  14  Electron Deficiency  Figure 2.10  Plot of in rate constants vs. electron deficiency for the primary ions from CH 2 CpCr(NO) . 3  2.5.2 Rate Constants of the Negative Ions CD 2 CpCr(NO) The rate constant measurements of the negative ions produced from 3 using the electron ionization, measured four times, are given in Table 2.9.  For  CH 2 CpCr(NO) comparison, the rate constants of the corresponding negative ions from 3 are given in Table 2.10. The decay rate constant of the excited state negative molecular ion, CD*, cannot be determined, since its reaction mechanism is not resolved as yet 2 CpCr(NO) (please see Eqs. (2.12a), and (2.21b)).  —53—  Table 2.7 The experimental rate constants of the positive condensation ions from CH (25 eV electron ionization) 2 CpCr(NO) 3  m/z  7 Torr•sec k x10  Positive product ion  1  2  3  Average  Error  185  O 2 CpCr  11.8  12.7  12.5  12.3  ±1.2  199  NO 2 CpCr  11.0  11.2  11.4  11.2  ±0.5  250  Cr Cp O 2 ’  9.28  9.06  9.21  9.2  ±0.2  251  C Cp O 2 4 H r  1.76  2.06  2.22  2.0  ±0.5  264  Cr,NO 2 Cp  1.63  1.78  1.90  1.8  ±0.4  265  C Cp O 2 CH r  1.82  1.63  1.48  1.6  ±0.4  266  Cr Cp O 2  0.40  0.35  0.47  0.4  ±0.1  267  Cr Cp W O 2  0.23  0.12  0.37  0.2  ±0.2  279  C Cp N 2 OCH r  0  280  C Cp C O 2 H r  0  285  0  294  C Cp N 2 O(CH r  0.11  0.16  0.53  0.3  ±0.6  309  C Cp ( C 2 NO) H r  0.034  0.15  0.23  0.1  ±0.2  312  C CP ( O 2 ) 3 CH r  324  C Cp ( ( 2 N0) CH r  332  C 2 Cp ( 3 NO)O r  7.36  8.32  7.23  7.6  ±1.5  347  C 2 Cp ( 3 NO)OCH r  2.35  2.27  1.89  2.2  ±0.6  397  C Cp ( 3 NO)O r  398  C Cp ( 3 NO)OW r  451  Cr 3 Cp O 4  6.15  6.89  4.08  5.7  ±3.6  465  C 3 Cp ( 4 2 NO)0 r  4.96  3.90  2.33  3.7  ±3.3  467  Cr 3 Cp O 4  3.63  3.66  2.94  3.4  ±1.0  479  C 3 Cp ( 4 0 2 NO) r  0  481  C 3 Cp C O 4 H r  0  598  C 4 Cp ( 5 3 NO)0 r  0  0 <<0.05  <0.05 <<0.05  —54—  Table 2.8 The experimental rate constants of the positive condensation ions from  cD (25 eV electron ionization) 2 CpCr(NO) 3  m/z  Positive  7 Torr•sec kx10  m/z  primary ion  Positive  7 Torr•sec kx10  product ion  52  Cr  23.3  185  O 2 CpCr  117  16.7  199  NO 2 CpCr  132  CpCr Cr 2 D 4 H 6 C  20.7  250  Cr Cp O 2  10.1  133  CpCrO  16.8  251  DCr 4 H 5 CpC O 2  10.3  147  CpCrNO  16.6  252  C Cp O 2 D r  2.00  165  CpCrNOCD  9.75  264  C Cp N 2 O r  177  CpCr(NO)  8.64  268  C Cp O 2 CD r  1.35 ¶  —  195  CD 2 CpCr(NO)  8.63  266  Cr Cp O 2  0.21  268  Cr Cp D 0 2  —  282  C Cp N 2 OCD r  0  282  C Cp C O 2 D r  0  286  12.1  8.83  ¶  300  C Cp N 2 O(CD r  0 0.21  312  C Cp ( C 2 NO) D r  0.098  318  C Cp ( 3 O 2 CD r  0  —  330  <<0.05 6.80  332  C 2 Cp ( 3 NO)O r  350  C 2 Cp ( 3 NO)OCD r  397  C Cp ( 3 NO)O r  <0.05  398  D 4 H 5 C 2 Cp ( 3 NO)O Cr  <0.05  399  C Cp ( 3 NO)OD r  451  Cr Cp 3 O 4  5.40  465  C 3 Cp ( 4 2 NO)0 r  4.08  467  Cr 3 Cp O 4  2.71  479  C 3 Cp ( 4 0 2 NO) r  0  483  C 3 Cp C O 4 D r  0  598  C 4 Cp ( 5 3 NO)0 r  0  2.64  <<0.05  ¶ Rate constants of these two ions are not given because of their same nominal mass.  —55  Table 2.9  —  The experimental rate constants of the primary negative ions from  3 C 2 CpCr(NO) D (1.2 eV electron ionization)  m/z  7 Torr•sec k x10  Negative ions 1  2  3 8.82  4  Average  149  CpCrO  7.95  9.29  165  CpCrNOCD  8.73  9.74  177  CpCr(NO)  8.86  195  CD 2 CpCr(NO)  195  CD 2 CpCr(NO)  0  183  CpCrNO(CD  0  342  3 C CP ( C 2 NO) D r  0  Table 2.10  10.1  10.1  10.6  Error  7.67  8.4  ±1.2  9.95  9.6  ±1.0  10.0  ±1.2  10.4  —  The experimental rate constants of the primary negative ions from 3 C 2 CpCr(NO) H (1.2 eV electron ionization)  m/z  Negative  7 Torr•sec kx10  ,,,  primary ions  Negative  7 Torr•sec kx10  product ions  8.2  192  CH 2 CpCr(NO)  0  CpCrNOCH  10.3  177  ) 3 CpCrNO(CH  0  177  CpCr(NO)  10.0  339  C Cp ( 2 C NO) H r  0  192  CH 2 CpCr(NO)  149  CpCrO  162  -  —56— 2.6  Proposed Ion Structures  2.6.1  Cluster Ions from the &-Molecule Condensation Reactions All these reactions are written in Eq. (2.5)  —  (2.10) of Section 2.3.1. CpCr O and 2  CpCr N 2 O are the smallest condensation complexes here, whose structures have been proposed in Fig. 2.7. A series of interesting oxo compounds are produced from these two ions, which are as follows: ( Cr 3 Cp O , 4 C 3 Cp ( Cr 3 Cp O NO)O, 3 C 2 Cp NO)Ot r 4 , r 4 2 C 3 Cp ( 4 O C 3 Cp C O NO) H, r r and ( , 4 5 C 4 Cp NO)O. r  The largest known neutral  chromium cluster complex is pentanuclear J [Cp ( ) 3 C 2 SCMe S r in which four sulfurs serve as J13 bridge ligands to connect five chromiums with a “bow-tie” frame.( ) If 83 3 C 4 Cp ( 5 NO)0 r has a similar structure, its precursors ( Cr 3 Cp O , and 3 C 2 Cp NO)Ot r 4 2 C 3 Cp ( 4 NO)0 r can be suggested as those in Figure 2.11. The four Cr atoms in both 4 C 3 Cp O r and ( 4 C 3 Cp NO)O r may have a  “i”  skeleton like a shovel to which can be  added one more Cr to form a bow-tie frame. In the shovel skeleton, the positive charge is localized on the central Cr with no Cp ligand as an active center. Successively, the three Cr atoms in ( 3 C 2 Cp NO)O r could be of a triangular skeleton. With reference to Bottomley’s work on oxo chromium cubanes O Cr and ) Cp 4 84 C Cp ( ) 4 H 5 , 3 C ( 0 r O 4 C 3 Cp r was never found in the mass spectra of these two oxo compounds, but a strong O Cr mass 3 Cp 4 peak was observed.( ) Bottomley’s work provides a clue to the structures of 8485  4 C Cp 3 O r and O 4 C 3 Cp r as found in our FT-ICR mass spectra: the four Cr atoms in 4 C 3 Cp O r would not form a tetrahedron, but the four Cr atoms in O 4 C 3 Cp r would. Thus, C 3 Cp C 4 O H r may have a cubane-like structure as its possible precursor, O 4 C 3 Cp r . The non-reactivity of O 2 C 3 Cp ( 4 NO) r shows that it may have a different structure from the shovel skeleton of 4 Cr and ( 3 Cp O 4 C 3 Cp NO)O. r O 2 C 3 Cp ( 4 NO) r is assumed to be tetrahedral. There may exist double metal-metal bonds in those clusters shown in Fig. 2.11. Before qualitative analysis of their molecular orbitals, the metal-metal bonds are indicated by single bold lines.  —57  —  CpC r.”C r  Cr Cp 3 O 4  3 C 2 Cp ( NO)O r  0  L  ;rCp  4 C 3 Cp ( NO)O r  5 C 4 Cp ( NO)O r 2 r=CH  Cp  Cp  Cr 3 Cp O 4  C 3 Cp C 4 O H r  2 C 3 Cp ( 4 0 NO) r Figure 2.11  Proposed structures of the cluster ions from the Cr+mo1ecule condensation reactions.  The metal-metal bonds are indicated by bold lines.  For  simplification, the metal-metal bonds in clusters O 4 C 3 Cp r and C 3 Cp C 4 O H r are not shown.  —58— 2.6.2 Cluster Ions from the CpCr-Molecule Condensation Reaction Reactions involved with CpCr are listed in Eq. (2.11)  —  (2.13) of Section 2.3.1 and  their proposed structures are drawn in Figure 2.12. 2 Cr and 2 Cp O C Cp N O r may have structures similar to those of CpCr O and CpCr 2 NO, shown in Fig. 2.7. Because the 2 rate constants of 2 C Cp O H r and N Cr by a factor Cp O C 2 Cp O r are lower than that of 2 of 5 (Tables 2.7 and 2.8), the 0 and H in 2 -bridging 2 C Cp O W r are assumed to be p  ligands (rather than a OH ligand) between the two chromiums in order to explain the lower reactivity of 2 C Cp O H. r  Then, the proposed structures of tertiary cluster ions  C Cp ( 3 NO)O r and ( C 3 Cp NO)OH r are presumed on the basis of those of 2 Cr and Cp O C Cp O 2 H. r 2.6.3 Other Bimetallic Cluster Ions The dimer chromium clusters are the major product ions in reactions described in Eq. (2.14)  —  Eq. (2.17). The oxo bimetallic 2 Cr is relatively inert toward condensation Cp O  (its rate constant is  -.  0.3 x  7 Torr). This Torr/sec at nominal pressure 6.5x10  a) 7 ( 2 9 and interesting ion has been found in the mass spectra of [CpCr(NO)(OMe)] C Cp . O 4 ) 85 ( r Bottomley et a!.( ) predicted that any cluster of form CpMetL, where L 86 is a ligand other than Cp, could exist, when the cluster satisfies the structural geometry of Euler’s polyhedron theorem:( ) 87 for every convex polyhedron the number V of its vertices plus the number F of its facesminusthenumberEofitsedgesisequalto2,i.e.,Vi-F—E  =  2.  0 forms a tetrahedron (Figure 2 Cr may be the smallest cluster, in which Cr CP O 2 2.13) similar to the non-planar four-membered ring Cr 0 in 2 2 C Cp ( O-t-Bu) r synthesized by Chishoim et a!. in 1979.(88) Furthermore, the oxidation states of the chromium atoms in Cr are +2 and +3. Although such a new i-i7 0 2 -dioxygen ligand has not yet been 2  —59—  reported, the u-n - ligands of other non-metallic elements (S, Se, and Te) in group VIB 2 have been ) 90 89 synthesized.( The most abundant metallic dimer in this work is the extremely stable ion, C 2 Cp N OCH. r  When two chromiums in a metallic dimer are Cr(II) and their  coordination numbers are five or six, a quadruple Cr-Cr bond is possible.( ) 91 C 2 Cp N OCH r probably possesses a quadruple metal-metal bond.  CPCr\/Cr  cPcr<>: Cr Cp O 2  C 2 Cp O H r  0 -  Cp CpCr\,/Crl +  C 3 Cp ( NO)O r  C 3 Cp ( NO)OH r Figure 2.12  3 C 2 Cp ( NO)OCH r  Proposed structures of the cluster ions from the CpCr-molecule condensation reactions.  -60--  cPcK:  Cft><;: p  pccp flT.1+ C _ _p _ 2 i .i i. r  C  Cr CP O 2  0  -r  3 H  2 H  C Cp N 2 OCH r  C Cp C O 2 H r  0  0  0 PCP 4 cPCr,  3 H  3 H  C Cp N 2 ) 3 O(CH r  C CP ( 2 23 N0) r  or  C Cp ( ( 2 ) 3 NO) CH r  CpCrZ Nrcp  C Cp ( O 2 ) 3 CH r  Figure 2.13  Proposed structures of all the other positive bimetallic cluster ions.  —61—  C Cp ( O 2 ) CH r are one containing a non-planar four Two plausible structures for 3 membered ring Cr 0 as in 88 2 C Cp ( , 2 ) O-t-Bu) ( r and one containing a planar four membered ring Cr 0 as in 92 2 -C [ M C 5 . ] 2 ) rO ( e The two proposed structures of this bimetallic ion are drawn at the bottom of Figure 2.13.  2.6.4 The Negative Ions Although the negative ion CpCrO is a fragment ion from 3 CH its 2 CpCr(NO) , structure is of interest. It can be an oxide, peroxide, or superoxide. The structure of the anionic dioxide Cr(CO) O was proved to be either a peroxide or a superoxide by the 3 Collision-Induced Dissociation (CD) method of FT-ICR.( ) The authors, Bricker and 93 Russell, preferred the superoxide structure. Therefore, the CpCrO probably is a superoxide, too, for the reason that mononuclear superoxo complexes are formed almost exclusively by metals of the first transition series because of formal one-electron oxidation requirement of the metal,( ) and the CpCrO-molecule reaction is an one-electron transfer 94 reaction (Eq. (2.18)). The proposed structure of the CpCrO is drawn in Figure 2.14 with C Cp ( C 2 NO) H, r the only bimetallic anion, in which every Cr has 16 electrons (assuming a Cr-Cr single bond).  00—i-  3 H CpCrO  Figure 2.14  C Cp ( C 2 NO) H r  Proposed structures of CpCr0 and 2 C Cp ( C NO) H. r  —62—  2.7  Discussion Cluster research is a very active field. Cluster structure has been recognized as a link  between gas and solid phases.( ) The enormous effort being focused on metal carbonyl 95 clusters is because such research pertains to applied areas, especially, to catalysis.( ) 97 Transition metal ion-molecule condensation chemistry in gas-phase is also valuable in metal vapor synthesis in organometallic 12 chemistry.( 98) Compared with carbonyl, nitrosyl ’ ligands show many unusual bonding features (Table 2.11, also see Appendix A1.2, Table A1.1 Coordination Modes of Carbon Monoxide and Table A1.2 Coordination Modes of Nitric Oxide). In contrast with the positive ion chemistry of transition metal carbonyls, the NO ligand can be broken up in the ion-molecule reactions of 3 CH Typically, 2 CpCr(NO) . oxygen is retained in the ionic product and nitrogen is lost as a part of the neutral (even dinitrosyl can be broken at the same time to form an oxo complex CpCrO). Evidently, transition metal nitrosyls are an important source for transition metal oxo complexes.( ) 99 Some results of this work have been presented at two international conferences.( ) 100 Gas-phase ions are difficult to characterize structurally and electronically. Even for transition metal carbonyls, the structure-reactivity relationship (electron deficiency model) suggested by Wronka and Ridge( ) on the basis of the 18-electron rule provides only an 29 indication of the number of metal-metal bonds or electron donation of the carbonyls. Because the condensation oxo complexes do not obey the 18-electron rule, and the ligands of these complexes vary greatly, their rate constants are very difficult to relate to electron deficiencies. No plot of rate constant vs. electron deficiency for the condensation complexes is attempted here. The ion-molecule condensation chemistry of 20 N0( is not as complicated 2 CpCr(CO) ) CH Since there is one more nitrosyl in CpCr(NO) 2 CpCr(NO) . as that of 3 3 C 2 H than in CpCr(CO) N 2 0, it is expected that more oxo compounds can be produced from the ion molecule condensation chemistry of . 3 C 2 CpCr(NO) H For example, oxo compounds  —63—  C 4 Cp ( NO)O r (Section 2.3), Cr Cp O 2 ,2 Cr Cp O ,4 Cr 3 Cp O ,2 C 3 Cp ( 4 O NO) r , and 5 CH 2 CpCr(NO) , which are produced from the ion-molecule condensation reactions of 3 were not found in those of 20 N0.( In the ion-molecule chemistry of 2 CpCr(CO) ) CH many unreacive bimetallic complexes were produced from the reactions 2 CpCr(NO) , 3 of primary ions CpCrNOCH, CpCr(NO), and molecular ion CpCr(NO) CH with the 2 parent molecules (Section 2.3.4).  In contrast, in the ion-molecule chemistry of  N0,( its molecular ion is unreactive, and only one unreactive bimetallic 2 CpCr(CO) ) 20 product, 2 C Cp ( C NO) O, r was observed. Advantages of gas-phase studies arise from the spatial dispersion of the organometallic species under high vacuum conditions employed. In gas-phase ionmolecule reactions, there is no solvent-shell interaction and associations by ion pairs. C N H 5 CH,( 101) 76 Cr,( H 2 Cp ) -C ( C 2 ) 6 r,( 102) and CpCr(NO) 8 C 2 H,( 1) which have been formed in solid or liquid state, were all found to be unreactive in the gas state without counter-ions or solvent-shell effect. Moreover, (r-C Cr produced from the reaction 2 ) H 6 of 6 (r ) H Cr C with 3 CH (Eq. (2.16)) shows that the Cp-ring may be 2 CpCr(NO) extended to form a benzene ring through the ion-molecule reaction. If possible, binary nitrosyls Cr(NO) 4 and Co(NO) , and cyclopentadienyl nitrosyls 3 , [CpFeNO] ] 2 [CpCr(NO) , and 4 2 M Cp ( 3 NO) n are recommended for further study of the ion-molecule condensation chemistry of nitrosyl transition metal complexes. Gas-phase ion chemistry of these nitrosyl complexes can be made a comparison with that of CH The intrinsic properties of coordination of NO ligand to transition metal 2 CpCr(NO) . 3 , and 2 , Co(NO) 4 , [CpCr(NO) 3 , [CpFeNOJ ] 2 atoms will be further revealed, since Cr(NO) M Cp ( 3 4 NO) n do not contain methyl or other ligands (except the inert Cp ligand in later three nitrosyl complexes). Nitrosyl behavior in transition metal complexes has been also studied recently by the CD method of FT-ICR( ) in which a triangular structure of 3 1031 ( Co NXJt OY’ j1 was proposed, and chemical ionization method in which carbonyl-nitrosyl condensation metal  -64-  clusters up to ( (NO)j 8 Fe ( CO) and pure nitrosyl metal clusters up to 8 ( 7 Co NO)a CO) and 3 (NO)j and Fe 6 Co (NO)j were reported.( 7 ) 105 Table 2.11 shows a summary of the important characteristics of carbonyl transition metal cluster ions and nitrosyl transition metal cluster ions. From carbonyl to nitrosyl, the coordination modes to metals become more complicated. 4-electron p -CO and multiple 2 metal-metal bond may be deduced from the ion-molecule condensation chemistry of transition metal carbonyls. For the nitrosyls, i -NO and 3-O are commonly proposed for 3 the metal clusters, and multiple metal-metal bonds possibly exist in the condensation products. Table 2.11  A summary of carbonyls and nitrosyls in ion-molecule condensation chemistry of transition metal complexes  Ligand  Carbonyl( 45) ’ 29  Proposed metal-metal bond  up to triple bond,  Proposed ligand bond type  terminal, p -CO, 2  mixed valent Nitrosyl  up to quadruple bond, mixed valent  -NO, u 2 /1 4 -NO, #2-° -dioxo, superoxo 2 -O, ?7 3 p  —65—  REFERENCES 1.  Dempster, A. 3. Phil. Mag. 1916,31, 438-443.  2.  Schumacher, E.; Taubenest, R. Helv. Chim. Acta 1964,47, 1525-1529.  3.  Hartmann, H.; Wanczek, K.-P. “Ion Cyclotron Resonance Spectrometry II”; Springer-Verlag: Berlin, 1982.  4.  McLafferty, F. W. “Tandem Mass Specirometry”; John Wiley & Sons: New York, 1983.  5.  Hanison, A. G. “Chemical Ionization Mass Spectrometry”; CRC Press: Roca Raton, Florida, 1983.  6.  Bowers, M. T. “Gas Phase Ion Chemistry”; Academic Press: Orlando, Volume 3, 1984.  7.  Ferreira, M. A. A. “Ionic Processes in the Gas Phase”; D. Reidel: Dordrecht, 1984.  8.  Lindinger, W.; Murk, T. D.; Howorka, F. “Swarms of Ions and Electrons in Gases”; Springer-Verleg: Wien, 1984.  9.  Fontijn, A. “Gas-Phase Chemiluminescence and Chemi-lonization”; North-Holland: Amsterdam, 1985.  10. Futrell, J. H. “Gas Phase Ion Chemistry and Mass Spectrometry”; John Wiley & Sons: New York, NY, 1986. 11.  Buchana, M. V. “Fourier Transform Mass Spectrometry: Evolution, Innovation, and Applications”; American Chemical Society: Washington, DC, 1987.  12.  Ausloos, P.; Lias, S. G. “Structure/Reactivity and Thermochemistry of Ions”; Plenum: New York, 1987.  13.  Suslick, K. S. “High-Energy Processes in Organometallic Chemistry”; American Chemical Society: Washington, DC, 1987, Chapters 2, 4, 18, 19.  14.  Farrar, J. M.; Saunders, W. H. Jr. “Techniques for the study of Ion-Molecule Reactions”; John Wiley & Sons: New York, 1988.  —66—  15.  Busch, K. L.; Gush, 0. L.; McLuckey, S. A. “Mass Spectrometry/Mass Spectrometry, Techniques and Applications of Tandem Mass Spectrometry”; VCH: New York, NY, 1988.  16.  Russell, D. H. “Gas Phase Inorganic Chemistry”; Plenum: New York, 1989.  17. Marks, T. 3. “Bonding Energetics in Organometallic Compounds”; American Chemical Society: Washington, DC, 1990, Chapters 2-5, and 18. 18. Foster, M. S.; Beauchamp, 3. L. J. Am. Chem. Soc. 1971, 93, 4924-4926. 19.  Comisarow, M. B.; Marshall, A. G. Chem. Phys. Lett. 1974,25, 282-283.  20. Parisod, G.; Comisarow, M. B. Adv. Mass Spectrom., 1980, 8A, 212-223. 21. Freiser, B. S. Chemiracts 1989, 1, 65-109. 22.  Sharpe, P.; Richardson, D. E. Coord. Chem. Rev. 1989, 93, 59-85.  23.  Marks, T. J. “Bonding Energetics in Organometallic Compounds”; American Chemical Society: Washington, DC, 1990, Chapters 3-5.  24.  Eller, K.; Schwarz, H. Chem. Rev. 1991, 91, 1121-1177.  25. Dunbar, R. C.; Ennever, J. F.; Fackler, 3. P. Jr. Inorg. Chem. 1973, 12, 27342736. 26.  Richardson, J. H.; Stephenson, L. M.; Brauman, J. I. J. Am. Chem. Soc. 1974, 96, 3671-3673.  27. Foster, M. S.; Beauchamp, 3. L. J. Am. Chem. Soc. 1975, 97, 4808-48 14. 28. Wronka, 3.; Ridge, D. P.  mt. J. Mass Spectrom. Ion Phys.  1982,43, 23-30.  29. Wronka, 3.; Ridge, D. P. J. Am. Chem. Soc. 1984, 106, 67-7 1. 30.  Allison, 3.; Ridge, D. P. J. Am. Chem. Soc. 1979, 101, 4998-5009.  31. Foster, M. S.; Beauchamp, 3. L. J. Am. Chem. Soc. 1975, 97, 48 14-4817. 32.  Corderman, R. R.; Beauchamp, 3. L. Inorg. Chem. 1976, 15, 665-668.  —67—  33.  Corderman, R. R.; Beauchamp, 3. L. Inorg. Chem. 1977, 16, 3135-3139.  34. Weddle, G. H.; Allison, J.; Ridge, D. P. J. Am. Chem. Soc. 1977, 99, 105-109.  35. Kappes, M. M.; Staley, R. H. J. Phys. Chem. 1982, 86, 1332-1333. 36.  Sallans, L.; Lane, K. R.; Squires, R. R.; Freiser, B. S. J. Am. Chem. Soc. 1983, 105, 6352-6354.  37.  Meckstroth, W. K.; Ridge, D. P. 152.  38.  Jacobson, D. B.; Freiser, B. S. J. Am. Chem. Soc. 1984, 106, 4623-4624.  39.  Jacobson, D. B.; Freiser, B. S. .1. Am. Chem. Soc. 1984, 106, 535 1-5352.  mt. J. Mass Spectrom. Ion Proc.  1984,61, 149-  40. Jacobson, D. B.; Freiser, B. S. J. Am. Chem. Soc. 1985, 107, 1581-1589. 41. Hettich, R. L.; Freiser, B. S. J. Am. Chem. Soc. 1985, 107, 6222-6226. 42.  Sallans, L.; Lane, K. R.; Squires, R. R.; Freiser, B. S. J. Am. Chem. Soc. 1985, 107, 4379-4385.  43.  Meckstroth, W. K.; Ridge, D. P.; Reents, W. D., Jr. J. Phys. Chem. 1985, 89, 612-617.  44.  Meckstroth, W. K.; B., F. R.; Reents, W. D., Jr.; Ridge, D. P. Inorg. Chem. 1985, 24, 3139-3146.  45.  Fredeen, D. A.; Russell, D. H. J. Am. Chem. Soc. 1985, 107, 3762-3768.  46.  Fredeen, D. A.; Russell, D. H. J. Am. Chem. Soc. 1986, 108, 1860-1867.  47.  Fredeen, D. A.; Russell, D. H. J. Am. Chem. Soc. 1987, 109, 3903-3909.  48. Jacobson, D. B.; Freiser, B. S. J. Am. Chem. Soc. 1986, 108, 27-30. 49. Huang, Y.; Bucker, S. W.; Freiser, B. S. in “Physics and Chemistry of Small Clusters”; Jean, P.; Rao, B. K.; Khanna, S. N., Ed.; Plenum Press: New York,  50.  1987; pp 891-896. Hettich, R. L.; Freiser, B. S. J. Am. Chem. Soc. 1987, 109, 3537-3542.  —68—  51.  Tews, B. C.; Freiser, B. S. J. Am. Chem. Soc. 1987, 109, 4433-4439.  52. Mullen, S. L.; Marshall, A. G. J. Am. Chem. Soc. 1988, 110, 1766-1774. 53.  Huang, Y.; Freiser, B. S. J. Am. Chem. Soc. 1988, 110, 4434-4435.  54.  Forbes, R. A.; Laukien, F. H.; Wronka, J. mt. J. Mass Spectrom. Ion Proc. 1988, 83, 23-44.  55. Houriet, R.; Vulpius, T. Chem. Phys. Lett. 1989, 154, 454-457.  56.  Strobel, F.; Ridge, D. P. J. Phys. Chem. 1989, 93, 3635-3639.  57.  (a) Comisarow, M. B.; Taylor, S. M. International Chemical Congress of Pacific Basin Societies, Honolulu, Hawaii, 1989, paper No.: ANYL 488. (b) Taylor, S. M.; Comisarow, M. B. Proceedings of the 38th ASMS Conference on Mass Spectrometry and Allied Topics, Tucson, Arizona, 1990, p 1258.  58.  Kerley, E. L.; Russell, D. H. J. Am. Chem. Soc. 1990, 112, 5959-5965.  59.  Operti, L.; Vaglio, G. A.; Gord, J. R.; Freiser, B. S. Organometallics 1991, 10, 104-111.  60.  Kan, Z.; Comisarow, M. B. Proceedings of the 39th ASMS Conference on Mass Spectrometiy and Allied Topics, Nashville, TN, 1991, pp 461-462.  61.  For recent reviews on nitrosyl complexes, see: (a) Richter-Addo, G. B.; Legzdins, P. Chem. Rev. 1988, 88, 991-1010. (b) Mingos, D. M. P.; Sherman, D. 3. Adv. Inorg. Chem. 1989, 34, 293-375. (c) Richter-Addo, G. B.; Legzdins, P. “Metal Nkrosyls”; Oxford U. Press: New York, 1992 (in press).  62.  Richter-Addo, G. B.; Legzdins, P. “Metal Nitrosyls”; Oxford U. Press: New York, 1992; pp 79-80 (in press).  63.  (a) Piper, T. S.; Wilkinson, G. Chem. and Idu. 1955,41, 1296. (b) Piper, T. S.; Wilkinson, G. J. Inorg. Nucl. Chem. 1956,3, 104-124.  64. Hoyano, J. K.; Legzdins, P.; Malito, J. T. J. Chem. Soc., Dalton Trans. 1975, 1022-1025.  —69—  65.  Lukehart, C. M. “Fundamental Transition Metal Organometallic Chemistry”; Brooks/Cole: Monterey, California, 1985; Chapter 2.  66. Johnson, B. F. G.; Haymore, B. L.; Dilmorth, 3. R. in “Comprehensive Coordination Chemistry”; Wilkinson, G.; Gillard, R. D. McCleverty, 3. A., Ed.; Pergamon Press: Oxford, 1987; pp 99-159. 67.  Bartmess, J. E.; Georgiadis, R. M. Vacuum 1983,33, 149-153.  68.  (a) Aroney, M.; Le Fèvre, R. 3. W.; Somasundaram, K. M. J. Chem. Soc. 1960, 1812-1814. (b) Le Fèvre, R. 3. W.; Murthy, D. S. N.; Saxby, 3. D. Aust. J. Chem. 1971, 24, 1057-1058.  69.  Waite, 3.; Papadopoulos, M. G. Z. Naturforsch. 1987, 42a, 749-752.  70.  Miller, T. M. in “CRC Handbook of Chemistry and Physics”; Lide, D. R., Ed.; CRC Press: Boca Raton, Florida, 1990-199 1; pp 193-209.  71.  Aroney, M. 3.; Cooper, M. K.; Pierens, R. K.; Pratten, S. J. J. Organomet. Chem 1985, 295, 333-342.  72.  Miller, K. J.; Savchilc, 3. A. J. Am. Chem. Soc. 1979, 101, 7206-7213.  73.  Aroney, M. 3.; Clarkson, R. M.; Klepetko, R. J.; Masters, A. F.; Pierens, R. K. J. Organomet. Chem 1990, 393, 37 1-378.  74.  Crabtree, R. H. “The Organometallic Chemistry of the Transition Metals”; John Wiley & Sons: New York, 1988; p 104.  75.  Busch, K. L.; Glish, 0. L.; McLuckey, S. A. “Mass Spectrometry/Mass Spectrometry, Techniques and Applications of Tandem Mass Spectrometry”; VCH: New York, NY, 1988; p 99.  76.  Robbins, 3. L.; Edelstein, N.; Spencer, B.; Smart, 3. C. J. Am. Chem. Soc. 1982, 104, 1882-1893 and references therein.  77.  Taylor, S. M. private communication, 1991.  78. Legzdins, P.; Wassink, B. Organometallics 1988, 7, 477-48 1.  —70  —  79.  (a) Legzdins, P.; Nurse, C. R. Inorg. Chem. 1985,24, 327-332. (b) Herring, F. G.; Legzdins, P.; McNeil, W. S.; Shaw, M. 3. J. Am. Chem. Soc. 1991, 113, 7049-7050.  80.  Mclver, R. T. Jr. in “Kinetics of Ion-Molecule Reactions”; Ausloos, P. Ed.; Plenum: New York, 1979, pp 255-270.  81.  Legzdins, P.; Wassink, B. Organometallics 1988, 7, 482-493.  82.  Powell, P. “Principles of Organometallic Chemistry”; Chapman and Hall: London, 1988; pp 148-152.  83.  Pasynskii, A. A.; Eremenko, I. L.; Orazsakhatov, B.; Gasanov, G. S.; Shklover, V. E.; Struchkov, Y. J. Organomet. Chem. 1984,269, 147-153.  84.  Bottomley, F.; Paez, D. E.; Sutin, L.; White, P. S.; Kohier, F. H.; Thompson, R. C.; Westwood, N. P. C. Organometallics 1990, 9, 2443-2454 and references therein.  85.  Bottomley, F. private communication, September, 1989.  86.  (a) Bottomley, F.; Grein, F. Inorg. Chem. 1982,21, 4170-4178. (b) Bottomley, F.; Paez, D. F.; White, P. S. J. Am. Chem. Soc. 1981, 103, 558 1-5582.  87.  Hazewinkel, M. “Encyclopaedia of Mathematics”; Kiuwer Academic Publishers: Dordrecht, Vol. 3, 1989, p 432.  88.  Chisholm, M. H.; Cotton, F. A.; Extine, M. W.; Rideout, D. C. Inorg. Chem. 1979, 18, 120-125.  89.  Herrmann, W. A.; Rohnnann, J.; Nöth, H.; Nanial, CH. A. J. Organomet. Chem 1985,284, 189-211.  90.  Herrmann, W. A.; Heinz-Josef, K. J. Organomet. Chem 1988,348, 193-197.  91.  Cotton, F. A.; Wilkinson, G. “Advanced Inorganic Chemistry”; John Wiley & Sons: New York, 1988; pp 679-686.  92.  Herberhold, M.; Kremnitz, W.; Abbas, R.; Schöllhorn, H.; Ulf, T. Angew. Chem. mt. Ed. Engi. 1985,24, 60 1-602.  —71—  93. Bricker, D. L.; Russell, D. H. J. Am. Chem. Soc. 1987, 109, 3910-3916. 94. Hill, H. A. 0.; Tew, D. G. in “Comprehensive Coordination Chemistry”; Wilkinson, G.; Gillard, R. D. McCleverty, 3. A., Ed.; Pergamon Press: Oxford, 1987; p 316. 95.  Shriver, D. F.; Kaesz, H. D.; Adam, R. D. “The Chemistry of Metal Cluster Complexes”; VCH: New York, NY, 1990.  96. Castleman, A. W. J.; Keesee, R. G. Acc. Chem. Res. 1986, 19, 413-419. 97. Jelski, D. A.; George, T. F. J. Chem. Ed. 1988, 65, 879-883. 98. Blackborow, J. R.; Young, D. “Metal Vapour Synthesis in Organometallic Chemistry”; Springer-Verlag: Berlin, 1979. 99. Bottomley, F.; Sutin, L. Adv. Organomet. Chem. 1988,28, 339-396. 100. (a) Comisarow, M. B.; Chen, S.-P. Proceedings of the 37th ASMS Conference on Mass Spectrometry and Allied Topics, Miami Beach, Florida, 1989, 323-324. (b) Comisarow, M. B.; Chen, S.-P. International Chemical Congress of Pacific Basin Societies, Honolulu, Hawaii, 1989, paper No.: ANYL 489. 101. Mosher, H. S. in “Heterocyclic Compounds”; Elderfield, R. C., Ed.; John Wiley & Sons: New York, Volume 1, 1950; pp 414-420. 102. Powell, P. “Principles of Organometallic Chemistry”; Chapman and Hall: London, 1988; . 135 p 103. Jacobson, D. B. J. Am. Chem. Soc. 1987, 109, 6851-6852. 104. Klaassen, 3. 3.; Jacobson, D. B. J. Am. Chem. Soc. 1988, 110, 974-976. 105. Torroni, S.; Foffani, A.; Innorta, G. Inorg. Chim. Acta 1991, 185, 103-117.  —72—  CHAPTER 3  REACTIONS OF THE FRAGMENT IONS OF CH WITH SMALL MOLECULES 2 CpCr(NO) 3 , 112, H 2 N 0, D 2 0, NH 2 , AND CH 3 4 IN THE GAS-PHASE  —73— 3.1  Introduction Study of the reactivity of gas phase ionic transition metal complexes with inorganic  and organic small molecules is another important application of FT-ICR mass spectrometry. When a second species such as H 2 is introduced with a transition metal complex into the FT-ICR mass spectrometer, reactions which differ from the condensation chemistiy occur (e.g., ligand substitution, oxidation-addition of atomic metal ions etc.). Of particular interest to chemists are the reactions between the second species and ions derived from the transition metal complex. The bare metal or bare metal cluster products from transition metal carbonyls reacting with small molecules have been studied frequently in order to understand metal-ligand and metal-metal binding energetics. Many studies have  been summarized in several recent reviews,( ) especially in a new review presented by 14 Eller and Schwarz in 1991,() where the literature has been covered up to early 1991. A trend this year in FT-ICR is to study transition metal reactivities with large molecules (molecular weight  300 Daltons), such as all the first-row transition metal ions and the  second-row transition metal ion Ag with tribenzocyclothyne,( ) Fe, Mn, and ligated Cr, 5 Fe, Co, Ni ions with porphine,( ) and Fe, Cow, Ni, Cu, Rh, La, and VO with 6 buckminsterfullerene (C),(7 ) 10 ) Fe with thpeptide val-pro-leu.( 9 CH (or 2 CpCr(NO) In this work, reactions of ions derived from the metal nitrosyl 3 , 3 ,H 2 0, NH 2 CD with small molecules were examined. Small molecules H 2 CpCr(NO) ) 3  and CH 4 have been chosen in the present work as the reactants, because all of these small molecules contain hydrogen atoms (from 2 to 4). N 2 was also used. In our experiments, 0, D 2 0, NH 2 , CH 3 , and N 4 2 were found not to react with the negative ions , H 2 H produced from 3 CH or 3 2 CpCr(NO) , CD Therefore, only the positive ion 2 CpCr(NO) . CH and 3 2 CpCr(NO) CD with the six small molecules are 2 CpCr(NO) chemistry of 3 discussed in this chapter. So far only a few studies have been carried out on gas-phase chemical behavior of ) When the cobalt nitrosyl ions reacted with 1114 cobalt nitrosyl ions with small molecules.(  —74—  small molecules CO or 02,(12_14) the NO ligand in the cobalt nitrosyl ions was completely replaced by CO or 02, or the N-O bond was broken. In the latter case, the nitrogen was retained by the cobalt atom, but the oxygen was lost. Obviously, gas-phase ion chemistry of metal nitrosyls with small molecules is a developing research area.  3.2  Experimental Section The instrumentation of the FT-ICR mass spectrometer has been described in Section  2.2.1 of Chapter 2. The ions were produced by electron impact on 3 CH at 25 2 CpCr(NO) eV energy. N ,H 2 , NH 2 , and CH 3 4 were commercial pure gases obtained from Matheson of Canada, Ltd. H 0 was obtained directly from distilled water and 99.9% purity heavy 2 water, D 0, was from Merck & Co., Ltd. Chromium nitrosyls 3 2 CH and 2 CpCr(NO) CD were provided by the research group of Professor P. Legzdins. 2 CpCr(NO) 3 Chromium nitrosyl 3 CH or 3 2 CpCr(NO) , CD was first introduced into 2 CpCr(NO) , the analyzer cell to a nominal pressure of 3.0 x iO- Torr. Then, a second molecular gas , H 2 H 0, D 2 0, NH 2 , CH 3 , or N 4 2 was added into the analyzer cell at a total nominal pressure 3.0 x 10 Torr. A high pressure of the second molecular species was used to obtain a high reaction efficiency and to avoid confusion with the condensation reactions discussed in Chapter 2.  