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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 OFCHROMIUM NITROSYL COMPLEX CpCr(NO)2H3,AND COULOMB INTERACTION BETWEEN IONS INFOURIER TRANSFORM ION CYCLOTRON RESONANCEMASS SPECTROMETRYbySHU-PING CHENB. Sc., Xiamen University, 1977M. Sc., Xiamen University, 1982A THESIS SUBMflTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of ChemistryWe accept this thesis as conformingto the reonired standardTHE UNiVERSITY OF BRiTISH COLUMBIAFebruary, 1992© Shu-Ping Chen, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives, It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ChemistryThe University of British ColumbiaVancouver, CanadaDate April, 1992DE-6 (2)88)—“ABSTRACTThis work is devoted to application and performance modifications of Fouriertransform ion cyclotron resonance mass spectrometry (FT-ICR): (1) gas-phase chemistryof chromium nitrosyl; (2) Coulomb interactions between ions in ion cyclotron motion.Chromium nitrosyl CpCr(NO)H3(Cp = ?7-C5H)produces a series of ions whichhas been observed to fifth kinetic order. The ions of CpCr(NO)2H3show many productsin which the oxygen of the NO ligand is retained and tl nitrogen is lost as part of a neutralproduct. An empirical method was proposed for calibrating nominal pressures of transitionmetal complexes to determine real rate constants. The Cr ions in an excited state can bequenched in collision with molecules N2,H2,H20, NH3, and CH4. The ground state Crions prefer charge transfer reactions which result in different products from those of thecondensation reactions of the excited state Cr ions. H2,H2O, NH3, and CH4 also canreact with the nitrosyl ligand in CpCr(NO)2H3to produce the ammine ligand.A point model and a line model, which correspond more closely to physical realitythan some prior models, are proposed to account for the Coulomb-induced frequency shiftsobserved in FT-ICR. The first model consists of two point charges which undergocyclotron orbits with the same orbit centers at their respective frequencies. The modelpredicts that each excited cyclotron motion should induce a negative frequency shift in theother’s cyclotron motion. The line model, created by extension of the point model, givesrise to a position-dependent frequency shift which is synonymous with inhomogeneousCoulomb broadening.A disk model for the Coulomb shifting, unlike the point model, has a finite averageradial Coulomb force. It consists of a uniformly charged disk, whose excited cyclotronmotion is perturbed by a second excited, uniformly charged disk. The average radial forceis found to be a function of ratio of the cyclotron radius to the disk radius. This allowscharacterization in terms of an “apparent Coulomb distance”. This distance, when applied-ifiin a charged-cylinder model, accounts for Coulomb-induced line broadening and frequencyshifting. The charged-cylinder model agrees with experiments. Absolute mass calibrationof FT-ICR spectra is enhanced by the Coulomb correction.The charged-point and charged-disk models are valid when the Coulomb interactionis much smaller than the Lorentz force. When the Coulomb force is comparable to theLorentz force, a strong coupling interaction arises. A strong coupling Coulomb interactionfor small spatial separations between two ion species is developed using a Taylor’sexpansion method based on two tetragonal ion clouds. Under strong coupling, the twoionmass peaks will merge.—iv—TABLE OF CONTENTSAbstract iiList of Tables xiiList of Figures xivList of Abbreviations xviiiAcknowledgements XXChapter 1 General Introduction 11.1 A brief history of ion cyclotron resonance mass spectrometry 21.2 The background of Fourier transform ion cyclotron resonancemass spectrometry 51.2.1 Basic operating principle of FT-ICR 51.2.2 Fourier transform of a time signal 81.3 Ion-molecule reactions and FT-ICR multiple resonance 121.4 Performance of FT-ICR 13References 16Chapter 2 Ion-Molecule Condensation Chemistry of(75-2,4-Cyclopentadien-1-yl)methyldinitrosylchromiumCpCr(NO)H3 212.1 Introduction 222.2 Experimental section 252.2.1 Experimental hardware 252.2.2 Multiple resonance technique 262.2.3 Pressure calibration 272.3 Condensation chemistry of the positive ions 322.3.1 Cr reactions 37—v—2.3.1.1 Charge transfer and electrophilicadditions of Cr 372.3.1.2 CpCrO reactions 392.3.1.3 CpCr2O reactions 412.3.2 CpCr reactions 422.3.2.1 Charge transfer and electrophilicadditions of CpCr 422.3.2.2 CprO,CpC5H4DCr2O,Cp2rODreactions 432.3.2.3 Cp2rNO reactions 432.3.3 CpCrNO reactions 432.3.4 CpCrNOCD, CpCr(NO), and CpCr(NO)2Dreactions 442.3.5 CH4D2r reactions 452.3.6 CpCrO reactions 452.4 Ion-molecule chemistry of the negative ions 462.5 Rate constants 492.5.1 Rate constants of the positive ions 492.5.2 Rate constants of the negative ions 522.6 Proposed ion structures 562.6.1 Cluster ions from the Cr-molecule condensationreactions 562.6.2 Cluster ions from the CpCr-mo1ecule condensationreactions 582.6.3 Other bimetallic cluster ions 582.6.4 The negative ions 612.7 Discussion 62— vi —References 65Chapter 3 Reactions of the Fragment Ions of CpCr(NO)2H3with Small Molecules