The ion-molecule reactions of each small molecule with  CH were monitored from 0 to 5 seconds. In order to eliminate all reactions 2 CpCr(NO) 3 between ions produced from H , H 2 0, D 2 O, NH 2 , CH 3 , or N 4 2 with the molecules, a double resonance was performed during the beam time (ionization time) in which the resonance frequency for ejection was set to the corresponding resonance frequency of O (mlz 18), or D 2 H O (mlz 20), or NH (mlz 17), or CH (m/z 16), or N (m/z 28). 2 Because the resonance frequency of H (mlz 2), 14.6 MHz, can not be reached by the equipment at 1.9 Tesla magnetic field, no ejection was applied for H. However, it was found that there was no signfficant difference between the spectra with and without the  —75—  double resonance because productions of these primary ions from H ,H 2 0, D 2 0, NH 2 , 3 , and N 4 CH 2 are limited at 25 eV ionization energy.  3.3  Calibrations of the Nominal Pressures In Section 2.2.3 of Chapter 2, the chemical sensitivity of an ion gauge was shown to  vary with different gases, and so the true pressure of a gas is not the nominal pressure taken from a vacuum meter. According to Bartmess and Georgiadis,( ) the true pressure 15 of a sample x  —  x  nominal pressure of x R  3 1 (.)  where R is the relative sensitivity of the ion gauge with respect to the sample x. The chemical sensitivity of 3 CH has been estimated, in Section 2.2.3 of Chapter 2, 2 CpCr(NO) tobe  RCpcr(NO)Cp  =  7.1.  The molecular polarizabilities of N ,H 2 0, NH 2 4 have been reported as 1.00, ,H 2 , and CH 3 0.44, 0.97, 1.12, and 1.62  , respectively.( 3 A ) 15  As the partial pressures of N ,H 2 ,H 2 0 2  (or D 0), NH 2 , and CH 3 4 were much larger than the partial pressure of 3 CH in 2 CpCr(NO) our experiments, the total sampling pressures in the analyzer cell are taken as the pressures of N ,H 2 ,H 2 0, NH 2 , and CH 3 , CH 3 . The calibrated pressures of N 4 ,H 2 , H 2 0, NH 2 , 4 and 3 CH are calculated from Eq. (3.1) and shown in Table 3.1. 2 CpCr(NO)  —76—  Table 3.1  Calibrations of the sampling pressures of N , H 2 , H 2 0, NH 2 , 3 , and 3 4 CH CH 2 CpCr(NO)  Sample  Nominal pressure  Chemical sensitivity  Calibrated pressure  2 N  3.0 x 10Toff  1.00  3.0 x 10Toff  2 H  3.0 x 10 Torr  0.44  6.8 x 10 Torr  0 2 H  3.0 x 10Toff  0.97  3.1 x 10Toff  3 NH  3.0 x 10Torr  1.12  2.7 x 10Toff  4 CR  3.0x10Toff  1.62  1.8x10Toff  CH 2 CpCr(NO) 3  3.0 x i0 Torr  7.1  4.2 x iO Torr  3.4  Ion-Molecule Chemistry of 3 CH with Hydrogen, H 2 CpCr(NO) 2  3.4.1  Collisional Quenching and Charge-Transfer Reactions  All small molecules used in this work can quench the primary ion Cr from its excited state to its ground state. As a result, high nuclear metal condensation products which mainly came from the successive condensations of Cr with 3 CH (as discussed 2 CpCr(NO) in Section 2.3.1 of Chapter 2) almost disappeared. The spectrum shown in Figure 3.1 (a)  CH and H 2 CpCr(NO) 2 given in Table 3.1 was obtained under the nominal pressures of 3 with no delay time. After a 100 ms delay time, none of the high nuclear condensation products such as 4 Cr 3 Cp O , 4 C 3 Cp ( NO)O, r 2 C 3 Cp ( 4 O NO) r , 4 Cr 3 Cp O , and C 4 Cp ( 5 NO)O r appeared in the spectrum of Figure 3.1 (b).  The product ions,  CpCrNOCH (mlz 162) and CpCr(NO) (m/z 177), produced from charge transfer of Cr  —77—  C.  C F  b9 U  I  z C U  U9  > ‘-4  F  a  I U i 0 r  iii  iiiiijiIIIjTI1I  I  I  I  I  CC 11P53 ]N P.M.U,  2CC  iCC  I  I  I  I  I  I  li’  5CC  I  6CC  (a) C ‘-4  F  Delay time =100 ms  CpCrNOCH (Product of the charge transfer)  (Product of the charge transfer)  U F  z ILiW I-’  F  a  I U  I C  i  1.  i  1  111 4C  M55  5CC  ]N p It-I. U (b)  Figure 3.1  (a) Spectrum of pure Cr obtained by the triple resonance. (b) After a 100 ms delay time, large amounts of product ions from charge transfer of Cr to H appeared. 3 C 2 CpCr(NO)  —78—  to the parent molecule, 3 CH followed by dissociation, appeared in large 2 CpCr(NO) , amounts. Nevertheless, some condensations products, such as CpCr O (mlz 185) and 2 NO (m/z 199), can be still observed in the spectrum of Fig. 3.1 (b). A comparison 2 CpCr with Fig. 2.4 (b) in Chapter 2, after the same 100 ms delay time, showed that the ionmolecule reactions of Cr with pure 3 CH were predominantly condensations. 2 CpCr(NO) The reactions of Cr with H 2 and 3 CH are summarized as follows, 2 CpCr(NO)  collisional quenching:  Cr + H 2  other reactions of Cr+* with H 2 :  Cr  charge transfer:  Cr  +  *  +  2 H  H,  (3.2a)  products,  (3.2b)  (i +  2 k  CH 2 ) CpCr(NO 3  k3  CpCrNOCH+NO+Cr 162  CpCr(NO) + CH 3  52  +  Cr  (3.2c)  177  condensation:  Cr+*  +  CH 2 CpCr(NO) 3  O+N 2 CpCr 0 2  k  CpCrNO + NO  52 ‘  where the superscript”  “  +  3 •CH  185 +  3 •CH  (3.24)  199  indicates the ion or the molecule in an excited state, k ,k 1 ,k 2 , 3  4 are the respective rate constants of the reactions in Eqs. (3.2), and mass-to-charge and k ratio of each ion is given just under the ion formula. The collisional quenching and other reactions of Cr+* with H 2 (Eqs. (3.2a) and (3.2b)) have been studied by Elkind and ) 16 Armentrout.(  These secondary ions in Eqs. (3.2c) and (3.2d), CpCrNOCH,  CH according to 2 CpCr(NO) CpCr(NO), CpCr O, and CpCr 2 NO, reacted further with 3 2 the reactions described in Section 2.3.4 of Chapter 2. Temporal behavior of the five ions in Eqs. (3.2) is shown in Figure 3.2. The temporal variation of Cr did not show a pseudo-first order kinetics as in the reactions of Cr with pure 3 CH 2 CpCr(NO)  —79—  —1  -3  -5 C  -7  -9 0.2  0.0  0.4  0.6  0.8  1.0  Time (second) Figure 3.2  Temporal behavious of Cr, CpCrNOCH, CpCr(NO), CpCr O, and 2 NO in H 2 CpCr 2 medium. The temporal variation of Cr obviously does not obey pseudo-first order kinetics.  (Fig. 2.5 of Chapter 2). The reactions of Cr with all of the other small molecules, H 0, 2 , CH 3 NH , and N 4 , showed the same kinetics (Appendix A2). 2  Since [H ] and 2  CH are much larger than the total ion intensity, they can be considered to be 2 [CpCr(NO) ] 3 constant concentrations in the reactions. Then, the reactions in Eqs. (3.2) can be reduced to pseudo-first order reactions. The rate laws would be d [Dr]  = —  =  1 [112] [Cr+*] k  —  ] [Cr+*] 2 2 [H k  _(kj+kj+k)[Cr+*].  —  CH [Cr+*] 2 [CpCr(NO) 1 4 3 k  (3.3)  —80—  and d Dr]  =  1 [112] [Cr+*] k  =  kj [Cr+*]  —  —  H [&] 1c3 ] 3 C 2 [CpCr(NO)  kj [Cr1  (3.4)  where  k  =  , [H ] 1 2 k  (3.5)  =  , 22 k [H ]  (3.6)  =  , H 3 ] k 3 C 2 [CpCr(NO)  (3.7)  =  . H 4 } k 3 C 2 [CpCr(NO)  (3.8)  Integration of Eq. (3.3) gives  [Cr+*]=[&+*]oe_++k4)t  (39)  where [Cr+*Jo is the initial ion intensity of Cr in the excited state. Substituting Eq. (3.9) into Eq. (3.4), and then rearranging the terms in Eq. (3.4),  d [Crj + k 3 [Cr] d t  =  [Cr+*] e 1 0 k’  —  “  +  +  k  ) d t.  (3.10)  After multiplying both sides of Eq. (3.10) by e  e  3 [Cr] e d [Cr] + ic’  dt  =  k [Cr+*jo e  —  —  —  k  td t.  (3.11)  —81—  Integration of Eq. (3.11) gives  [Cr1  e’  =  where C 1 is the integration constant. Since at t =0, [Cr1  =  +Ce’.  (3.12)  , initial ion intensity of 0 [Cr1  Cr in the ground state by electron ionization, the integration constant  (3.13)  C= 0 [Cr1 k —k—k— k —  Thus, the total rate equation of Cr 4 is the sum of Eq. (3.9) and Eq. (3.12)  [Cr+*] k  —  +  —  [Cr1 k  —  —  “1  /  —  k  b —  —(k+k÷k)t [r 0 ] e (÷*  i,., —  k’ [Cr+*]  (3.14)  +  The right-hand of Eq. (3.14) is a double exponential function of time  t  which has the  following form  fQ)  =  e_b2 + b b 1 ’. 1 e 3  (3.15)  where  fQ)  =  [Cr+*]  +  [Cr],  (3.16a)  —82—  1 b  =  , 0 [Cr+*]  k—k—k—k  2 b  +  =  k  (3.16b)  (3. 16c)  +  k[Cr+*]o 3 b  =  [Cr]o  k  —  4 b  (3.16d)  k  =  (3.16e)  k.  Such a function was designed and input into the Macintosh “Igor” program for curve fitting of the ion intensity ([Cr+*1  [Cr*]  where  t  =  0  —  +  +  [Cr1) vs. time t. The curve fitting equation was  [Cr]  =  0.00612 e  0.4 s. The curve of ([Cr+*]  +  46.43t  +  0.01832 e  9.130t  (3.17)  [Cr1) vs. t fitted very well (sum of squared  error = 1.2 x 10 only), as shown in Figure 3.3. The rate constants are then found to be =  9.130 sec 1 and k  +  k  +  k  =  46.43 sec . 1  The calibrated pressure of  CH is 4.2 x 1O Torr (Table 3.1). 2 CpCr(NO) 3  3 k  =  =  CH 2 [CpCr(NO) ] k/3 2.2 x 108 1 •sec Torf  =  =  9.130 sec 1 / 4.2 x 108 Torr •molecules 3 cm • sec 6.7 x i0 1  The k 3 value is very close to the decay rate constant of Cr in last chapter (Table 2.6).  ,k 1 , and k 2 4 can Theoretically, if the initial ion intensities [Cr+*]o and [Cr] 0 are known, k be evaluated. Accuracy of the rate constants solved from the curve fitting method was estimated at ±30%.(1718)  —  83  —  0.025 0.020  0.015 0.010  0.005  0.000 0.1  0.0  0.2  0.4  0.3  0.5  Time (second) Figure 3.3  Curve fitting for ion intensity ([cr+*J  +  [CrJ) vs. time  Experimental  t.  data were obtained in 112 medium.  3.4.2 Ligand Substitutions 2 reacted to replace oxygen of the ligand NO in both CpCrNOCH (m/z 162) and H 3 ligand. CpCr(NO) (mlz 177) to form a NH  CpCrNOCH +2 112  -  2 CpCr(N0) +2 H 177  CpCrNH  +  2 + 112. 0C11  (3.18)  NO + •OH.  (3.19)  134  162  —>  CpCrNH  +  134  N H 5 H+ C CpCrNOCH N 3 CH 2 ) CpCrN + CpCr(N0 3 134  +  Cr0  179  H 3 C H 5 H+ C Cr(NO) N 2 194  +  Cr  (3.20)  —84—  , or the nitrosyl ligand, NO, of 3 3 CH was 2 CpCr(NO) In Eq. (3.20), the methyl ligand, CH NH (m/z 179) and 3 replaced by the ammonia ligand. Temporal variations of CpCrNOCH NH (m/z 194) are shown in Figure 3.4. 2 CpCr(NO)  Both the product ions  NH and CpCr(NO) 3 CpCrNOCH NH are unreactive. This behavior can be explained by 2 their electron structures. The metal in CpCrNOCH NH has a 16-electron structure and 3 Using deuterated  the metal in CpCr(NO) NH has a 18-electron structure. 2  CD the methyl ligand was found to be a hydrogen source for ammonia 2 CpCr(NO) , 3 ligand, too. 2 CpCrNOCD+2H  -  165  +HD. 2 CpCrNH+OCD  CpCrNOCD+ H 2  -*  CpCrNDH + OCD . 2  NH + C 3 (CpCrNOCD N H 5 +  Cr0  +  182  CD 2 CpCr(NO) 3  NH 2 1 CpCr(NO) 194  134  (3.21b)  135  165  CpCrNH  (3.21a)  134  (3.22a) +  CD H 5 C 3  NDH + C 3 (cpcrNOCD N H 5 CpCrNDH + 3 CD 2 CpCr(NO)  —  Cr  +  +  NDH + 3 2 CpCr(NO) CD H 5 C  135  Cr0  183  (3.22b) +  Cr  195  3.5  Ion-Molecule Chemistry of 3 CH with Water and Heavy Water, 2 CpCr(NO) 0 and D 2 H 0 2 CH 2 CpCr(N0) H2° is also a good hydrogen donor in the ion-molecule reactions of 3  with H 0. 2  All these ion-molecule reactions are exactly equivalent to those of  CH with 2 CpCr(N0) 3  H2 including the collision quenching, the ligand substitutions, and  —  85  —  -3  -4  -5  -6  -7 0.0  0.2  0.4  0.6  0.8  1.0  Time (second) Figure 3.4  Temporal variations of the chromium ammine complexes produced from the ion-molecule reactions of 3 CH with H 2 CpCr(NO) . 2  additions. These equations are not duplicated here. D 0 was used to verify that the 2 substituted ligand was ND , not OD, because masses of NH 3 3 and OH are both 17 Daltons and there is a 2 Dalton mass difference between ND 3 and OD. CpCrND (m/z 137), CpCrNOCD N 3 D (m/z 183), and CpCr(NO) ND (m/z 197) were products in the ion2 molecule reactions of 3 CD with D 2 CpCr(NO) 0. 2  3.6  Ion-Molecule Chemistry of CpCr(NO) 3 C 2 H with Ammonia, NH 3 From the above analysis, we expected that ammonia would react with CpCrNOCH  and CpCr(NO) directly to form CpCrNOCH NH and CpCr(NO) 3 NH. The experiment 2 fulfilled this expectation exactly. No intense mass peak of the intermediate CpCrNH (m/z  —86—  134) was observed. Therefore, the reactions between ammonia and CpCrNOCH and CpCr(NO) are additions, not substitutions: CpCrNOCH + NH 3  -,  CpCrNOCH N 3 H,  162  CpCr(NO ) + NH 3  —*  CpCr(NO ) NH. 2  177  3.7  (3.23)  179  (3.24)  194  Ion-Molecule Chemistry of CpCr(NO) 3 C 2 H with Methane, CH 4 4 was a good quencher of Cr in the excited state, produced from CpCr(NO) CH 3 C 2 H  by electron ionization. However, CH 4 is not a good hydrogen donor for formation of the ammonia ligand.  Although the chromium ammine complexes CpCrNH,  CpCrNOCH N 3 H, and 2 CpCr(NO) N H were all observed from ion-molecule reactions of 3 C 2 CpCr(NO) H with CH , their relative intensities as shown in Figure 3.5 were weaker 4 than those from the ion-molecule reactions of H 2 or H 0 with CpCr(NO) 2 3 C 2 H (Fig. 3.2). Reactions of Cr produced from chromium carbonyl with methane have already been studied in detail.( ) It had been observed that Cr could react with CH 1920 4 to give CrCH  and the carbonyl ligand of CrCO could be substituted by 19 .( Because the yield of 4 CH ) the CrCH is very low, it will not be considered here. In our experiment, no ligand substitution by direct addition of CH 4 was observed.  3.8  Ion-Molecule Chemistry of CpCr(NO) 3 C 2 H with Nitrogen, N 2 In all six small molecules used here, nitrogen was the only molecule without a  hydrogen atom. No ligand substitution occurred between N 2 and . 3 C 2 CpCr(NO) H Therefore, nitrogen is a poor quencher of the excited Cr ions. Consecutive condensations CH were still observed. 2 CpCr(NO) of Cr with 3  —  87  —  -4  -5  -  -6  C  -7  -8 0.0  0.2  0.4  0.6  0.8  1.0  Time (second) Figure 3.5  Temporal variations of the chromium ammine complexes produced from the ion-molecule reactions of CpCr(NO) 3 C 2 H with CH . The measured 4 relative ion intensities of CpCrNH, 3 CpCrNOCH N H,  and  CpCr(NO) N 2 H are obviously weaker than those from ion-molecule reactions of 3 CH with H 2 CpCr(NO) 2 (Fig. 3.4).  3.9  Discussion  3.9.1 The Electron Transfer Reaction At least two states of Cr ions produced by electron ionization, the excited state and ground state, were already discovered by Ridge et al. from the ion-molecule chemistry of .( Using FT-ICR mass spectrometry they found that variation of the 6 Cr(CO) ) Cr with 19 Cr ion intensity with reaction time did not obey simple pseudo-first-order kinetics. The  —88—  reactions of Cr, in the excited state or in the ground state, with Cr(CO) 6 were considered to be condensations:( ) 19  Cr  where m  =  1  —  3.  +  6 Cr(CO)  -,  (CO)L 2 Cr  +  m CO  (3.25)  This difference from our ion-molecule chemistry of Cr with  CH can be understood by noting that the chromiums in the metal nitrosyl ions 2 CpCr(NO) 3 CpCrNOCH and CpCr(NO) have even electron structures, 14-electron and 16-electron, respectively. However, the chromiums in monometallic carbonyl ions Cr(CO)m (m = 1  —  5) always have odd electron structures. The ground state, 6 S, of Cr (3d ) has a half-filled 3d shell. An important result 5 from our experiments on the ion-molecule chemistry of 3 CH is that ground 2 CpCr(NO) state Cr can undergo charge transfer reactions. In Section 3.4.1, the mechanism of Cr reactions were shown in Eqs. (3.2)—(3.17). Because it was verified experimentally here that the ion-molecule reaction of Cr ions in the ground state produced from CH was charge transfer and the ion-molecule reaction of the & ions in the 2 CpCr(NO) 3 excited state produced from 3 CH was condensation, the fractions of [Cr] and 2 CpCr(NO) [Cr+*] could be estimated from yields of their product ions. For example, according to the data shown in Fig. 2.4 of Chapter 4, 40% of Cr ions were in the ground state and 60% of Cr ions were in the excited state by 25 eV electron ionization. Comparing with 25% of Cr ions in the ground state and 75% of Cr ions in the excited state by 70 eV electron ) populations of Cr ions in the ground and excited states are obviously a 19 ionization,( function of the ionization energy. The temporal variation of the logarithmic ion intensity of CpCr in H medium,  shown in Figure 3.6, can be perceived to be non-linear (i. e., not a pseudo-first order kinetics). Thus, CpCr ions should exist in at least two states, ground state and excited  —89—  state.  Reactions of CpCr ions in the excited state and the ground state were not  distinguished as clearly as those of Cr ions in the excited state and the ground state.  3.9.2 The Condensation Reactions Since the ion-molecule condensation chemistry of 3 CH has been 2 CpCr(NO) discussed in Chapter 2, the collisional quenching and the ligand substitution reactions are focus points in this chapter. However, due to coffisions between large amount of the sinall molecules and small numbers of ions, the condensation rates of the ions were different in different media. As an example, rate constants, k, of the reactions of CpCrN0CH and CpCr(N0) in H , CH 3 , and N 4 ,H 2 0, NH 2 2 media, given in Table 3.2, show that these two ions are more reactive in the media of polar molecules 1120 and NH 3 than in the media of nonpolar molecules H , CH 2 4 and N . 2  Table 3.2  Rate constants of CpCrN0CH and CpCr(NO) in H ,H 2 0, 2 , CH 3 NH , and N 4 2 media  Medium  CpCrNOCH  CpCr(NO)  7 Torr•sec k x10  7 Torr•sec k xlO  2 H  12.3  12.3  1120  38.5  46.8  3 NH  33.2  40.8  4 CR  29.6  30.0  2 N  26.5  24.1  ¶ The method for calculating rate constants has been given in Section 2.5.1 Calibrated pressure of 3 CH was 4.2 x 10 Torr (Table 3.1). 2 CpCr(N0)  of Chapter 2.  —90—  Il /  I 0.01  0.001  0.0  0.2  0.4  0.6  0.8  1.0  Time (second) Figure 3.6  Temporal variations of CpCr and the secondary ions Cp Cr 2 2 Cr Cp O , C Cp ( 3 NO)O r in I12 medium. Cp Cr, 2 2 Cr Cp O , and 3 C Cp ( NO)O r were produced from the ion-molecule reactions of CpCr with CH For simplification, the variations of condensation ions 2 CpCr(NO) . 3 C Cp O 2 W, r2 C Cp N O, r 3 C Cp ( NO)OH, r which were produced also from CpCr with 3 CH are not shown here. The plot of 2 CpCr(NO) , logarithmic relative ion intensity of CpCr against reaction time is non-linear.  —91—  3.9.3  Further Work H with the 3 C 2 The ion-molecule chemistry of the metal nitrosyl complex CpCr(NO)  small molecules discussed in this chapter has demonstrated that it is an interesting research area. On the basis of this work, many studies should be done. The collision quenching and charge transfer of Cr has been verified experimentally. The mechanism of these reactions can be identified from the kinetic data of Cr. In order to get good ion intensities to compare with the results given in Chapter 1, a 20 ms beam time was used as in the experiments of Chapter 1. Because some ion-molecule reactions already occurred during the beam time, the primary ion initial intensities in this experiment were all contaminated. Thus, when the curve fitting method is used to determine the rate constants of the Cr reactions (Section 3.4.1), a shorter ionization beam time (for example, 5 ms) is advantageous to increase accuracy of the determinations. However, this will sacrifice the ion intensity. One important result of our research is that the ground state Cr ions prefer CH and result in different products from 2 CpCr(NO) , charge transfer to the metal nitrosyl 3 those of the condensation reactions of the excited state Cr ions. Populations of Cr ions in the ground state and the excited state, therefore, can be determined according to respective yields of the charge transfer products and condensation products for various ionization energies. The pressures of the small molecules were all controlled at a constant nominal 3.0 x 10 Torr. The collision quenching effects of the small molecules could be changed with different partial pressure ratios of the small molecules to the metal nitrosyl. The effects of 2 to the condensation rate constants are also worth , and N 4 ,H 2 0, NH 2 , CH 3 the media H studying further. Reduction of metal nitrosyl to metal ammine in acidic solution has been known since 1973.(21) In the present study, it was found that the nitrosyl ligand in 3 CH can 2 CpCr(NO) be reduced to the ammonia ligand by gas-phase ion-molecule reactions, if the reductants  —92—  contain hydrogen. This is a new type of reaction never observed before for metal nitrosyls. Other small molecules also can be used to react with the metal nitrosyl, especially small molecules containing hydrogen. It is possible that many new discoveries will come with more work in this area.  —93—  REFERENCES  1.  Freiser, B. S. Chemtracts 1989, 1, 65-109.  2.  Sharpe, P.; Richardson, D. E. Coord. Chem. Rev. 1989, 93, 59-85.  3.  Marks, T. J. “Bonding Energetics in Organometallic Compounds”; American Chemical Society: Washington, DC, 1990.  4.  Eller, K.; Schwarz, H. Chem. Rev. 1991,91, 1121-1177.  5.  Dunbar, R. C.; Solooki, D.; Tessier, C. A.; Youngs, W. J.; Asamoto, B. Organometallics 1991, 10, 52-54.  6.  frikura, K. K.; Beauchamp, 3. L. J. Am. Chem. Soc. 1991, 113, 2767-2768.  7.  Roth, L. M.; Huang, Y.; Schwedler, J. T.; Cassady, C. J.; Ben-Amotz, D.; Kahr, B.; Freiser, B. S. J. Am. Chem. Soc. 1991, 113, 6298-6299.  8.  Huang, Y.; Freiser, B. S. J. Am. Chem. Soc. 1991, 113, 8186-8187.  9.  Huang, Y.; Freiser, B. S. J. Am. Chem. Soc. 1991, 113, 9418-94 19.  10.  Speir, J. P.; Gorman, G. S.; Amster, I. J. Proceedings of the 39th ASMS Conference on Mass Spectrometry and Allied Topics, Nashville, TN, 1991, pp 455456.  11. Weddle, G. H.; Allison, 3.; Ridge, D. P. J. Am. Chem. Soc. 1977, 99, 105-109. 12.  Jacobson, D. B. J. Am. Chem. Soc. 1987, 109, 68 15-6852.  13.  Klaassen, J. J.; Jacobson, D. B. J. Am. Chem. Soc. 1988, 110, 974-976.  14.  Gord, 3. R.; Freiser, B. S. J. Am. Chem. Soc. 1989, 111, 3754-3755.  15.  Bartmess, 3. E.; Georgiadis, R. M. Vacuum 1983,33, 149-153.  16. Elkind, J. L.; Armentrout, P. B. J. Chem. Phys. 1987, 86, 1868-1877. 17.  Kerley, E. L.; Russell, D. H. J. Am. Chem. Soc. 1990, 112, 5959-5965.  —94—  18.  Pan, Y. H.; Sohlberg, K.; Ridge, D. P. J. Am. Chem. Soc. 1991, 113, 2406-2411.  19. Reents, W. D. J.; Strobel, F.; Freas, R. B. I.; Wronka, J.; Ridge, D. P. J. Phys. Chem. 1985, 89, 5666-5670. 20. Armentrout, P. B. in “Gas Phase Inorganic Chemistry”; Russell, D. H., Ed.; Plenum Press: New York, 1989; pp 24-26, and references therein.  21. Armor, 3. Inorg. Chem. 1973, 12, 1959-1961.  —95—  CHAPTER 4  SIMPLE PHYSICAL POINT AND LINE CHARGE MODELS FOR COULOMB-INDUCED FREQUENCY SHIFT AND INHOMOGENEOUS BROADENING IN FT-ICR MASS SPECTROMETRY  —96—  4.1  Space Charge Effects in FT-ICR Mass Spectrometry  4.1.1  Space Charge Effects and Mass Measurement The simplest theoretical treatment of ion cyclotron resonance (ICR) presumes that the  ion motion is controlled by exclusively applied forces, such as those from the radiofrequency field, the static electric field, and the static magnetic field.  In a  homogeneous static magnetic field B=Bk  (4.1)  where k is the unit vector in the z direction, and the Lorentz force on an accelerated ion is (4.2)  F=qvXB  where q is the charge of the ion and v is the ion velocity. The solution to the motion equation (4.2) predicts that the ion of mass m will travel in a circular orbit of radius r with a cyclotron frequency( ) 1 °o  =q,  (4.3)  and the instantaneous position of the ion related to the Cartesian coordinates, x and y, is given by t+) 0 x=rcos (a t+ 0 y=—rsin(w  where  •)  (4.4x) (4.4y)  is the initial phase of the ion circular motion at t =0.  To be precise, Eqs.(4.3) and (4.4) are correct only if the motion of a given ion is independent of the motions of other ions. It is well-known that in ICR spectrometry and  —97—  FT-ICR spectrometry, spectral lineshapes and positions are functions of the ion number density. That is, the resolution and mass measurement of FT-ICR will become degraded with growth of the ion number density. This experimental phenomenon apparently comes from space charge effects in ICR and FT-ICR. Therefore, a Coulomb interaction term should be added to the right-hand of Eq. (4.2). We define the ions of same m/q ratio as like ions and the ions of different m/q ratio as  unlike ions. The space charge effects should include: Coulomb interaction of the ions with the external electhc fields, Coulomb interaction between like ions, and Coulomb interaction between unlike ions. 4.1.2 Prior Research on the Space Charge Effects During the past decade, a number of authors have studied the space charge effects in FT-ICR mass spectrometry. Ledford et al. experimentally measured the frequency shift with ion number, for like ions in the cubic ion-trapped cell of FT-ICR, and first offered an  empirical equation to calibrate the frequency shift due to the trapping electrostatic field and ion number dependence.( ) Their procedure gave mass measurement errors averaging 3 2 ppm. Jeffries, Barlow, and Dunn first gave a theoretical quantitative study on the Coulomb interaction of ions in ICR and FT-ICR mass spectrometry for various ICR cells.( ) Franci 3  et al. used the space charge theory of Jeffries et al. to calibrate mass measurements of both scanning ICR spectrometry and FT-ICR spectrometry and gave accurate mass determinations with errors less than 1 ppm.( ) The basis of Francl et al.’ s procedure is that 4 a mass peak can be calibrated by using two other different mlz reference ions. However, Jeifries et al.’s model closely corresponds to the scanning ICR experiment, in which ions are formed along the z-axis in the center of ICR cell and then the ICR motion of these ions is sequentially excited as the spectrum is scanned. This is inappropriate for the FT-ICR experiment where all the ions in the cell are undergoing excited cyclotron motion during the FT-ICR detection period (see the section 1.1 of Chapter 1). Ledford, Rempel, and Gross  —98—  improved Jeffries et al.’s model by using more reference ions (for example, six reference ) Wang and Marshall studied spectral lineshape broadening due to ion-ion Coulomb 5 ions).( interaction in Fr-ICR by using numerical analysis.( ) On the basis of the model derived by 6 Ledford et al.,( ) Meek et al. gave an average mass error of less than 0.5 ppm for m/z range 5 from 119 to 556 Daltons in a 7 Tesla magnetic field.( ) Yang, Rempel, and Gross 7 examined the effects of the electron emission current and time on mass errors in FT-ICR.( ) 8 They pointed out that systematic errors in mass measurements increase with ion number, owing to the Coulomb interaction amongst the ions. Rempel and Gross revised their own ) to include the time averaged interaction of ions, but no detail was given.( 5 model( ) Yang, 9 Rempel, and Gross incorporated relative ion intensity into the mass calibration coefficients according to Jefferis et al.’s model.( ) Herold and Kouzes simply used an infinitely long 10 line charge model to calibrate the space charge 1 effects( 1) and their result was in keeping  ) Smith has noticed that ion-ion Coulomb interactions 4 with the experiment of Franci et al.( from different m/q ions which are called unlike ions can be averaged and their intensities should be included in the calibration of space charge effect.( ) There is, therefore, an ion 12 intensity term in Smith’s calibration formula based on the models of Jeifries et al.( ) and 3  ) After ten reference mass peaks were used, Smith shown that calibrating 5 Ledford et al.( FT-ICR mass spectra by inclusion of unlike ion intensities gives superior calibration accuracy. Marshall and Verdun concluded in 1990:(13) “Unfortunately, there is no simple analytical treatment for either the static or dynamic effect of multiple-ion Coulomb interactions, and one must resort to computer-intensive trajectory calculations of each of 10,000 or more ions in a trap of given geometry.” Grosshans, Shields, and Marshall have  realized that any accurate calibration of space charge effect in FT-ICR mass spectrometry must employ the interaction potential between ions rather than the simpler Coulomb potential model usually used previously.( ) Until now, no theoretical quantitative 14  —99—  discussion has been developed on Coulomb interactions between different ion masses in FT-ICR. Recently, we have proposed simple physical point and line charge models which more closely correspond to physical reality than prior models.( ) Coulomb interaction 15 between two or more different m!q ions can be calculated by averaging the separation between two charges (either points or lines).  The Point Charge Model  4.2  —  Coulomb Shifting of Unlike Ions  For a simple two-dimensional picture as shown in Figure 4.1, two excited ions with different mass m 1 and m 2 are undergoing cyclotron motion with common orbit centers in a homogeneous static magnetic field B and their cyclotron radii are r 1 and r . If each ion is 2 influenced only by the Lorentz force, the ions will undergo cyclotron motion at their natural cyclotron frequencies, c for m . Now the instantaneous magnitude, D, 2 1 and c002 for m of the spatial separation between the ions, D, is given by  ID  I  =  D  =  1 (x  —  + (y ) 2 x 1  —  ) 2 y  (4.5)  where x 1 and y 1 (x 2 and y ) are the instantaneous Cartesian coordinates of ) 2 2 ( 1 m m . Substituting Equations (4.4) into Equation (4.5) gives D If r 2  =  2 + 1 ‘\Jr  —  2 r 1 2 r cos[(co )t 0 — 0 2  + (1—2)1  •  (4.6)  , Eq. (4.6) then reduces to 1 r D  =  1 \J2 —2 cosP r  =  2 r  I sin (cIi /2) I  (4.7)  where c1 = (c0 ) 02 —w 01  +  (—)•  (4.8)  -100-  Figure 4.1  , are 2 1 and m Point model for Coulomb effects. Two positive ions, m undergoing excited cyclotron motion at their respective cyclotron frequencies,  (Ooi  and  , 2 1 and m Eqs. (4.7) and (4.8) give the instantaneous spatial separation between two ions m , the two cyclotron frequencies, c- and 1 as a function of their equal cyclotron radii, r the two initial phase angles,  and  2’  %2’  and the time t. This separation varies sinusoidally  as indicated by Equation (4.7). The mean value, Dme, of the separation between the ions is given by averaging D for ‘ =0— it (or by averaging D for  CP =  It  —  2it)  —101—  Dme  =  1  -  =  J  1 sin (/2) d 2r  (I)=O  4r =  (4.9)  1.2732 r . 1  It  Because the Coulomb force is inversely proportional to the square of the separation, the root mean square value,  Dims  =  =  of the separation between the ions is given by  0.5  (  r?(2—2cos) thP)  =  1.4142 r . 1  (4.10)  Thus, from the point of view of the first ion mass, the second ion mass will appear on average a distance 1.2732 r 1 away, as indicated in Figure 4.2, if we chose the “average” 1 away if we choose the root distance to be the mean distance, Dmean or a distance 1.4 14 r mean square distance, D . If we choose Dm to be the “average” distance, the Coulomb 5 force between the ions is given by Equation (4.11), 2 q 1 kq Frad  =  D 2 mean  k q 2 1 =  1 6211  Ti2  (4.11)  9 N2 where k is the Coulomb constant, 8.9876x10 2 are the respective /C q m , 1 and q charges of m 1 and m , and Frad is a radial Coulomb force on m 2 1 which subtracts from the 1 of m , 1 Lorentz force of Eq. (4.2). In terms of the cyclotron frequency w and radius r the Lorentz force on m 1 from Eq. (4.2) can be written 1 F  =  o. q B 1 r 01 .  (4.12)  —102—  Frad  Figure 4.2  The average distance approach. The figure gives the location of ion in 2 as seen by in 1 by viewing the system in a coordinate frame which rotates at the frequency  co.  The “average” distance between m and in 1 is taken to be  2 1 if r the mean distance Dmej and is 1.2732 r  =  . This distance is used 1 r  1 due to in . If the 2 to calculate the average radial Coulomb force on m distance r 1 is used, corresponding to the average laboratory-frame position of in 2 as in scanning ICR mass spectrometry, the radial Coulomb force is 1.27322  =  1.6211 times as great.  —  103  —  If any change in the net radial force is assumed to be small with respect to the Lorentz force, then from Eq. (4.12), we can write for m , 1 1 AF  =  (4.13)  . E% B r q 1  with the change in radial force,  Eq. (4.13) relates the change in cyclotron frequency,  , which caused that change in frequency. Substituting Eq. (4.11) into Eq. (4.13) and 1 AF rearranging, gives k q 2  =  =  k ,  (4.14a)  or equivalently,  01 Af  =  1.4137 x i0  2  B 1 r  (4.14b)  Eq. (4. 14b) gives Af(fl, the change in experimental cyclotron frequency in Hz, of an ion , the 2 2 of equivalent cyclotron radius, as a function of q , due to the presence of an ion m 1 m , the ion radius in cm; and B, the magnetic 1 2 in units of the electronic charge; r charge on m field strength in Tesla. Eq. (4.14a) is the equivalent equation in SI units. 1 is in a coordinate frame 2 on m The easiest way to visualize the Coulomb effect of m 1 and the .( In this frame, both the position of m 1 m ) which rotates at the frequency of 16 2 appears as shown in Figure 2 are static and the average position of m average position of m 4.2. It follows from Eqs. (4.5)  —  (4.7) that both the spacing, D, and the frequency shift,  , caused by ions of mass identical to the ion mass being monitored, are zero. This 01 Af ) 6 ) and numerical analysis.( 1718 result has been previously noted from both theoretical( For purposes of illustration, the Coulomb shift due to a particle with 106 charges on the cyclotron frequency of an ion of cyclotron radius 1 cm in a magnetic field of 2 Tesla  —104-  1 which collectively are taken to be a point charge. would correspond to 106 ions of m Substitution of these parameters into Eq. (4. 14b) gives a negative frequency shift of 70.7 Hz.  4.3  The Line Charge Model  —  Coulomb Shifting and Broadening of  Unlike Ions In conventional electron ionization experiments, the ions are formed in a long cylindrical shape at the center of FT-ICR cubic cell. When the radius of this cylindrical rod is small relative to its length and excited cyclotron radius, the ion cylindrical rod can be treated as a line charge. The cyclotron radius r of an ion during excitation is only a function , and excitation time t, without 0 of magnetic field strength B, rf electric magnitude E ) 1920 dependence of its mass and charge( roc  Et .  (4.15)  No matter what the mlq ratios of the ions are, their cyclotron radii are all the same when a uniform sweep rate of excitation frequency is used. A physical model which better approximates the actual ICR experiment can be derived from Figure 4.2 by dispersing the ion charges into the z-dimension to create a rotating line charge of length 1, for each excited ion mass. Figure 4.3 shows this model. Consider the effect of line charge 2 upon the cyclotron frequency of line charge 1. As for the model of Figure 4.2, the average distance, Dm of line charge 2 from line charge 1 is given by Eq. (4.9), and we assume that the Coulomb interaction between the charges follows from this distance. Now the radial electric field at point P of line charge 1 from a differential l) 2 element, dz of line charge 2, j(  —105—  line charge 1 —z  line charge 2  Figure 4.3  z = 1/2!  Line model for Coulomb effects. This model is derived from the model of Fig. 4.2 by uniformly distributing the charges in the z direction. The length of the charges is 1 for both m 2 ions. The mean distance of line 1 and m charge 2 from line charge 1 is Dme  =  . The radial Coulomb 1 1.2732 r  electric field on point P of line charge 1 from the differential element dz of line charge 2 is rad = dE x cos 8.  —106--  rad  =  dExcosO  =  2q k N 2  Drnean dz ) 1 (z—z  (4.16)  3/2  +  mean  ) 22 and so the total radial electric field created by line charge 2 at point P is(  z=l/2 —  E  q 2 kN  DmeandZ z=—l/2  —  =  2q kN 2 mean  [ J(l/2  )2 + 1 [(z_z  1/2 —  Dan  1 z  —  2 ) 1 z  ]3/2  1/2  +  Dan  +  J(l/2  +  + Z 1  2 ) 1 z  +  1(4.17) Dean  Eq. (4.17) gives the radial electric field at position P in an ICR experiment modelled by Figure 4.3, as a function of N , the total charge on line charge 2; 1, the length of line q 2 charge 2;  Dmean,  the mean distance between the two line charges. Each element, dz, of line  charge 1 (not shown) will have a mass N /l dz, and the radial q 1 /l dz and charge N m 1 Coulomb force from line charge 2 on this element is given by the product Fj  =  (4.18)  /l dz q 1 Erad N  where Erad is given by Eq. (4.17). Now Eq. (4.13), which was derived for the point model in Figure 4.2 and which relates the change in frequency and the radial Coulomb force, is also valid for a differential element of line charge 1 in Figure 4.3. Substituting the charge of the differential element, N /l dz, and the radial force in Eq. (4.18), into Eq. q 1 (4.13) and solving for the frequency change gives &Ooi —  kN q 2 1 r mean  [ J(l/2  1 l/2—z —  2 ) 1 z  +  1 l/2+z Dean  sJ(l/2  +  2 ) 1 z  +  Dean  ]  —107—  or equivalently,  Afoi  =  2.2918x10N q 2 B I Dmean 1 r  [ (1/2 +  For z 1  =  1 1/2—z —  2 ) 1 z 1/2  +  +  mean  1 z  (4.19b)  2 g (1/2 + z 2 + Dmean ) 1  0, Eq. (4.19) reduces to  1  kN q 2 B Dmean 1 r  [ J(l/2)2 1  +  (4.20a)  2  Dmean  or equivalently,  01 ‘V  =  2.2918x1ON q 2 B Dmean 1 r  [  1 l/22  +  (4.20b) 2 Dmean  In the limit (i. e., an infinite long line charge),  112 D mean  —  00,  (4.21)  Eq. (20) reduces to 2 k N 2q 2 r 1 Dmean B 1  (4.22a)  4.5836 x 10N2q 2 1 1 Dmean r  (4.22b)  =  or equivalently,  ‘vol  =  Equation (4. 19b) gives the frequency shift in Hz for a differential element of line charge 1 at position z 1 in Figure 4.3 as a function of B, the magnetic field in Tesla; N 2q , the total 2 charge on line charge 2 in units of the electronic charge; r , the cyclotron radius of line 1 charge 1 in cm; 1, the lengths of line charges 1 and 2 in cm; Dmean, the average distance between the lines (1 .2732 r ) in cm; and z 1 , the position in cm of the differential element. 1  —108--  Eq. (4. 19a) is the equivalent equation in SI units. Note that the shift is a function of the position, z , in line charge 1. This model predicts that different ions of mass 1 wifi have 1 different cyclotron frequencies because of their different positions in line charge 1. In other words, the experimental spectral peak will be broadened as well as shifted by the Coulomb interaction with line charge 2. Eq. (4.20a) gives the Coulomb shift in SI units for ions at the center of line charge 1. This is also the maximum Coulomb frequency shift for the peak. Eq. (4.20b) is the equivalent equation in the same units as for Eq. (4.19b). Eq. (4.22) gives the shift for infinitely long lines and has been noted previously, with Dmean 01 . 1 r replaced by ) Figures 4.4, 4.5, and 4.6 show the effects of Coulomb shifting and broadening due to the model illustrated in Fig. 4.3. Figs. 4.4, 4.5, and 4.6 give the Coulomb shift , the z-axis position in line charge 1, and 1, 1 calculated from Eq. (4. 19a) as a function of z the length of the line charges, forB = 2 Tesla, Dmean  =  1.2732 cm,  =  cm, r 1  =  1.0  , the number of m 2 cm, and N 2 ions. Figure 4.4 gives the shift for 1=2.4 cm which would correspond to a cubic ICR cell, where the ions have been excited to a radius of 1 cm. Figures 4.5 and 4.6 give the corresponding shifts for elongated ICR cells of length 8 cm and 15 cm, respectively. The charge density in line charge 2 is constant at 4.167 x iO ions/cm for all graphs in Figs. 4.4, 4.5, and 4.6. This line model predicts that there will be no Coulomb-induced frequency shift or Coulomb-induced peak broadening for like ions.  4.4  Discussion  4.4.1  Choice of Model One prior physical model for Coulomb shifts in ICR was developed by Jeffries,  ) Their model had a charge cloud on the z axis of Fig. 4.3 with an ion 3 Barlow, and Dunn.( undergoing cyclotron motion in the x-y plane. This model closely corresponds to the  —109—  85 65 ‘vol  (Hz)  45 25 -1.2  -0.8  -0.4  0.0  0.4  0.8  1.2  1 (cm) z  Figure 4.4  Coulomb shifting and broadening for the line model of Fig. 4.3. The frequency shift for m 1 ions, calculated from Eq. (4. 19b), is plotted as a function of z , the position of the m 1 1 ion. The parameters used for these calculations are: average ion separations Dme = 1.2732 cm and Drms cm, corresponding to a cyclotron radius, r 1  =  1.0 cm; the magnetic field,  B =2 Tesla; and the z-axis length, 1=2.4 cm. N , the number of m 2 2 ions, is adjusted so that the ion number density is constant at 4.167 x 10 ions/cm. The upper curve gives the Coulomb shifts for the “average position” model, and was calculated from Eq. (4. 19b) by replacing Dme with r 1  =  1.0 cm. The lower curve gives the Coulomb shifts for the  “average distance” model, using Drms (Eq. (4.10)) as the “average distance”. The Coulomb shifts arising from using Dme (Eq. (4.9)) as the “average distance” lie between the “position” and “D” curves.  —110—  110 90 (H z) ° 7 50 30 -4 Figure 4.5  -3  -2  -1  0 1 (cm) z  1  2  Same graph as that in Fig. 4.4 for 1= 8 cm and N 2  =  3  4  3.33x 106 ions. (N ]! 2  is constant at 4.167x10 5 ions/cm).  110 90 01 Lf  (Hz)  70 50 30 -7.5  Figure 4.6  -5.0  -2.5  0.0 1 (cm) z  2.5  5.0  7.5  Same graph as that in Fig. 4.4 for 1= 15 cm and N 2 = 6.25x10 6 ions. (N J1 2 5 ions/cm). Note that although the magnitude of the is constant at 4.167x10 Coulomb shift is greater for longer 1 , because of the greater number of ions, the dispersion in cyclotron frequencies for most m 1 ions is less.  —111—  scanning ICR experiment, in which ions are formed along the z axis and then the ICR motion of these ions is sequentially excited as the spectrum is scanned. The experimentally observed negative direction of the Coulomb frequency shift was correctly predicted by this model. This model is inappropriate, however, for the Fr-ICR experiment where all the ions in the sample are undergoing excited cyclotron motion during the FT-ICR detection period, as indicated in Figs. 4.1 and 4.2. Condensation of the charge cloud into a charged line gives a static line charge on the z-axis( ) which again, is applicable only in the 11 scanning ICR experiment. Why then does this model work for FT-ICR? An examination of Fig. 4.2 shows the reason. A two-dimensional Jeffries model can be created by condensing all of the cloud charges to a point at the origin of the coordinate frame of Fig. 4.2. Thus, in terms of Fig. 4.2, the two-dimensional Jeffries model predicts a negative frequency shift which is (1.2732)2  =  1.6611 times greater than that predicted by the  “average distance” model of Eq. (4.9). The shifting of the cyclotron frequency of a particular ion in the presence of other ions can be qualitatively explained as being due to an average radial force on the particular ion caused by the other ions. Attempts to quantitatively derive the magnitude of this frequency shift within the framework of the simple physical models of this work give rise to three possible approaches. What is desired is the average Coulomb radialforce on the particular ion, which we assume gives the frequency shift via Eqs. (4.13) and (4.14).  1 Unfortunately, the average radial force is infinite for r  =  2 (Fig. 4.1), due to the r  momentary superposition of the two ions during one complete cyclotron orbit. A second approach, which is developed here, is to derive the average distance of the perturbing ion from the particular ion and then assume that the “frequency-shifting radial force” is equal to the radial force from the “average distance”. The average distance can be either the mean distance, Dme Eq. (4.9), or the root mean square distance, Drms Eq. (4.10).  —112—  A third approach is to use the average position of the perturbing ion, which is x  =  y=  0, and then taken the “frequency-shifting radial force” as being equal to the radial force ’ 5,7—12) have 4 from the average position. This, in effect, is the approach which others( used. For point charges, the “average distance” model predicts lower shifts by a factor of 2.0 for  and 1.6211 for Dme from the “average position” model. The predictions of  the average-position model can be derived from the average-distance equations in this in any of Eqs. (4.16)  chapter by replacing  —  . 1 (4.22) by r  As discussed above, for a specific case, the two-particle model in Fig. 4.2, predicts a  2 ions. When the 106 ions are spread out over a Dme Coulomb shift of 70.7 Hz for 106 m distance of 2.4 cm (Fig. 4.4), to create the line model, the corresponding Coulomb shift is reduced to a maximum (Eq. (4.20b)) of 51.4 Hz. This reduction is not surprising, since  1 ions in the line model than in 2 are on average, farther away from m the perturbing ions m the point model. The average distance models are not only for two line charges, but also can be extended to arbitrary numbers of line charges. When the ion-ion Coulomb effects  1 are studied, all other line charges can be averaged to the distance Dmen on line charges m or Drms provided their separations vary sinusoidally.  1 In another example, where ion masses are very similar, let m =  =  2 1000 Daltons and m  01 1001 Daltons in a 2 Tesla magnetic field, in which their resonance frequencies arev  4 Hz. In a detection time T 4 Hz and v = 3.06813x10 3.07120x10  =  =  8.398 ms (fairly  short), they oscillate for 257.92 cycles and 257.66 cycles, respectively. Their difference is much less than 1 cycle, so that their separation can not be averaged from 0 to 2ir. ) is 23 However, the theoretical resolution of the magnitude mode(  m % 50 Am  —  gBT 7.582 m  —  19 x 2 x 0.008398 1.60x10 27 7.582 x 1000 x l.66x10  —  213  —  It is impossible to detect the Coulomb interaction between these two ion species at such a poor mass resolution. Therefore, this case can be ignored in our models.  —113—  It is common in the development of physical models for certain approximations or simplifications to be made.() The point model of Figs. 4.1 and 4.2 treats the problem as a two-dimensional one, with no radial electric field. Inclusion of a z-axis distribution of charge, as in Fig. 4.3, gives rise to a position-dependent frequency shift. In addition, the models in Figs. 4.4  —  4.6, while including axis position, omit an electrostatic trapping  field, and omit any z-axis motion.  4.4.2 Inhomogeneous Broadening in FT-ICR ICR spectral peaks can be broadened by two different types of processes; called ) The corresponding processes in the 25 homogeneous and inhomogeneous broadening.( time domain are called homogeneous and inhomogeneous relaxation. Homogeneous relaxation and broadening result from any process which limits the lifetime of the oscillation. Inhomogeneous relaxation and broadening result from a dispersion in the ) From these 25 resonant frequency of the individual components of a macroscopic sample.( definitions, it follows that the position-dependent Coulomb-induced frequency shifting process of Fig. 4.3 and Eq. (4.18) gives rise to an inhomogeneous broadening/relaxation mechanism. ) have discussed Coulomb broadening in FT-ICR and concluded 6 Wang and Marshall( that a two-dimensional model, such as those in Fig. 4.1 or 4.2, cannot account for Coulomb broadening.  These authors concluded that z-axis oscillations could give  Coulomb broadening if the Coulomb force was comparable to the Lorentz force. This conclusion is supported here. Fig. 4.3 and Eq. (4.19) show that it is the dispersion in z axis position, not z-axis oscillations, which causes Coulomb broadening; and that small Coulomb forces, which are much less than the Lorentz force, can create Coulomb broadening. For example, for B =2 Tesla and m  =  100 Daltons and the other parameters  —114—  as in Fig. 4.4, the maximum Coulomb force for the average position model, the force at z = 0, is smaller than the Lorentz force by a factor of 2.389 x iO.  4.4.3 Coulomb Shifting and Broadening of Like Ions ’ 17-18) predicts no 6 As noted above, our model as well as the work of others( frequency shift for ions of identical mass. Yet, such shifts have been observed when the ’ 6) Prior treatments make the 4 ICR experiment is conducted on just a single ion mass.( quadrupolar approximation in which it is assumed that all ions of a particular mass are subjected to an identical radial component of the trapping field. However, it is well known that in practice, the observed cyclotron frequency depends upon the z-coordinate, as the experimental field is quadrupolar only at z  =  0. We believe that these actual non  quadrupolar electrostatic fields may give rise to a Coulomb-induced frequency shift as follows: consider m 1 ions located at z  =  ; 1 0 which have been excited to some radius, r  consider next ions of identical mass located at ± z whose cyclotron motion has also been excited. These ions will have a different cyclotron frequency from the ions at z =0, due to ions 1 the non-quadrupolar radial component of the trapping electric field. Compared to m at z  =  2 ions in the 0, they will appear to be ions of differing mass, and can be treated as m  present model, and will produce a Coulomb-induced frequency shift and a Coulomb-  induced broadening. Consequently, a Fl’-ICR experiment conducted with a single ion mass can give an ICR lineshape which is both Coulomb-shifted and inhomogeneously Coulomb-broadened.  4.4.4 ICR Cell Design and Coulomb Effects ) ICR cells 30 ) and elongated( 2829 ) segmented-trap-electrode,( 27 ) shaped,( 26 Screened,( have been developed to create what approaches zero trapping fields in much of the interior of ICR cells. The advantage of these cells is that the smaller radial component of the  —  115  —  trapping field creates a smaller dispersion in trapping field shifts and concomitant higher mass resolution than is present in the cubic ICR cell.( ) The analysis in this chapter 3132 suggests that there is afurther advantage to these cells, namely that the lower trapping field dispersion gives rise to lesser Coulomb effects, both shifting and broadening, because a particular ion will “see” the Coulomb effects from other ions, but only to a lesser extent from ions of the same mass. Hence, the like-ion contamination can be minimized in these  ICR cells with negligible trapping fields. For either the point charge model or the line-charge model, the average radial Coulomb force is infinite for r 1  =  , due to the momentary superposition of the two ions 2 r  during one complete cyclotron orbit. This, of course, is an inherent flaw in these models. In the next chapter, we will show the development of a disk charge model and then a cylinder charge model (33) which can overcome this limitation of the point charge model  and the line charge model.  —116—  References  1.  David, A. D. “Classic Mechanics”; Academic Press: Orlando, 1986; pp 135-137.  2.  Ledford, E. B.; Ghaderi, S.; White, R. L.; Spencer, R. B.; Kulkarni, P. S.; Wilkins, C. L.; Gross, M. L. Anal. Chem. 1980,52, 463-468.  3.  3. B.; Barlow, S. E.; Dunn, G. H. 54, 169-187.  4.  Franci, T. 3.; Sherman, M. G.; Hunter, R. L.; Locke, M. 3.; Bowers, W. D.; Mclver, Jr., R. T. mt. J. Mass Spectrom. Ion Proc. 1983,54, 189-199.  5.  Ledford, E. B.; Rempel, D. L.; Gross, M. L. Anal. Chem. 1984,56, 2744-2748.  6.  Wang, T.-C. L.; Marshall, A. G. 301.  7.  Meek, 3. T.; Millen, W. G.; Franci, T. 3.; Stockton, G. W.; Thomson, M. L.; Wayne, R. S. 35th ASMS Conference on Mass Spectrometiy and Allied Topics, 1987, 1122-1123.  8.  Yang, S. S.; Rempel, D. L.; Gross, M. L. Proceeding of 36th ASMS Conference on Mass Spectrometry and Allied Topics, 1988, 586-587.  9.  Rempel, D. L.; Gross, M. L. Proceeding of 37th Conference on Mass Spectrometry and Allied Topics, 1989, 1222—1223.  10.  Yang, S. S.; Rempel, D. L.; Gross, M. L. Proceeding of the 37th ASMS Conference on Mass Spectrometry and Allied Topics, 1989, 1224-1225.  Jeffries,  11. Herold, L. K.; Kouzes, R. T.  mt. J. Mass Spectrom. Ion Proc.  mt. J. Mass Spectrom. Ion Proc.  1983,  1986, 68, 287-  mt. J. Mass Spectrom. Ion Proc. 1990, 96, 275-289.  12. Smith, M. 3. C. 2nd Amer. Soc. for Mass Spectrom. Sanibel Conf. on Ion Trapping in Mass Spectromerry, 1990. 13. Marshall, A. 0.; Verdun, F. R. “Fourier Transforms in NMR, Optical, and Mass Spectrometry”; Elsevier: Amsterdam, 1990; p 243.  —117—  14.  Grosshans, P. B.; Shields, P. 3.; Marshall, A. G. J. Chem. Phys. 1991, 94, 53415352.  15.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1991,5, 450455.  16.  Wang, M.; Marshall, A. G.  mt. J. Mass Spectrom. Ion Proc. 1990, 100, 323-346.  17. Walls, F. L.; Stein, T. S. Phys. Rev. Len. 1973,31, 975—979. 18.  Wineland, D.; Dehmelt, H. mt. J. Mass Spectrom. Ion Phys. 1975, 16, 338—342.  19.  Comisarow, M. B.  mt. J. Mass Spectrom. Ion Phys.  20. Grosshans, P. B.; Marshall, A. G.  mt.  1978,26, 369-37 8.  J. Mass Spectrom. Ion Proc. 1990, 100,  347-379. 21.  Nayfeh, M. H.; Brussel, M. K. “Electricity and Magnetism”; John Wiley and Sons: New York, NY, 1985; p 35.  22. Gradshteyn, I. S.; Ryzhik, I. M. “Tables of Integrals, Series and Products”; Academic Press: New York, 1965; p 83. 23.  Comisarow, M. B. In “Transform Techniques in Chemistry”; Griffiths, P. R., Ed.; Plenum Press: New York, 1978; pp 266-267.  24. Feynman, R. P. “The Feynman Lectures on Physics”; Addison Wesley: Reading, MA, 1964; Vol. I, p 25—5. 25.  Comisarow, M. B. In “Lecture Notes in Chemistry, Vol. 31, Ion Cyclotron Resonance Spectrometry II”; Hartmann, H. Wanczek, K. P., Ed.; Springer-Verlag: Berlin, 1982; pp 484—513.  26. Wang, M.; Marshall, A. G. Anal. Chem. 1989,61, 1288-1293. 27. Hanson, C. D.; Castro, M. E.; Kerley, E. L.; Russell, D. H. Anal. Chem. 1990, 62, 520-526. 28.  Rempel, D. L.; Grese, R. P.; Gross, M. L. 100, 38 1-395.  mt. J. Mass Spectrom. Ion Proc.  1990,  —118—  29.  Naito, Y.; lnoue, M. Proceeding of 36th ASMS Conference on Mass Spectrometry and Allied Topics, San Francisco, CA, 1988, 608—609.  30. Hunter, R. L.; Sherman, M. G.; Mclver Jr., R. T. Phys. 1983,50, 259—274.  mt.  J. Mass Spectrom. Ion.  31. Comisarow, M. B. Adv. Mass. Spec. 1980,8, 1698—1706. 32.  Comisarow, M. B. In:. J. Mass Spectrom. Ion Phys. 1981,37, 251—257.  33.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1992,6, 1-3.  —119—  CHAPTER 5  SIMPLE CHARGED-DISK MODEL AND SIMPLE CHARGED-CYLINDER MODEL FOR COULOMB SHIFTING AND COULOMB BROADENING IN FT-ICR MASS SPECTROMETRY  —120—  5.1  Introduction In Chapter 4, we developed two simple physical models (the point and line charge  ) which explained Coulomb-induced frequency shifting and Coulomb-induced 1 models)( inhomogeneous line broadening in FT-ICR mass spectrometry. The first model consisted of two point ion masses, in 1 and m , which were undergoing excited cyclotron motion. 2 When viewed in a coordinate frame which rotated at the frequency of m , the model 1 showed that the motion of in 2 gave rise to a radial Coulomb force of repulsion at ml which subtracted from the Lorentz force on m 1 (because of its opposite direction) and lowered the cyclotron frequency of m 1 and m 2 shifted the . Thus, the Coulomb interaction between in 1 given by the simple SI formula  frequency of m . We derived the frequency shift, 1  O1  kN q rB’ 2 D  (5.1)  where k is the Coulomb constant, 2 2 is the number of in 2 ions, B is mN , q is the charge on 2 1 and the magnetic field strength, r is cyclotron radius of m  , 2 in  and D is the “apparent  Coulomb distance” of in 2 from in . By extending the point model into the magnetic-field 1 (z) direction, a charged-line model( ) for Coulomb shifting in FT-ICR was developed, 1 whose Coulomb-induced frequency shift was given by the SI formula  —  q 2 kN  where z 1 is the  1 1/2 z ( (l/2_z )2+D2 1  z-axis position  1/2  —  +  +  (l/2÷zi)2+D2  )  ‘  5 2) .  of a particular m 1 ion, 1 is the length of the line charge and  the other parameters are as in Eq. (5.1). Note that for a macroscopic sample of ions disthbuted along the z direction, Eq. (5.2) predicts a Coulomb-induced line broadening. For z 1  =  0, Eq. (5.2) reduces to  —121— q 2 kN  —  01 —  rB  53  q (112)2+ D 2  which is the maximum Coulomb shift for the line-charge model. Eqs. (5.1) and (5.2) were derived as follows. We assumed that the motion of the perturbing m 2 ions gave rise to an average force, AF, on in 1 which gave rise to a frequency shift, &o given by ,  =  &o  1 rB. q  (5.4)  Eqs. (5.1), (5.2), (5.3), and (5.4) are equivalent to Eqs. (4.14), (4.19), (4.20), and (4.13) in Chapter 4, respectively. How then, can the average perturbing force, AF, be calculated? For either the point-charge model or the line-charge model, the average Coulomb interaction is infinite because of the momentary superposition of the ions during one complete cyclotron cycle of the frequency difference °kn  —  (002.  This, of course, is an  inherent flaw in the model. So, instead of attempting to calculate an average Coulomb force for the point-charge or line-charge models, we calculated an apparent distance of m 2 as seen by m 1 and then modeled the apparent Coulomb force as the force from the “apparent Coulomb distance”. If we took the average location of m 1 in the laboratory frame as the appropriate distance model, the “apparent Coulomb distance” was r, the cyclotron radius of m 1 and m , and if we took the mean distance between m 2 1 and in 2 as the appropriate distance model, the “apparent Coulomb distance” was 4 nit (or  r for the  root mean square distance). These models gave rise to simple analytical formulae for both Coulomb shifts and Coulomb broadenings.O) Moreover, the simplicity of these models provided the physical basis for both Coulomb shifting and Coulomb broadening. For each of the point charge and line charge models, quantitative prediction of the magnitude of the shifts and broadenings was less clear and the absence of a definitive “apparent Coulomb  —122—  distance” in these models was unsatisfying. We noted that further work would be necessary to choose more defmitively an appropriate “apparent Coulomb distance” model. It is the purpose of this chapter to provide this model. Some aspects of the present chapter have appeared in communication form.( ) 2  5.2  The Charge-Disk Model  5.2.1 Two Charged Disks in a Rotating Laboratoiy-Frame Figure 5.1 shows the charged-disk model for Coulomb frequency shifting in FT ICR. This 2-dimensional model takes the distribution of m 1 ions to be a uniformly charged disk of radius r’, which is undergoing excited cyclotron motion at a laboratory-frame frequency  with a cyclotron radius of r. A second ion mass, m , also uniformly 2  distributed over a disk of radius r’, is also undergoing excited cyclotron motion with cyclotron radius r, but at a laboratory-frame frequency  %2.  The distribution of m 1 and m 2  ) at the laboratory-frame 3 ions in a coordinate frame is shown in Fig. 5.1, which is rotating( frequency co 1 disk is stationary and the m 2 disk rotates . In this coordinate frame, the m 01 at a frequency  —  At any instant of time the relative positions of the disks are characterized by an angle cP (Fig. 5.1), which varies linearly with time. Disk 2 will exert on Disk 1 a Coulomb force, the radial component of which is of interest here because the averaging over a cycle of its tangential component wifi be zero as shown Figure 5.2. Calculation of this radial Coulomb force depends upon whether or not the disks are overlapping.  uch’  the angle  when the two disks just touch, is given by  touch  =  2arcsin(---)  and separates the overlapping from the non-overlapping values of Cl).  (5.5)  —123—  F  Figure 5.1  Uniformly-charged-disk model for FT-ICR.  The figure gives the  instantaneous location of Disk 1 and Disk 2 in a rotating frame which rotates in the x-y plane at the cyclotron frequency of Disk 1. In this rotating frame, the location of Disk 1 is constant and Disk 2 rotates on the circular path shown, at a frequency (O  —  %• The cyclotron radius of both disks  is r. The disk radius is r’ for both disks.  ‘  is the instantaneous angular  coordinate for Disk 2. Disk 2 exerts an instantaneous radial Coulomb force, Frad,  on Disk 1, the magnitude of which is given by either Eq. (5.13) or Eq.  (5.29). In the limit where r’ —* 0, this model reduces to the charged-point model of reference 1. Dispersal of the charges of Disk 1 and Disk 2 into the z-direction creates the charged-cylinder model of the text.  —124—  F  or F  Figure 5.2  F’d  F’  The averaging of tangential components of the Coulomb interaction between two ions m 1 and m . Whether their cyclotron radii, r 2 1 and r , are the same 2 or different in the rotating laboratory-frame of Fig. 5.1, when m 2 moves from the position (r , )) to its mirror position m, (r 2 , —), their tangential 2 components of Coulomb forces F and F’ on m 1 are: F  _F’tan•  =  Therefore, the average of F and F is zero, that is, the averaging over a cycle of F is zero. However, their radial Coulomb forces on =  F.  are:  —  125  —  5.2.2 Radial Coulomb Force on Disk 1 for Non-Overlapping Disks (0  > 0 uch)  For values of the angle 0, (Fig. 5.1) for which the disks do not overlap, the instantaneous radial force can be evaluated with aid of Figure 5.3 as follows: Consider the xdy of Disk 2 located at (x th differential element 2 ,y 2 ). This element will exert a 2 Coulomb force dF on the differential element 1 xdy of Disk 1 located at (x dx ,y 1 ). This 1  differential force is given by  2 2 U  —  1  1 2 c7 dy  1 dy  2 1 Y  2  where k is the Coulomb constant, o is the charge density of Disk 1 and  2 U  is the charge  density of Disk 2. The charge densities are given by N . 2 ( 1 )q ) 2 ( 1 2 i r’  —  O’l(2) —  where N 1 and q 2 and q ) are the number of and charge of the ions which make up Disk 2 1 (N 1 (Disk 2). The radial component, dF, of the total differential force dF is given by  dF ra d  =  k  Udxdy 2 2 Udxdy 1 1 2 (  X2 + j 2 ‘  —  2  (  sin a, e  (5.8)  —  where ae, the angle of elevation (Fig. 5.3), which gives the direction of the total  differential force, is given by  ae  =  arcsm (  2 1 Y  i  /  (5.9)  —126—  y  +2 2 1 x — 1 (y = r’ r) Disk 1 —r sin 2 (x  1)2+  r cos) 2 = r’ 2  Disk 2  x  0  Figure 5.3  Differential elements of the charged-disk model for Coulomb shifting in F1’-ICR. The differential element at P 2 of Disk 2 exerts a radial Coulomb force on a differential element of Disk 1 located at P . This force is given 1 by Eq. (5.8). Disk 1 is static in the rotating frame of the figure and is located at 01. Disk 2 can rotate on the circular cyclotron path shown. 