N2,H2,H20, D20, NH3, andCH4 in the Gas-Phase 723.1 Introduction 733.2 Experimental section 743.3 Calibrations of the nominal pressures 753.4 Ion-molecule chemistry of CpCr(NO)2H3with hydrogen, H2 763.4.1 Collisional quenching and charge-transfer reactions 763.4.2 Ligand substitutions 823.5 Ion-molecule chemistry of CpCr(NO)2H3with water andheavy water, H20 and D20 843.6 Ion-molecule chemistry of CpCr(NO)H3with ammonia, NH3 853.7 Ion-molecule chemistry of CpCr(NO)2H with methane, CH4 863.8 Ion-molecule chemistry of CpCr(NO)H3with nitrogen, N2 863.9 Discussion 873.9.1 The electron transfer reaction 873.9,2 The condensation reactions 893.9.3 Further work 91References 93Chapter 4 Simple Physical Point and Line Charge Models forCoulomb-Induced Frequency Shift and InhomogeneousBroadening in FT-ICR Mass Spectrometry 954.1 Space charge effects in FT-ICR mass spectrometry 964.1.1 Space charge effects and mass measurement 964.1.2 Prior research on the space charge effects 97-vi’-4.2 The point charge model — Coulomb shifting of unlike ions 994.3 The line charge model—Coulomb shifting and broadeningof unlike ions 1044.4 Discussion 1074.4.1 Choice of Model 1074.4.2 Inhomogeneous broadening in FT-ICR 1134.4.3 Coulomb shifting and broadening of like ions 1144.4.4 ICR cell design and Coulomb effects 114References 116Chapter 5 Simple Charge-Disk Model and Simple Charged-CylinderModel for Coulomb Shifting and Coulomb Broadeningin FT-ICR Mass Spectrometry 1195.1 Introduction 1205.2 The charge-disk model 1225.2.1 Two charged disks in a rotating laboratory-frame 1225.2.2 Radial Coulomb force on Disk 1 for non-overlappingdisks (‘I’ > touch) 1255.2.3 Radial Coulomb force on Disk 1 overlapping disks(‘ < 1295.2.4 Instantaneous radial Coulomb force 1355.2.5 Average radial Coulomb force 1405.2.6 Apparent Coulomb distance, D 1405.2.7 Validity of the charged-disk model 1435.2.7.1 Assumption of small like-ion interactions 1435.2.7.2 Assumption of small unlike-ioninteractions 147— viii—5.2.7.3 Assumptionofhighfrequencyperturbations 1485.3 The charged-cylinder model 1505.3.1 Apparent Coulomb distance and charged-cylindermodel 1505.3.2 Validity of the charged-cylinder model 1525.2.7.1 Assumption of small like-ion interactions 1525.3.2.2 Assumption of small unlike-ioninteractions 1555.3.2.3 Assumption of high frequencyperturbations 1565.4 The Coulomb-induced frequency shift and inhomogeneousbroadening of the charged-cylinder model 1575.5 Experimental tests of the charged-cylinder model 1595.5.1 The unlike-ion Coulomb-induced frequency shiftingfrom experiment of Francl et al 1595.5.2 Unlike-ion Coulomb-induced frequency shiftingin FT-ICR 1625.5.3 Experimental inhomogeneous broadening in FT-ICR 1685.6 Discussion 170References 174Chapter 6 Taylor’s Expansion Approximation of Ion-IonStrong Coupling Coulomb Interaction in FT-ICRMass Spectrometry 1766.1 Introduction 1776.2 Potential energy of a single tetragonal ion cloud 1816.3 Potential energy of two tetragonal ion clouds 184— ix —6.3.1 Taylor’s expansion approximation of the Coulombpotential energy 1846.3.2 Integration of the Coulomb potential energy 1876.4 Motion equations and solutions of two tetragonal ion clouds 1916.5 Strong coupling Coulomb interaction between two ion clouds 1936.5.1 Strong coupling condition 1936.5.2 Two oscillations under the strong coupling condition 1946.5.3 Strong coupling critical curve 1956.5.4 Strong coupling condition for N1 N 1976.6 Weak coupling Coulomb interaction 2006.6.1 Weak coupling condition 2006.6.2 Validity of the weak coupling condition 2016.7 Electric trapping potential and frequency shifts 2036.8 Discussion 2086.8.1 The charge-cylinder model and Taylor’s expansionapproximation 2086.8.2 The electric quadrupole trap and analyzer cell ofFT-ICR 2096.8.3 Intermediate coupling condition 2106.8.4 Ion distribution functions 2126.8.5 Further work 214References 216Appendixes 218Al Appendixes of Chapter 2 219A1.1 The “twenty-second” ion-molecule reaction massspectra 219—x—A 1.2 Coordination modes of metal carbonyls and metalnitrosyls 220A1.3 Known oxo chromium complexes 224A2 Appendixes of Chapter 3Kinetic behavior of the Cr produced from CpCr(NO)2H3in H20, NH3, CH4, and N2 media 228A3 Appendixes of Chapter 5 230A3.1 The FORTRAN program for calculating ion-ionCoulomb interaction between two charged disksbased on double Gaussian numerical computation 230A3.2 The FORTRAN programs for calculating ion-ionCoulomb interaction between two charged disksbased on double Romberg numerical computation 234A3.2.1 Program “DISK.ROMB” for two chargeddisks without overlapping 234A3.2. 1 Program “OVER.ROMB” for two chargeddisks with overlapping 236A3.3 The results from the double Romberg numericalcomputation 240A3.3. 1 Average radial Coulomb force for= 1 mm and r = 1 cm 240A3.3.2 Average radial Coulomb force for= 0.5 mm and r = 1 cm 241A3.3.3 Average radial Coulomb force forr’ = 0.1 mm and r = 1 cm 244A3.4 Curve fitting for “apparent Coulomb distance”vs. r/r’ 246— xi —A4 Appendixes of Chapter 6 248A4. 1 Four particular integrals applied in the Taylor’s expansionapproximation of ion-ion Coulomb interaction 248A4.2 Integrations of the Coulomb interaction coefficient 252A4.2. 1 Integration for = 252A4.2.2 Integration for z1 = z2 253A4.2.3 Integration fory’1 = y 256A4.2.4 Value of the Coulomb interactioncoefficient, C 266References 269-xl’-LIST OF TABLESTable 1.1 Summary of FT-ICR line shape formulas of a time signal 11Table 1.2 Comparison of typical mass spectrometer performanceparameters 15Table 2.1 The important electric and Fourier transform parameters 26Table 2.2 A comparison between literary data of average a valuesand theoretical calculations of a for some transition metalcomplexes 33Table 2.3 The primary positive ions from CpCr(NO)2D3 34Table 2.4 The identification for the ions from CpCr(NO)H 38Table 2.5 The primary negative ions from CpCr(NO)2D3 46Table 2.6 The experimental rate constants of the primary positiveions from CpCr(NO)2D3 51Table 2.7 The experimental rate constants of the positive condensationions from CpCr(NO)2H3 53Table 2.8 The experimental rate constants of the positive condensationions from CpCr(NO)2D3 54Table 2.9 The experimental rate constants of the primary negativeions from CpCr(NO)2D3 55Table 2.10 The experimental rate constants of the primary negativeions from CpCr(NO)2H3 55Table 2.11 A summary of carbonyls and nitrosyls in ion-moleculecondensation chemistry of transition metal complexes 64Table 3.1 Calibrations of the sampling pressures of N2, 1I2, H20, NH3,CH4, and CpCr(NO)2H3 76— xlii—Table 3.2 Rate constants of CpCrNOCH and CpCr(NO) in H2,H20, NH3, CH4, and N2 media 89Table 5.1 Apparent Coulomb distance for Coulomb shifting andCoulomb broadening in FT-ICR 142Table 5.2 The validities of Eqs. (5.70b), (5.71b), and (5.73b) forB = 1 Tesla, r = 1 cm, 1 = 2.4 cm, N1 = N2 = iO, q1 = q2= 1 electron charge, m1 = 250 Daltons, m2 = 251 Daltons,and D = 1.327 r and 1.215 r 159Table 5.3 The effective cyclotron frequency,fff, of CpMn with thechange of electron emission current, EMC 168Table 5.4 The peak width at 50% height of CpCr with the changeof electron emission current, EMC 169Table A1.1 Coordination modes of carbon monoxide 220Table A1.2 Coordination modes of nitric oxide 222Table A1.3 Known oxo chromium complexes 224Table A3.1 Comparison of the values of D’ calculated from Eqs.(A3.12)— (A3.15) with theoretical values ofD 247— xiv —LIST OF FIGURESFigure 1.1 The first ICR spectrometer—the Omegatron massspectrometerA conventional cubic trapped-ion cell of FT-ICR(a) Ion excitation; (b) Ion detectionThe elementary experimental sequence of FT-ICRThe absorption, the dispersion, and the magnitude Fouriertransform spectra of a time domain signal K0 cosThe vacuum system of the FT-ICR mass spectrometerA positive ion mass spectrum of CpCr(NO)2D3The triple resonance procedure of FT-ICR mass spectrometry(a) A pure Cr÷ spectrum by using the triple resonance technique;(b) The spectrum under the same conditions after a reactiondelay time of 100 msThe temporal behaviors of products from the condensationreactions of Cr with the parent molecule CpCr(NO)2H3The temporal behaviors of products from the condensationreactions of Cr with the parent molecule CpCr(NO)2D3Two proposed mechanisms for the bent nitrosyl ligand inthe Cr-molecule condensation reactionThe temporal behaviors (solid lines) ofCp3r4OandCp3r4ODThe temporal variations of the negative ions fromCpCr(NO)2D3Plot of in rate constants vs. electron deficiency for theprimary ions from CpCr(NO)2H3Figure 1.2Figure 1.3Figure 1.4Figure 1.5Figure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Figure 2.8Figure 2.9Figure 2.1046791125262630353640414852— xv —Figure 2.11 Proposed structures of the cluster ions from theCr-molecule condensation reactions 57Figure 2.12 Proposed structures of the cluster ions from theCpCr-molecule condensation reactions 59Figure 2.13 Proposed structures of all the other positive bimetalliccluster ions 60Figure 2.14 Proposed structures of CpCrO andCp2r(NO)H 61Figure 3.1 (a) Spectrum of pure Cr obtained by the triple resonance(b) After a 100 ms delay time, large amounts of product ionsfrom charge transfer of the Cr to CpCr(NO)2H3appeared 77Figure 3.2 Temporal behaviors of Cr, CpCrNOCH, CpCr(NO),CpCr2O, and CpCr2NO in H2 medium 79Figure 3.3 Curve fitting for ion intensity ([Cr+*]+[cr+J) vs. time t.Experimental data were obtained in H2 medium 83Figure 3.4 Temporal variations of the chromium animine complexes producedfrom the ion-molecule reactions of CpCr(NO)2H3with H2 85Figure 3.5 Temporal variations of the chromium ammine complexes producedfrom the ion-molecule reactions of CpCr(NO)2H3with CH4 87Figure 3.6 Temporal variations of CpCr and the secondary ionsCp2rt Cp2rO,Cp3r(NO)O in H2 medium 90Figure 4.1 Point model for Coulomb effect 100Figure 4.2 The average distance approach 102Figure 4.3 Line model for Coulomb effect 105Figure 4.4 Coulomb shifting and broadening for the line model of Fig. 4.3 109Figure 4.5 Same graph as that in Fig. 4.4 for 1= 8 cm andN2 = 3.33x106 ions 110— xvi —Figure 4.6 Same graph as that in Fig. 4.4 for 1= 15 cm andN2 = 6.25x 106 ions 110Figure 5.1 Uniformly charge-disk model for FT-ICR 123Figure 5.2 The averaging of tangential components of the Coulombinteraction between two ions m1 and m2 124Figure 5.3 Differential