02 is the instantaneous location of Disk 2. ) is the instantaneous angular coordinate of Disk 2. ae is the angle of elevation of the total differential Coulomb force from P . 1 2 to P  —  127  —  Now for Disk 1, 1 (y  +  —  r  =  , 2 r’  (5.10)  and for Disk 2,  —rsin) 2 (x  +  —rcoscl.’) (y 2  =  . 2 r  (5.11)  The total radial Coulomb force on Disk 1 is then given by the integral of Eq. (5.8) over all relevant values of the coordinates x 2 ,y 1 , andy 2 ,x 1  fYimx  Fj  =  J 2 kirirJ Yi  fYzxn  J  y  fX  fX  sin a 1 dx 1 2 dx dy 2 dy (x 2 1 y x + (y ) ) 2  J  j  1 x  (5.12)  —  —  Substituting Eq. (5.9) into Eq. (5.12) gives  Yi  1 F=ku  Y  1 X  X  i Lm L  ÷(y ) 2 —x 1 1 [(x  2 dx dX , 1 1 dy 2 dy  >  312 ) 2 _y 2j  (5.13)  The integrals overx 2 in Eq. (5.13) can be analytically evaluated,( 1 andx ) reducing Eq. 4 (5.13) to  (Yi  cY 1 =ku J Fr 2 Yi  (Y2  J  Yi  I  1 d 2 dy , y  (5.14)  —128—  where  =  Y1  2 Y  (q (Ximin  2 j 1 _g(X j _X )2  Ylma r +  1 y  Xlmjn  1 2 + (y X2max)  +  ) 2 1 —y (Y  +  b’  —  2 Y2)  + •\I(Xlmax_X2min)2 + ()‘i  _y ) 2 2);  ) 2 Y  (5.15)  ?;  (5.16)  (5.17)  =  r  =  rcos+r’;  (5.18)  rcoscl—r’;  (5.19)  Ym=  Xlmax  —  Ir’ 2_  =  =  —  —  4r’ 2  (y  —  1 (y  (5.20)  2 ; r)  —  —  (5.21)  2 ; r)  _rcosø)2 2 x= r02_  +  rsin;  (5.22)  sin  (5.23)  and 2 x  =  —  r’ 2  —  2 (Y  —  r  cos  )2  + r  ci’.  Eq. (5.14) gives the radial Coulomb force on Disk 1 arising from Disk 2, for values of D> eLouch.  —129—  5.2.3 Radial Coulomb Force on Disk 1 for Overlapping Disks (ci) < Figure 5.4 shows the two disks when 0  <Otouch•  touch)  Because each of the two  overlapping disks has the same cyclotron radius r, both crossing points, P and P_, will make an angle 0i2 with y-axis. Let a be the distance between P÷ and the origin, 0.  P+O  =  (5.24)  a.  Then, the coordinates of P are given by Eqs. (5.25).  =  a sin (0/2),  (5.25x)  =  a cos (0/2).  (5.25y)  Substituting Eqs. (5.25) into Eq. (5.10), or (5.11), gives  =  rcos (0/2)±’sJ r’  2_  2 (0/2). 2 sin r  Eq. (5.26) with the positive term gives the polar coordinate, a÷, for the point,  (5.26)  (Fig.  5.4). Eq. (5.26) with the negative term gives the poiar coordinate, a_, for the point, P_. The Cartesian coordinates (x, y÷) of  are given in terms of the polar coordinates r, r’,  and 0 by Eqs. (5.27)  x  =  [r cos (0/2)  +  r’ 2  =  [r cos (0/2)  +  r’  —  2 sin r 2 (0/2) 1 sin (0/2),  (5.27x)  —  2 sin r 2 (0/2) 1 cos (0/2).  (5.27y)  2  —130—  y  (x, .v+)  Disk 1 Disk 2  x 0  Figure 5.4  Overlapping charged disks. When 1 is less than  ouch  (Eq. (5.5)), the  disks overlap, creating the hatched region shown, which does not contribute to the Coulomb force between the disks.  —131—  Similarly, the Cartesian coordinates (x_, y_) of P_are given by Eqs. (5.28).  x  =  [r cos (0/2)  y_  =  [r cos (0/2)  r’ 2 r2 sin 2 (0/2) ]  —  sin (0/2),  (5.28x)  r’ 2 r2 sin 2 (0/2) 1 cos (0/2).  (5.28y)  —  —  —  Evaluation of the integral for the Coulomb force between two overlapping disks is facilitated by dividing each disk into three sections based upon the crossing points, Pf and P_, as shown in Figure 5.5. Disk 1 is divided into the sections , 11 , A 12 and . A 13 Disk 2 A is divided into the sections A, , 22 and . A 23 The overlapping area (shadowed sections A in Figs. 5.4 and 5.5) does not contribute to the Coulomb interaction. Therefore, the total  radial Coulomb force on Disk 1, arising from an overlapping Disk 2, is given by Eq. (5.29), as a sum of six integrals.  1 F  where F 1  =  1 F  +  2 F  +  3 F  +  4 F  +  5 F  +  , 6 F  0  (5.29)  <Ouch  6 are given by Eqs. (5.30), (5.35), (5.38), (5.40), (5.42), and (5.43), F  —  respectively. 11 and A 1 is the Coulomb force between regions A F 21 of Fig. 5.5.  Yi=Y+  1 F  =  k  ( u = 1 J 2  I  =y 2 y  ty. 2 iy  X  I  =X 1 x 12  WiD 1 JX  I JXzX  dx d 2 . 1 x y  1 [(x  —  ) 2 x  +  1 (y  —  )2] 2 y 312  (5.30)  —132—  P (x÷, y..) 22 A  Disk 1  13 A  Disk 2 P_ (x_, yJ  Figure 5.5  Sections of overlapping disks used for piecewise evaluation of the total radial Coulomb force from Disk 2 to Disk 1. The regions are used in Eqs.  (5.30), (5.35), (5.38), (5.40), (5.42), and (5.43).  fY1Y+  1 F  =  1a a 2  k  (Y2=Y+  J J YiY-  YzY  ‘2  (5.31)  2 dy dy 1  where 12  1 y  12 —(X  2 Y  —  \I(Ximin  ((X1min_X2max)2 +  ) 2 y  2 + X2max)  1 (y  2 ) 22 —X  1 —Y (y ) 2  +  —  +  )  1 —Y (Y ) 2  12 (X  —  22 X 2 )  +  1 (y  —  (5.32)  —133—  and  12 X  =  22 X  =  r’  —  2_  r cos  —  2_  2 (y  r sin ;  +  (5.33)  (5.34)  .  —  2 is the Coulomb force between regions A F 11 and A 22 and between regions A 13 and 21 of Fig. 5.5. A  2 F  = 2k  =X 1 x 12  y  py=y+  Limb  2JYiY  ,,, 2 x  I  Yi kxi  —  —  +  Y2 1 (y  —  X . dx 1 d 2 x y  PYiY+  2 F  =  cT 1 2kcT J 2 “YiY-  (5.35)  fYm,  J  3 I d 2 1 y  (5.36)  YzY.  where  13  1 =  12  —  X2max  )2 +  1 (y  ) 2 y  +  —  ) 2 y  —  )2 +  —  (Ximin  —  X2max  —  J(Ximin  —  min) + X 2  —  (Yi  —  ) 2 Y  )  1 (X  —  2 + X2min)  (  —  (5.37)  —134—  3 is the Coulomb force between regions A F 11 and A and between regions A 12 and 21 of Fig. 5.5. A  12 X 1 x  YY-  YiY+  , 2 x  =2ku a 3 F 1 1 2 1  L  [(x ) _ 1 ) 2 312 2+(y 2] x y  X dx dx 1 dy 2 dy . 1  (YiY+  =2ku a 3 F 1 ) 2 Yi—Y-  (5.38)  (Y2Y_  )  13 1 dy 2 dy .  (5.39)  4 is the Coulomb force between regions A F 13 and A 23 and between regions A 12 and 22 of Fig. 5.5. A  fYi  F4 —2k —  1 U  I U 1 j 2  fY  fXi  I  I  1 =y+ Jx 2 Jy  fX  I 2 Jx  1 [(x  —  ) 2 x  + (y 1  X 1 th dx d 2 . y  —  )2] y 2 / 3  (5.40)  (Yi (Y  4 F  =  ) 2 a 1 2ka YiY.  J  I d 2 . 1 y  YzY.  13 and A 22 of Fig. 5.5. 5 is the Coulomb force between regions A F  (5.41)  —135—  Y1Y-  F—k _  1 X  Y2max  I I I  O °2J,  2 X  =y, Jxj, 2 Jy  1 [(x  +  —  1 (y  —  )2] y 2 / 3  X 1 dx d 2 . x y  (5.42)  . 1 2 dy Ii dy  (5.43)  (Ya=Y (YZm  5 F  =  J  ko Yimin  Yz=Y+  6 is the Coulomb force between regions A F 12 and A 23 of Fig. 5.5.  F—k 6 —  i  (Yi  fyy-  I  I  = 1 °2J  1 fX  I  I  111 JX  (Yi  =ku 6 F ) 2 Y 1 YiY+  ,, 21 Jx  1 [(x  —  ) 2 x  + (y 1  —  )2] 2 y 312  X 1 dx d 2 . x y  (5.44)  I dydy . 1  (5.45)  (YzY  J  5.2.4 Instantaneous Radial Coulomb Force The instantaneous radial Coulomb force is given by Eq. (5.13) for (1)> Eq. (5.29) for  CX) < touch  ouch  and  The integrals in these equations were numerically evaluated  ) on an IBM 3081 mainframe computer 7 5 using a multiple Gaussian integration algorithm( (the corresponding Fortran program is given in Appendix A3.1).  Several of the  —136—  calculations were duplicated using a Romberg integration algorithm( ) on a Nicolet 1180 5 minicomputer (the corresponding Fortran program is given in Appendix A3.2), with agreement to 3 significant figures with the Gaussian results (Appendix A3.3). Figures 5.6 —  5.10 show the results of many calculations with these equations. Fig. 5.6 shows the instantaneous radial Coulomb force for a system of two disks,  each of charge 10 electronic charges, undergoing excited cyclotron motion, as a function of the position angle, ‘1 (Fig. 5.1). The force curve in Fig. 5.6 is the plot of the value of  Eq. (5.13) or Eq. (5.29) for a total of 291 different values of 0. The ordinate shows the force in Newtons for r  =  1 cm, r’  =  5 mm, for which the ratio of cyclotron radius to the  disk radius is 2. When the calculations were repeated for other values of r and r holding the ratio r/r’ constant, the shape of the force curve remained the same if the ordinate scale was adjusted by a factor of hr . For example, if r = 2 cm and r’ = 10 mm, rir’ still equals 2 2, and the force curve superimposes with that of Fig. 5.6, if the ordinate scale extends from 0  —  16 Newtons, rather than 0 0.5x10  —  16 Newtons as in Fig. 5.6. The 2.0x10  overall shape of the force curve is readily explained. As Disk 2 traverses its cyclotron orbit (Fig. 5.1), the Coulomb force is a minimum when 0 = it, and reaches a maximum at 0 touch 0  For  touch’  =  the overlap area (Fig. 5.3) increases as 0 decreases and the  radial Coulomb force becomes zero when the disks completely overlap (0 = 0). Figure 5.7 shows the instantaneous radial Coulomb force over the range 0 for rir’  =  <  0<  it,  5. The general features of Fig. 5.7 are similar to those of Fig. 5.6, but the  maximum in the instantaneous radial Coulomb force occurs at the smaller touching angle appropriate to r/r’ = 5. Figures 5.8, 5.9, and 5.10 show the force curves for r/r’ = 10, 20, and 100, respectively. Note that the cyclotron radius was held constant for Figs. 5.6  5.10, and the ratio r/r’ varied by changing the disk radius, r’.  —  —137—  2.0 1.6  ‘ad  1.2  (10_lB N) 0.8 0.4 0.0 0  Figure 5.6  it/4  lt/2 cJ (radian)  32t/4  it  Radial Coulomb force on Disk 1, a uniformly charged disk of m 1 ions, due to a uniformly charged disk of m 2 ions, as a function of the position of Disk 2. This position is characterized by the angle 0 (Fig. 5.1). The force is calculated from Eq. (5.13) if 0  <  < 0 uch  and from Eq. (5.29) if  <0< it, where ouch is the touching angle, the value of 0 where the two disks touch (Eq. (5.5)). The shape of the curve depends upon the ratio nT’, the ratio of the cyclotron radius to the disk radius. For this figure this ratio  equals 2. The ordinate gives the force in Newtons for r, the cyclotron radius =  =  1 cm, r’, the disk radius  =  5 mm, and N , the number of m 2 2 ions,  i0. The average radial Coulomb force, <Fave  Eq. (5.46). Figs. 5.7  —  >,  was calculated from  5.10 are analogous curves for larger values of n/n’.  Note that both the maxima in the force curve and the average force, <Fave>  are lower than their counterparts in any of Figs. 5.7  —  5.10.  —138—  Frad  (1O_16 N)  0 Figure 5.7  it/4  lt/2 (I) (radian)  3it/4  it  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  r/r’  =  5. This figure was derived as was Fig. 5.6, except that r’ = 2 mm.  Both the maxima in the force curve and the average force, <Fave>, are larger than their counterparts in Fig. 5.6, but smaller than their counterparts in Figs. 5.8  —  5.10.  10 8 ad  (1O_16  6 N) 4 2  0 0 Figure 5.8  7t12 0 (radian)  37t/4  it:  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  rir’ = 10. This figure was derived as was Fig. 5.6, except that r’ = 1 mm. Both the maxima in the force curve and the average force, <Fave>, are larger than their counterparts in Fig. 5.6 and 5.7, but smaller than their counterparts in Figs. 5.9 and 5.10.  —139—  20 16 ad  (1O_16 N)  12 8 4 0 0  7t/4 ‘  Figure 5.9  31t/4  1t12  it  (radian)  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  rir’ = 20. This figure was derived as was Fig. 5, except that r’ = 0.5 mm. Both the maxima in the force curve and the average force, <Fave> are larger than their counterparts in of Figs. 5.6  —  5.8, but smaller than their  counterparts in Fig. 5.10.  100  r/r’= 100 r = 1 cm r’ = 0.1 mm  80 Frad  60  N) 16 (10  ouch  =  0.0200003  40  <‘ad>  =2.175  20 0 0 Figure 5.10  it/4  it/2 cJ (radian)  3n/4  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  r/r’ = 100. This figure was derived as was Fig. 5, except that r’ = 0.1 mm. Both the maxima in the force curve and the average force, <Fave>, are larger than their counterparts in any of Figs. 5.6—5.9.  —140—  Comparing Figs. 5.6—5.10 shows that the maximum in the radial force appears at a progressively smaller touching angle, 0 touch’ as r/r’ increases and further, that the magnitude of the maximum in the force curve is proportional to r/r’. In the limit rir’  —  00,  the disk model of this work approaches the point model( ) with its infmite radial Coulomb 1 force at 0=0.  5.2.5 Average Radial Coulomb Force Eqs. (5.13) and (5.29) give Frad the instantaneous radial Coulomb force, as a function of the angle 0. The average-over-one-cycle radial Coulomb force, <Frad> which we assume accounts for Coulomb shifting, was evaluated using Simpson’s rule( ) 8 from the integral f7  fusi  <Fraj>  =  -  77.;  (IJo  Eq. (5.13) dO +  J  Eq. (5.29) dO  The value of < Frad > for the force curve in each of Figs. 5.6 figure. Note that in Figs. 5.6  —  —  ).  (5.46)  5.10 is indicated in the  5.10, the only parameter which was varied was the disk  radius, r’ and therefore the ratio r/r’. Note that as this ratio increases, <Frad> the average radial force, increases and in the limit r/r’  —  oo,  where the disk model becomes the point  model, the average radial Coulomb force becomes infmite.  5.2.6 Apparent Coulomb Distance, D Of greater interest than <F>, which is characteristic of N , the number of m 1 1  , the number of m 2 ions, and N 2 ions, is the “apparent Coulomb distance”, D. This distance, which would be a function only of the model, can be obtained as follows. Combining Eqs. (5.1) and (5.4) gives  —141—  AF  =  1 q 2 kN  (5.47)  Equation (5.47) gives the average radial Coulomb force on a single perturbed ion. For N 1 ions of mass m , the force would be 1  AF  =  2q 1 q kN 2N 1  (5.48)  Now if we assume that the average Coulomb force (Eq. (5.46)) for the charged-disk model of this work can be used with Eq. (5.1), which was derived from the charged-point model, we can substitute the average Coulomb force given by Eq. (5.48) into Eq. (5.1) to give  Dc  1 N q 2 /kN  ‘4  49  <Frad>  1q Eq. (5.49) gives the “apparent Coulomb distance”, D, as a function of N , the total 1 charge on Disk 1, N 2q , the total charge on Disk 2, and <Frad 2  > (Eq.  (5.46)), the average  radial Coulomb force for the uniformly charge-disk model. D is the distance at which the charges of Disk 2 appear to be seen by Disk 1. Hence, it is termed “apparent Coulomb distance”. Table 5.1 lists values of D for many possible situations. D is proportional to the cyclotron radius, r; and the most insightful way of tabulating values of D is as a function  , as is done in Table 5.1. When tabulated this way, D varies modestly, from 1.792 r t r/r when rlr’ =2 to 1.030 r when rir’  =  100. The dependence of D upon r/r’ was confirmed  for the 30 cases indicated by the pound signs. For example, calculation of D from Eq. (5.49) for nT’ =10 gave a constant value of 1.327 for DJr, for rir’ = 1 cm/i mm, 2 cm/2 mm, 3 cm/3 mm, 4 cm/4 mm, and 5 cm/5 mm. As indicated in the footnotes of Table 5.1,  —142—  Table 5.1 Apparent Coulomb Distance for Coulomb Shifting and Coulomb Broadening in FT-ICR  Model  Apparent Coulomb Distance Dc*  Average Position  Reference  1  Mean Distance  1.273 (4/it) r  1  Rms Distance  1.414  (‘J) r  1  Charged disk and charged cylinder; rir’ = 2#  1.792 r  this work  Charged disk and charged cylinder; r/r’ = 3@  1.629 r  this work  Charged disk and charged cylinder; r/r’ = 4 Charged disk and charged cylinder; rir’ = 5#  1.539 r  this work  1.478 r  this work  Charged disk and charged cylinder; rir’ = 8  1.371 r  this work  Charged disk and charged cylinder; rir’ = 10  1.327 r  this work  Charged disk and charged cylinder; r/r’ = 15@  1.257 r  this work  Charged disk and charged cylinder; r/r’ = 20#  1.2 15 r  this work  Charged disk and charged cylinder; r/r’ = 25  1.183 r  this work  Charged disk and charged cylinder; rir’ = 50#  1.102 r  this work  Charged disk and charged cylinder; rir’ =100#  1.030 r  this work  *) Calculated from Eq. (5.49). D gives the Coulomb-induced frequency shift when used  with Eqs. (5.1)  —  (5.3).  $) This distance was termed an “average distance” in ref. 1. #) The apparent Coulomb distance, D, was calculated for r = 1 cm, 2 cm, 3 cm, 5 cm, and 10 cm. @) The apparent Coulomb distance, D, was calculated for r =3 cm.  )  The apparent Coulomb distance, D, was calculated for r = 1 cm.  —143—  for some of the values of D only a single calculation with Eq. (5.49) was performed.  Figure 5.11 shows a plot of the values of D from Table 5.1 vs. rir’. We were able to fit these values to a double-exponential equation of logarithmic rir’  =  [0.5504  +  1.128  bOl08 3 _  +  0.5338 e 2511  losfr/r’)]  r.  (5.50)  Several other curve fitting equations are also available (Appendix V. 4), For example,  =  {  2.172  +  0.05836 [log(r/r’)] 4  —  1.535 log(rIr) + 1.014 [log(r/r’)] 2  }  —  0.3827 [log(r/r’)] 3  r.  (5.51)  Eq. (5.50) is the best one among those curve fitting equations for calculating values of D  with any r/r’ ratio between 2 and 100 (Appendix A3.4). There is no physical meaning to be associated with Eqs. (5.50) and (5.51).  5.2.7 Validity of the Charged-Disk Model 5.2.7.1 Assumption of small like-ion interactions There are several approximations which have been made in the above modelling in  addition to those which already have been identified above as assumptions. The first additional assumption is that each of Disk 1 and Disk 2, the distributions of perturbed ions  and perturbing ions respectively, maintain their integrity as each traverses its cyclotron orbit. For Disk 1, this integrity will be maintained if the Coulomb forces within the disk are negligible with respect to the Lorentz force on the elements of the disk. Consider a differential circular element of radius dr’ and area dA in Disk 1 whose charge density is o, disk radius is?, and cyclotron radius is r. The mass and number of  -144-  2.Or  1.8  1.6  (cm) 1.4  1.2  1.0  r / r’  Figure 5.11  Normalized apparent Coulomb distance parameter, D, as a function of TI?, the ratio of the cyclotron radius to the disk radius, for the charged-disk  model. The data points shown as circles were calculated from Eq. (5.49). The data points for rir’ from 2 to 100 can be fitted to the double exponential logarithmic equation shown. This equation can be used to predict values of D for any rir’ ratio between 2 and 100. No physical meaning can be  attached to this equation.  —145—  ions in Disk 1 are m . Consider next an identical differential element which is 1 1 and N adjacent and radially aligned with the first differential element. Since r’  >>  dr’, the two  differential circular elements can be approximated to two point charges. The radial Coulomb force, dFr&d on the first element, arising from the second element would be  k 01 dA 1 dA 4 (dr’) 2  r&d  =  k o  2 it  k  =  (dr’) 01 2 4 (dr’) 2  1  it  2 (dr’)  2 (dr’)  (5.52)  The differential Lorentz force on the first differential element would be  dFi..orentz  =  o.j it (dr’) w 01 r B.  (5.53)  If we assume that the criterion for maintaining the integrity of the charged disks is  differential like-ion Coulomb force  <<  differential Lorentz force,  (5.54)  then, substituting Eqs. (5.52) and (5.53) into Inequality (5.54) yields  k o 4  it  rB,  <<  (5.55)  or  rB 01 4w  or equivalently,  =  rB 1 4q 2 ,  (5.56a)  —146—  <<  8.52 x 108  rB 1 q 2  (5.56b)  Since Disk 1 is uniformly charged,  (5.57)  01 =  Substituting Eq. (5.57) into Inequality (5.56a)  1 N  4 <<  r B 2 r’ 2 km  (5.58a)  or equivalently,  1 N  <<  2.68 x iO  T  2 B  ‘2  (5.58b)  miT  Inequalities (5.56b) and (5.58b) describe the criterion for validity of the assumption of negligible like-ion Coulomb interactions, contained in the models of this chapter, in terms of  o,  the charge density of Disk 1 in electronic charges per square mm; N , the ion 1  number in Disk 1; r’, the disk radius in mm; m , the ion mass in Dalton; r, the cyclotron 1 radius in cm and B, the magnetic field strength in Tesla. Inequalities (5.56a) and (5.58a) are the corresponding expressions in SI units. For example, for N 1  r = 1 cm, B  =  1 Tesla and m 1  =  =  10 ions, r’  =  1 mm,  100 Daltons, the left hand side of Inequality (5.58b) equals  iO, whereas the right hand side of Inequality (5.58b) equals 2.68 x  and the  inequality is satisfied. Nevertheless, when all the other parameters are the same but r’  =  0.1 mm, the right hand side of of Inequality (5.58b) is changed to 2.68 x iO which is  —147—  barely satisfied. Analogous inequalities, of course, apply for Disk 2. The criterion expressed by Inequalities (5.58) is violated by any point-charge model or any line-charge model, where by defmition, r’ =0. This violation is a characteristic of almost all prior treatments of FT-ICR motion. In these prior treatments, the violation is implicitly made and then implicitly ignored. Inequalities (5.58) provide the criterion for when it is safe to ignore like-ion Coulomb interactions.  5.2.7.1 Assumption of small unlike-ion interactions The derivation of Eq. (5.4) from the cyclotron equation implicitly requires that the perturbation of the motion of Disk 1 by Disk 2 is small. That is  I  the average value of unlike-ion Coulomb Force  I  <<  Lorentz force.  (5.59)  By symmetry and noting that the average value of the tangential component of the Coulomb force is zero, the average value of the disk-model Coulomb force over one cycle, 0  <  ci, <  2it, equals the average value of the radial component of the Coulomb force. Substituting Eq. (5.47) into Eq. (5.59) gives  2q 2 kq 1 N <<  1 c rB. q  (5.60)  Solving for N 2 gives  2 N  or equivalently,  <<  D rB w 2 kq  —  (D’) r q 2 2 B 3 k q2 m 1  (5.61 a)  —  2 N  148  6.70 x 1010  <<  —  (D’) r q 2 2 B 3 2 1 q  (5.61b)  ,  where D’ is the numerical component of D (Table 5.1). Inequality (5.61b) gives the criterion for validity of our uniformly charged-disk model as a function of N , the number 2 of m 2 ions, q 1 and q 2 in units of the electronic charge, m , the charges of m 2 1 and m , the 1 ion mass of Disk 1 in Dalton, r, the cyclotron radius of the disks in cm, Do’, the numerical component of D from Table 5.1, and B, the magnetic field strength in Tesla. Eq. (5.61a) is the corresponding equation in SI units. If Inequalities (5.61) are satisfied, it is implicit in our model that the disks are rigid and will maintain their integrity in the presence of the Coulomb perturbation. For example, for N 2 1 q  =  2 q  =  1 electronic charge, B  =  =  10 ions, D’  1 Tesla and m 1  =  =  1.327 (r/?  =  10, Table 5.1),  100 Daltons, the left hand side of  Inequality (5.61b) is 10, whereas the right hand side is 1.18 x iO, and the Inequality is satisfied.  5.2.7.3 Assumption of high frequency perturbations Another assumption implicit in our treatment is that the perturbation frequency, —  k2’  is large with respect to the perturbation, A 0 That is,  —C0 I% 1 02  (5.62)  >>  Substituting Eq. (5.1) into Inequality (5.62) gives  >>  kN q 2 B 2 D C  ‘  —  149  —  or alternatively, 1 2 m m 1 2 m  I  kN (D’) r 2 B 3  >>  (5.64a)  or equivalently,  Im-mi 2m m 1  f‘  =  >>  1.49 x  lO  N  (5.64b)  3B 2 2 r (D’)  . Inequality (5.64b) gives the “high frequency criterion” for the validity of the 2 q  disk model as a function of m 1 and m , the masses in Dalton of the perturbed and 2 perturbing ions, Do’, the numerical component of D in Table 5.1, N , the number of m 2 2 ions, B, the magnetic field strength in Tesla, and r, the cyclotron radius in cm of Disk 1  and Disk 2. For example, for B  =  1 Tesla, r  =  1 cm and N 2 = i0, and D’  =  1.327 (r/r’  =  10, Table 5.1), the right hand side of Inequality (5.64b) is 2.62 x iü-. For a mass difference of 1 Dalton, the left hand side of Inequality (5.64) would be 10” at 100 Daltons, and the inequality is satisfied. However, at 1,000 Dalton, the left hand side of (5.64b) would be 10-6, and the criterion of high frequency perturbations is only barely satisfied. At masses above 1,000 Dakons, we would expect that greater mass separations, higher magnetic fields or lower numbers of perturbing m 2 ions would be needed for the models of this work to be valid. Perturbation treatments, where an oscillatory perturbation causes a sample to “see” only the average value of the perturbation over one perturbation cycle, are known. For example, Inequality (5.62) has its counterpart in NMR where magnetic field gradients which would broaden the signal from a macroscopic sample are averaged to zero by spinning the sample, if the sample-spinning frequency exceeds the inhomogeneously broadened linewidth. The same process applies to FT-ICR, where the cyclotron motion  —150—  ) The physical basis for these 9 itself averages x, y magnetic inhomogeneities to zero.( averagings of oscillatory perturbations is the uncertainty principle.  5.3  The Charge-Cylinder Model  5.3.1 Apparent Coulomb Distance and Charged-Cylinder Model In many Fr-ICR experiments, ions are formed in a cylindrical volume whose axis coincides with the z axis of the ICR cell. The cylinder has a radius r’ and extends from —112 to +1/2, where l is the z-axis length of the cell. After Fr-ICR excitation, which we will assume is uniform throughout the spectrum, the dynamic disthbution of ions in the cell would be a sum of cylinders, one for each ion mass, each cylinder rotating around the z  axis with cyclotron radius r. For purposes of calculating Coulomb effects between ions of different masses, we can examine the system in a rotating frame, just as was done for the charged-point, charged-line and charged-disk models. This charged-cylinder model can be created from the charged-disk model by dispersing the charge for each of Disk 1 and Disk 2 (Fig. 5.1) into the z direction. The charged-cylinder model is shown in Figure 5.12. ) gave 1 Derivation of the Coulomb-induced frequency shift for the charged-line model( Eq. (5.2) which was a function of the same “apparent Coulomb distance”, D, as was present in Eq. (5.1), the frequency shift equation for the charged-point model. We therefore assume that our “apparent Coulomb distance” D for the charged-disk model of this work, would also be applicable to Eqs. (5.2) and (5.3). That is, one could substitute “apparent Coulomb distances” from Table 5.1 into Eqs. (5.2) and (5.3) to calculate Coulomb-induced frequency shifts and Coulomb-induced line broadenings for the chargedcylinder model. Rigorous conformation of the legitimacy of this assumption would have to come from the evaluation of 6-dimensional integral equations, analogous to the 4dimensional integral Eqs. (5.13) and (5.29).  —151—  F  Cylinder 1 2r’ Cylinder 2  x  0 I I I I I  S  I I  S  S  S  S  S  S  z  S  S S  S  S  S  S  S  I S  S  S  S  S  S  S  S  0  544 t  0  4%  Figure 5.12  The Charge-cylinder model. The cyclotron radius  r, cylinder radius r’, and  z direction length 1 of Cylinder 1 and Cylinder 2 both are the same. The directions of applied magnetic field B and their cyclotron motions are shown as indicated.  —152—  5.3.2 Validity of the Charged-Cylinder Model The charged-line model predicts smaller Coulomb-induced frequency shifts than does model.( That is, Eq. (5.2) (Eq. (4.19) in Chapter 4) gives smaller ) the charged-point 1 frequency shifts than does Eq. (5.1) (Eq. (4.14) in Chapter 4). This is not surprising since dispersing the m2 ions into the z direction will reduce the Coulomb force seen by any single  1 ion in line 1. We would similarly expect that the Coulomb-induced frequency shifts for m the charged-cylinder model would be less than those of charged-disk model. Accordingly, the validity criteria, Inequalities (5.58), (5.61), and (5.64), derived above for the chargeddisk model, will also apply to the charged-cylinder model. However, for any example characterized by a particular N , r/r’, and r, etc., the criteria will be more easily satisfied for 2 the charged-cylinder model. The charged-cylinder inequalities-of-validity are given below. These were derived using Eq. (5.3), the equation for the maximum frequency shift for the charged line and charged-cylinder models. The corresponding charged-disk expressions were derived using Eq. (5.1).  5.3.2.1 Assumption of small like-ion interactions Consider a differential disk element of radius dr’ and thickness cii at coordinate z 1  =  0  in Cylinder 1, whose cylinder radius is r’, z-length is 1, and cyclotron radius is r. The mass, charge, and number of ions in Cylinder 1 are m ,q 1 , respectively. Consider 1 1 and N another differential cylinder element of radius dr’ and z-length 1 which is adjacent and radially aligned with the first differential disk element. Since Cylinder 1 is uniformly charged, the ion numbers in the differential disk element, dNdl, and the ion number in the differential cylinder element, dNa, are  dN  =  N  (dr’) dl 2 itr 1  =  N  (dr’) dl 2 r 1  (5.65a)  —153—  and =  1 N  d  =  . 2 1 (d?/r’) N  (5.65b)  As ci,’ <<r’, the line-charge model (Eq. (4.17)) presented in Chapter 4 can be applied to  calculate the radial Coulomb force, dFr on the differential disk element at  =0 due to  the differential cylinder element. Then,  =  —  —  k dNdl dN q 2 11 1 2 dr l 4(l/2)2 + (2 dr’) 2 3 dl 1 1 q (dr’) 2 ) k (N 2 l 4 r’  The Lorentz force on the differential disk element would be  ‘‘Lorentz  =  1 N  dl 2 (dr  1 oi r B. q  (5.67)  If we assume that the criterion for maintaining the integrity of the charged disks is  I  like-ion Coulomb force  I  <<  Lorentz force,  (5.68)  then, substituting Eqs. (5.66) and (5.67) into Inequality (5.68) yields k N q dr’ <<  o rB. 0  Since dr’ <<r’, Inequality (5.69a) can be simplified by setting 10 dr’  (5.69a)  r’ in order to get  —154—  the desired result, kNq 10 ‘  rB,  (5.69b)  10 2 rB r’ 1 k  (5.70a)  <<  or alternatively  1 N  <<  or equivalently  1 N  <<  6.70 x 1010  r Br 1  (570b)  Inequality (5.70b) gives the criterion for validity of the assumption of negligible like-ion Coulomb interactions, contained in the charged-cylinder model of this chapter, in terms of N , the ion number in the cylinder; m 1 , the ion mass in Dalton; r’, the cylinder 1 radius in mm; 1, the z-direction length of the cylinder in cm; r, the cyclotron radius in cm, and B, the magnetic field strength in Tesla. Inequality (5.70a) is the corresponding expression in SI units. For example, for N 1 B  =  1 Tesla and m 1  =  =  10 ions, r’  =  1 mm, 1= 2.4 cm, r  =  1 cm,  100 Daltons, the left-hand side of Inequality (5.70b) equals io  whereas the right-hand side of Inequality (5.70b) equals 1.61 x iO, and the inequality is satisfied. Analogous inequalities, of course, apply for Cylinder 2. Note that for the example quoted, the ratio between the right and left hand sides, 1.61 x iO: iOn, of , whereas in the corresponding example for the disk 5 Inequality (5.70b) is 1.61 x i0 model, the ratio between the right and left hand side, 2.10 x i0: 10, of Inequality (5.58b) is 2.10 x i0. These examples illustrate the lesser sensitivity of the charged cylinder model to violations of the “small like-ion interaction” assumption. The Coulomb  —155—  force on the differential disk element due to the differential cylinder element is a maximum value at  =  0. Therefore, for the differential disk element at  0, Inequalities (5.70a)  and (5.70b) still hold.  5.3.2.2 Assumption of small unlike-ion interactions Inequalities (5.61) gave the criterion that the Coulomb interaction between the disks should be small with respect to the Lorentz force on either disk. The corresponding expression for the charge-cylinder model from Eqs. (5.3) (the maximum Coulomb shift)  and (5.4) is  2 N  <<  D C J(l/2)2+D2 2 q 1 rB C 1 2m q  (5.71a)  ,  or equivalently,  2 N  <<  6.70 x 1010  D C .