elements of the charge-disk model forCoulomb shifting in FT-ICR 126Figure 5.4 Overlapping charged disks 130Figure 5.5 Sections of overlapping disks used for piecewise evaluationof the total radial Coulomb force from Disk 2 to Disk 1 132Figure 5.6 Radial Coulomb force on Disk 1, a uniformly charged diskof m1 ions, due to a uniformly charged diskm2ions, as afunction of the position of Disk 2 137Figure 5.7 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for r/r’ = 5 138Figure 5.8 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for r/r’ = 10 138Figure 5.9 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for r/r’ = 20 139Figure 5.10 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for r/r’ = 100 139Figure 5.11 Apparent Coulomb distance, D, as a function of nT’, the ratio of thecyclotron radius to the disk radius, for the charge-disk model 144Figure 5.12 The charge-cylinder model 151Figure 5.13 Coulomb shifting and broadening for the charge-cylindermodel of Fig. 5.12 158— xvii —Figure 5.14 Drift in the 1.9 Tesla magnetic field used in the experimentafter preheating 30 mm 165Figure 5.15 The effective cyclotron frequency,fff, of CpMn vs. numberof ions with and without unlike-ion Coulomb interactions 167Figure 6.1 A cylinder in cylindrical coordinates (p, q, z) 178Figure 6.2 The charge-tetragon model 180Figure 6.3 The coordinate expression of singular points for =y =y, z1 = z2 188Figure 6.4 Strong coupling critical curve 198Figure 6.5 The cubic ion-trapped cell configuration employed in the FT-ICRmass spectrometer 204Figure A1.1 The “twenty-second” ion-molecule reaction mass spectra(a) Parent molecule CpCr(NO)2H3;(b) Parent moleculeCpCr(N0)2D3 219Figure A2 Kinetic behavior of Cr produced from CpCr(NO)2H3in1120, NH3 CH4, and N2 media(a) Temporal behavior of Cr in H20 228(b) Temporal behavior of Cr in NH3 228(c) Temporal behavior of Cr in CH4 229(d) Temporal behavior of Cr in N2 229Figure A3.1 Radial Coulomb force on Disk 1, a unifonxily charged disk ofm1 ions, due to a uniformly charged disk of m2 ions, as afunction of the position of Disk 2 242Figure A3.2 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for rir’ = 20 243Figure A3.3 Radial Coulomb force on Disk 1 as a function of the positionof Disk 2 for r/r’ = 100 245— xviii —LIST OF ABBREVIATIONSA Angstromarnu atomic mass unitBu Butyl (-CH2C3)C CoulombCU) Collision-Induced DissociationCp Cyclopentadienyl(5-CH)Eq. EquationEt Ethyl (-CH2C3)eV electronVoltFig. FigureFT-ICR Fourier Transform Ion Cyclotron Resonance mass spectrometryFT-JR Fourier Transform InfraRed spectroscopyFT-MS Fourier Transform ion cyclotron resonance Mass SpectrometryFT-NMR Fourier Transform Nuclear Magnetic Resonance spectroscopyHz HertzICR Ion Cyclotron Resonance mass spectrometry3 Joulek kilokHz kiloHertzK KelvinMe Methyl (-CH3)MHz MegaHertzm metermm minutem/q, m/z mass-to-charge ratio of ion-xixms millisecondMS-MS Mass Spectrometry-Mass SpectrometrypA microampereN NewtonnA nanoAmperePa PascalPh Phenyl (-C6H5)ppm parts per millionif radiofrequencysee secondSI units Standard International unitsV Volt— xx —ACKNOWLEDEGMENTSMy sincere gratitude goes to my research supervisor, Professor Melvin B.Comisarow, for being a constant source of support to me (both academically andpersonally) over years. Without question, whatever I have achieved academically over thelast five years has been largely due to his great inspiration and energetic enthusiasm forhard work.I also want to thank the occupants of E61/63, Sandra M. Taylor, and Ziyi Kan, forproviding the friendly research environment, and special thanks to Ms. Sandra M. Taylorfor proofreading the manuscript of this thesis.My gratitude goes to Professor Peter Legzdins and Dr. George Bannerman RichterAddo for their kindness in providing chromium nitrosyl samples and the information ontheir new development in metal nitrosyls; to Dr. Xingru Zhang (Department ofMathematics, UBC), Mr. Peter Yueqiang Zhu (Department of Physics, UBC), and Dr.Wenzhu Zhang for their helpful discussions in mathematics, physics, and computerscience, respectively.I would also like to thank the people working in the Electronic shop and Mechanicalshop 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 andencouragement.— xxi —This dissertation is dedicated tomy youngest brother, Shu-Huan Chen—1—CHAPTER 1GENERAL INTRODUCTION—2—The first English book on mass spectrometry which I read in China in 1976 wasMcDowell’s “Mass Spectrometry”.(’) At that time I did not expect that 10 years later Iwould study mass spectrometry in the Department of Chemistry formerly headed by thisvery professor. In 1980, when my Master’s Degree supervisor, Professor Ou Ji (at thattime an associate professor), introduced Fourier Transform Ion Cyclotron Resonancemass spectrometry (FT-ICR or FT-MS)(2)to me, I was very surprised and had manyquestions. How could a mathematical method enhance the resolution of ion cyclotronresonance mass spectrometry? Was there another transform method better than the Fouriertransform? What were the elementary differences between Ion Cyclotron Resonance(ICR) mass spectrometry and FT-ICR mass spectrometry? How was mass spectrometrymass spectrometry run in a single FT-ICR? I even argued with my former supervisor. Ihad thought that the high resolution capability of FT-ICR was due to the long relaxationtime of ions and Ou Ji thought that the Fourier transform itself produced the highresolution. In a word, because I was attracted by its beautiful “first name” — Fouriertransform, I travelled to the birthplace of FT-ICR. This work is devoted to the applicationof FT-ICR and its performance modifications. There are two parts: (1) gas-phasechemistry of chromium nitrosyls; (2) Coulomb interaction between ions in ion cyclotronmotion, including treatment for averaging the Coulomb interaction and treatment for theCoulomb interaction within small separations.1.1 A Brief History of Ion Cyclotron Resonance Mass SpectrometryThe fundamental basis for ICR mass spectrometry was established in the early 1930’sby Ernest 0. Lawrence. He was the first to demonstrate that charged particles could beaccelerated to high energies by an oscillatory electric field having the same frequency astheir cyclotron frequency.(3) In a uniform static magnetic field B, an excited chargedparticle is constrained to move in a circular orbit with a frequency given by—3—%= qB/m (1.1)where Co0 is termed the natural cyclotron frequency,(4)q and in are the charge and mass ofthis particle. The cyclotron particle accelerator, in which a charged particle can be excitedto very high translational energy, is a powerful tool for high-energy nuclear physics, but itis not very useful in chemistry.The first mass spectrometer based on the cyclotron resonance principle wasdeveloped in 1949 by Hipple and co-workers.(5)At that time the device shown in Figure1.1 was called the Omegatron mass spectrometer. After the gas in the Omegatron is ionizedby an electron beam (Figure 1.1), an electric radiofrequency (ii) field is applied between apair of planar electrodes to excite the newly-formed ions, and the field is homogenized byfeeding a stack of guard rings from a voltage divider. If the frequency of the rf field isequal to the cyclotron frequency, w0, of an ion, the ion formed in the center of the devicemoves 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. Theplot 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 effectsfrom the electron beam severely degrade the performance of the Omegatron, its resolutionwas very poor above m/q 50. In 1960’s Wobschall et a!. detected ICR ion signals bymeasuring the ion cyclotron resonance power absorption from the rf field rather than fromthe direct ion current,(6)and Varian Associates produced the first commercial ICR massspectrometer, in which a three section ICR drift cell was developed to separate theionization, resonance, and detection procedures.(7)The three section ICR drift celleliminated the ion cyclotron frequency shift from space charge effects of the electron beam.Mclver made important innovations in 1970 for ICR mass spectrometry: the pulsingtechnique and a one-region trapped-ion analyzer cell.(8) The pulsing technique allowed—4—BGuardrings ‘—Filament===EN]) VIEW SIDE VIEWFigure 1.1 The first ICR spectrometer — the Omegatron mass spectrometer. Afterions are formed in the Omegatron, a rf field is applied across the top andbottom 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 therf 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 ofICR experimental events were separated in a time sequence rather than in three spatialregions. This innovation combined the advances of power absorption detection and thethree 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 thefrequency of the if field. Thus, the m/q of an ion had to be calculated from the frequencyof the if field. A dramatic development in 1974 was Fr-ICR, invented by Comisarow andPath of ionsat resonance7’ /“h, ‘,‘Ion collectorTrappingvoltage—5—Marshall in our Department of Chemistry.(2)In FT-ICR, all ions are excited and detectedsimultaneously. The ion image signals on the receiver plates are detected all at once, bothfor ion intensities and ion cyclotron frequencies.(9)The interest in FT-ICR is illustrated bymany reviews on the topic. So far, there have been 51 reviews (in four languages:English, German, Chinese, and Japanese) on the theory, instrumentation, and applicationsof FT-ICR.(1060)FT-ICR has been shown to be an active area in spectrometry.(61)1.2 The Background of Fourier Transform Ion Cyclotron ResonanceMass Spectrometry1.2.1 Basic operating principle of FT-ICRA cubic trapped-ion cell(17’62) is shown in Figure 1.2. When a sample to beanalyzed is introduced into this cubic cell, it can be first ionized by various ionizationtechniques, such as electron ionization, photo-ionization, or chemical ionization. Theseions are confined in the cubic cell by a static trapping voltage. In order to measure thecyclotron frequencies and the intensities of an ensemble of ions, the experiment is dividedinto two steps (see Figure 1.3). At first, a broadband rf rapid sweep is applied across thecubic cell to excite all the ions successively. When the radiofrequency is equal to thenatural cyclotron resonance frequency of an ion, this ion will be excited into its orbitalradius of cyclotron motion. This process is the ion excitation step of FT-ICR.Secondarily, after the if sweep is turned off, the excited cyclotron motion goes into acircular orbit by maintaining its resonance frequency. The orbiting ion induces analternating image cunent on two opposing parallel receiver plates and the voltage inducedacross this circuit is the FT-ICR signal.