\/(112)2+D2 2 q 1 rB C 1 2m q  •  (5.71b)  Inequality (5.71b) gives the criterion of small intercylinder Coulomb interactions for our uniformly charged-cylinder model as a function of N , the number of m 2 2 ions, q 1 and q , 2 1 and in 2 in units of the electronic charge, m the charges of m , the ion mass of cylinder 1 in 1 Dalton, r, the cyclotron radius of the disks in cm, DC, the “apparent Coulomb distance” (Table 5.1), and B, the magnetic field strength in Tesla. Eq. (5.71a) is the corresponding equation in SI units. Note that unlike Inequalities (5.61) which is a function of Do’, the numerical component of D, Inequalities (5.71) are a function of D itself. If Inequalities (5.71) are satisfied, it is implicit in our model that the cylinders are rigid and will maintain 2 their integrity in the presence of the Coulomb perturbation. For example, for N  =  iO  —156—  ions, D  =  cm, B  1 Tesla and m 1  =  1.327 r (r/r’  10, Table 5.1), q 1  =  =  2 q  =  1 electronic charge, r  =  1 cm, 1  =  10  100 Daltons, the left hand side of Inequality (5.71b) is i0,  =  whereas the right hand side is 4.60 x iO, and the inequality is satisfied. Note that for the example quoted, the right hand side of Inequality (5.7 lb) was 4.60 x 10’°, whereas in the corresponding example for the disk model, the right hand side of Inequality (5.61b) was 1.18 x iO. These examples illustrate the lesser sensitivity of the charged-cylinder model to violations of the “small unlike-ion interaction” assumption.  5.3.2.3 Assumption of high frequency perturbations As for the charge-disk model we assume in our treatment that the perturbation frequency,  %  —  is large with respect to the perturbation, tco on the charged  cylinders. Substituting Eq. (5.3) (the maximum Coulomb shift) into Inequality (5.62) gives  IW  —  (002  I  kN 2 q 2  >>  (5.72)  ,  rB D q(l/2)2+D  or alternatively,  Im-mi 2 1 1 2 m  kN 2  (5.73a)  rB2Dc4(l/2)2+Dc2  or equivalently,  I  =  “2  1 — 2 m m 1 2 m  I  >>  1.49x 10-11  2 N ,  (5.73b)  rB2Dc4(l/2)2+D  Inequality (5.73b) gives the “high frequency criterion” for the validity of the  —157--  charge-cylinder model as a function of m 1 and m , the masses in Dalton of the perturbed 2 and perturbing ions, D, the apparent Coulomb distance in cm, N , the number of m 2 2 ions, r, the cyclotron radius in cm of cylinder 1 and cylinder 2, 1, the length of the cylinders in cm, and B, the magnetic field strength in Tesla. For example, for B 1  =  10 cm, N 2 = iO, and D  =  1.327 cm (r/r’  Inequality (5.73b) is 2.17 x  =  =  1 Tesla, r = 1 cm,  10, Table 5.1), the right hand side of  For a mass difference of 1 Dalton, the left hand side of  Inequality (5.73b) would be 10 at 100 Dakons, and the inequality is satisfied. However, at 1,000 Daltons, the left hand side of (5.73b) would be 10, and the criterion of high frequency perturbations is only barely satisfied. At masses above 1,000 Daltons, we would expect that greater mass separations, higher magnetic fields or lower numbers of perturbing m 2 ions would be needed for the models of this work to be valid. Note that for the example quoted, the right hand side of Inequality (5.73b) was 2.17 x i0-, whereas in the corresponding example for the disk model, the right hand side of Inequality (5.64b) was 2.63 x 10_6. These examples illustrate the lesser sensitivity of the charged-cylinder model to violations of the “high-frequency perturbation” assumption.  5.4  The  Coulomb-Induced  Frequency  Shift  and  Inhomogeneous  Broadening of the Charged-Cylinder Model On the basis of the above discussion in Section 5.3, we can predict the Coulombinduced frequency shift and inhomogeneous broadening of the charged-cylinder model  directly from Eq. (5.2) if this cylinder model is valid. For a certain rir’ ratio, the apparent Coulomb distance, D, can be found from Table 5.1 or calculated from Eq. (5.50). Consider two charged cylinder ion clouds in which every ion is singly charged. For B Tesla, r =  =  1 cm, 1  =  2.4 cm, N 1  1.327 cm and 1.215 cm (r/r’  =  =  2= N  1 m  =  =  1  250 Daltons, m 2 = 251 Daltons, and D  10 and 20, Table 5.1), the validities of Eqs. (5.70b),  (5.71b), and (5.73b) are judged in Table 5.2. All these three Inequalities are satisfied.  —158— 15  13 11 01 tf  9 (Hz) 7 5 -1.2  -0.8  -0.4  0.0 1 (cm) z  0.4  0.8  1.2  14 12 10 8 (Hz) 6 4 -4  Figure 5.13  -2  0 1 (cm) z  2  4  Coulomb shifting and broadening for the charged-cylinder model of Fig. 5.12. The frequency shift for m 1 ions, calculated from Eq. (5.2b), is plotted as a function of the position of of the m 1 ion. The parameters used for these calculations are listed in Table 5.2 and z-axis length of the cylinder as indicated. The upper curve in each figure gives the Coulomb shifts for  rir’  =  20 (D = 1.2 15 cm), and the lower curve in each figure gives the  Coulomb shifts for rIr’ = 10 (D = 1.327 cm).  —159—  Coulomb-induced cyclotron frequency shifts and inhomogeneous broadenings calculated  from Eq. (5.2) are shown in Figure 5.13(a) for these parameters. As a comparison, for the same parameters, except that 1 is changed to 8 cm, the same calculations axe processed and shown in Figure 5.13(b). The longer length of the cylinders will effectively reduce both the Coulomb-induced cyclotron frequency shift and inhomogeneous Broadening. Without doubt, the validities of Eqs. (5.70b), (5.7 lb), and (5.73b) for Fig.  5.13(b) are satisfied.  Table 5.2  The validities of Eqs. (5.70b), (5.71b), and (5.73b) for B r  =  1 cm, 1 =  =  2.4 cm, N 1  =  2= N  1 q  =  250 Daltons, n. = 251 Daltons, andD 2  =  1 Tesla,  2 = 1 electron charge, q =  1.327 rand 1.215 r  Eq. (5.70b) 1 N right-hand  Eq. (5.71b) right-hand 1 N  1.327 r  8 6.43x10  8 6.36x10  5 1.59x10  7 6.28x10  1.215 r  8 3.22x10  8 5.49x10  5 l.59x10-  7 7.18x10  D  5.5  5 i0  Eq. (5.73b) left-hand right-hand  Experimental Tests of The Charged-Cylinder Model  5.5.1 The Unlike-ion Coulomb-Induced Frequency Shifting from Experiment of  Franci et al.( ) 10  Francl et al.(lO) in 1983 measured the experimental ICR frequency vs. ion number in a pulsed (no Fl’) ICR mass spectrometer with an elongated cell (3.18 cmx3.18 cmxl5.34 cm) and a 3.4 Tesla magnetic field.  Four compounds (thmethylamine, carbon  —160—  tetrachioride, benzene, and bromobenzene) were used in the study of Franci et al. The dependences of ICR effective frequencies of these four compounds on ion numbers were experimentally all linear. The slope of the ICR effective frequency vs. ion number for benzene, which has only one mass peak, was measured to be —58 HzJV, where V is voltage of the ion Coulomb potential in the analyzer cell. According to the data given by Franci et al., we deduce  1 million positive ions  =  +  0.186 V (ion Coulomb potential).  (5.74)  Since there is no mass dependence in the ion Coulomb potentia1,( 11 1) the slope of benzene curve can be used as a standard for frequency shifting because of the trapping electric field. In Chapter 4 and this chapter, the unlike-ion Coulomb-induced frequency shift was solved by the perturbation method (Eqs. (4.13) and (5.4)). It has been proved that the experimental ICR frequency approximately was equal to the ion natural cyclotron frequency minus the frequency shifts induced by the trapping electric field and the ion Coulomb ) Therefore, these frequency shifts are superimposed on the ion cyclotron 0 potential.(’ motion. When there exist two ion species simultaneously in the analyzer cell, the unlikeion Coulomb-induced frequency shifts can be calculated by Eq. (5.75),  fi41ei4t  =  —(—58Hz/V)x(N [slopexN + 1 ) 2 N ] xO.186V  (5.75)  1 is the unlike-ion Coulomb-induced frequency shift component in Hz, of an ion where bf species m , due to the presence of an ion species m 1 ; 2  ; 1 experimental cyclotron frequency in Hz, of m  4f  Afei  is the total change in  is the frequency shift component  1 is the ion number of m 1 in unit of induced by the trapping electric field in unit of Hz, ; N  —  161  —  2 is the ion number of m million ions; and N 2 in unit of million ions. The slopes of bromobenzene curves of two mass peaks m/z 156 Br) 79 and 5 H 6 (C  m/z 158 Brj 81 were measured to be —131 Hz/V and —146 Hz/V, respectively. 5 H 6 (C Although Francl er at. did not give the relative ion intensities of these two ion peaks, the intensity ratio of rnlz 156 to mlz 158 should be 100 : 98 from isotopic abundances of 79 Br and 81 Br. If there are totally 2 million ions of m/z 156 and m/z 158 in the analyzer cell, 1.01 million ions of m/z 156 and 0.99 million ions of m/z 158, the unlike-ion Coulomb induced frequency shift of mlz 156 due to the presence of m/z 158 is  =  —  (131 Hz/V x 1.01  —  58 Hz/V x 2) x 0.186 V  =  —3.0 Hz,  (5.76a)  and the unlike-ion Coulomb-induced frequency shift of mlz 158 due to the presence of mlz 156 is  =  —  (146 Hz/V x 0.99  —  58 Hz/V x 2) x 0.186 V  The parameters in the experiment of Franci et at. are: r elongated cell 3.18 cmx3.18 cmxl5.34 cm), B  =  =  =  —  5.3 Hz.  1.5 cm, 1  3.4 Tesla, q 1  =  2  =  =  (5.76b)  15 cm (for an  1 electron charge,  and D = r (for scanning ICR), the maximum unlike-ion Coulomb-induced frequency shifts expected from the charged-cylinder model (Eq. (4.20b), or Eq. (5.3)) are:  —  =  2.2918x10 x 0.99x10 6  —  =  3.9 Hz,  (5.77a)  —4.0 Hz.  (5.77b)  —  x 3.4 x 2 1.5  1  =  2 7 J 5 +  1.52  6 2.2918x10x 1.01x10 =  x 3.4 x 2 1.5  2 7 J 5 +  1.52  —162—  Comparing Eqs. (5.76) with Eqs. (5.77) shows that the charged-cylinder model agrees well with the experimental results. Even though the error range is ±25  —  30%, for such a  small frequency shift and without calibration for the Coulomb effects from like-ion contamination, the agreement seems acceptable.  5.5.2 Unlike-ion Coulomb-Induced Frequency Shifting in FT-ICR In our experiment, CpMn(CO) 3 (Cp  =  -C was chosen as the parent molecule ?7 ) H 5  to study the unlike-ion Coulomb interactions, because only four primary ions, Mn (mlz 55), CpMn (m/z 120), CpMnCO (mlz 148), and CpMn(CO) (m/z 204), were produced from CpMn(CO) 3 by electron ionization, and these mass peaks are separated well in the mass spectrum. The experiments were run in a 2.54 cm cubic trapped-ion cell of a homebuilt FT-ICR mass spectrometer combined with a Nicolet FTMS 1000 console which has been described in Section 2.2.1 of Chapter 2. Typical experimental conditions were: nominal pressure 4.3 x iO Torr, beam duration 5 ms, beam voltage —90 V, data points 64 k, zero filling 1, trapping voltage +1 V, and strength of the magnetic field 1.9 Tesla. It is central in the study of Coulomb effects in FT-ICR to know the number of ions in each cylindrical ion cloud. A method for measuring ion current (ion number then can be calculated) was described by Hunter, Sherman, and Mclver.( ) The total ion number in 12 the analyzer cell was measured using an electrometer (e. g., Keithley, Model 616) and a small modification of the trapping bias was made to separate the ion current from the electron 12 current.( We studied the unlike-ion Coulomb interactions by monitoring ) electron emission current, because for a pulsed electron beam such as employed in FT ICR, the number of positive ions, N, produced during the beam pulse or duration tb is proportional to electron emission current ‘e from the filament used for ionization( ) 13  N  QI P1 t qk Tb  (5.78)  —163--  where  1 Q  is the total ionization cross section of the gaseous molecules at a specified  ionizing potential,  e 1  is the path length that ionizing electron travel, P is the pressure of the  molecules, kB is the Boltzmann’s constant (kB temperature. The total ionization cross sections, number of compounds. There is no available  =  1.380658x 10—23 J K— ), and T is the 1  , have been measured for a veiy limited 1 Q  . 3 Q. value of CpMn(CO)  Nevertheless, in  Section 2.2.3 of Chapter 2, the three hypotheses have been suggested, and can be applied to calculate the theoretical molecular polarizabilities of transition metal complexes following 18-electron rule and to calibrate the nominal pressure. Hence, the molecular polarizability of CpMn(CO) 3 would be  ) 3 a(CpMn(CO)  =  ) +3 x a(carbonyl) H 5 a(C  =  8.60  =  19.54  +  3 x 2.82  +  +  a(Kr)  2.4844  ) 3 (A  (5.79)  where a(C ),a(carbonyl), and a(Kr) are the polarizabilities of Cp ligand, carbonyl H 5 ligand, and Kr atom, respectively; and their values have been given in Section 2.2.3 of Chapter 2. According to Eq. (2.2) in Chapter 2, the calibrated pressure of CpMn(CO) 3 is the nominal pressure of CpMn(CO) 3 3 “CpMn(CO) —  —  —  =  0.36 x a(CpMn(CO) ) 3  +  0.30  4.3 x i0 Torr 0.36 x 19.54 +0.30 5.9 x 10 Torr.  (5.80)  Total ionization cross sections for molecules could be linearly related to the molecular ) For general cases, the empirical relationship between total ionization 4 polarizabilities.(’  -164-  cross sections and molecular polarizabilities can be estimated as( ) 14  1 Q where  Q is in A2 and a  in  A.  (5.81)  =2.2 a  Total ionization cross sections generally are constant at  ) Assuming that this relationship in Eq. (5.81) can 15 electron energy around 75— 100 eV.( be extended to the organometallic complexes whose molecular polarizabilities are around 20 3 will be at electron energy 90 eV, total ionization cross section of CpMn(CO)  ) 3 Q(CpMn(CO)  =  2.2 a(CpMn(CO) ) 3  =  2.2 x 19.54  =  43 (Az).  (5.81)  Ionizing electrons travel the analyzer cell in a helical trajectory because of the longitudinal ) le can be assumed to be slightly less than twice the length of the ICR 15 magnetic field.( ) (le = 4.8 cm for the 2.54 cm cubic analyzer cell). When the pressure P 12 cell( 10-8 Torr (7.8 x 10-6 Pa) and the temperature T  =  5.9 x  =  300 Kelvin, for a 100 nA electron  3 at current in aS ms beam time, the number of the positive ions produced from CpMn(CO) 90 eV electron ionization, according to Eq. (5.78), is  N+  —  —  =  ) x (5.OxlO9 ) 3 ) x (lOOxlO6 ) x (7.8x10 2 ) x (4.8x1020 (43x10 ) x 300 23 ) x (1.380658x10 19 (1.6022x10  1.2 x i0 (ions).  (5.82)  After the FT-ICR mass spectrometer with a water cooling system was turned on for preheating for 30 minutes, drift of the magnetic field was monitored at the cyclotron frequency of m/z 204 for 40 minutes until the drift continuously increased as indicated in Figure 5.14. The single resonance and the triple resonance, described in Section 2.2.2 of Chapter 2, were used to study the frequency shifts in the FT-ICR mass spectrometer.  —165—  143.62  fen (kHz)  :o 143.59 1 43.58  0  5  10  15  20  25  30  35  40  Time (mm) Figure 5.14  Drift in the 1.9 Tesla magnetic field used in the experiment after preheating 30 mm. The cyclotron frequency of CpMn(CO) (m/z 204) was monitored with time by using emission current = 335±20 nA.  Using the triple resonance, CpMn was retained in the cubic analyzer cell, but the other three ions, Mn, CpMnCO, and CpMn(CO), were ejected. The effective cyclotron frequency of CpMn was monitored as the nominal electron emission current was raised from 100 nA to 1 pA. Then, the experimental procedure was transferred from the thple resonance to the single resonance. The three ion species Mn, CpMnCO, and CpMn(CO), which were ejected before by the triple resonance, were added in the cubic analyzer cell. Their ion intensity sum was about 1.5 times the ion intensity of CpMn. The ion intensity of CpMn remained constant with or without the ejection pulses. The effective cyclotron frequency of CpMn was monitored again while the nominal electron emission current was scanned from 100 nA to 1 iA. The measurements are shown in Figure 5.15 and are listed in Table 5.3. The unlike-ion Coulomb induced frequency shift is greater in the 2.54 cm cubic cell and 1.9 Tesla magnetic field than in the 3.18 cmx3. 18 cmx 15.34 cm elongated cell and 3.4 Tesla magnetic field used by Francl et al.( ) The 10  —166—  kHz/million ions) of the plot of the effective cyclotron frequency of CpMn vs. electron emission current with unlike-ion Coulomb interaction is about 3.5 times the slope (0.0555 kHzlmilhion ions) of that of single ion species CpMn vs. electron emission current without unlike-ion Coulomb interaction, as shown in Fig. 5.5 (rather than 2.5 times, if the frequency shifts were caused only by the ion Coulomb potential in the trapping electric field). Since the magnetic field steadily increased (Fig. 5.14), the frequency shifts (toward lower frequencies) in Fig. 5.15 were caused undoubtedly by ion Coulomb interactions including the ion Coulomb potentials in the trapping field, the unlike-ion Coulomb interactions, and the like-ion Coulomb interactions. If there are 2.5 million ions of Mn, CpMn, CpMnCO, and CpMn(CO) in the analyzer cell, total number of Mn, CpMnCO, and CpMn(CO) will be 1.5 million ions and ion number of CpMn will be 1 million ions. According to experimental slopes in Fig. 5.15, the unlike-ion inducedCoulomb frequency shift from Eq. (5.75) is  fcpM  =  — 0.194 x 1 — (— 0.0555 x 2.5)  =  —55 Hz.  =  — 0.055 (kHz) (5.83)  The unlike-ion induced-Coulomb frequency shift according to the charged-cylinder model forD  =  1.327 r (Eq. (4.20b) or Eq. (5.3)) is  AfcpMfl+  =  —  1.5x10 2.2918x1Ox 6 1.327 x 1 x 1.9 x 1.22 + 1.3272  q  =  —43 Hz.  (5.84)  Comparing Eq. (5.83) with Eq. (5.84), the charged-cylinder model produced the desired value.  —167—  244. 244.50 fett  244.46  (kHz) 244.42  244.38 0.0  0.1  0.2  0.3  Number of ions  Figure 5.15  0.4  0.5  0.6  (million ions)  The effective cyclotron frequency, feff’ of CpMn  vs.  number of ions  with and without unlike-ion Coulomb interactions. The data indicated by “0” were obtained by ejecting all ions except CpMn, and there was no  unlike-ion Coulomb effect on CpMn. The data indicated by  “•“  were  obtained without triple resonance, and the effective cyclotron frequency of  CpMn was affected by unlike-ion Coulomb interaction.  —168—  Table 5.3  The effective cyclotron frequency feff of CpMn with the change of electron emission current, EMC. The experimental parameters are given in the text.  Without unlike-ion Coulomb interaction EMC (i)  feffOZ)  With unlike-ion Coulomb interaction  EMC (jiA)  feffOZ)  0.085  244.492016  0.095  244.505997  0.170  244.491344  0.160  244.501626  0.330  244.486645  0.270  244.493118  0.490  244.481521  0.400  244.478988  0.595  244.479028  0.515  244.468298  0.665  244.478110  0.615  244.460204  0.760  244.474919  0.765  244.443997  0.905  244.467348  0.850  244.426488  1.045  244.467348  0.925  244.411289  1.135  244.464552  1.100  244.396246  5.5.3 Experimental Inhomogeneous Broadening in Fr-ICR The mass peak width at 50% height for CpCr (mlz 117) as a function of electron emission current was monitored, and the experimental data are listed in Table 5.4. The other four unlike-ion species in the analyzer cell were: C Cr (m/z 130), CpCrO (mlz H 6 133), CpCrNO 1 (m/z 147), and CpCrNOCH (mlz 162). The sum of their ion intensities was double that of Cp&. The experiment was run after the FT-ICR mass spectrometer was turned on for 2 hours, so that the drift of the magnetic field was minimized. The mass peak widths at 50% height in Table 5.4 were measured at a constant excitation power.  —169—  Typical experimental parameters are: nominal pressure 2.0 x i0 torr, electron beam duration 20 ms, beam voltage —25 V. data points 64 k, zero filling 1, trapping voltage +1 V, and strength of the magnetic field 1.9 Tesla.  Table 5.4  The peak width at 50% height of CpCr with the change of electron emission current, EMC. The experimental parameters are given in  the text. Left frequency*  Right frequency*  Peak width  at 50 % height (kHz)  at 50% height (kHz)  at 50% height (Hz)  0.125  250.6785  250.5368  141.7  0.230  250.6605  250.5197  140.8  0.330  250.6423  250.5001  142.2  0.435  250.6327  250.4900  142.7  0.565  250.6251  250.4819  143.2  0.665  250.6155  250.4724  143.2  0.805  250.6073  250.4631  144.2  0.965  250.6059  250.4610  144.9  1.125  250.5958  250.4518  144.0  1.300  250.5901  250.4459  144.2  1.555  250.5831  250.4385  144.6  EMC (uA)  *  Because these frequencies at 50% height were read from the raster display of the Fl’  ICR, only four digits after the decimal point were displayed on the raster rather than the six digits after the decimal point as given by the Nicolet computer in Table 5.3.  —  170  —  In the charged-line model( ) and the charged-cylinder model,( 1 ) the maximum 2 frequency shift is at z 1  =  0 and minimum frequency shift is at z 1 =1/2. The broadening,  however, is not a simple subtraction of the minimum frequency shift from the maximum frequency shift, because the frequency spectrum of FT-ICR is given by the Fourier transform of the time domain signal. Therefore, the inhomogeneous broadening (the time domain signal) induced by unlike-ions can be expected from Eq. (5.2), or Fig. 5.13, but the inhomogeneously broadened FT-ICR peak shape, such as the experimental data given in Table 5.14, can not be evaluated directly from Eq. (5.2). Further work is needed to formulate the inhomogeneously broadened FT-ICR peak shape for Eq. (5.2), or Fig. 5.13.  5.6 Discussion  In this chapter, a two-dimensional charged-disk model is proposed to explain Coulomb-induced frequency shifting in FT-ICR mass spectrometry.  This model  corresponds more closely to the actual FT-ICR experiment than does the charged-point ) which has recently been developed. The model consists of a uniformly charged 1 model,( disk of ions of mass m , whose excited cyclotron motion is perturbed by a second 1 uniformly charged disk of ions of mass m , whose cyclotron motion is also excited. 2 Apparently, the second disk creates a radial force on the first disk which lowers the cyclotron frequency of the first disk. This is most easily seen in a rotating frame which rotates at the cyclotron frequency of the first disk. This radial force is numerically evaluated and found to be a function of the ratio of the cyclotron radius to the disk radius, as well as the charge of the second disk. Unlike the charged-point model,( ) which has an 1 infinite average radial Coulomb force, the average radial Coulomb force for the charged disk model is finite. This average Coulomb force allows use of formulae which permit characterization of a given set of model parameters in terms of an “apparent Coulomb  —  171  —  distance”, for which a model consisting of point charges with a fixed location in a rotating frame would give the same frequency shift. It is argued that the same “apparent Coulomb distance” would apply in the case of a charged-cylinder model, which accounts for Coulomb-induced line broadening in addition to Coulomb-induced frequency shifting. The “apparent Coulomb distances” are tabulated for convenient use with simple analytical expressions which quantitatively give the Coulomb-induced frequency shifts and the Coulomb-induced line broadenings in any experimental situation covered by the chargeddisk or charged-cylinder models. The assumptions of the disk model are examined and analytically defined by formulae in terms of like-ion Coulomb interactions, unlike-ion Coulomb interactions and the magnitude of the Coulomb interaction relative to the frequency difference between the perturbed and perturbing ion. From our numerical analysis, the “apparent Coulomb distances” is found to be a function of ratio of the ion cyclotron radius to the cylindrical radius of the ion cloud, rir’. The experimental tests for unlike-ion Coulomb-induced frequency shifting showed that the charged-cylinder model worked fairly well for either the elongated analyzer cell or the cubic analyzer cell. In our experiment, the number of ions was estimated theoretically with respect to the electron emission current (Eq. (5.78)). We assumed: (1) the method suggested in Section 2.2.3 of Chapter 2 can be used to evaluate the molecular polarizabiity , and then, to calibrate the sample pressure; (2) the total ionization cross 3 of CpMn(CO) . 3 section of CpMn(CO) 3 was linearly related to the molecular polarizabiity of CpMn(CO) These assumptions has not been proved experimentally yet.  Further study on  polarizabilities and total ionization cross sections of organometallic molecules is, therefore, of considerable interest. Like-ion Coulomb-induced broadening effects in FT-ICR were studied by Wang and Marshall in 1986.(’) Theoretically, through numerical analysis of the ion motion in Fr  ICR, and experimentally, through examining positive benzene ions, they proved that as the ions were excited to larger cyclotron radii, the like-ion Coulomb-induced broadening was  —172—  negligible, even for more than 1O ions in the 2.54 cm cubic analyzer cell. The charged-  line model developed in Chapter 4 and the charged-cylinder model developed in this chapter can be used to explain the unlike-ion Coulomb-induced broadening in FT-ICR. The unlike-ion Coulomb-induced broadening effect on CpCr due to the presence of four unlike-ions CpCrO, C Cr, CpCrNO, and CpCrNOCH has been studied here. H 6 Broadening effects were indeed observed, even for larger cyclotron radii (Table 5.4). Either theoretically (Fig. 4.4 and Fig. 5.13) or experimentally (Table 5.4), the inhomogeneous broadenings caused by the unlike-ion Coulomb interactions in FT-ICR were apparently not serious. As a matter of further interest, it should be noted that when the electron emission current was increased to a critical point, such as —1 jiA in Table 5.4, the broadening was terminated. When ion densities were large, the radii of cylindrical ion clouds could be expanded because of the ion-ion Coulomb repulsion; that is, the rir’ ratio was decreased, and the D value was increased (Table 5.1). From Eq. (5.2), this change in D should lessen the Coulomb-induced frequency shifting. Compared to the superconducting magnetic field, the magnetic field drift in this experiment was serious as indicated in Fig. 5.14, because a water cooling system was used. Even after the magnets had been turned on for 2 hours, the ICR frequency shift due to the magnetic field drift can be almost 10 Hz within 30 mins. For a superconducting magnetic field of 1.9 Tesla, ICR frequency shift due to the magnetic field drift was reported less than 1 Hz per dáy( ) Consequently, the FT-ICR mass spectrometer in our laboratory 14 should be equipped with superconducting magnets in the future. The procedures for calibrating Coulomb-induced frequency shifts, proposed by either ) or Ledford, Rempel, and Gross,( 11 ) all make use of 16 Jeffries, Balow, and Dunn,( calibrant masses to correct the trapping field effect, geometrical shapes of ion clouds, and ion-ion Coulomb interactions. In the charged-line model (Chapter 4) and charged-cylinder model (Chapter 5), the only important parameter is ion number in the analyzer cell. Therefore, for many FT-ICR experiments conducted with elongated(’ 1) and screened( ) 1720  —173—  ICR cells in which the trapping fields are reduced so as to be negligible, the results of this work could allow absolute mass calibration of FT-ICR spectra, independent of any known masses, only requiring a knowledge of the magnetic field strength and a calibration of the FT-ICR signal strength vs. number of ions.  —  174  —  REFERENCES  1.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1991, 5, 450— 455.  2.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1992, 6, 1—3.  3.  Wang, M.; Marshall, A. G. In:. J. Mass Spectrom. Ion Proc. 1990, 100, 323-346.  4.  Gradshteyn, I. S.; Ryzhik, I. M. “Tables of Integrals, Series and Products”; Academic Press: New York, 1965; pp 82—83.  5.  Davis, P. J.; Rabinowitz, P. “Methods of Numerical Integration”; Academic Press, Inc.: Orlando, 1984;.  6.  Hammer, P. C.; Wymore, A. W. Math Tab. Aids Comput. 1957, 11, 59—67.  7.  Hildebrand, F. B. “Introduction to Numerical Analysis”; McGraw Hill: New York, NY, 1974; p 387.  8.  Atkinson, K. “Elementary Numerical Analysis”; John Wiley & Sons: New York, NY, 1985; pp 163—165.  9.  Comisarow, M. B. In “Lecture Notes in Chemistry, Vol. 31, Ion Cyclotron Resonance Spectrometry II”; Hartmann, H. Wanczek, K. P., Ed.; Springer-Verlag: Berlin, 1982; pp 484—513.  10.  Franci, T. J.; Sherman, M. 0.; Hunter, R. L.; Locke, M. 3.; Bowers, W. D.; Mclver Jr., R. T. mt. J. Mass Spectrom. Ion Proc. 1983, 54, 189-199.  11.  Jeifries, 3. B.; Barlow, S. E.; Dunn, 0. H.  mt. J. Mass Spectrom. Ion Proc.  1983,  54, 169-187.  mt.  12.  Hunter, R. L.; Sherman, M. 0.; Mclver, R. T. Jr. Phys. 1983, 50, 259—274.  J. Mass Spectrom. Ion.  13.  White, R. L.; Onyiriuka, E. C.; Wilkins, C. L. Anal. Chem. 1983,55, 339-343.  —175—  14. Lampe, F. W.; Franklin, J. L.; Field, F. H. J. Am. Chem. Soc. 1957, 79, 61296132.  15. Wang, T.-C. L.; Marshall, A. G. In:. J. Mass Spectrom. Ion Proc. 1986, 68, 287301. 16.  Ledford, E. B.; Rempel, D. L.; Gross, M. L. Anal. Chem. 1984, 56, 2744-2748.  17.  Wang, M.; Marshall, A. G. Anal. Chem. 1989, 61, 1288-1293.  18.  Hanson, C. D.; Castro, M. E.; Kerley, E. L.; Russell, D. H. Anal. Chem. 1990, 62,  520-526. 19.  Rempel, D. L.; Grese, R. P.; Gross, M. L.  mt. J. Mass Spectrom. Ion Proc.  1990,  100, 38 1-395. 20. Naito, Y.; Inoue, M. 36th ASMS Conference on Mass Spectrometry and Allied Topics, San Francisco, CA, 1988, 608—609.  —  176  —  CHAPTER 6  TAYLOR’S EXPANSION APPROXIMATION OF ION-ION STRONG COUPLING COULOMB INTERACTION IN FT-ICR MASS SPECTROMETRY  —177--  6.1  Introduction The charged-line model and charged-cylinder model developed recently  (1-2)  explain  ion-ion Coulomb-induced frequency shifting and ion-ion Coulomb-induced broadening in FT-ICR mass spectrometry. The charged-cylinder model( ) corresponds more closely to 2 ) However, there are 1 the actual FT-ICR experiment than does the charged-line model.( three essential prerequisites in the charged-cylinder model, which have been discussed in Chapter 5: (1) Lorentz force on the ions  >>  like-ion interaction;  (2) Lorentz force on the charged-cylinder>> average unlike-ion interaction; (3) resonance frequency difference between perturbed and perturbing ions  —  O2  >>  (), perturbation frequency shift. 