(9) This process is the ion detection step of FTICR. The ion cyclotron motion is detected as a time domain signal. Then, the signal isdigitally sampled and Fourier transformed to give a frequency spectrum in a computer.—6—yFigure 1.2 A conventional cubic trapped-ion cell of FT-ICR. The cell is composed ofsix square plates. Two T’s are trapping plates to which is applied a directvoltage, typically, + 1 V for positive ions and 1 V for negative ions. Nand S are receiver plates for detection of ion cyclotron motion. E and Ware transmitter plates for rf excitation. C is an electron collector forattracting and monitoring the electron current. G is a grid for shielding thestrong ionization voltage. F is a thermionic filament. P is a preamplifierfor ion detection. A is a transmitter amplifier for the if field. B is amagnetic field applied across the cubic cell.C—7—GB___I I.I I(a) (b)The radiofrequency is on The radiofrequency is off(a) Ion excitation. When a radiofrequency field is applied across the cubictrapped-ion cell, the ion is excited and its cyclotron radius increases in aspiral orbit. (b) Ion detection. When the rf field is off, the ion cyclotronradius is constant, and the ion cyclotron motion induces an image current1(t) that is converted to an alternating voltage signal VQ) and amplified bythe preamplifier.I IFigure 1.3—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 iondetection, broadband circuitry is used, which is different from the narrow-band inductive-capacitive circuitry used in Fourier Transform Nuclear Magnetic Resonance spectroscopy(FT-NMR).(9)All the experimental procedures of FT-ICR consist of a sequence of pulses controlledby a computer. The FT-ICR experimental pulse sequence is indicated in Figure 1.4.1.2.2 Fourier transform of a time signalAny periodic well-behaved function can be expressed as a sum of Fourier series, thefamiliar 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,f(t). For an aperiodic function, the summation becomes an integration(63a)00 00f(t)= I A(co) cos (cot) dw + J B(o)) sin (cot) dco. (1.3)Jo JOA(w) is called the pure cosine frequency content off(t), and B(w) is called the pure sinefrequency content off(t). These two frequency spectra are given by00A(w) =- )f(t)cos(wt)dt (1.4)00B(co) =- JfQ)sin(wt)dt. (1.5)—9—Figure 1.4 The elementary experimental sequence of FT-ICR. Neutral atoms ormolecules are ionized in the first pulse, ion formation. These ions can betemporally retained in the cubic cell and then excited in the second pulse, orimmediately excited without any delay. The ion cyclotron motion ismonitored 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 toenhance the ion signal.—10—When the lower limits of the integrals in Eqs. (1.4) and (1.5) are zero (i.e., t> 0), thenA(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 K0 cos w0t, 0 <t < T (1.6)where K0 is a constant proportional to the number of excited ions and T is the signalacquisition time or the duration of the detection period. The basic Fourier transformformulas of this time domain signal are:(16)A() = f(t)cos tdt= K0sin[(w—w)TJ (1.7)7t 0 (OW1 . K0{1—cos[(o.—w)TJ}B(o.) =— I f(t)srnwtdr= (1.8)ltJO (0—CL)2 2 112 sin[(a— w,.) T/2]C(co) = {[A(co)1 +[B(w)] } = 2K0 (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 kindsof 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 = (1.10)From the line shapes of Eqs. (1.7) — (1.9), the full linewidths of the absorption, thedispersion, and the magnitude spectra are solved and summarized in Table 1.1. Thetheoretical spectral linewidths are all inversely proportional to the the acquisition time T at— 11—Figure 1.5 The absorption, the dispersion, and the magnitude Fourier transform spectraof a time domain signal 2t K0 cos %t.Table 1.1 Summary of FT-ICR line shape formulas of a time signalf(t) = 2itK0 cos0< t < T, and their extrema and full linewidthsSpectral Zero-pressuredisplay mode line shape’2urn (ct—w0)—* 0 Full hnewidth w50Absorption sin[(a — ci.0)T] 3.791A(w)= Tco.-o TDispersion 1 — cos[(w — w0)TJ 0.7246 b 4.662 cB(oi) =w— TMagnitude sin[(0 — o,0)T/2] I T 7.582(Absolute value) C(a) 2 Ta The scaling coefficient K0 has been omitted for simplicity.b These are the extrema of B(w) at w— = ± 2.331/T.C This value is the distance based on the extrema on either side of w.Time, in secöiö 0 D 20 -20 -10 0 0 20(w — in see-’— 12—the low-pressure limit. Thus, if there is no damping force, the relaxation time of theexcited ion cyclotron motion will become arbitrarily long and the resolution will bearbitrarily high. The longer relaxation times (at a lower sample pressure) of ions are, themore precisely the cyclotron frequencies of these ions can be measured, which is theuncertainty principle in FT-ICR.(16) Table 1.1 shows that the absorption mode has thenarrowest linewidth.Fourier analysis (the use of Fourier series and Fourier transform in analysis) is alsocalled frequency analysis.(63b) The use of Fourier transform in FT-ICR actually is atechnique for measuring the frequency content of a time signal, and frequency can bemeasured very accurately.(17)Therefore, high resolving power of FT-ICR comes not fromthe mathematical method, but rather from precise electronic techniques for the measurementof frequency.1.3 Ion-Molecule Reactions and FT-ICR Multiple ResonanceAn important application of ICR mass spectrometry is gas phase ion-moleculechemistry. The trapping direct voltage applied on the cubic ion-trapped cell has thecapability to confine ions. After the ions are formed, they can be maintained within thecubic cell for a while (from microseconds to tens of seconds) to allow the ions to react withthe neutral molecules whose pressure is typically iO — iO Torr. Then, the products ofthese ion-molecule reactions are detected by FT-ICR. The ICR multiple resonancetechnique that was invented for conventional ICR by Anders et at. in l966,(M) and thentransplanted to FT-ICR by Comisarow et at. in 1978(65) can help to monitor the pathwaysof these ions. If the amplitude of a specific if sweep of FT-ICR is strong or the duration ofthe if sweep is long, the cyclotron resonance motion of the corresponding ions in the cubiccell will be excited enough that these ions hit the walls of the cubic cell and their chargesneutralized. 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 ionejection is used to eject a desired mass range of ions, it is called double resonance becausethere are two resonance events in this sequence: ion ejection and ion excitation. If two ionejections are used to eject two desired mass ranges of ions, it is called triple resonance, andso on. Double resonance can be used to eject only one ion mass, leaving all other ions, andtriple resonance can be used to eject all the ions except ions of one mass. Triple resonanceexperiments 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 ofions. 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 theneutral molecules. These processes will be discussed in Chapter 2 in more detail. The ICRmultiple resonance is a Mass Spectrometry-Mass Spectrometry (MS-MS) technique inwhich 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 fromfragmentation or ion-molecule reaction of those initially selected parent ions. Multipleresonance of FT-ICR is executed in only a single analyzer cell rather than in multiple massanalyzers as is the MS-MS experiment in other tandem mass spectrometers.1.4 Performance of FT-ICRFr-ICR has the same advantages as other Fourier transform spectroscopies, such asFourier Transform InfraRed spectroscopy (FT-IR) and FT-NMR. These advantages arethe Fellgett (multiplex) advantage for great speed and great signal-to-noise ratio. Thewhole spectrum can be detected simultaneously rather than one mass at a time by scanningas in a conventional ICR mass spectrometer, and a low signal-to-noise ratio can beenhanced by accumulation. FT-ICR has other advantages, such as ultrahigh mass— 14—resolution, wide mass range, trapped-ion ability, and multiple resonance capability. Itstransmission and mass resolution are better than other commercial mass spectrometers. Acomparison of Fr-ICR features with those of other conventional mass spectrometries(66)isgiven in Table 1.2.The performance of FT-ICR has undergone much development in the last threeyears. The resolution of FT-ICR has increased amazingly to over 4x108 for‘1Ar ions in a7 Tesla magnetic field.(67) Furthermore, by using high-precision techniques of trapconstruction, operation, and analysis, very high accuracy in mass measurement has beendemonstrated as being possible. A mass ratio accuracy of up to 4x10—°has been obtainedfor the light ions (CO compared with N2, 1H, 2H, and 3He with‘2C).(68-9) In order toobtain greater accuracy, these light ions were detected one by one,(68-9)without regard tothe 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 Daltonshas been measured,(70)but this is far from the theoretical prediction that a 2.5 cm cubic cellat a trapping voltage 1 V in a magnetic field of 13 Tesla is capable of storing an ion havingan m/q as large as 950, 000 Daltons.(28)It is well known that the relative peak heights inFT-ICR do not always reflect the ion abundances accurately. The most important errorsources are that ion signal intensity of FT-ICR not only depends on the ion abundance, butalso the cyclotron radius of an ion(9)and the ion z-mode ejection(71). Because FT-ICR isnot usually applied to quantitative chemical analysis, this problem is not of much concernyet. Recently, Koning et al.(72) have used a segmented Fourier transform method toenhance the accuracy of ion abundance measurements of FT-ICR. In their method, thetime domain signal is divided into two parts for t from 0 to T/2 and t from T12 to T, whereT is the acquisition period. The individual components of the signal are separated byFourier transform, and then the amplitudes of ion intensities are calibrated against the decayof the ion cyclotron motion. For a narrow mass range, such as among xenon isotopes, themeasurement of ion abundances with accuracies of about 0.1% was demonstrated.