1 A%  The second prerequisite is related to the third prerequisite, because they both depend on the number of unlike ions, N 2 (Eqs. (5.61) and (5.64)). Coulomb interaction between two ion species will be sharply increased for small separations (non-overlapping or overlapping a little) as described in Figs. 5.6  —  5.10 of Chapter 5. It is desirable to know what will  happen if the unlike-ion Coulomb interactions for small separations are comparable to the Lorentz force on the ions, i. e., the second prerequisite does not hold; with increase in the number of unlike-ions, N , the unlike-ion Coulomb interactions will become strong 2 enough to overcome the Lorentz force. Also, the perturbation frequency shift will become large enough to nullify the relationship in the third prerequisite. Obviously, it is impossible to solve the effects of strong unlike-ion Coulomb interactions for small separations by using the averaging method in Chapter 5. In this chapter, a Taylor’s expansion approximation is used to study the strong Coulomb interactions between two ion species within small separations in FT-ICR. A preliminary  —178—  p  I  r  0  112  1/2  0  Figure 6.1  1  z  A cylinder in cylindrical coordinates (p. q,, z). Radius and z-length of the cylinder are? and 1, respectively. Its center is at (r, £2), off the origin of the cylindrical coordinates.  report of this study was presented in 1989,() in which it was showed that the strong unlike-ion Coulomb interaction could lead to a strong coupling oscillation. An ion cloud is a collection of ions with the same mass-charge ratio m/q. Integration must be used to develop equations for the ion-ion Coulomb interactions of a large number of ions. The ion cloud in the FT-ICR experiment assumes the shape of an approximately  cylindrical rod. Assume that this charged-cylinder has a radius r’ and length 1 and is moving circularly with a cyclotron radius r. The analytical equations for a cylinder in cylindrical coordinates (q, p, z) of Figure 6.1 are  —179—  2 p  =  r’  —  +  —112  2 p r cos(q  —  (6. la)  £2),  (6.lb)  112  z  where £2 is the angular coordinate of the center of the cylinder. When the cylinder is moving in a circular orbit, the angular position £2 of the cylinder is given by  (6.2)  £2 =co t+ 0  where w- is natural resonance frequency of the ions, t is time, and the cylinder at t  =  is initial phase angle of  0. Coulomb interaction between two charged cylinders will be a six  dimensional integral equation. Evaluation of such a six dimensional integral requires application of a numerical method. However, it is most desirable to obtain the analytical answers with certain physical meanings, which can be compared to numerical solutions. Therefore, a simpler case will be considered: two uniformly charged tetragonal ion clouds in the analyzer cell of FT-ICR, since integration is easier relative to the interaction between two uniformly charged cylinders, and will be similar to the latter. The edge lengths of both 0 and both z-axis lengths are I as shown in Figure 6.2. square cross sections both are 2 a 2 ions in Tetragon 2 in Cartesian The ion densities for N 1 ions in Tetragon 1 and N coordinates are  )Z fl X 1  Yz n x 2  A  A  N 02i a  ‘  N 02 a  Applying Taylor’s expansion approximation, the Coulomb interaction between two  —180—  Tetragon 1 0 2a  Tetragon 2  I x  0  S S S  z  S I I S  V S  I  I —  Figure 6.2  I  Il_Il  The charged-tetragon model. Both tetragons have the same cyclotron radius, R; the same edge width and height, 2 a ; and the same z-axis 0 length, 1. The directions of applied magnetic field B and their cyclotron motions are shown as indicated.  —181—  tetragonal ion clouds within a range of small separations is integrable. In order to distinguish the cyclotron motion of an individual ion and the collective cyclotron motion of an ion cloud, coordinates of an individual ion in an ion cloud will be symbolized as r, and cyclotron radius of the ion cloud, as R. Note that these two symbols are different from those used in Chapter 4 and 5. Potential energy is a scalar quantity and force is a vector quantity. Thus, evaluations of potential energies of ion clouds will be simpler than those of forces on the ion clouds. Therefore, we start with the potential energies of ion clouds. The forces can be found later, since the gradient of potential energy is force.  6.2  Potential Energy of a Single Tetragonal Ion Cloud When an excited tetragonal ion cloud, Tetragon 1, in which each ion has a mass m 1  and a charge q 1 1 is moving circularly in a cyclotron radius R energy  =  (xRl, YR1’  0), its potential  will be the sum of the external potential where the ion cloud is located and its  internal Coulomb potential,( ) 4  (xRI +  )  ) 1 U(R  XRI  + (a0  —  ao  0 a  (a0  (YR1 +  )  YRI  faa  —  o 1  (112  0 a  —1/2  (O  (1/2  0 i-a i—a 0 i—a 0 J-a 0  q1 N 1’ dx dy dz 4 a 02 I (1/2  N dx dx’ dy dy’ d zdz’ 2 1 k q J—l/2 J-l/2 16 2 12 2 I r r’ I  ‘  64  —  where f’is the external potential; Coulomb constant k  =  8.98756 x i0 2 /C rand r’ Nm ;  are coordinates of two individual ions in the tetragon; and the distance between two ions,  r  —  r’  ,  is multiplied by 2 (the denominator in the second term), because every ion is  counted twice in the integration. The second term in Eq. (6.4) must be a constant after  —182—  being integrated, because its integral limits are all constants (no variables XR1 and YR1)• The external potential in the first term in Eq. (6.4) is  V= V’—A.v  where  iy,  (6.5)  A, and v are electric potential, magnetic vector potential, and ion velocity,  ) The electric potential, 5 respectively.(  which is the trapping potential of the analyzer  i;  cell, is another perturbation with respect to the Lorentz potential. For the sake of calculation, ywi1l be separated first and discussed later. Thus, Eq. (6.5) reduces to  =  -A.v.  (6.6)  In vector analysis, a vector, A, in cylindrical coordinates (q,, p, z) is related to the same vector in Cartesian coordinates (x, y, z) by  A  =  Aq, e + A ep + A c  i  =  +  A,, j  +  k  where e, es,, and e are unit vectors in cylindrical coordinates, and e  (6.7)  =  k. The magnetic  ) 6 vector potential of an uniform static magnetic field in cylindrical coordinates (  where p = Ix2  +  . Hence, 2 y  Aq,  =  A  =  A  Bp,  (6.8a)  0  (6.8b)  =  —183—  A,e,  Ai.  +  Ai  =  (6.9)  The cylindrical unit vector e, is related to the Cartesian unit vectors i and j by  e  =  —  i sin q’  j cos  +  (6.10)  q’,  so, the three components of the magnetic vector potential in Cartesian coordinates are  A  =  —-  =  (6.lla) (6.llb)  Bx,  =  A  By,  0.  (6.llc)  The elementary motion equation of an ion in ICR mass spectrometry, which was given in Section 4.1 of Chapter 4 is  x=Rcos (w t+), 0  (6.12x)  ).  (6.12y)  t+ 0 y =—R sin (w  The derivatives of Eqs. (6.12) are dx =  =  y 0 w x 0 —w  =  =  v,  v,.  Substituting Eqs.(6.1 1) and (6.13) into Eq. (6.6) gives the external potential  (6.13x) (6.13y)  —184—  =  ) 22 BwO(  Since the ions in Tetragon 1 are taken as a collection, x  (6.14)  =  XR v , 1  YR1’  2  1  and co 0 = c• The  first integral in Eq. (6.4) is solved to be  ) 1 U(R  =  rBeo q [ N 1  (6.15)  Eq. (6.15) gives the the potential energy of Tetragon 1 as a function of the ion number in Tetragon 1, N , the strength of the magnetic field, B, the cyclotron frequency of Tetragon 1 1,  the edge width and height of Tetragon 1, 2a , and central coordinates of Tetragon 0  l,XR1 and YR1  6.3  Potential Energies of Two Tetragonal Ion Clouds  6.3.1 Taylor’s Expansion Approximation of the Coulomb Potential Energy If there are two tetragonal ion clouds of masses m 1 and m 2 whose cyclotron radii are 2 in the analyzer cell of F1’-ICR, the potential energy of each tetragon should be 1 and R R its own potential energy plus the Coulomb potential energy due to another tetragon. For Tetragon 1,  ) 2 (R R U , 1  =  ) + (I 1 01 R U ( 12 R U —R 1 2  I)  (6.16)  12 is the where U 01 is the potential energy of Tetragon 1 given by Eq. (6.15) and U 1 ions of charge q , 1 Coulomb potential energy due to Tetragon 2. If Tetragon 1 contains N Tetragon 2 contains N 2 ions of charge q , and their geometric sizes are as indicated in Fig. 2  —185—  6.2, the ion densities of the two tetragons are those of Eq. (6.3). The Coulomb potential energy of Tetragon 1 due to Tetragon 2 is  12 R U (I —R 1 2 I)  =  N 2 q 1 kq  1d r 3 d 2 r 3  1 r  Tetragon 1 Tetragon 2  —  —ao 2 )XR  fXRI+ck  JXR1_CO  (YRZ+ao JYR22O  2 F  kq N 2 q 1 X 2 1 4 0 16a  —  +aO 2 (XR  —  (YRI+ao )yRI—ao  ( 1/2 ( 1/2  dx dz d 1 2 y z  J—l/2 )—1/2  — ‘V’(x + + — 1 ) 2 -y (y (z x z  (6.17)  12 is a six-dimensional integral. Because two particles cannot occupy the same position U in space, i. e., for x 1  =  , 2 X  Yi  =  Y2’  and z 1  =  , 2 z  1  .%J 2 —z 1 (z ) —x + (Y1—Y2) 1 (x ) 2 + 2 Thus, the singular points (x 1  =  , Yi 2 x  =  Y2’  and z 1  =  =  0.  (6.18)  ) must be deleted from Eq. (6.17). 2 z  The following substitutions are used to simplify the upper and lower integral limits of Eq. (6.17):  =  —Y2 1 Y  where  =  X 2 —XR 1 _Xj+XR ,  (6.19x)  Y—YjYR1--YR2  (6.l9y)  x =x 1 —XR1,x2 =X2—XR2,yj =Y1YR1’ andy =Y2—YR2’ such that there is no  variable in the upper and lower limits of Eq. (6.17).  —186—  12 R u (I —R 1 2 I) —  —  kq N 2 q 1 2 1 4 0 16a  (xR2+ao  JXp2  X  (YR2+ao  0 (x+a  JYiwao  JXRrao  —  Jyp.i—ao  kq N 2 q 1 2 J 16 a j  (1/2  y z dx dz 2 d 1  .11/2  112 .L  z (z x y 2+(y .v’(x ) _ + _ 1 2 )  (1/2  0 p 0 0 p p  (ao  —  (1/2  (YRI+ao  J J J.  (1/2  2 J_i/2 J_i/  1 dz dz 2 dy 1 dy 2 dx 1 dx 2 2+ (x—Xj+xR1—xR2)  (6.20) +  7i_y2+yR1_yR2)  —z 1 (z ) 2  The magnitude of Coulomb interaction for a small separation AR  1 R  —  2 between two R  ion clouds in FT-ICR is of great importance to this study. Therefore, the integrand in Eq. (6.20) is expanded in a Taylor’s series about AR YR2)O =0.  =  0, i. e., (xRl  —z 1 (z ) 2+ 2 2 + (yj-y+yR1yR2) .‘.J (xj—xj+XR1--xR2) 1 +  2 (y—y)  + (zl_z2)2  x—xj (xRl  XR2)  [  2 (xj—xj)  2 (Yj—y)  +  +  —z 1 (z ) 2  j3/2  y —y —  (YR1  2  xR2)o  =  0 and (YR1  Taylor’s expansion of the integrand in Eq. (6.20) to second-order terms is 1  —  —  YR2)  [  (XR1 XR2 )2  2 (xj—xj)  [  +  2 (y—y)  + 2 —z 1 (z )  2 (yj—y)  +  (zl_z2)2  2 (xj—xj)  +  2 (yj—yj)  —  j3/2  2  + 2 —z 1 (z )  j52  —  —187—  1 2  +  Ri  1 (Xj  ‘ l 2 1  2 YR2  [  ‘12’  2 (x—4)  +  ‘ —  ‘  —z ]5l2 1 (z ) 2+ 2 (y—y)  (xj—4)(y—y) —  XR2)(YR1  —  YR2)  r , ‘ 2 + ) 2 — 1 (z z 2 15/2 2 — 1 (x x2 + ) 2 — 1 (y y j I )  Since the expansions were made at (xRl—xR2)o =0 and (y1—Y2) =0, x  1 and z  =  =  x, yj  =  y,  2 are still singular points and must be deleted. As shown in Figure 6.3, the z  singular points in the area (x,  4) are composed of a straight line xj  =  4,  and also the  ,z 1 ). All these singular points are composed of a parallelepiped 2 areas (y, y) and (z singular subspace in six-dimension space.  6.3.2 Integration of the Coulomb Potential Energy At first glance, the integration for Eq. (6.21) appears very complicated. However, the integrals of the zero-order term, the first-order terms, and the cross term  (XR1—  in Eq. (6.21) can be ruled out. The zero-order term comes from complete  overlapping of the two ion clouds. Because two completely overlapped ion clouds are equivalent to a single ion cloud (Eq. (6.4)), the integral of the zero-order term is a constant.  Because the characteristic frequency solution is only related to homogeneous second-order ) any constant can be ignored in the 7 linear differential equation of ion motion in FT-ICR,( linear differential equation (or simultaneous linear differential equations). The first-order terms and the cross term in Eq. (6.21) are odd functions of (—4) and (y—yj). After the singular points in their integral areas (Fig. 6.3) are deleted, the integrals of the first-order terms and the cross term should be zero. Only the second-order terms should be considered for a small separation R. The expansion coefficient of  2 (XR1—XR2)  in Eq.  (6.21) actually has the same form as the expansion coefficient of (YR1—YR2) Therefore,  —188—  1 x,  a Y2  1 z  —1/2  1/2  —112  Figure 6.3  2 z  I  The coordinate expression of singular points for x 1 z  =  =  x, y  =  y, and  . Three coordinates x vs. x, y vs. y, and z 2 z 1 vs. z 2 are used.  The integral limits of x,  4, yj,  and yj are from  2 are from —1/2 to 1/2. limits of z 1 and z  —  0 to a a . The integral 0  —189—  the Coulomb potential energy of Tetragon 1 due to Tetragon 2 is predicted to be  —R 1 2 I) 12 R U (I  12 U (0)  NN 2c  +  [  (xRl  + (YR1  —  (6.22)  YR2)2]  —  where (0) 12 is the Coulomb potential energy when the two ion clouds are overlapping U completely and the coefficient C can be obtained from the integrations of the second-order terms. We define C as the Coulomb interaction coefficient between two ion clouds. From Eqs. (6.20) and (6.21),  — — —  kq q 1 16 a 4 0  (ao  (ao  0 p  fao  (1/2  (112  J—a, i-a 0 J.ao i—a 0 J112 p1/2  (yj  [  (x  —  —  +(z 2 y) 1  2+ x)  ) 2 z  —  (yj  —  —  2 (x  1 2 + (z y)  —  —  2 x)  ) j512 2 z  x dz 1 dz dyl dy dx dxj. 2  (6.23)  It is important in evaluating C that singular points are deleted by talcing the left and right limits for xj  =  x, y  =  1 y, and z  =  . The details of the integrations, in which several 2 z  particular integral formulae must be used are given in Appendix A4. After integrations,  C  =  hi 1 4 72  _2al+8alth[l+Jl2+ 3 0 { 5.9464a  8  21 0 a  _8al1n[l+gl2+ 4 0 2 a 1 + 21n[2a ]_6a 8 a q12÷ 2] 0 0 1 2 +6aol + n[2a il2 +l2 0 —8aln [2a  +  +  ] a 1 2 1 2 1 21n(12+4a _a )+2a n1 4 0  8 a ] +4a(l2+4a)+4agl2 2 0  +  8 a 2 0  —  2Il2 0 _4a  +  4 a 2 0  I  190  —  2 (12+4a2) (12+8a2)m+ 2 ” 3  13  }•  (6.24)  The results of Eq. (6.24) have been checked by numerical evaluation for a given a 0 and 1 (Appendix A4). The first two terms in brackets of Eq. (6.24) arise from the Coulomb interaction in x-y plane, and all the other terms arise from the Coulomb interaction for separation 1 (Appendix A4). Of course, the Coulomb interaction in the x-y plane is predominant. The coefficient C can be approximated as  q 1 kq  21) 0 (5.94Mao3_21ra  (2.9732_ i).  (6.25)  =  For a ) the error in C given by 8 0 = 1 mm and a 2.54 cm x 2.54 cm x 2.54 cm cubic cell,( Eq. (6.25), i. e., the difference from the exact value of C given by Eq. (6.24), is less than 0.2%. Furthermore, since 1>> 2a 0 generally, Eq. (6.24) reduces to  —  ——  itlcq 2 q 1 1 2 0 2a  626 (.  The Coulomb potential energy of Tetragon 2 due to Tetragon 1 is the same as U 12 but of opposite sign  2 — 1 R 21 I R U (  I)  =  —  2 — 1 R 12 I R U (  I).  The potential energies of Tetragons 1 and 2 in the magnetic field from Eq. (6.15)  (6.27)  —191—  01 U ) 1 (R  =  1B w 1N q  1 + (4 + y  02 U ) 2 (R  =  qN 2 B w02  +y 2 ( 2+  2), 0 a  (6.28a)  (6.28b)  Total potential energies of Tetragons 1 and 2 are  (R U , 1 ) 2 R  =  +  (R R 2 U , 1 ) 2  1N N 2 C [(xRl  B 002 N 2 q  =  —  6.4  + (q + 1 2 0 (O) 12 )+U y a  B N 1 q  NN 2C  —  2 + (YR1 XR2)  —  2) 0 2 +y ( 2+a  1 [(XR  —  2 + (YR1 XR2)  —  (6.29a)  YR2)l  —  12 U (O)  (6.29b)  YR2)2].  Motion Equations and Solutions of Two Tetragonal Ion Clouds After the total potential energies of the two ion clouds are known, their FT-ICR  equations of motion can be solved. The force field of a velocity-dependent potential (S) aLl Fg=  aii  g=x,y,z.  (6.30)  Because the ion velocity is replaced by the angular frequency in Eqs.(6.12), the force field  in FT-ICR is independent of time and is still conserved. That is,  —192—  F=—VU  where the operator V  .a + j.a  i  +  k  a  .  (6.31)  For simplicity, settmg  1 N = 2 , 0 N  (6.32)  the simultaneous equations of motion of the two tetragonal ion clouds are  1 F  =  NOm1idu1  =  -  (R —VU , 1 ) 2 R (6.33)  2 F  =  ddR2 2 NOm  =  (R —VU , 1 ) 2 R  Substituting Eqs.(6.29) into Eq. (6.33), C 0 N D 2XR1 + (Ooi2 XR1 + m (xRl 1  yRl+ayRl+ 2 D  XR2)  =  0  (—‘)  =  0  —  (6.34) 2 D XR2 + (0022 XR2 —  yR + °O2YR2 D 2 where  D  =  —  N C 0 . 2 (XR1 — XR2) m  =  0  2 1 —y) m  =  0  d/dt, the time derivative operator. The reduced matrix of Eq. (6.34) is 2  C 0 N 2  N C 0 — m1  1  NC 0  2  D+w+ ‘  NC 0  (6.35)  —193—  From Eq. (6.35), two determinantal solutions of the operator D are obtained. These  two  solutions are the eigenfrequencies, w and c of Eq. (6.34), where 1 =  +  2 2  C\ 0 N  If  /  1.Vo?+ in 1 )  ri  C 0 N  LP)o?+ m 1  + V1)02+  —CO—  C 0 N  2 m  C2 0 N  1  +  11/2 C 2 0 4N  1 J m  (6.36a)  C 0 C 0 N 1 1/ 2 N 2 2 ) 1 _2O1+m +,P)O2+m  —  C 0 N 2 [(  NOC)2 2  +  (6.36b)  Many important conclusions can be derived from the frequency solutions a and a, which are discussed in the following section.  6.5  Strong Coupling Coulomb  6.5.1  Strong Coupling Condition  Interaction between Two Ion Clouds  The relationship  O2  can be used to rewrite the terms in the square root in Eq. (6.36a) f  01  NC 0  1 m  _2_ 02  NC\2 0  +  (6.37)  =  C 2 4N 0  2 1 m  (or  Eq. (6.36b))  —  194  —  2 \2 1 / 4 rn + rn nz 1 rn 2  =a01  {  m) ]2 +  2T0  m  rn2) rn  —  rn2 +  2C 0 N 2  }•  (6.38)  Making a comparison among the three terms in Eq. (6.38), a strong coupling condition between the two ion clouds, Tetragon 1 and Tetragon 2, for a small separation is defined as  C 0 N  2 ( rn — 1 ) rn rn rn  ‘0i  (6.39)  or alternatively,  C 0 N  (rn—mi 1 2  2 qB  2 1 m  (6.40)  6.5.2 Two Oscillations under the Strong Coupling Condition Under the strong coupling condition of Inequality (6.39), two oscillations in F1’-ICR due to the Coulomb interaction between the two ion clouds are obtained from Eqs. (6.36):  (02  =  rn+m\ 1, o?÷+2NoC m’ ) 2 m 1  ( ). +  (6.41)  (6.42)  Note that w is smaller than co , since the coefficient C in Eq. (6.41) is negative 2 (Eq.(6.26)). The oscillatory motion in Eq. (6.41) probably is undetectable electronically. Jungmann, Hoffnagle, DeVoe, and Brewer recently investigated ion motions of two ion  —195—  clouds in an electric quadrupole trap.( ) Theoretically, two oscillatory motions were also 9 found in the electric quadrupole trap, whose modes are similar to those in Eqs. (6.41) and (6.42). Only the oscillatory motion similar to the a mode in Eq. (6.42) was observed in their experiment. Jungmann et a!. thought that in the  01  mode, the distance between the  two ion clouds was stretched and compressed about an unchanging center-of-mass position, and hence this collective oscillation cannot give rise to the electric signal. If this explanation is correct, the  mode will not be observed in Fr-ICR, either.  The co mode in Eq. (6.42) is a cyclotron motion of the two ion clouds with a fixed 2 distance. The Lorentz forces on two ion clouds are balanced by the ion-ion Coulomb interaction between them.  Consequently, only one mass peak will be observed,  corresponding to a root-mean-square value of the natural cyclotron frequencies of the two ion clouds, under a strong coupling condition of Inequality (6.40).  6.5.3 Strong Coupling Critical Curve Substituting the value of C (Eq. (6.26)) into Inequality (6.40), and q 1  =  2 q  =  unit  charge (If q 2 are not unit charges, they can be set to unity by changing ion masses 1 and q  2 to their mass-charge ratios.) 1 and m m ickN 2 2 1B 0 a  >>  Im-mi  (6.43)  In a first-order approximation, when a physical quantity is ten times greater than another, this quantity can be defmed as “much larger”. The strong coupling condition is redefined as itkN x 0.1 2 a 2 1 82 0  Im-mI 1m m 2  (6.44)  —196—  Solving for N 0 gives  20a21B21m 0 1 —m 2  0 N  it  I  (6.45a)  ,  1 m km 2  or equivalently,  0 N  9 4.27x10  2lB2Im U 1 -m 2 1m m 2  I  (6.45b)  Inequality (6.45b) gives a criterion of the strong coupling Coulomb interaction between two (tetragonal) ion clouds in terms of N , ion number in each ion clouds; a 0 , half edge 0 length of the square cross section of the tetragonal ion cloud in mm (approximately as the radius of a cylindrical ion clouds); 1, the z-axis length of the ion clouds in cm; m , ion mass 1 of an individual ion in Tetragon 1 in Dalton; m , ion mass of an individual ion in Tetragon 2 2 in Dalton; and B, strength of the magnetic field in Tesla. Inequality (6.45a) is the corresponding expression in SI units. Resolution of any mass spectrometer can be expressed as the ability to distinguish two adjacent mass peaks,  2 m  =  1 m  +  1 (Dalton).  (6.46)  Substituting Eq. (6.46) into Inequality (6.45b)  0 N  9 4.27x10  (6.47)  0 Typical parameters in a 2.54 cm cubic analyzer cell are as follows: 1= 2 cm, a  and B  =  2 Tesla. N 0 as a function of m 1 (m 2  =  1 m  +  =  1 mm,  1 (Dalton)) for a mass range from  100 to 600 Daltons is plotted according to the left ordinate in Figure 6.4. When the ion  —197—  density, n, of the ion cloud, as in Eq. (6.3), is used to define the strong coupling condition, Eq. (6.47) will be independent of the size of the ion cloud.  n  =  N 1 0 4a  D2  1.07x1O  (6.48)  m2 1  where the unit of a 0 is changed to cm, so that n is in units of ion number per cubic cm. Eq. (6.48) is plotted according to the right ordinate in Fig. 6.4. For ions of m 1 = 100 Daltons and m 2 = 101 Daltons in a 2 Tesla magnetic field and a conventional 2.54 cm cubic analyzer cell, if their ion numbers N 1 and N 2 are both greater than 3.4 million ions, the strong coupling Coulomb interaction between ions m 1 and m 2  can occur, that is, only one mass peak will be observed. When ion masses are increased to =  300 Daltons and m 2  =  301 Daltons, the ion numbers are reduced to 3.8x 10 ions.  Therefore, no matter how long the acquisition time of FT-ICR is, when the Lorentz force is unable to overcome the strong coupling Coulomb interaction between two ion clouds, the resolution of FT-ICR will deteriorate.  6.5.4 Strong Coupling Condition for N 1 When the ion numbers N 1  (01  2  2 N  , the solutions in Eq. (6.36) are changed to 2 N  N C 2 \ N C 1 1 1/ 2 ( 2 )+i%+ oi+ 1 m 2 m  rr  +  I  VOo?  +  C 2 N —  —  [(  C2 1 N —  2 ) m  +  4N 1 C 2 N 1 1/2 J J 1m m 2  (6.49a)  N C 1 N C 2 \ I 1 )+O)o+ in m 2  11  1  2 02  +  —  —  +  I }. ‘  (6.49b)  —198—  4  3  0 N (million)  n 3  2  (1 3 /cm 7 0 )  2 1  0 100  200  300  400  500  600  1 (Dalton) m  Figure 6.4  Strong coupling critical curve. When the ion number N 0 or ion density n, as a function of the ion mass m 1 (m 2=m 1  +  1 (Dalton)), is in the shadowed  area, the strong coupling region, the FT-ICR resolution deteriorates. The left ordinate is for the ion number N 0 in a 2 Tesla magnetic field and a conventional 2.54 cm cubic cell 0 (a = 1 mm and 1  =  2 cm). The right  ordinate is for ion density in a 2 Tesla magnetic field without dependence of geometrical size of an ion cloud. The strong coupling critical curve is calculated from Eqs. (6.47) and (6.48).  —199—  The terms in the square roots of Eqs. (6.49a) and (6.49b) can be rewritten as C 2 N  /  w  =  +  {  _2_  N 1 C\2 ma-’  4N C 2 N 1 nz 1 m 2  +  [ml(m 2 C ) N _ 1 ) 2 ml(m (mlNl_m m + 2 m  (m C + 1 ) N 2 m  }  (6.50)  The strong coupling condition for N 1  ICI  2 is defined as N  +m 1 (m ) N 2  Substituting Eq. (6.26) and c  =  >>  2I. 2 Imj2_m  B 1m 1 q 1 into Inequality (6.51) and  (6.51)  setting q 1  =  , the 2 q  strong coupling condition becomes  1 1N m  +  N 2 m  >>  2  2 0 a  1B 2 t  2=m Form 1  +  I m1  m 221  1m km 2  (6.52)  1 Dalton, and rn >> 1 Dalton, Inequality (6.52) reduces to 1  1 N  +  2 N  >>  8.53 x 108  a 2  2 B  m2 1  (6.53a)  or alternatively using the ion densities of Eqs. (6.3),  nl  +  2 n  >>  2.13x lOb  2 B  (6.53b)  2 are the ion numbers in Tetragon 1 and Tetragon 2, respectively; a where N 1 and N 0 is half  —  200  —  edge length of the square cross section of the tetragonal ion cloud in mm; 1 is the z-axis length of the tetragonal ion cloud in cm; B, strength of the magnetic field in Tesla; m 1 is mass of the ions in Tetragon 1 in Dalton; and n 1 and n 2 are the ion densities of Tetragon 1 and Tetragon 2 in units of ions per cubic cm. If the left-hand sides of Inequalities (6.53a) and (6.53b) are set to be at least ten times greater than the right-hand sides, then, 2,  1 N  +  2 N  8.53x iO  2 B  m2 1  (6.54a)  or alternatively, 1 + fl  When N 1=N , i. e., n 2 1  =  11 2.13x10  (6.54b)  , Inequalities (6.54a) and (6.54b) reduce to Inequalities (6.47) 2 n  and (6.48), respectively.  6.6  Weak Coupling Coulomb Interaction  6.6.1 Weak Coupling Condition When the inequality sign of Inequality (6.39) is reversed C 0 N  2 ( m — 1 ) m m 2 in  (6.55)  or alternatively,  —m 1 m I I2 2 1 m  2tkN 0 1B 0 2a  Inequality (6.56) is the weak coupling condition for the Coulomb interaction between the  —201—  two ion clouds. The frequency solutions of Eqs. (6.36) become N m +m C 0 2 ’ 2 1 m  01  1  =  +  N 2 C 0 +m 1 m 2 2 1 m  •  (6.57b)  The squares of the natural resonance frequencies, w and w, of the two ion clouds are both shifted by a term N 0 C (m 1  +  )I2m 2 m 1m , i. e., shifted to lower frequencies, 2  because C is negative (Eq. (6.26)). It has been known that Coulomb-induced frequency shifts for two weak-coupling ion clouds in an electric quadrupole trap are both shifted to higher frequencies,( ) wlich is exactly opposite to the Coulomb-induced frequency shift in 9 FT-ICR (Chapters 4, 5, and 6). This difference is understandable.  In an electric  quadrupole trap, the radial force of Coulomb interaction between two ion clouds has the same direction as that of the quadrupole electric field. In FT-ICR, the radial force of Coulomb interaction between two ion clouds has an exactly opposite direction to the Lorentz force. Because the Lorentz force can overcome the Coulomb weak coupling interaction between two ion clouds, the distance between two ion clouds will change periodically. ) discussed in Chapters 4 and 5 can be applied to explain and 12 Then, the average models( predict the Coulomb-induced frequency shifts in FT-ICR.  6.6.2 Validity of the Weak Coupling Condition In Section 5.3.2.3 of Chapter 5, we gave a criterion for high frequency perturbations  —m 1 Im I 2 2 1 m  2 kN rB2Dc4s,I(l/2)2+Dc2  (6.58)  —  202  —  where D is the apparent Coulomb distance. It has a similar form to Inequality (6.56). Considering Inequalities (6.56) and (6.58), which one should be used as a criterion for weak Coulomb interaction between two ion clouds? In a conventional 2.54 cm cubic cell where N 1  =  2=N N , 1=2 cm, r = 1 cm, D = 1.327 r, and a 0 0 = 1 mm, the ratio of the weak  coupling condition from the Taylor’s expansion approximation (the right side of Inequality (6.56)) compared to the weak coupling condition from the high frequency perturbations (the right side of Inequality (6.58)) would be 7rkN 02 lB 2a  kN 2  =  173  :  1.  (6.59)  rB2D.\I(l/2)2+D2  We know that Inequality (6.56) arose from small separations between two ion clouds. When two ion clouds are just touching, their Coulomb interaction is a maximum. Therefore, Inequality (6.56) is more critical than Inequality (6.58), which arose from the average Coulomb interaction between two ion clouds. Eq. (6.57a) and Eq. (6.57b) can be rewritten as NCm +m  =  — (w + 1 )(o, ) 01 w a  =  =  ((02  02 + %2)  =  ,  (6.60a)  and  Noting that w  o and  C02  ‘°&  (6.60b)  c, the frequency shifts from Eqs. (6.60) are  =  and  NCm +m 1m m 2  2’  (6.61a)  —  =  203  —  +m 1 m C 2 0 N o2  2 1 m  .  (6.61b)  Eqs. (6.61a) and (6.61b) are only valid at the instant when the separation between two ion clouds is small, from zero (overlapping disks) to 2 a 0 (just touching). After the two ion clouds are separated further, Eqs. (6.61a) and (6.61b) become invalid. Therefore, the charged-cylinder model presented in Chapter 5 is still correct for calculation and prediction of the Coulomb-induced frequency shifts in FT-ICR. The Coulomb-induced frequency shift predicted from Eqs. (6.61a) and (6.61b) would be about one hundred times the frequency shift predicted from the charged-cylinder model. The shifts predicted from Eqs. (6.61a) and (6.61b) do not tally with the experimental results (Section 5.5 of Chapter 5). This error is anticipated, because whole Coulomb interaction was included in the Taylor’s expansion approximation, not just the radial component of the Coulomb force as was the case in the charged-cylinder model.  6.7  Electric Trapping Potential and Frequency Shifts The electric trapping potential of the analyzer cell of FT-ICR, whose radial  7 1041) also can be component induces a negative resonance frequency shift to the ions.( included in the Taylor’s expansion approximation. For example, if the electric trapping potential, r, of a cubic ion-trapped cell shown in Figure 6.5 is taken as a quadrupole approximation (12)  =  where y,  iç  (VT+V0)+(VT—Vo){y— _(x2+y2_2z2)]  (6.62)  and 1 are geometric factors of the cubic ion-trapped cell, VT is the trapping  —  Receiver plate or Transmitter plate  204  —  p  =  x,ory  Trapping plate  plate  B z  Figure 6.5  The cubic ion-trapped cell configuration employed in the F1’-ICR mass spectrometer, where the trapping voltage, VT, is applied across the two trapping plates, and the voltages of the other four plates are set at zero. This applied electric field produces a quadrupole trapping field, Eq. (6.62), near the center, 0, of the cubic cell. The magnetic field, B, is along the z-axis.  voltage across the two trapping plates, V 0 is the trapping voltage on the other four plates (Fig. 6.5). The geometric factors are as follows: y  =  0.166667,  =  1.38686, and 1  =  0.0254 m for the 2.54 cm cubic cell.( ) Conventionally, the trapping voltage VT = 1 V, 12 and 1/ =0 V. Substituting these values into Eq. (6.62)  —  205  —  =5.166667—2.15x ) (x 3 10 + — 2 2z y .  i’  (6.63)  Substituting Eqs. (6.63) and (6.14) into Eq. (6.5), the external potential of an ion in the cubic ion-trapped cell is  =  5.166667 —2.15 x ) (x 3 10 + — 2 2z y —A. v  =  5.166667  +  2.15 x i& (x 2+y ) 2 2 —2 z  —  0 Bo.  2+y (x ). 2  (6.64)  The simultaneous ion motion equations of two tetragonal ion clouds, Eq. (6.34), is changed to  xR + W XR 2 D 1 1  —  yR + W 2 D 1  —  4.310 XR1  4.3Q<10 iyR1  1 (x  XR2)  =  0  (YR1 —YR2)  =  0  +  (6.65)  The  xR + (OXR2 D 2  —  yR + WOYR2 D 2  —  3 2xR2_’, (xRl 4.3Ox10  xR2)  =  0  4.3O10 YR2—,, 2 (YR1 —YR2)  =  0  reduced matrix of Eq. (6.65) is C 2 N 1  1 02  (6.66) 2 m  —  206  —  The eigenfrequencies of the determinant solutions are  2 co  [  =  + [(2  +  2C N  —  4.30x iO q 1  1  {  (2  +  2_  2C N  —  112 C 2 N 1 4N  1m m 2  —  4.30x 2 m  )2 2 q  (6.67a)  3 4.30x10  1)  + (2  2  +  1C N  1  2 1 m  =  —  +  1 C N  —  4.30x10 q 3 ) 2 2 m  )2 2 3 NlC_4.3Ox10  }.  (6.67b)  Defining  =  —  4.30x10 q 3 1 1 m  (6.68a)  and =  2  4.30x  2 q  —  the eigenfrequency solutions in Eqs. (6.67a) and (6.67b) reduce to  (6.68b)  —  {  =  + [((W’ )2 1  +  +  N C) 2 1 m  N C 2 1 m  +  + ((w12  — —  207  (co2)2  —  C) 1 N  N C 1 )2 2 m  +  4N 11121 C 2 N 1 ‘ 2 1 m -  (6.69a)  and C)+(( (02)2 + NC + 2 ) 1 {((w 2 m  2_’  —  [((&)2  +  2C N  — —  2 (WI)  1 C)2 + 4 1 N NN 2C 2 2 1 m  ] } 112  (6.69b)  The following relationship can be found from Eqs. (6.68a) and (6.68b) 1 m =  X  —  (6.70)  ‘i  where the coefficient  —  —  2B2_4.3ox1o3m 2 J 22 B 2 1 q  —  3 m 4.30x10 1 q 1  =  1B 2 4.30x 10 m ) 1 1 (q q 4qq2(q2B4•30x’om2)  (6.71a)  —  Since generally q 2 2B  >>  2 4.30x i0 m 2 and q 1B  z  1.  >>  4.30x 103 m , 1  (6.71b)  —  208  —  Then,  (02  (6.72)  w’.  Analogous to Eq. (6.50), the strong coupling condition with the quadrupole trapping potential correction for the Coulomb interaction between two ion clouds is found to be  ()2  1 (in  1 N  + in 2 N) >>  2 1 Im  —  2  (6.73)  Inequality (6.73) is comparable to Inequality (6.51).  6.8  Discussion  6.8.1 The Charged-Cylinder Model and Taylor’s Expansion Approximation In Chapters 4 and 5, only the radial component of the Coulomb interaction between two ion clouds was taken into account in the average models.( ) In the Taylor’s 12 expansion approximation, the whole Coulomb interaction between two ion clouds is accounted for. The different treatments arise from different sources. When the separation between two ion clouds in FT-ICR varies sinusoidally, as is conventional, the frequency shifts induced by their Coulomb interaction can be predicted by the charged-cylinder model on the basis of the average Coulomb effect. Thus, since the average of tangential Coulomb interaction is zero, only the radial Coulomb interaction is involved in the averaging procedure. However, the charged cylinder model does not predict the strong coupling Coulomb interaction, i. e., two mass peaks would virtually be merged into one when the ion numbers (or ion densities) in each ion cloud exceed the strong coupling critical curve  (  —  209  —  Fig. 6.4). Further, the requirement of the weak coupling condition from the Taylor’s expansion approximation, Inequality (6.56), overrides that of the weak coupling condition from the charged-cylinder model, Inequality (6.58).  In the Taylor’s expansion  approximation, the magnitude of the Lorentz force was compared to the magnitude of the total Coulomb force within small separations, especially in strongest interactions, rather than averaging the interactions. When the Lorentz force cannot overcome the strong coupling Coulomb interaction, the charged-cylinder model will become ineffective. However, the Taylor’s expansion approximation cannot be used to evaluate the unlike-ion Coulomb-induced frequency shifts (Section 6.6.2) and the inhomogeneous Coulombinduced broadening. Therefore, the charged-cylinder model and the Taylor’s expansion approximation compensate each other.  6.8.2 The Electric Quadrupole Trap and Analyzer Cell of FT-ICR There are many similarities between an electric quadrupole trap and FT-ICR. Both experiments use radiofrequency fields to excite ions and detect ion resonance frequencies. Ion motion in the electric quadrupole trap is a complex superposition of axial oscillation and magnetron motion. Ion motion in FT-ICR is a complex superposition of axial oscillation, magnetron motion and cyclotron motion. The electric quadrupole trap and the cubic iontrapped cell of FT-ICR both have quadrupolar trapping fields. The Coulomb-induced frequency shifts of stored ions in a radiofrequency electric quadrupole trap have been investigated by Jungmann et al. for both the strong coupling condition and weak coupling condition.( ) These authors presented a two-particle model 9 and a two-cloud model to explain the strong coupling Coulomb interaction, which forced two mass peaks of Ho (mlz 165) and Er (mlz 167) to merge into one; and the weak coupling Coulomb interaction which shifted the resonance frequencies of Ho (mlz 165) and Xe (m/z 131) to higher frequencies. The two-cloud model presented by Jungmann et  —210—  a!. also was derived from a Taylor’s expansion approximation for small separations between two ion clouds. Although cyclotron motion is detected in Fr-ICR rather than axial oscillation as in the quadrupole trap, their Taylor’s expansion approximations have a similar form except that the direction of the Lorentz force applied in FT-ICR is opposite to the direction of the trapping force in the quadrupole trap (over a time average). For their special purpose (to maintain a small separation between ion clouds), Jungmann et a!. used an electric quadrupole trap in which the narrowest space between the electrodes was orily5 mm. The hole diameter for transmitting an electron beam was not stated. If the hole diameter is estimated to be 2 mm, the largest separation between two ion clouds was close to that of touching in Jungmann’s quadrupole trap. Therefore, the strong coupling condition is easily attained in Jungmann’s quadrupole trap. On the other hand, the conventional analyzer cell used in Fr-ICR is the 2.54 cm cubic ion-trapped cell, and largersized analyzer cells are common. The strong coupling condition in Fr-ICR is, therefore, much more rigorous. The Coulomb interaction coefficient C is negative from the analytical solution in Eq. (6.26), and the numerical estimation of Jungmann et al.( ) The physical reason for the 9 negative C is not only because two ion clouds can overlap as explained by Jungmann et a!., but also because the z-axis length of the ion cloud is larger than its cylinder radius (Eq.  (6.25)).  6.8.3 Intermediate Coupling Condition Obviously, a transition exists between the strong coupling regime and the weak coupling regime  —  intermediate coupling. Jungmann et a!. defmed this transition as the  “minimum resolvable condition”.( ) That is, when the intermediate coupling condition is in 9 effect, two mass peaks will not merge completely. This assumption was experimentally  —211—  verified by using the two ions Ho (m/z 165) and Sm (m/z 150).(9) In FT-ICR, the intermediate coupling condition is defined as  (6.74)  From Eq. (6.74), and setting m 2  >  0C N  2 m  —  —  (6.75)  1 cEoo?. m  Substituting Eq. (6.75) into Eqs. (6.36),  2  +‘a  =  (Oo2+W02((O01_COo)’\/1  =  %O)o2(COo1(Oo2)’\J1 +  ,  (6.76a)  and  22  If m 2  .  (6.76b)  , the frequency solutions of Eqs. (6.76a) and (6.76b) are reduced to 1 m  2 Wi  =  c+Jo ) 2 —% 01 (w 02 ,  (02  =  o2  (6.77a)  and  —  %2  (ooi  —  %2)  (6.77b)  —  212  —  The solutions in Eqs. (6.77a) and (6.77b) indicate that one ion cloud will be accelerated and another will be decelerated. If so, this would produce a transition regime in FT-ICR, which could rapidly be reduced to the weak coupling interaction (Eq. (6.55)).  6.8.4 Ion Distribution Functions At the beginning of this chapter, the ions in the analyzer cell of FT-ICR were assumed to have uniform distributions, i. e., the ions were assumed to have zero initial thermal velocity. The cylindrical symmetric pseudopotential of a if electric quadrupole trap corresponds to a static quadrupole potential averaging over time.( ) It has been proved 13 that ion spatial distribution in such a potential is nearly uniform at —300 K for non-zero ion thermal velocities.( ) Because a stronger static magnetic field is applied in FT-ICR along 14 the z-direction, the ions in an analyzer cell of the ICR could have a Maxwellian distribution along the x- and y-axes.(’ ) For a cylindrical ion cloud containing N 5 1 ions, in which each ion has a mass m 1 and charge q , its Maxwellian distribution in the center of the analyzer 1 ) will be 15 cell(  1 1 n ,y (x , ) z  =  “‘‘ 1 ( exp[_13 2 2 +y x )J  where 1 is the z-axis length of the cylinder and  1 co / 2 kB fl = m  (6.78)  T, kB being the  Boltzmann constant, 1.380658x10 J•K, and T being the absolute temperature. After 1 = (xRl, YR1)’ Eq. (6.78) becomes this ion cylinder is excited to its cyclotron radius R  ,R 1 ) 1 1 (r n  =  i”  ‘‘ exp  {  —  f3 [(x 1 —XR1) + (yl —YR1)i 2  }.  (6.79)  —213—  Analogously, a second cylindrical ion cloud containing N 2 ions, in which each ion has a mass in 2 and charge q , has a Maxwellian distribution after excitation 2  ) 2 ,R 2 2 (r n  where f2  =  =  2 I2 N  exp  —  132  2 kx  —  2+ XR2)  2 (Y  —  YR2)]  }  (6.80)  2 co / 2 m kB T. The Coulomb potential energy of Cylinder 1 due to Cylinder  2 will then be  12 U  =  —R 1 (1R 1 2 )  xf 2 kqjq  ff  ,R (r ) 2 2 1 (r,R) n n  1 r  Cylinder 1 Cylinder 2  poo —  2 j2  e Pi —  —  ) 2 x  pee  fOe  p1/2  J_oe  2 + (Y1—YR1)J ) 1 [(Xl—XR  (x  l/2  pee  I J_eo I J—l/2 I JI I J—l/2 I  N q kq f 1 2 3  —  2 1 r 3 d  —  +  1 (y  —xp) + [(x 2 e P2 —  —  ) 2 y  +  1 (z  —  (Y2YRZ)J  ) 2 z  dz 2 dy dy dx d . z 1 1 dx  (6.81)  The transformsxj =X1—XR1,4 2 X =X , Yj R =Y1—YR1’ andy =Y2—YR2 can be used, and the integrand of Eq. (6.81) can be expanded into Taylor’s series. After the masses of two particular ion species are preset, numerical integration can be used to solve the integral in Eq. (6.81). Because the integral of Eq. (6.81) has the same form as that  ), the solution of Eq. (6.81) is expected to be 9 derived by Jungmann et al.( —R 1 2 I) 12 R U (I  12 + U (0)  1N N 2 C’  [ (xR1  —  2 + (YR1 XR2)  —  2 YR2)  1  (6.82)  —214--  where the Coulomb coefficient C’ should be negative. Then, the strong coupling, weak coupling, and intermediate coupling conditions, like those in Eqs. (6.39), (6.55), and (6.74), may be found. That is, the ion distribution has no effect upon the Taylor’s expansion approximation.  6.8.5 Further work The most important conclusions in this chapter are those obtained first for the strong coupling regime; and secondarily, those regarding the weak coupling condition in Eq.  (6.55) or (6.56). In order to test experimentally the strong coupling Coulomb interaction between ion clouds in FT-ICR, using the analysis in Section 6.4.3, two ion species of masses 300 Daltons and 301 Daltons are a suitable ion pair. For the 2.54 cm cubic cell, ion 5 from Fig. 6.4. Conventional electron ionization can produce numbers must exceed 5x10 this number of ions. Also, these two ion species must have similar ion intensities. In order to obtain analytical solutions for the strong and weak coupling conditions, the tetragonal ion cloud model was used as an approximation of cylindrical ion clouds in the Taylor’s expansion approximation. If a model of the cylindrical ion clouds can be applied in the Taylor’s expansion approximation, the results obtained here would be more precise. The eigenfrequency solution  co in Eq. (6.41), which is the oscillation about an  unchanging center-of-mass position, is interesting. Verification of its physical regime is required, both theoretically and experimentally. Undoubtedly, Taylor’s expansion approximation can be applied for more than two ion species, but this will need more time. In Eq. (6.64) of Section 6.7, the electric trapping field was not calibrated for the ion Coulomb potential in the cubic analyzer cell. Jeifries, Barlow, and Dunn have examined the ion Coulomb potential of a prolate ion cloud in the center of ICR analyzer cell, and  —215—  found it to be harmonic.( ) More recently, the ion Coulomb potential of a spherical ion 10 cloud at cyclotron radius r in the hyperboloidal Penning cell of FT-ICR was solved using an image charge model developed by Vogel, Kluge, and Schweikhard.( ) In the image 16 charge model, the ion Coulomb potential of the ion cloud was increased not only as the number of ions increased, but also as its cyclotron radius r increased, which agreed with the experimental results measured by Vogel et al.( ) In reality, ion clouds in the analyzer 16 cells of FT-ICR are cylindrical or prolate, and revolve in the circular orbits. Therefore, these two models, developed by Jeffries et al.( ) and Vogel et al.( 10 ), cannot be applied to 16 the cubic analyzer cell of FT-ICR. The application of these two models with respect to the cubic analyzer cell remains to be solved in the future.  —216—  References  1.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1991, 5, 450— 455.  2.  Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1992, 6, 1—3.  3.  Comisarow, M. B.; Chen, S.-P. The 1989 International Chemical Congress of Pacific Basin Societies, 1989, PHYS 0414.  4.  Nayfeh, M. H.; Brussel, M. K. Electricity and Magnetism; John Wiley & Son: New York, 1985; p 183.  5.  Goldstein, H. “Classical Mechanics”; Addison-Wesley: Reading, MA, 1980; pp 21-23.  6.  Ximen, J. “Aberration Theory in Electron and Ion Optics”; Academic Press: Orlando, 1986; p 264.  7.  Ledford, E. B.; Rempel, D. L.; Gross, M. L. Anal. Chem. 1984,56, 2744-2748.  8.  (a) Comisarow, M. B. Adv. Mass. Spec. 1980, 8, 1698—1706. (b) Comisarow, M. B. mt. J. Mass Spectrom. Ion Phys. 1981,37, 251—257.  9.  Jungmann, K.; Hoffnagle, J.; DeVoe, R. G.; Brewer, R. G. Phys. Rev. A 1987, 36, 3451-3454.  10. Jeifries, J. B.; Barlow, S. E.; Dunn, G. H. Int. J. Mass Spectrom. Ion Proc. 1983, 54, 169-187. 11. Francl, T. 3.; Sherman, M. G.; Hunter, R. L.; Locke, M. 3.; Bowers, W. D.; Mclver Jr., R. T. mt. J. Mass Spectrom. Ion Proc. 1983,54, 189-199. 12. Hunter, R. L.; Sherman, M. G.; Mclver, R. T. Jr Int. J. Mass Spectrom. Ion Phys. 1983,50, 259-274.  13. Dehmelt, H. G. Adv. At. Mol. Phys. 1967, 3, 53-72.  —217—  14.  Cutler, L. S.; Flory, C. A.; Giffard, R. P.; McGuire, M. D. Appi. Phys. B 1986, 39, 251-259.  15.  Sharp, T.; Eyler, J. R.; Li, E.  mt. J. Mass Spectrom. Ion Phys.  1972, 9, 421-439.  16. Vogel, M., diploma thesis, Johannes Gutenberg-Universitat Mainz, 1990.  —218—  APPENDIXES  —219—  Al Appendixes of Chapter 2  A1.1  The “Twenty-Second” Ion-Molecule Reaction Mass Spectra  C  U F  Ju I— ci I U  n C  (b) Parent molecule 2 3 CpCr(N C D O)  I—  Delay time =20 sec LI I— Lfl 11 I-  a  I U  L. C  ‘  100  I  I  I  I  I  I  1 200  I  I  I  I  [ 1 600 ‘-iOO F1P55 ]N 1 P.M.U I  1  I  I  l[l  11 I I1TlFT1F  500  600  —  220  —  A 1.2 Coordination Modes of Metal Carbonyls and Metal Nitrosyls Table A1.1 Coordination Modes of Carbon Monoxide  Mode  Terminal  Formal Charge  Electron Donated  Structure  0  2  M—CO  Character  r(M—C): —1.85A  Ref.  1.1  LMCO: 165—180° Symmethcal  0  2  0  2  cc77-90°;/3=y  1.1  13C0±M—M  1.1  13cC0not±M—M  1.1  #201’)-  Asymmetrical  0  MM  Semibridging  0  0  2  II M  0  ‘M  0  2  (no M-M bond) M  Linear  (C, 0)2 p  0  4  (0) (C,Oj  4  r(M....M): 3.1—3.4A a: 104—120°  II  1.2  M  /C\ M(orM) M M—C0—M’  13: 165—175°  1.2, 1.3  ZCOM’: 135—180°  1.2, 1.3  r(O—M’): 2.05—2.15A 0  iL3(7)-  0  2  1.2, 1.3  M-)M(or M  (2or)3 p  0  4  1.2, 1.3  —221  —  Table A 1.1 Continued Formal Charge  Electron Donated  p ( 3 i’,’i-  0  6  (2C, 0)3 p  0  (3C, 0)4 j.t  0  M ode  Structure  Character  Ref.  1.2, 1.3  ’j:;’co__r’ 1 4  ZCOM’: 165_1800  1.2,1.3  r(0—M): 1.80—2.05A  1.2, 1.3  M\ /CO_M M 14(371,772)  0  /_ ç 7 \ (orM)Mç M(OrM)  1.2, 1.3  M (or M) it-Carbene with two carbonyls’  1  M MO”  1.4  (1r’)CO, p 4 (’)-CO, p 5 (’)-C0 are not included in this table. M is used to represent 6 ¶ The hypothetical p a metal atom and M’ hetero-metal atom. 4.  The p-Carbene is listed here, since it can be produced directly from two carbonyl ligands.  1.1 Lukehart, C. M. Fundamental Transition Metal Organometaflic Chemistry, Brooks/Cole, Monterey, 1985, Chapter 2. 1.2 Cruz, C. D. L.; Sheppard, N. J. Mol. Struc. 1990,224, 141-161. 1.3 Kawaguchi, S. Variety in Coordination Modes of Ligands in Metal Complexes, Springer-Verlag, Wien, 1988, Chap. 3. 1.4 Berry, D. H.; Bercaw, 3. E.; Jircitano, A. 3.; Mertes, K. B. J. Am. Chem. Soc. 1982, 104, 47 1247 15.  —  222  —  Table Al .2 Coordination Modes of Nitric Oxide  Mode  Formal Charge  Linear terminal  Electron Donated  Structure  3  (M N=O)  r(M—N): —1.76±0.1A LMNO: —180°  2.1  1  M—N  r(M—N): —1.83±0.15A LMNO: 120—170°  2.1  fl=-Y r(M—N): —1.85  2.1  111  Bent terminal 711  Symmetrical  3  Character  Ref.  0  V  A  MM Asymmetrical  0  3  2.1 MM r(M....M): —3.2 A a: —102°  3  (no M-M bond) M #2(1’  2.1  M  ,ij2)  2.2  Isonitrosyl bridge#  M—NO-4M  3  113(1?’)-  #4(112,172)-  —1  ZWNO = 169° ZNOMg = 135.6°  2.3  0 2.1  7  2.1 M’’M  #4(311  ,i72)-  2.1  —  223  —  Table A1.2 Continued Mode  Formal Charge  Terminal ’ 0 2 cis-N Hyponitrite bridge’  Electron Donated 2  Structure  Character  Ref.  N°\ M II / 0 N  2.4  M/N\ .M  2.5  b (a/ic)-NO, i 3 (’)-NO are not included in 4 ¶ The hypothetical ji(’)-NO linked to heteronuclear metals, j.i this table. # The crystal structures have not been identified completely. *  These two modes are listed here, since they can be produced directly from two nitrosyl ligands.  2.1  Richter-Addo, G. B.; Legzdins, P. “Metal Nitrosyls”; Oxford U. Press: New York, 1992 (in press).  2.2  Legzdins, P.; Rettig, S. J.; Veltheer, 3. E. J. Am. Chem. Soc. 1992 (in press).  2.3  (a) Legzdins, P.; Rettig, S. J.; Sanchez, L. Organometallics 1988, 7, 2394-2430. (b) Christensen, N. 3.; Hunter, A. D.; Legzdins, P. Organometallics 1989, 8, 930-940.  2.4  McCleverty, J. A. Chem. Rev. 1979, 79, 53-76.  2.5  Hoskins, B. F.; Whillans, F. D.; Dale, D. H.; Hodgkin, D. C. J. Chem. Soc., Chem. Commun. 1969, 69-70.  —  Al.3  224  —  Known Oxo Chromium Complexes Table A1.3  Known oxo chromium complexes  (in order of oxidation state of chromium) Compound  Oxidation state  Electron counting  Coordination number  Ref.  Monomer: 1 2 [Ci0  +3  11  4  3.1  (CO) Cr0 2  +4  16  6  3.2  2 Cr0  +4  10  4  3.1  2 CrOF  +4  10  6  3.1  CrO(Porphrins)  +4  14  6  3.1  CrO(Pc)  +4  14  6  3.3  ] 4 [Cr0  +4  18  8  3.1  ] 5 [Cr0  +4  3.1  j 6 [Cr0  +4  3.1  [(CrTPP)OJ  +5  13  6  3.1  (CF] C 2 Cr0[O  +5  13  6  3.1  OR C Cr0[O ] 2  +5  13  6  3.1, 3.3  [Cr0(sa1en)j  +5  13  6  3.1  CrO(MEC)  +5  13  6  3.3  3 (X=F, Cl) CrOX  +5  11  5  3.1  L (L=bipy, phen) 3 CrOC1  +5  15  7  3.3  ] (X=F, Cl, Br) 4 [CrOX  +5  13  6  3.1  ](X=F,C1) 5 [Cr0X  +5  15  7  3.3  Cp*CrOBr  +5  15  7  3.4  4 CrOF  +6  12  6  3.1  J 5 [CrOF  +6  14  7  3.1  X (X=F,Cl) 2 CiO  +6  12  6  3.1  (N0 0 2 ) 3  +6  16 or 20  6 or 8  3.1, 3.5  (O Cr0 C 2 ) 3 CF  +6  16  8  3.1  ),py 2 Ci0(O  +6  14  7  3.1  (1, 10-phen) ) 2 CrO(0  +6  16  8  3.1  3 Cr0  +6  12  6  3.1  XJ(X=Halogens, OH) 3 [CrO  +6  14  7  3.1  —  225  —  Table A1.3 continued  Electron  Coordination  Compound  oxidation  (py) 3 Ci0 2  ÷6  16  8  3.6  2 and its derivatives ] 4 [Cr0  +6  10  4  3.1  [(H C O 2 ] 5 zOCr(H )  +3, +3  17, 17  6, 6  3.7  CaOCr(NH [(NH 4 ] 5 ) 3  +3, +3  17, 17  6, 6  3.7  0 2 [Cr(NCSXTPyEA)1  +3, +3  15, 15  6, 6  3.8  C 4 [(2-picetam) O 2 3 (OH)J r  +3, +3  15, 15  6, 6  3.9  Cr(O)(OH)Cr(bipy) 2 [(bipy)  +3, +3  15, 15  6, 6  3.10  j 2 [(ph(O)(OH)Cr(phen)  ÷3, +3  15, 15  6, 6  3.10  C [(bipy) C ] 2 r(bipy) r(O)  +3, +3  16, 16  6, 6  3.10  C [(phen) C ] 2 r(phen) r(O)  +3, +3  16, 16  6, 6  3.10  CHPh 2 (CrOC1)  +41+4  4, 4  3.11  0 2 (CrC1TPP)  6, 6  3.12  2 (CrPcO)  6, 6  3.13  6, 6  3.14  4, 6  3.15  5, 5  3.11  counting  number  Ref.  Dimer:  i4  O(OCMe [(CpCr) ] 2 ) 3  +4, +4  15k,  (HO)OCrOCrOCH (C1) P 2 h(C1)  +4, +6  10, 12  O [(C1) C O 2 HPh Cr]  +5, +5  1 2 [Sfp*CrcJ  +5, +5  16, 16  6, 6  3.16  ( C 2 [OCrO 4 ) 3 p-O] cF  +5, +5  13, 13  6, 6  3.17  (HO)OCrJ [(C1) C 2 HPh  +6, +6  6, 6  3.11  O? 2 [Cr  +6, +6  7, 7  3.1  14, 14  Trimer: 0(O Cr C H 2 p 6 3 CF y )  222 +2, +2, +2-  6, 6, 6  3.18  0 Cr C H 2 ( 6 3 4-cynopy) (O CF )  222 +2, +2, +2  6, 6, 6  3.18  15, 15, 15  6, 6, 6  3.18  17, 17, 17  6, 6, 6  3.14  6,6,6, 6  3.19  0H [Cr C ( 6 ) 0 2 1 3 (O CH )  +3, +3  O(OCMe [(CpCr) 1 ) 3 Tetramer: O(OH) 4 [Cr ( 5 O 2 ] 10 H )  average +3  4 (CpCrO)  ÷3, +3, +3, +3  18, 18, 18, 18  6, 6, 6, 6  3.20  4 (Cp’CrO)  +3, +3, +3, +3  18, 18, 18, 18  6,6,6, 6  3.20  ( 4 (Cp’Cr) ( ) 2 SCuBr S) O  +3, +3, +3, +3  6, 6, 6, 6  3.20  P (Ph [ C 3 ) J 4 O)(SO MeCN Cr(ji 1 )  +3, +3, +3, +3  6,6,6, 6  3.21  —  226  —  Table A1.3 continued Electmn counting  Coordination number  Compound  Oxidation state  (TJv1EDA) Cr 10 (OPh) ( 4 p-O)Na  +3, +3, +3, +3  6, 6, 6, 6  3.22  (S) 4 [(Cp’Cr) O 3 ]  +3, +3, +3, +4  6, 6, 6, 6  3.20  C CP C 3 O 5 ) 4 H 2 r(T1 C r  +3,  6, 6, 6, 7  3.20  +4  +3k,  18,  i4, 16, 18  Ref.  or+4-  ¶ The abbreviations of ligands can be found in the following references and are not listed here. 3.1  Cotton, F. A.; Wilkinson G., “Advanced Inorganic Chemistry”, John Wiley & Sons: New York., 1988, pp 719-736 and references therein.  3.2  Poliakoff, M.; Smith, K. P.; Turner, 3. 3.; Wilkinson, A. J. J. Chem .Soc. Dalton Trans. 1982, 3, 651-657.  3.3  Nag, K.; Bose, S. N. Struct. Bonding 1985, 63, 153-205.  3.4  Morse, D. B.; Rauchfuss, T. B; Wilson, S. R. J. Am. Chem. Soc. 1988, 110, 8234-8235.  3.5  Marsden, C. 3; Hedberg, K.; Ludwig, M. M.; Gard, 0. L. bzorg. Chem. 1991, 30, 4761-4766.  3.6  Cainelli, G; Cardillo, G. “Chromium Oxidations in Organic Chemistry”, Springer-Verlag: Berlin, 1984, pp 151-161.  3.7  Holwerda, R. A.; Petersen, J. S. Inorg. Chem., 1980, 19, 1775-1779.  3.8  Vaira, M. Di; Mani, F. Inorg. Chem., 1984,23, 409-412.  3.9  Michelsen, K.; Pedersen, B.; Wilson, S. R.; Hodgson, D. 3. Inorg. Chim. Acta, 1982, 63, 141150.  3.10 (a) Josephsen, 3.; Pedersen, B. Inorg. Chem., 1960,38, 2137-2142. (b) Josephsen, 3.; Schäffer, C. B. Acta. Chem. Scand., 1970,24, 2929-2942.  3.11 Wheeler, 0. H. Can. J. Chem., 1977, 16, 2534-2538. 3.12 Nill, K. H.; Wasgestian, F.; Pfeil, A. Inorg. Chem., 1979, 18, 564-567. 3.13 Creager, S. E.; Murry, R. W. Inorg. Chem., 1985, 24, 3824-3828. 3.14 Nefedov, S. B.; Pasynskii, A. A.; Eremenko, I. L.; Orazsakhatov, B.; Ellert, 0. 0.; Struchkov, Yu. T.; Yanovsky, A. I. I. Organomet. Chem., 1990, 385, 277-284. 3.15 Wheeler, 0. H. Can. J. Chem., 1964, 42, 706-707. 3.16 Herberhold, M.; Kremnitz, W.; Razavi, A.; Schôllhorn, H.; Thewalt, U. Angew. Chem. EngI., 1985, 24, 601-602.  mt.  Ed.  —  227  —  3.17 Nishino, H.; Kochi, 3. K. Inorg. Chim. Acra 1990, 174, 93-102. 3.18 Cotton, F. A.; Wang, W. Inorg. Chem., 1982,21, 2675-2678. 3.19 StUnzi, H.; Marty W. Inorg. Chem. 1983, 22, 2145-2150. 3.20 Bottomley, F.; Sutin, L. Adv. Organomet. Chem. 1988, 28, 339-396. 3.21 Clegg, W.; Errington, 3. Hockless, D. C. R.; Glen, A. D.; Richards, D. 0. J. Chem. Soc. Chem. Cornmun. 1990, 1565-1566. 3.22 Edema, 3. 3. H.; Gambarotta, S.; Smeets, W. 3. 3.; Spek, A. L. Irtorg. Chem. 1991, 30, 13801384.  —  A2  228  —  Appendixes of Chapter 3  Kinetic Behavior of Cr Produced from 3 CH 2 CpCr(NO) , CH 3 , and N 4 in H 0, NH 2 2 Media  The experimental parameters are given in Section 3.2 of Chapter 3. The kinetic behavior of Cr produced from 3 CH in H 2 CpCr(NO) , and N 4 0, NH 2 , CH 3 2 Media all are not pseudo-first order, like the behavior of Cr in H 2 medium shown in Fig. 3.2.  -3 -4 -5 -6 -7 -8 -9 0.00  0.05  0.15  0.10  Time (second) -3 -4 N  -5 -6  -7 0.00  0.02  0.04  0.06  Time (second)  0.08  0.10  0.12  —  229  —  -3 -4 -5 -6 -7 -8 0.00  0.05  0.10  0.15  Time (second) -3 -4 -5 -6 -7 0.000  0.025  0.050 Time (second)  0.075  0.100  —  230  —  A3 Appendixes of Chapter 5  A3. 1  The FORTRAN Program for Calculating Ion-Ion Coulomb Interaction between Two Charged Disks Based on Double Gaussian Numerical Computation This program requests the user to input the charged-disk radius, ion cyclotron radius,  ion number in each disk, data points for calculating non-overlapping disks, data points for calculating overlapping disks and increment from point to point. The program will list touch angle of two disks, the instantaneous distance, angle, radial Coulomb force between the two disks, average radial Coulomb force, and apparent Coulomb distance.  1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 23 24 .5 26 27 28 29 30 31 32 33 34 35 36 .7  C: C C C C C C C C C.. C C C.. C C C C  1  THIS PROGRAM FOR CALCULATION OF RADIAL COULOMB FORCE BETWEEN TWO DISK ION CLOUDS WITH OVERLAP AND WITHOUT OVERLAP BASED ON DDBLGS (DOUBLE GAUSSIAN INTEGRATION) OFMTS. r(or A0) radius of the disk R(or RC) = ion cyclotron radius N0(orDN) =ion number in each disk DIO = initial separation between two disks with overlap DIN initial separation between two disks without overlap N =..data..points...for..ca1culatin. f overlap M = data points for calculation of non-overlap Z1,22 = increments in N and M do loop ANG = angle between two disks ifl.. a..yloton....(radi.an) ANGTOUCH = “touching angle” (radian) FORC radial Coulomb force for each calculation FAV = average Coulomb force Dc = Apparent Coulomb distance DOUBLE PRECISION A0,DAI,BA1 ,BA2,BA3,UA1,UA2,UA3,TOT1, ...DA 1 I.M.1.,PAIM P AIMEN4,.PA EN2.,PAIME IN.ENS., N3. SUM,TQT.,DN,D.I DOUBLE PRECISION Z1,Z2,RC,Q,COUL,API,S0,FAVE,TOD,TEV, + ANG,ANGTOUCH,DAIMEN6,DAIMEN7 COMMON .,IFLAG/CCf,BA.1. BA3,UA1.,U 3LPD/.AQ.,.ANG,, RC DIMENSION SUM(512) PARAMETER (Q=1.6021892D—19,COUL=8.987551787D+09, ..A!’I=3.1415.92 535897.932.4.P.O) WRITE(6,*) ‘r = READ(5,*) A0 *. 1 WR.IT.E.6 R = READ(5,*) RC WRITE(6,*) ‘NO READ (.5., “.) DN DIN=2.D0*AO WRITE(6,*) ‘DIN = ‘,DIN DIO = O.D0 WRITE(6,*) ‘DIO = ‘,DIO WRITE(6,*) ‘M = (odd) READ(5,*) H  —231— 38 39 40 41 42 .4.3 44 45 46 47 48 .9 50 51 5.2 53 54 5.5 56 57 58 .5 60 61 .62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 .83 84 85 .86 87 88 89 90 91 92 93 94 5 96 97 .9.8 99 100 11  3 4 6 8 C: C C  21 22 23  24 25 26  30 10 C:  32  33 34 27  ANGTOUCH2 .DO*DASIN (AO /RC) Z2 (API-ANGTOUCH) /FLOAT (H-i) WR.I.T.E(6,..*) N (vcn) READ(5,*) N WRITE(6,*) ‘Zi RE?.D.(.5 .) Zi IF (Z1*FLOAT(N)_ANGTOUCH) 6,4,4 WRITE(6,*) ‘Zi SHOULD BE SMALLER !‘ GOTO 3 WRITE(7,8) AO,RC,DN,A.NGTOUCH FORNAT(1X,’r ‘,1PG7.1,’, R ‘,1PG7.1,’, NO 1.PG7.1., ANGTOUCH ..1PG 12 6) 1 CALCULATION OF THE COEFFICIENT OUTSIDE OF INTEGRAL. COUL THE COULOMB CONSTANT. ION ARGE.. Q SO=COUL*Q*Q*DN*DN/ (API*API*AO*AO*AO*AO) DO 10 I=1,N ANG.?.i*FLQAT ) DS1=2.DO*RC*DSIN(O.5D0*ANG) UA1=RC+A0 BA1RC-A0 D.COS(ANG)A EA2=RC*DCOS fANG) -A0 UA3=(RC*DCOS(O.5D0*ANG) +DSQRT(AO*A0_RC*RC*DSIN(O.5D0*ANG) *D.SI.N(0....5D0*ANG).).).*.DCOS.(O.,5D0ANG) + BA3=(RC*DCOS(0.5D0*ANG)_DSQRT(A0*A0_RC*RC*DSIN(0.5D0*ANG) + *DSIN(0.5DD*ANG)))*DCOS(0.5D0*ANG) IFLAG=1 DAI=DDBLGS(BA3,UA3,48,48) DAIMEN1=DAI IFLAG=2 DAI=DDBLGS(UA3,UA2,48,48) DAIMEN2=DAI*2.D0 IFLAG=3 DAI=DDBLGS(BA2,BA3,48,48) DAIMEN3=DAI*2.D0 IFLAG=4 DAI=DDBLGS(UA3,UA2,48,48) DAIMEN4=DAI*2.D0 IFLAG=5 DAI=DDBLGS(UA3,UA2,48,48) DAIMEN5=DAI IFLAG=6 DAI=DDBLGS(BA2,BA3,48,48) DAIMEN6=DAI SUM (I).=S0*. (DAIMEN1.DAH,N PAl N34DAIMEN .PIMEN.5.+.PAIMEN6) WRITE(7,30) DS1,ANG,SUM(I) FORMAT(1X,’D = ‘,1PG1O.4,.’ ANG = ‘,1PG12.6, FQRC ,2.PG12 6) CONTINUE SIMPSON’S RULE WILL BE PERFORMED FOR OVERLAP CASE. TOD=0.D0 DO 32 I=1,N—1,2 TOD=TODSUM(I) CONTINUE TEVtO.D0 DO 33 I=2,N—2,2 TEV=TEV .SUM (.1.) CONTINUE TOTtZ1*(SUM(N)+4.DO*TOD+2.D0*TEV) /3.D0 WM.TEfl,..34) N,TQT ‘,13,’ FORMAT(1X,’N TOT = ‘,1PG12.6) IFLAG=7 DC 17 K1,,.M  —  102 103 .1.0.4 105 106 1.0.7 108 109 .1.1 111 112 113 114 115 116 .117 118 119 .120 121 122 .123 124 125 .126 127 128 .129 130 131 132 133 134 .135 136 137 138 139 140 .141 142 143 144 145 146 147 148 149 150 151 152 .153 154 155 .15 157 158 159 160 161 162 163 164 I 5  35 17 C:  36  37  38 39 99 C  232  —  ANGLANGTOUCH+Z2*FLOAT(K_1) DS62.D0*RC*DSIN(0.5D0*ANG) C.+A0 BA1RC-A0 +A0 O.S..(AN A0 DAI=DDBLG5(BA2,UA2,48,48) St.IM(N+K) =DAI’SO WRITE (.7.35.) PS6.,.AN,SUM (N+K) FORMAT(1X,’D ‘,1PG1O.4,.’ ANG ‘,1PG12.6, + ‘ FORC ‘,1PG12.6) CONTINUE SIMPSON’S RULE WILL BE PERFORMED FOR NON-OVERLAP CASE. TEVO.D0 DO 36 I=N+3,M+N—2,2 TEY=TEV.SUM (.1) CONTINUE TOD=0.D0 DO 37 I.2.,M.N —1,,2 TOD=TOD4SUM(I) CONTINUE TOTI=Z2*(SUM(N+1),SUM(N+M)+2.D0*TEV+4.D0*TOD)/ DO FAVE (TOT+TOT1+0.5D0* (SUM(N) SUM(N+1)) + *(ANGTOUCN_Z1*FLOAT(N)))/API WRITE(7.,38).M,TO ,FAVE FORMAT(1X,’M ‘,I3,’ TOT1 ‘,1PG12.6,/1X, • ‘FAVE ‘,1PG12.6) 39) p.$Q.{cQUL*.pN.pN.*Q.*.Q/FA ).../.R.C 1 WRITE(7 FORMAT(1X,’Dc = ‘,1PG13.7) STOP END FUNCTION BLIM(Y) DOUBLE PRECISION... BM.,.BA uAl, UA COMMON IFLAG/CC/BA1,BA3,UA1,UA3 IF (IFLAG.EQ.1) BLIM=BA3 IF...(IFLAG.EQ.2) BLIM=BA3 IF (IFLAG.EQ.3) BLIM=BA3 IF (IFLAG.EQ.4) BLIM=UA3 IF..(IFLA.G.EQ.5)....BLIM=BA IF (IFLAG.EQ.6) BLIM=UA3 IF (IFLAG.EQ.7) BLIM=BA1 RETURN END  C FUNCTION ULI$CY DOUBLE PRECISION BA1,BA3,UA1,UA3 COMMON IFLAG/CC/BA1 ,BA3,UA1,UA3 IF (.IFLAG.EQ.1.) Z.M=UA3 IF (IFLAG.EQ.2) ULIM=UA3 IF (IFLAG.ZQ.3) ULIM=UA3 IF (IFLAG.EQ...4) UUM=UA1.. IF (IFLAG.EQ.5) ULIM=BA3 IF (IFLAG.EQ.6) ULIM=UA1 .F (IFLAG.EO.7) ULIM=UA1 RETURN END -  C FUNCTION GFUN(Y) GFUN=1 .ODO RETURN END C UNCTIO  —  166 167 169 170 .1.7 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 1.90 191 192 .193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214  41  42  44  88  233  —  DOUBLE PRECISION A0,X,Y,Y0,Y11,Y12,Y21,Y22,R0,ANG,RC COMNON IFLAG/DD/A0,ANG,RC EQ..1.) OTO 41 IF IF (IFLAG.EQ.2) GOTO 42 IF (IFLAG.EQ.3) GOTO 42 IF (.IFI.AG.EQ..4.) OTO 44 IF (IFLAG.EQ.5) GOTO 44 IF (IFLPG.EQ.6) GOTO 44 IF (IFLAG.EO.7) GOTO 44 Y0=X-Y Y11=DSQRT (A0*A0_ (X_RC*DCOS (ANG) ) **) Y12=DSQRT (A0*A0_ (X—RC) **2) G)).2.) Y22=DSQRT (A0*A0_ (Y—RC) **2) RO=RC*DSIN(ANG) IF (Y0 .EQ.O.D0) THEN AUXIN0.D0 ELSE AUXIN=..(_DSQRT(.(Y11+Y21).*.*2+Y0.*Y0)+DS.QRT.((RO+Y12+Y21)*.*2 +yQ*yO)+DSQRT((RO_y11_y22)**2+yO*yQ)_DSQRT((y12+y22) **2+yO*yO))/yO + ENDIF GOTO 88 YO=X—Y Yll DSQRT (Ao.*A0..(X.pcOS.(ANG)..)..z) Y12=DSQRT(A0*AO_ (X—RC4 **2) Y21=DSQRT(A0*AO(Y_RC*DCOS(ANG) )**2) RO=RC*PSIN(ANG) IF (Y0.EQ.0.D0) THEN AUXIN=0.DO ELSE AUXIN=(_DSQRT((Y11+Y21)**2+YO*Y0)+DSQRT((R0+Y12+Y21)**2 +yQ*yO)+DSQRT((y11_y21)**2+yQ*yO)_DSQRT((R04y12_y21) + **2+Y0*YOfl./YQ + ENDIF GOTO 88 Y0=X—Y Y11=DSQRT(A0*A0_(X_RC)**2) Y21=DSQRT(A0*A0_(Y_RC*DCOS(ANG))**2) RO.=RC*PSIN.(ANG) IF (Y0.EQ.0.D0) THEN AUXIN=0.D0 ELSE AUXIN=(_DSQRT((R0_Y11.Y21)**2+Y0*Y0).DSQRT((R0+Y11+Y21)**2 +yQ*yO)+DSQRT((RO_y11_y21)**2.yQ*yO)_DSQRT((RQ+y11_y21) *.*2+.Y0YO)..)..,tY Q ENDIF RETURN END  —  234  —  The FORTRAN Program for Calculating Ion-Ion Coulomb Interaction between  A3.2  Two Charged Disks Based on Double Romberg Numerical Computation  A3.2.1  Program ‘DISK.ROMB” for Two Charged Disks without Overlapping  This program requests the user to input the charged-disk radius, ion cyclotron radius, ion number in each disk, data points for the numerical computation, and number of iterations. The program will list instantaneous angles between the two disks (from the angle of the two disks just touching to the maximum angle it), and radial component of the instantaneous Coulomb force.  C C C C C C C C C C C  C 8 C  *  DISK.ROM8  *  r (Dr AG) = radius of the disk R (or RC) = ion cwclotron radius NO (or DN) = ion number in each disk ANGTOIJCH = ‘touching angle (radian) N = number of iteration (maximum 15) 10 = data points for calculation of non—overlap Zi = increment in TO do loor ANG = anle between two disks in a cclotror, (radian) PARAMETER (Q=1,6021892E—19,CDUL=8.987551787E+09, + API=3.14159265358979324) PRINT*,’r = READ*AO PRINT*p’R = READ*,RC PRINT*,’NO = READ*,DN ANGTOUCH=2. *ASIN C AO/RC) PRINT*,’ANGTOUCH = ‘,ANGTOUCH Special coefficient in Romber numerical interatior. AL=1 .55 PRINT*,’AL = ‘,AL Tolerance in this roram E1 .E—14 PRINT*,’E = PRINT*v’N = READ*rM PRINT*,’IO = READ*, TO  —  235  —  Z1=(API—ANGTOUCH)/(FLOAT(I0) )**1.6 DO 9 J0r10 IF (J.EQ,0) THEN ANG=ANGTOUCH ELSE ANG=ANGTOUCH+Z1*(FLOAT(J) )**1 .6 END I F PRINT*,’ANG = ‘,ANG CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY) COEFF=COUL*O*Q*DN*DN/ (API*A0*A0 ) **2 PRINT 200, FORC*COEFF 200 FORMAT(/1X,’Fore ‘,E16.9//) 9 CONTINUE 99 C  END SUBROUTINE NZIIMRI (A0,RC,ANG,ANGTOUCHrALvErM,RESU,KEY) DIMENSION K(15),AA(15),V(15),X(2) DATA K(1),K(2) /1,2/ DATA TOL /1.E—15/  10 21  22  31  30  40  43 44 45  H=A0/4. KEY=O IF (M,LT.1.OR,AL.LT.1,5.OR,AL.GT.2.) RETURN IF (H.LE,0,,OR.H.GT.1.) RETURN EE=AMAX1 (E,TOL) MM=MINO(M, 15) L1 CC=2.*A0 U=0. KT=0 G=1000.*H/K(L) NN=1 ./G+.5 P=CC*G BA2=RC*COS (ANG )—A0 BA1=RC—A0 NN2=NN DO 30 I2=1,NN2 X(2)=BA2+P*( 12—.5) NN1=NN DO 30 I1=1,NN1 X(1)=BAI+P*(I1—.5) U=U+F(X,ANG,A0,RC) DO 40 1=1,2 LJ=U*P PRINT*, ‘U’ ,U V(L)=U IF (L—1) 43,43,44 AA(1)=V(1) L=L+1 GO TO 21 EN=K(L) DO 45 LL=2,L I=L+1—LL V(I)=V(I+1)+(V(I+1.)—V(I) )/( (EN/K(I))**2—1.) RESU=V(1) KEY=1 IF (ARS(RESU—AA(L—1)) ,LT.A8S(RESU*EE)) RETURN KEY=—1 IF (L.EQ.MM) RETURN  —  236  —  AA(L)=RESU L=L+1 K(L)=AL*K(L—1) GO TO 21 END C FUNCTION F(XrANG,AOrRC) DIMENSION X(2) Y0=X(1 )—X(2) IF (Y0.E0.0.) THEN F=0. ELSE Y1=SORT(AO*AO—(X(1 )—RC)*(X(1)—RC)) Y2=SORT(AO*A0—(X(2)—RC*COS(ANG) )**2) RO=RC*SIN (ANG) F=(—SGRT((R0—Y1+Y2)**2+Y0*yO)+SQRT( (R0+Y1+Y2)**2+Y0*YO) +SQRT((R0—Y1—y2)**2+yO*yO)—SQRT((Ro+yl—y2)**2+yo*yo Z ))/YO ENDIF RETURN END  A3.2.2 Program “OVER.ROMB” for Two Charged Disks with Overlapping This program requests the user to input the charged-disk radius, ion cyclotron radius, ion number in each disk, data points for the numerical computation, and number of iterations. The program will list instantaneous angles between the two disks (from 95% overlapping to 5% overlapping), and radial component of the instantaneous Coulomb force. C C C C C C  C 8  *  OVER.ROME  *  All smbo1s used in this roram are the same as those in roram ‘DISK.ROMB’ except: 10 = data points for calculation of overlap PARAMETER (0=1 .6021892E—19,COUL8,987551787E+09r + API=3. 14159265358979324) COMMON IFLAGvAO,RC,ANG PRINT*,’r = READ* , A0 PRINT*,’R READ*,RC PRINT*,’NO = READ*,DN ANGTOUCH=2 .*ASIN ( A0/RC) ,ANGTOUCH PRINT*,’ANGTOUCH = Special coefficient in Ronher numerical integration AL=1 .55 = ‘,AL PRINT*,’AL  —  237  —  Tolerance in Romber numerical interatiori E=1 .E—14 ‘E PRINT*,’E PRJNT*i’ READ*rM = PRINT*,’IO READ*, 10 Z1=ANGTGUCH*0.95/(FLOAT( 10) )**1 .6 liD 9 J=1,I0 ANG=Z1*(FLOAT(J) )**1 .6 ‘,ANG PRINT*,’ANG COEFF=COUL*Q*Q*DN*DN/ (API *A0*A0 ) **2  C  IFLAG=1 CALL NEIIMRI (A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY, IFLAG) FO1=RESU IFLAG=2 CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,pE,M,RESUpKEY,IFLAG) F02=RESU*2, IFLAG=3 CALL NDIMRI (A0,RC,ANG,ANGTDUCHALrEMRESUKEYr IFLAG) F03=RESU*2. IFLAG=4 CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY,IFLAG> F04=RESU*2, IFLAG=5 CALL NL’IMRI (AOpRCpANGpANGTOUCHpALpE,MpRESU,KEYp IFLAG) FO5RESU I FLAG =6 CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY,IFLAG) FO6=RESU FORC=COEFF* (FOl +F02+F03+F04+F05+F06) PRINT 200, FORC 200 FORMAT(/1X,’Force =‘,E16.9//) CONTINUE 9 99 END C SUEIROUTINE NDIMRI (AORCpANGANGTOUCH,ALpE,MpRESU,pKEY, IFLAG) + DIMENSION K(15),AA(15) V(15)X(2),C(2) ,D(2),F(2) DATA K(1),k(2) /1,2/ DATA TOL /1.E—15/ H=A0/16. = PRINT*,’H KEYO IF (M.LT.I..OR.AL.LT.1.5.OR.AL,GT.2,) RETURN IF (H.LE.0,..OR.H,GT,1.) RETURN EE=AMAX1 (ETOL) MH=MINO(M, 15> UA1=RC+A0 A1=RC—A0 (JA2=RC*COS (ANG ) +A0 BA2=RC*CDS(4N0)—A0 UA3= (RC*COS (0 • 5*ANG ) +StRT C A0*A0—RC*RC*S INC 0 • 5*ANG) + *SIN(0.5*ANG)))*COS(0.5*ANG) A3= C RC*COS (0 • 5*ANG ) —SORT (A0*A0—RC*RC*SIN( 0 • 5*ANG) + *SIN(0,5*ANG)))*COS(0.5*ANG) 10  L1  —238—  22  IF (IFLAG.LE.3) IF (IFLAG.EQ.4.OR.IFLAG.EE.6) IF (IFLAt3.EO,5) IF (IFLAG.EQ.1) IF (IFLAG.EO,2) IF (IFLAG.EO.3.OR, IFLAG.EO.6) IF (IFLAG.EO.4,OR,IFLAG.ED,5) U=0. KT=0 G=1000.*H/K(L) NN=1 ./G+.5 DO 22 1=1,2 F’(I)=C(I)*G  31  NN2=NN  21  IF IF IF IF IF IF IF  30 40  43 44  45  (IFLAO..LE,3) (IFLAO,EQ,4,OR. IFLAG.EQ.6) (IFLAG.EO,5) (IFLAG.EO.1) (IFLAG.EO.2) (IFLAG,EQ,3.OR.IFLAG.EQ.6) (IFLO,EQ,4.OR. IFLAG.EQ.5)  C(1)=U3—BA3 Ccl )=UA1—UA3 C(1)=EA3—EA1 C(2)=C(1) C(2)=UA2—UA3 C(2)=E3—12 C(2)=UA2—UA3  D(1)=UA3 EI(1)=BA1 D(2)=EA3 D(2)=U3 D(2)=I’A2 D(2)=UA3  DO 30 12=1,NN2 X(2)=DC2)+P(2)*(12—.5) NN1=NN tiD 30 11=1,NN1 X( 1 )=tI(1 )+P(1 )*(I1—.5) U=U+F(X,ANG,A0,RC,IFLAO) DO 40 1=1,2 U=U*P(I) PRINT*, ‘U’ U V(L)U IF (L—1) 43,43,44 AA(l)=V(1) L=L+1 GO TO 21 EN=K(L) £10 45 LL=2,L I=L+1—LL V(I)=V(I+1)+(V(I+1)—V(I))/((EN/K(I))**2—1,) RESU=V( 1) KEY=1 IF <ABS(RESU—A(L—1)).LT.AES(RESU*EE)) RETURN KEY=—1 IF (L.EQ,MM) RETURN AA(L)=RESU L=L+1 K(L)=AL*K(L—l) GO TO 21 E Nt’  C FUNCTION F(X,ANG,A0,RC,IFLAG) DIMENSION X(2) IF (IFLAG.EO,1) GOTO 71 IF (IFLAG.EQ,2,OR,IFLAG.EO.3) GOTO 72 IF (IFLG.GE.4) GOTO 73 71  Y0=X(1)—X(2) IF (Y0.EQ.0,> THEN F=0.  —  239  —  ELSE Y11=SORT(AO*AO—(X(1)—RC*COS(ANG) )**2) Y12=SORT(AO*AO—(X(1 )—RC)**2) Y21=SORT(AO*AO—(X(2)—RC*COS< ANG) )**2) Y22=SQRT(AO*AO—(X(2)—RC)**2) RO=RC*SIN(ANG) F=(—SORT((Y11+Y21)**2+YO*YO)+SQRT((RO+Y12+Y21)**2 + +YO*YO)+SORT((RO—Y11—Y22)**2+YO*YO)—SOF<T((Y12+ Y22)**2+YO*YO) )/YO + ENDIF GOTO 88 72  YO=X(1)—X(2) IF (YO.EO,O,) THEN F=O. ELSE Y11=SORT(AO*AO—(X(1)—RC*COS(ANG))**2> Y12=SQRT(AO*AO—(X( 1 )—RC)**2) Y21=SQRT(AO*AO—(X(2)—RC*COS(ANG) )**2) RO=RC*SIN(NG) F=(—SORT( (Y11+Y21 )**2+YO*YO)+SQRT( (RO+Y12+Y21 )**2 + +YO*YO)+SQRT((Y11—Y21)**2+YO*YO)—SQRT((RO+Y12— Y21)**2+YO*YO) )/YO + END IF GOTO 88  73  YOX(1)—X(2> IF (YO.EQ,O.) THEN F0. ELSE Y12=SORT(AO*AO—(X( 1 )—RC)**2) Y21=SQRT(AO*AO—(X(2)--RC*COS(ANG) )**2) RO=RC*SIN(ANG) F=(—SORT((RO—Y12+Y21)**2+YO*YO)+SDRT((RO+Y12+Y21) **2+YO*YO)+SORT((RO—Y12—Y21)**2+YO*YO)—SQRT((RO+ + +Y12—Y21)**2+YO*YO))/YO + EN III F RETURN EN P  88  —  A3.3  240  —  Results from the Double Romberg Numerical Computation The programs “DISK.ROMB” and “OVER.ROMB were used to calculate the  instantaneous Coulomb forces between two charged disks for disk radii of 1 mm, 0.5 mm, and 0.1 mm, respectively. The other parameters are: cyclotron radius of both disks cm, ion number in each disk  =  =  1  i0 ions. Both programs were run on a Nicolet 1180  minicomputer. Because its operating speed was not fast enough, only 9 points were calculated for overlapping disks; and for non-overlapping disks, 20 points for r’ 22 points for r  =  0.5 mm, and 26 points for r’  =  =  1 mm,  0.1 mm. All these data were input the  RIgor” program of a Macintosh microcomputer. Then, curve fitting was used to fit these data and the average radial Coulomb forces were solved.  A3.3. 1  Average Radial Coulomb Force for?  =  1 mm and r  1 cm  Two curve equations, Cl and C2, from the “Igor” program, conform well with the plot of the instantaneous radial Coulomb force, Fr vs. the angle between the two disks for r’ For  =  ‘  1 mm and r = 1 cm, as shown in Figure All.  (radian) from 0 to 0.200335:  Cl  =  1O_’  —  x (0.02833  —  3.985 c1  —  97.17  5 Ti÷ 8.516x10 1.373x10 5 t  —  +  9.908x  6 t); 1.867x10  (A3.1)  for CI (radian) from 0.200335 to it:  C2  =  1016  x (0.6161 + 5.758 e 2302  +  117.5 e 1725  ).  The average-over-one-cycle radial Coulomb force was evaluated from the integral  (A3.2)  —  —241  0.20035  <Frad  C1d+I0.200335 C2d)  0  =  6 (Newton) 1.311x10’  where the answer is very close to  <Frad>  (A3.3)  16 Newton in Fig. 5.8. The latter 1.310x10  =  value was calculated from the double Gaussian numerical computation for the same parameters.  A3.3.2 Average Radial Coulomb Force for r’  =  0.5 mm and r = 1 cm  Three curve equations, C3, C4, and C5, from the “Igor” program, conform well with the plot of the instantaneous radial Coulomb force, Frad vs. the angle between the two disks for r’  =  0.5 mm and r = 1 cm, as shown in Figure A3.2.  For 0 (radian) from 0 to 0.100042:  C3  =  1016  x  (— 0.1697  — 4.273x10 6 0  51.46 0  +  +  —  7 5.089x10  3 3.746x10  —  02 +  8 2.258x10  5 1.926x10  0);  (A3.4)  for Cl) (radian) from 0.100042 to 0.406 165:  C4  =  1016 x (2.147 + 19.44 e 7986  856.3 0 ” 5 e 1 7  +  <T);  (A3.5)  for ci) (radian) from 0.406165 to it:  C5  =  1016 x (0.5465  +  2.897 e 516  +  8.518 e 5880  ).  (A3.6)  —  242  —  10  8  Curve fitting Cl Curve fitting C2 o ad from Romberg numerical integration  r/r’= 10 r =1cm  —  r’=lmm Frad 6 (1O_16 N)  4  2  0 7t/4  0  cJ  Figure A3.1  3it/4  it/2  it  (radian)  Radial Coulomb force on Disk 1, a uniformly charged disk of m 1 ions, due to a uniformly charged disk of m 2 ions, as a function of the position of Disk 2. This position is characterized by the angle 1 (Fig. 5.1). The force is calculated from Eq. (5.13) if 0 <c1  <it,  where  touch  <  1’ <  touch  and from Eq. (5.29) if  is the touching angle (Eq. (5.5)), and  touch  is the  instantaneous angle between the two disks. The shape of the curve depends upon the ratio r/r’, the ratio of the cyclotron radius to the disk radius. For this figure this ratio equals 2. The ordinate gives the force in Newtons for r, the cyclotron radius number of m 2 ions,  =  =  1 cm, r’, the disk radius  =  1 mm, and N , the 2  . The average radial Coulomb force, <Fave> 4 i0  was calculated from Eq. (A3.3).  —  20  243  —  -  16-  r/r’= 20 r 1 cm r’ = 0.5 mm  “  —  —  h  12-  ad  Curve fitting C3 Curve fitting C4 Curve fitting C5 ad from Romberg numerical integration  N) 16 (10 8-  .  0  it/4  lt/2  3it/4  P (radian) Figure A3.2  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  r/r’ = 20. This figure was derived the same way as was Fig. A3. 1, except that r’ = 0.5 mm.  The average-over-one-cycle radial Coulomb force was evaluated from the integral  0.100042 r  0  =  0.406165  It  C3d+I0.100042 C4d÷J0.406165 C5d)  6 (Newton) 1.566x10  (A3.7)  —  244  where the answer is very close to  =  —  16 Newton in Fig. 5.9. The latter 1.564x10—  value was calculated from the double Gaussian numerical computation for the same parameters. A3.3.3 Average Radial Coulomb Force for r’  =  0.1 mm and r = 1 cm  Three curve equations, C6, C7, and C8, from the “Igor” program, conform well with the plot of the instantaneous radial Coulomb force, Frad vs. the angle between the two disks for r’  =  0.1 mm and r = 1 cm, as shown in Figure A3.3.  For ci) (radian) from 0 to 0.0200003: C6  =  l0_16  —  x (1.169  435.6  +  —  5 l.788x10  +  7 5.210x10  l.606x1O9);  (A3.8)  for ‘i) (radian) from 0.0200003 to 0.257698: C7  =  10_16 x (3.739  34.34 e 1445  +  for cJ) (radian) from 0.257698 to C8  =  1016  x (0.5556  381.3 e 9982  +  (A3.9)  t:  +  3.317 e 1635  +  11.44 e 7270  ).  (A3.l0)  The average-over-one-cycle radial Coulomb force was evaluated from the integral  0.0200003 ra  0  =  0.257698  C6d÷J  6 (Newton) 2.167x10’  0.0200003  d+J  0.257698  C8d)  (A3.ll)  —  where the answer is very close to  245  =  —  6 Newton in Fig. 5.10. The latter 2.175x10—’  value was calculated from the double Gaussian numerical computation for the same parameters.  100  -  80  rlr’= 100 r = 1 cm r’ = 0.1 mm  -  rad  60-  —  Curve fitting C6 Curve fitting C7 Curve fitting C8 ad from Romberg numerical integration  N) 16 (10 40-  20-  0-it/4 It 37/4 0 7t12 cJ (radian)  Figure A3.3  Radial Coulomb force on Disk 1 as a function of the position of Disk 2 for  r/r’ = 100. This figure was derived the same way as was Fig. A3.l, except that r’  =  0.1 mm.  246  —  A3.4  —  Curve Fitting for “Apparent Coulomb Distance” D vs. rir’  All the curve fitting equations for “apparent Coulomb distance” D vs. rir’ are obtained from “Igor” program on a Macintosh microcomputer. The following four equations in which the fitting coefficients are expressed to four significant figures give a good fit forD vs. nT’:  D  =  =  {  1.535 log(r/r’)  2.172  —  4 0.05836 [log(r/r)]  [2.341  —  r 0 7142  =  0.4310 1og(rIr  +  +  =  1.128 e  [0.5504  [  1.171 exp(—  +  1.292  }  +  0.5338 e  1.014 [log(r/r)]2  +  —  2.511 1o(rIr’)]  r.  (A3.12)  3 0.3827 [log(rlr’)1  r.  (A3.13)  1.54) —  e log(rlr’)  0.2177 3 exp(_ ) ] r.  (A3.14)  i  j  r.  (A3.15)  In Table V.4.1, the numerical components of D, Dr’, calculated from the above four equations, Eqs. (A3. 12)  —  (A3. 15), are listed together with the theoretical values of D’ in  Table 5.1.  D  =  r.  (A3.16)  Their quality of fit to the theoretical value of D’ is judged by their deviations:  Deviation (sum of squared errors)  =  1 [(D’)  —  (Dc’)theo  ]2  (A3.17)  —  247  —  where (DC’)fit is the calculated values from Eqs. (A3.12)  —  (A3.15) and (Dc’)theo is the  theoretical values in Table 5.1. From their deviations listed in Table A3.1, Eq. (A3.12) is the best fitting equation.  Table A3.1  Comparison of the values of D’ calculated from Eqs. (A3.12)  —  (A3.15) with theoretical values of Do’.  D’ from Eq. (A3.12)  D’ from Eq. (A3.13)  D’ from Eq. (A3.14)  from Eq. (A3.15)  rir’  Theoretical D’  2  1.792  1.792  1.792  1.792  1.739  3  1.629  1.630  1.632  1.634  1.614  4  1.539  1.538  1.540  1.539  1.537  5  1.478  1.477  1.478  1.476  1.482  8  1.371  1.370  1.370  1.369  1.379  10  1.327  1.327  1.327  1.328  1.336  15  1.257  1.258  1.259  1.261  1.266  20  1.215  1.215  1.217  1.217  1.221  25  1.183  1.184  1.187  1.185  1.188  50  1.102  1.100  1.103  1.096  1.100  100  1.030  1.030  1.033  1.034  1.026  5 1x10  5 3.3x10  5 8.5x10  3 3.3x10  Deviation (sum of squared errors)  — 248 — Appendixes of Chapter 6  A4 A4.l  Four Particular Integrals Applied in the Taylor’s Expansion Approximation of Ion-Ion Coulomb Interaction Most integral formulae used to calculate the value of the Columb interaction  coefficient, C, have been taken from “Table of Integrals, Series, and 1 Products”( and ) “Integrals and 2 Series”( ) . The following four particular integral formulae which will be used to integrate the value of C have been developed as follows.  (1)  J  in (a+2+) dx, where the constants a >0, b >0, and b > a 2 . 2  +x 2 Lettinga+J  =  J J  =  u,x  =  —b 2 ‘I(u—a) .  ln(a+b2÷x2)  — b 2 lnudg(u—a) 2  =  —  =  — a) —b 2 2  2 in —b 2 a)  in u  —  J  — b 2 2 — 2a u+ a 2 u  du  — b 2 — 2a u + a 2 u 2  —  — 2 =(u—a) i nu— b  u  Iu2_ 2a  I ?—  J  u +  du  — b 2 a 2  U  2a u + a — b 2 2  4u  —  =  —b mu g(u—a) 2 —ainu +  —  2 —a 2 ‘1b  a  + u2_  —b (u—a) 2 +  2a  u +  — 2 u + a  2 b ‘  mu  2 arcsin —a 2 gb  — 2 ‘J4 a  —  a  2 b  +  + u2_  a  —  2a  u + a — 2  b21  integral constant.  4 (a — b 2 ) 2  +ainI  —b J(u—a) 2  —  —  — b21+2ainu 2 a  2 2au+2(a  arcsin u  =  2a  — 2 —‘Ju  249  — 2 u +a  2 b  u  + J(u_a)2_  a  —  2 b  integrat constant  +  =xln(a+b2+x2)_x+ainx+Jb2+ x 2  +  gb 2 a 2 arcsin  —  b The integral  J  (2)  =  =  in  =  (a  +  2+ X ‘J b  +  integral  )  2 + x 2 ‘Jb  constant.  (A4. 1)  0.  (a + a2+x2) dx, where the constant a >0. =  u,x  =  — 2 q(u—a)  . 2 a  —a 2 a) 2  am  ln(a÷ga2÷x2)  — a 2 inud(u—a) 2  g(u  +  a  —  formula (A4. 1) is valid for x  ÷x 2 Lettinga+ga  I I  2 b  —  —  —a 2 a) 2 in  u  —  (u  —  +  Iu  —  a  +  a) a2 — ‘J (u 2 —  integral constant  x in (a  +  Ia x2) + 2  —  x  +  a in  —  x  +  J a+  +  integral constant.  (A4.2)  —  J  (3)  =  {  =  =  ,x 2 u  2 ln(a+u)du  uin(a+u)  =  =  —  +  }  2 2 +a 2 u—a du a+u  ln(a+u) 2 u  du  u_a+a —  t?ln(a+u) 2 (b  ) 2 +x  2 in (a + a  {  =  u du  a —  (  { { {  . 2 — b 2 4u  =  x(a÷gb2+x2) dx  2 =iuin(a+u)  =  —  x(a+b2+)dx,wheretheconstantsa>O,b>O.  Letting b -f x 2 2  I I  250  (b  —  —  (u_a)2_a2ln(a+u)} +integralconstant. + x2)  in (a + + x2)  2+x a ) in (a 2  integral constant.  }  + integral  +  (qb2  —  +  —  2 a)  constant  + x2)  —  (gt2  +  2 x  —  2 a)  } (A4.3)  —251—  (4)  I  b  b  I  • x arcsin  (  • x arcsin  2 b b  =  • xarcsin  •  a  ab2+ X +  2 + x 2 /b  a + Jb 2 ÷ x 2  (  +  a  +  ÷ x 2 Ib 2  ) •  )  a b2+ X +  2 + x 2 ‘Jb  — a 2 +b 2  )  +  b (a  +  ÷ x 2 Jb 2  Jb2_a2J  )  a b2+ X  2 + x 2 b  )  xdx b+ x 2  +  )  .Ib2_a2J +x d(b ) 2 2 b2+x2(a+b2+x2)  b2+aJb2+x2  +  +  2 (a + x 2 gb  b(a+b2+x2)  2 b  b2+aVb2+x2 b (a  —I  +aJb2+x2 2 b  2 b(a+Jb2÷x • xarcsin  th  xd[arcsm  =xarcsin  =  dx,wherea>O,b>O,and b>a.  )  +ab2+x2 2 b b (a  =  (  +  +aJb b + 2 x  arcsin  =  2 b  arcsin  +x 2 dgb (a÷b2÷x2  —a in [a+gb +b 2 + x 2 2  )  The integral formula (A4.4) is valid for x  1.  ) (A4.4)  0.  All of these four integral formulae have been verified by two ways: either by taking the differential of the four antiderivatives or by the Romberg numerical integration of single  ) 3 variation.(  —  252  —  Integrations of the Coulomb Interaction Coefficient  A4.2  C (Eq.  In Section 6.2.2, the integral form of the Coulomb interaction coefficient (6.23)) was found to be  kqq 2 16 a 4 0  — — —  0 a  i  J-0  0 a  0 a  0 a  fff  1/2  1/2  £112 £112  { x  When  2 (y—y)  +  —z 1 (z ) 2  2 (xj_4)  +  2 (y-y)  A4.2.1  =  4, y  =  2 (xj—xj) 2  + (zl_z2)2  j512  2 d 1 dz d yjdydxjd4. z  this multiple integral is solved, the singular points  and right limits for  —  1 y, and z  =  can  (A4.5)  be deleted by taking the left  . 2 z  and 4  Integration for  For convenience, x and  4 are integrated first.  The integration order of (A4.5) has  been changed to 0 a  f  1/2  a0  1/2  J  Lao £1/2 £1/2  2 (yj—y)  [ 0 a  =  ‘a0  fa0  +1  =  2+ x  d 1 dxd4dz d 2 ydy z j512 —z 1 (z ) 2+ 2 (y—y)  112  1/2  £112  2  ‘a0  L f L1 J  £  ‘  0  {  =  —z ]3/2 1 (z ) 2+ 2 [ (44)2 + (Yj—Y)  x-4  +  [  (x—4) +  £  + (zl_z2)2_ 2(x4)2  (x—4) +  0 a  —C 2 1 X  lim(f  2 (yj—y)  +  —z 1 (z ) 2  j3/2  2 d 1 dz d ydy z  I Xi  =  X2 +  a  -  —  112 =  11/2 11/2 +  a0  1/2  Lo  a0  253  a0  f  lim £  o  —2£  [  e2 +  2 (a +x) 0  +4)2 0 [(a  —z ]3/2 1 (z ) 2+ 2 (YjY)  +  —  2+ (y—y)  —z ]3/2 1 (z 2 )  (A4.6)  d 1 4dz d ydy z } 2  where £ is a small variant for the left and right limits of  —,  x. The first integrand of Eq.  (A4.6) is obviously zero for either singular points or non-singular points. Only the second integrand is needed in order to integrate further, and singular points 4 and z 1  =  ao  1/2  1/2  ao  J—a + e f Lo 11/2 0 £112  Fa0  —2  , yj 0 a  =  y,  2 (a d 1 d4dz d ydy z +xj) 2 0  0 J-a  0 a  1/2  f  [(a  11/2 £112  {  The singular points y  2 (yj—y)  +  —z 1 (z ) 2  1  =  d 1 th d 2 ydy. z  2 must be deleted from the second integrand in Eq. 1=z y, and z  A4.2.2 Integration for z 1 and z 2  ao  Lf  (A4.7)  —z 1 (z ) 2+ 2 (YjY)  (A4.7) in the following integration for z ,z 1 , y, and y. 2  ao  ]3/2  4a —z 1 (z ) + (yjy) 2 2+ 2  0 e2 +  £  +4)2 +  1/2  -lim  —2  —  2 must be deleted. z  urn  =  =  1/2  1/2  11/2  11/2  {  4  + 2 0 a  —z 1 (z ) 2+ 2 (yj—y)  —  254  —  1  d 1 dz d 2 ydy z  —z 1 (z ) + 2 2 (Yj—Y)  = —2  ‘a0  1/2  0 J-a  -a £12  in  —z 1 (z ) 2  +  4 a 2 0  {  +  2 (Yj—Y)  +  —z 1 (z ) 2  1/2  —1n 2 —z 1 (z )  0 a  0 pa  = —2  -  —  (1/2  (1/2  in  + in  —  =  f  —21  +  ) 2 z  +  +  ) 2 z  1/2  +  {  £/2  1n  (1/2  ) 2 z  +  4 a + 2 0  2 (yj-y)  + Z2) +  (1/2  0 a  JL  dydy 2 th  1/2  12 { f f  1  —1n  2 (y—y)  +  + 2 _ 1 (z ) 2 z  2 (yj—y)  +  2 (yj—y)  2in (1/2  + 4a 2+ 0  +  (1/2  +  (l/2  +  +  ) 2 z  ) 2 z  + z2)2  (1/2  ) 2 z  +  + (1/2 2 (yj—y)  +  ) 2 z  + 4a 2+ 0  }  dydy 2 dz  + 2 (yj—y)  (1/2  +  ) 2 z  21  +2  in  Iy—y I  }  dydy 2 dz  —  fao =  _2f°  —2  =  44 a 2+ 0  2  ln (l/2+z ) 2  +  2 (yj—y)  (l/2  +  ) 2 z  ) +2 z 2 z 2  fao 2  2 lin[l+[i2  2 + (yj-y)  J’7° {  —lin[4a?+  +  4  +  }  +  1,  + (yj_y)2]  dy dy  +  4 0 a 2 + (yj—y)  + 2 +24a 2 (y—y) 2 0 (yj—y)  +2lln  Iy—yI  developed for integration of y  the first four integrands, which contain  I  4 0 a 2 + (yj—y)  _y)2I_2lin[l+l2  I  ) 2 z  dydy  4 0 a 2 + 2 (yj—y) ]_2Jl2  (y_y)2j+2l2  are  1/2  +  j  2 (y—y)  + (yj—y) 2 0 a 2 ]_2l2 +  _y)2J_2(y;_y)2  where eight integrands  I y—y I  2 +21 in I y-y (yj-y)  2 un {l+/l2  _2lin[l+l2  +  in  —lln[4a2+  -2  4 a 2+ 0  }  +  +2  [  1/2  +2 [_y)2 + (1/2  + (yj—y) 2 0 a 2  2 in z  ) 2 z  ) 2 z  {  —  + (1/2 2 (yj—yj)  + q(y_y)2 + (1/2 +  +  fao jao 2  —  [ (l,2+z )+’!4 a 2 + 2 0  in (1/2 —2Ufl+z ) 2  +244  =  {  255  and  ) y.  dydy  It  (A4.8)  is worth noting that  the z-iength of the tetragonal ion cloud, in  square roots, represent the Coulomb interaction for separation  1 in  the z-direction;  their  and the  —  256  —  last four integrands, which do not contain 1 in their square roots, represent the Coulomb interaction in x-y plane. These eight integrands will be integrated methodically in eight steps, as shown in A4.2.3 following.  A4.2.3 Integration for y and y (1) The integral formulae (A4.1), (A4.3), and (A4.4) have been used to evaluate the following integral.  fo  2l1nl+\Il2÷4aQ2+ (y1_y)2dy’dyt  2  =  =  —41  {f° f  +  4 a 2 + (yj—y) 0 2  I dydy  4°  in [1  +1 f  2+ (y_y)2] dydy 0 ln[l+l2+4a  +  in [1 +  J 12  J  [  —4!  { 2L  + in [1 + i 12  =  ln [1 + Jl2  —8 1  +  J { f°  +  l2  ln  +  +  4 a 2 + (y—yj) 0 2  4 a 2 + (y j—y) 2  .f + 2 1  4 a + 2 0  +  I dy dy }  (yj+y)2]  4 a o2 + (y jy)2] dy{dy  in [1 + g12  I dydy  }  4 a 2 + (yj+y)2] dy 0  -‘  +  a,  ci  a-SI  oj +  IC  sf-I  t’3  1.  0  -  ‘—I  ‘-I  I  ‘1  +  -  +  I  1’31  I +1  0  a—’  +1  I °I  •  +1  I  0+  00  :I  +  ::‘  +  I+I  01+1  0  I0  III  +iI I4I  •.•1.31  +  0  .I+ 4  )  I  II  t.3I  +  I  1.3  C  I  %_-‘  --I  -  I I  —a.’  II  +  0  II  +I+ 1.3  +  .)  +  +I) I  -i  i-a -  a-’  +1  oj  •  +  -  I  S....-.  I  I  I  -a-i  1  .  I I  I  0  L.J  1.3  -  ‘.  +  0  +  +  I  .  i  +  I  —  ‘.  t’.)  -  ,—I  I  +1  I I’.) I  +  +  +I  +  ‘—  00  t..  1.  +  +  t)  +  I  +  . -a- , — a, •— -1  +  I  i  I  I  -  -a.  I  +  ±  1.3  +  _‘-I  -I±i -  1.31  +  I I—’’ I —  a__I  +  0  +  +I  0  —-., a, I  I  ‘ ci +1  +  +  —-  I  00  3I  +  S  -  i  -I  +1  a-.,  -.  +1  o  I  I o  I +i  +  ‘  I  —I  _II  i-a-I  a-SI  +1  I  I + I  +l  -‘  +  +  U  +1 I oI + I .  +  -a’  +  +  +  ‘  1+1  I—i  131  ‘—I  +1  4I I 01.) I  1.31  I’13I  I + I I13-I  o+  I  Io  —I  oI  •  I I Ici  +1 i4.iI  ;i+I I,-j II  + II  I  +1 I  4I  I  -  +  +  ‘—‘  -  ‘•1  J  +1  icI 131 I  +  +  r11.3  0  I  +  I1  +  I  +1  3I  +  —.  r—I  +  0  C  I  131  •_.  +1  0131  +1  1.31  -  +  +  cL  II  -  +  ,  I  I  .  —  +  0  +  131  13’  I I I I13-I  io cli’  ci 131 i+I  +  1..)  +  ..  0  +  I ) 0  I  1.—i  I 131  9!  +1  + I  I 1.31  9 +  +  I  -  0  131  ci  01 I + I  I  4I  +  i  +  00  .  131 -  ‘-1::)  I I  -  +  1.3+  +  ol  +1  .d&  +  .  1.3  I+  t’)I I  +  a  0  I  +1  I  +1  .  i  +  +  i  13  13’  13-I  0  +1  .irJ  +  i  0)  +  .,  I  00  .-  I  I I  )  I  ‘I1.31  ‘i+I  L-I I I  I  +1+1 II  131  -I-  .  +  H  +  t.)  0  +  I  1.3-I ‘—‘I 1.31  +1  +  +  .  —  i  +  I—i  1.3-I  çI  +1  +  +  I  00  ‘  +  cci  I  l—  I  I.  3  +1  +  cxi o,J  C)I  uI  +1  +1  ..I  +  ;i  +  C  )  C  I 1’3IU  +  1%)  +1  •  .-‘  C  I  I t’.)  C)I  ci o  +  +  I  +  4I’•.  i_i  t31  ±1 ,I  t.31 ±1  +  h) 0  I  1 t3  VI CI  +  -  +  ••  t)  —  ...  co  II  t•3 -  +  t’) 0  +  t)  +  + +  t’)  t’) t’3  -  +  C  +  +  C  IJ  C  t%)  +  F-’  +  C  1  -  -  t’.)  -  +  -a  -I-  ?1  .—  ‘—J  +  Ct)  .  + I)I’—’  I  C  +  F-’  C  +  •  -.-  ‘.3  +  C  .  + +  t’.)  I.-.  Cl)  p  +  o  C  I  01.3  +  -  +  +  +  +  o  ‘•‘  +  C  +  C  i—i  I  t’.)  0  F-’  -I  0  +  +  E  I—J  %_  3-  C  s’  +  C)  II  ‘-  %—  0  +  c9  I  +  +  ••  Ct)  +  1)  ‘.3  t)  1’.)  -  +  0  +  1.3  +  C’.)  +  .1  —  12 —4aarcsin  +  4 j2 [1 + J  + 2 0 4 a  4—;-;:a  1  +  i  12 + 4 a + I 2 0  +2aarcsin  260  —  8  +  Ji +  a  0  2 _4ao21n[l+Ji2  J  8 a 2 0  5 a 2  ]  4i2÷4ao2 [j+Jl2+5ao2  0 4 2 1n a [  i+4  5  a]—  [+  2 0 a  —2aln[  l+Jl2÷ 4  a2]+  [[i-  4 a 2 0  a + [12+ 5 a 1 in[ 0 0 +a 2 0  —2  a arcsin  1—i[+  +  2 0 8a  ]  +4afl[l+’[i+5(bO2l  _112  j)2  5 ao2+h[i+ 4 a 2 0  l2+4a2+j)j2+5a2 0 —______  __  [i+4ao [l+4ll2+5ao2  I  2] 0 —4a in[i+[i2+ 5a ]+4aç ln[l+\[12+ 4 a 2 _a 2 0 )+ olln 2 (1 lt 2+ ao 4a  }  (_ 2 +4 6 ao a l + —41 0 0 1n [2a 8 a  arcsin  —8 a 2 0  —2 a 0  ]_ifi+  0 12 + 2 +4a lIl2+8ao2  8  a+l[+ 4 a 2 0  0 — I n +4 [ a l+\f4ao2I  J+4a [l+Il2÷8ao2]  l in (12+4 a) + 4it a  }  (A4.9)  faojao  (2)  _2)  3  —2  + 4 a 2 + (y jY 0  dydy  (ao’ =  2)  0 -a  1.  4 a + (yj— 0 2 y)  2 (yj Y)  ]}  lao  dy  —261—  =  2f  -i-yj) !l2 0 { 2 (a  +  4 a + 2 2 0 +y) 0 (a  +ln +(1 ) 2 0 4a 2 +)+l 0 [(a _(l2 + 4 a ) in 2 0  =  =  2f  gi2 +  } dy  12  +yj) -[l + + 0 j 2 4a (a  +  40 -i-) 0 ) 2 a t-yj) (a in {(a  —2 (l 2  +  4a ) 2 0  2  12 (l2  t  -  +  8a ) 2 0  3t2  2 —  +  4 a 2+ 2 0 ÷y) 0 (a  +  4i2  4 a 2+ 2 0 +y) 0 (a 2 +4 a (j ) 2 0  ( +4a + ) 2 0 in 4a l2 +g1 0 {2a 2)[l2 0 +2(l2+4a  +  I  3/2  2 2 (j  gi2 +  } dy  4 a + 2 2 +y) 0 (a  —(1 ) + ) 2 0 ln(l 4a  +  =  +yj) 0 { 2 (a  +  t  ]  -i-yj) + Ii2 + 4 a 0 + 2 2 +y) j 0 (a [— (a  j +4 a +2 2 ) in {(a 2 0 +) 0  2  4 a( + 2 2 +y) 0 (a  +  +  +  —  4 a + 2 2 0 ÷yj) 0 (a  I  (12+4 a ) in (12+4 a) y 2 0  }  3/2  8 ao2]_2(l2+4ao2)Il2  4 a 2 0 0 ( —2a + ) 2 4a l in (12 +4a ) 2 0  +  }.  8 a 2 0 (A4.1O)  (3) The integral formulae (A4.2), (A4.3), and (A4.4) have been used to evaluate the following integral.  _2J  f  —2lln 1  +  12  +  2 (yj—y)  dydy  —  =  =  41f  f  if {  in[l+Ji2  =  4lf +  2 ;-yin[i+l  =  4  =  if { 2 (a +y) in [i 0  4i  +  {  -i-y) in [i 0 (a 2  41  +  +  +  Jl2  +  + i2 +  +  j  —  un  +) + .\1i2 + 2 0 -f-y) I 0 (a [— (a  +y) 0 (a 2  I  —  dyj  [i2 +  +y) —1 0 (a 2  +  j2 —  0y 2a  +y) 0 (a ] 2  }  a 2 j_[2i2+4a 2 4 0 _2l4i  lin{2a —4a+4a + 0 2 4i  } dy  ]— 2a  1 —21 in i }  -f.y) 0 (a 2  +y) —2l1n(l) 0 (a 2  [ 4ao21n[l÷.Jl2  dy  + (a ] 2 0 —2a y)  +y) 0 (a 2  2+ Ji  2i (a +yj) in [(a 0 +) 0  _2ll2  =  +  +  -t-y) 0 (a 2  + Ji2  }  ] 2 (yjy)  + Ji2 +  +21 in [(a +) 0  ]--y) 2 (yj-yj)  +  +y)in[l+4i 0 2(a {2  Un [(a +y) 0  —  ] dydy 2 (yj—y)  +  +  262  4 a 2 2 ]_2ii 0  +  +  4 a 2 0  ]  4 a 2 0 0 — 2 +2l 1 1n1 4a  }  2+l2 0 =4i{4ao2in[l+l2+4ao2]_6a_ll2+4a lin 2 0 +4a -t-Jl 0 [2a  +  4 a linl 0 2 ]—4a 0  }  (A4.11)  —  0 pa  (4)  —2)  0 pa =  —2)  0 pa =  —2)  0 -a  _l2 in  =  —2  J  2 2(l  {  +))!l2 0 2(a  1  —2  {  +  + g2  +) 0 (a  2 (a +) Il2 0  +2 12 in [(a +) 0 =  +  {  0 ‘—a  —  pa  (y’—y’)  [—  263  [12 +  +  2 dydy (yj—y)  2 (YY)  + j2  in [(y;—y)  4  +  +y) 0 (a 2  +  4 a + 2 0 +y) 0 (a  .Jl2 + 2 -t-y) 0 (a  +yj)2j +2 0 (a 3  j2  ] 2 (yj—y)  +l2in[(a 2 (aO+y) + 0 ))+l2 +  +  a+  + gi2 +  ]}  }  I  lao  dy  -i-y) 0 (a j 2  dy  ] —2 j2 in l }  dy  +y) in [(a 0 (a +y) + gi2 + 0  +y) 0 (a 2  ]  =  —2  {  _l3+4aol2in[2ao+/l2 + 4 ao2J_2l2,/l2 + 4 a 2 / 3 (l2+4ao2) 2 0  — 3 +2l l 2 l 0 nl 4a  (5)  =  —21v—a 0  —2)  0 -a  {  J  0 —a  }.  (A4.12)  + (Yj—Y) 2 0 2\J4a 2 dydy  (y—y)  \I 4 a(2 +  (y j—y) 2  —  +  =  4ao2ln [j-y)  —2  J  {  +  =  _2J +  =  —2  {  }  +  g4  +  in 2 +y) + 4 a 0 (a 2  +y) J 0 (a +2 2 0 a  [4  +  4a 0+ 2 -fy) 0 (a  —2  {  +yj)2] 0 a+ (a  —2  {  +  8a 2 (a 0 +yj) in 0  =  21  =  21  }  dyj  +) + ‘4 a 0 [(a -- 2 0 ÷y) I 0 (a  }  }  3 [16qa 3 3 0 ]+16a _8a 0 +16a 1n(1+1)—16’Ja 3  ff f° f f {  (6) -2  dy  3 [16qa 3 3 0 ]_1oa _8a 0 )+16a 0 ln(2a 3 i-2Ja 1n(2a )  +16a —16Ja 3 0 =  }  +yj) + J4 a 0 2+ 2 0 ÷y) I 0 (a [ (a  2 in (2 a 0 ) 0 j —8 a  4 —8a + + — i 2 ) 0 n(2a y y) (a 8a a  =  dy  +yj)4a 2(a 2 0 + 2 +y) 0 (a  8a 2 in [(a 0 +y) 0  {  —  4 a2 + (yj-y) ] 2  2 (a +) 4 a 0 2  —4a1n [—(a -l-)) 0  264  2 -1 in [4 a + (y—y)  I dydy  In [4 a 2 + (yj—y) 0 ] dydy 2  (yj-y) in [4 a + (yj-y) 2  j  -  2(y-y)  }.  (A4.13)  —  265  —  lao  cnY2J 0 +4a dy 1  0 2a  =  21  J {  =  21  +2 2 0 +y) 0 (a ] { [4a  2 (a +) in [4 a + 2 0 ÷yj) 0 (a ]  y+8a 0 —4a  0 a  =  21  -a -a 0  0 pa  =  -a0-ao  I y—y I  2 ) 0 (y+a  {  2  =  Tao.  32  (A4.14)  dy dy  2 -ao L {- *P 0 a0 f {  =  i  a0  0 j’a  J J  2  +y) 0 (a ] 2  _2((yj_y)2 dydy  a0  =  dy  0 pa  J  (7) _2J  }  }  )_12ao2÷47ao2} 0 8ao21n(2a  0 pa  4  + 2 0 a  +yj 0 a  ( 2a 0  1 2 arctan (ao+Y2)82th[1(ao+Y2)  2l{ 8a 1n(8a)—4a  =  4a + 8 a 0 arctan  in [4a +2 2 0 + 0 (a j — y) [4  =  {  —  3 ) 0 (y+a  +  -  +  0 a  2 ) 0 (yj-a  (y—a  2 (yj-y)  +  }  }  J  dy  dy  Ia  (A4.15)  —  fQ (8) —2  f  —41  f f  =  =  =  {  _4lf  { —41 { —41  =  —  in  I y—y I  I yj—y 2I  —  dy dy  dy dy  { -y)1nIy-yI  =4lf  =  2 1 in  266  -(y-y)  2(a + 0 y)—2a yj)in(a  +y) in (a 0 (a 2 +y) 0  —  }  +y) 0 (a 2  2 0 in(2a—2a —4a 4a 2 0  —  }  dy  dy  }  0 2a  }  16 a 1 in (2 a) +24 a 2 1. 0  (A4. 16)  All of these eight integrals are verified by Romberg numerical integration of multiple ) 4 variabies.(  A4.2.4. Value of the Coulomb Interaction Coefficient, C Eqs. (A4.9), (A4.1O), (A4.1 1), and (A4.12) represent the Coulomb interaction for  separation 1. Their sum is  —2  f J {  2 tin [i + 412  +  4 a 2+ 0  2 1—2 .4i2 (yj—y)  _y)2j 1 _2lln[l+gl2+(y  }  dydy  +  4 a 2 + (y—y) 0 2  —  =  —  l2+ 8 a ln[2a + 2+4a 0 0 {_6a —41 1 2 0  [  2 0 Il2+4a  } { +2  Il2 [2a + l0 4a)ln 2 ( 0 +4a + 4ao +/l2 2 +2(l )  +  +  ) _(l2+4a 3 (l2+8a) 2 0 2 /  1_2(l2+4a)gl2 2 8 ao  l 4a) 4a 4 a ln(l ( —2a + ) 2 0 2 + 0  41{ 4aln{l+/l2  +  4 0 ]_6a _lil2 2 a  +  4 a 2 0  Il2 ln[2a 2 l 0 +4a +  +  12 ]_2l 2 0 a 4 i  8 a 2 0  }  l3 (l2+4a) _ 3  —2  +  +  4 a 2 +12 0  +  } {  1n1 ]—4a 1 0  l2 [2a + 1n 0 +4a 1 0  {  a 2 +li,il2+ 4 a 8 0 2 0  ] +4aln[j+fr+4a 2 0  ]  2 0 l+.q12+8a  ln(+4a)÷4ia 0 —2a 2  4  ]_i4i2+  2+l,Il2+8a 0 l2+4a 2  rcsin —8a a 2 0  =  267  4 a 4a n1 — 3 +21 1 2 1 2 0 0  }  ) +l2+ 8 2 ]+a l2ln[2a 0 _2a 2 2 a 1 0 21n(12+4a  2+hfl2+8a 0 l2+4a 2 larcsm 2 +8a  2 0 Jl2+4a  [  2 0 l+Jl2+8a  ]  l 2 —4ita  +  +  4a)_4a44l2 _4aln(l + 2 Jl2 ln[2a 2 l 0 +2a +  +  +  2 +4a,/l2 0 8 a  1_2aol2lnl 2 0 4 a  +  8 a ] 2 0  4 a 2 0  }.  (A4.17)  10 m 3 For 1=2.4 cm and a 0 = 1 mm, the analytical solution of Eq. (A4.17) is 6.65923x10  and the numerical solution of the sum of Eq. (A4.9), Eq. (A4.1O), Eq. (A4.l 1), and Eq. . 3 )is 6.65892x1(r’° m 4 (A4.12) from the Romberg multiple integration( Eqs. (A4.13), (A4.14), (A4.15), and (A4.16) represent the Coulomb interaction in x y plane. Their sum is  —  jao  f  {  —2  {  —  2+ (yj-y) 0 2J4a 2 -lin[4a÷  +21 in I yj—y  =  268  I  }  dy dy  3_8a3]+1oa 0 0 [16qa ÷16a n(1+q)—16qa 3  }  + 16 a 1 2 0 2 un (2 a 0 )— 24 al+ 87t al+a?— l6alln (2 a0) + 24 a 0  }  =  4  {  =  4  3 + 2it a 0 21 0 (— 5.9464 a  16J  [— 3 0 a  For 1=2.4 cm and a 0  =  (A4.18)  3 1 mm, the analytical solution of Eq. (A4.18) is 5.79400x i- m  and the numerical solution of the sum of Eq. (A4.13), Eq. (A4.14), Eq. (A4.15), and Eq. )is 5.79079x10 4 7m . 3 (A4.16) from the Romberg multiple integration( Substituting Eqs. (A4.17) and (A4.18) into Eq. (A4.5),  c= 4’”72  {  2+l2+ 5.9464ao3_2ta P 2 0 ln[2a l+2a 8 2  2  2 0 q12+4a  [  1n(1 1 —a + ) 2 0 4a —8 a larcsin 2 0  _.  21 0 a 2  2 0 l+sJl2+8a  (j2 3 2) 0 0 8a + 2 (l2+4a _ 2 / 3 3 ÷ l3_8a ln[2a .\Jl2 ) + 8  2 0 +4a)+4a\Il + 8 a +4aln(l 2  _4astqIl2  +l2 + 4 aj +2a 0 1n1 2 1 0 1n [2a 2 1 0 —2a  }.  ]  1 2 0 +4ica  21 0 a  + 4 a 2 0  (A4.19)  —  269  —  Reference  1.  Gradshteyn, I. S.; Ryzhik, I. M. “Tables of Integrals, Series and Products”; Academic Press: New York, 1965.  2.  Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, 0. I. “Integrals and Series”; Gordon and Breach Science Publishers: New York, 1986.  3.  Yakowitz, S.; Szidarovszky, F. “An Introduction to Numerical Computations”; Macmillan: New York, 1989; pp 216-221.  4.  Davis, P. J.; Rabinowitz, P. “Methods of Numerical Integration”; Academic Press: Orlando, 1984; pp 506-508.  

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