—15—Table 1.2 Comparison of Typical Mass Spectrometer Performance ParametersMinimum #IonsTransmission Mass Mass RangeInstrument Type (%) Needed to beFumd Resolving Power (1)alton)Quadrupole 0.01 102_103 i03 *Magnetic Sector—low resolution 10 10 10—high resolution 0.001 i05Time of Flight 10—50 2—10 10 104_b? **FT-ICR (— 100) 102 io4—io5 104_to? “Wien ExB 1—10 b0_102 104_to?* Actually, the mass range of a mass spectrometer may depend on the ionization approach. A newionization 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 massrange of the quadrupole was only b0 (T. Nohmi and J. B. Fenn, Proceedings of the 38th ASMSConference 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 ionsource (F. Hillenkarnp, M. Karas, Proceedings of the 37th ASMS Conference on Mass Spectrometry andAllied Topics, 1989, 1168-1169). The resolution of Time of Flight mass spectrometry has reached 35,000for 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, transmissionof 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 singleion 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 to4x108 for 40Ar (K. P. Wanczek, Proceedings of the 7th International Dynamic Mass SpectrometrySymposium, Salford, 1989). The current highest mass/charge ratio is 31830 Daltons with an accuracy of2.26% (C. B. Lebrilla, D. T.-S. Wang, R. L. Hunter, and R. T. Mclver, Jr., Anal. Chem., 62, 1990, 878-880).—16—References1. 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, NewYork, 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, 1565-1571.7. The first commercial ICR mass spectrometer was designed and constructed under thedirection of Dr. P. Llewellyn at Varian Associates for Professor 3. D. Baldeschwielerat 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 263-273.14. 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 IonCyclotron 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: NewYork, 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, 389-415.33. Land, D. P.; Pettiette-Hall, C. L.; Hemminger, J. C.; Mclver, R. T. Jr. Acc. 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M. in “Mass Spectrometry in the Analysisof Large Molecules”; McNeal, C. 3., Ed.; John Wiley & Sons: Chichester, 1986;p 39.67. Wanczek, K.-P. Proceedings of the 7th International Dynamic Mass SpectrometrySymposiwn, 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. mt. J. Mass Spectrom and Ion Proc. 1990, 101,65-78 and many references therein.72. Koning, L. 3. de; Kort, C. W. F.; Pinkse, F. A.; Nibbering, N. M. M. mt. J. MassSpectrom and Ion Proc. 1989,95,71-92.—21—CHAPTER 2ION-MOLECULE CONDENSATION CHEMISTRY OF(5-2,4-CYCLOPENTADIEN-1-YL)METHYLDINITROSYLCHROMIUM CpCr(NO)2H3—22—2.1 IntroductionGas-phase ion-molecule reactions in hydrogen were discovered by Dempster as earlyas 1916.() After almost fifty years, the gas-phase ion-molecule condensation reactions ofthe transition metal complexes ferrocene and nickelocene were first reported by Schmacherand Taubenest in 1964.(2) These early studies were performed on magnetic sector massspectrometers and the reaction yields were very limited. Only in the past decade, thedevelopment of gas-phase ion chemistry has been promoted intensely by newinstrumentation and new experimental methods.(317)Just twenty years ago (in 1971),Foster and Beauchamp first used ICR mass spectrometry to investigate the ion-moleculechemistry of iron pentacarbonyl.(18)FT-ICR,(19)with its greater mass range, higherresolution, and more efficient double resonance technique, was a significant improvementfor studying ion-organometallic molecule chemistry. FT-ICR mass spectrometry was firstused in 1980 by Parisod and Comisarow to study organometaffic chemistry.(20)Studies inion-molecule chemistry of transition metal complexes by ICR and FT-ICR can be classifiedinto three types:(1) Bond energetics and thermochemistry of transition metal complexes.(2) The chemistry of the ions (especially bare metal ions) generated from transitionmetal complexes with small inorganic and organic gas molecules. This will be discussed inChapter 3.(3) Condensation reactions of transition metal complexes. This will be dealt with inthe present chapter.There are several recent reviews for these three applications of Fr-ICR.(211)Thestudies of the condensation chemistry (type 3) by ICR are clustering of Fe(CO)5,18’25.-29)Cr(CO)6,25)Ni(CO)4,25’30) Cp2Fe (Cp is the abbreviation of Cyclopentadienyl),(31)Cp2Ni,(32)CpCo(CO),33CoNO(CO),34Ti, V÷, Cr, Mn, Fe, Cot, Ni withCr(CO)6and Ni(CO)4,35)andRe2(CO)10.(36 Most investigations involving transition—23—metal condensation chemistry by Fr-ICR have been concerned with transition metalcarbonyls, such as CpMn(CO)3,CpCrNO(CO)2,and CpCrNS(CO)2,(0)Cr withCr(CO)6,37)Fe, Cot, FeCH, and FeOH with Fe(CO)5,38)Fe with Co2(CO)8,(39)Co with Fe(CO)5,CoC4Hwith Cp2Fe,(40)V with Fe(CO)5,41)V with V(CO)6,Crwith Cr(CO)6,Fe— with Fe(CO)5,Co with Co2(CO)8,Mo with Mo(CO)6,42)Re2(CO)10,(43)Mn2(CO)10and ReMn(CO)10,44Cr(CO)6and Fe(CO)5,45)Ni(CO)4andCoNO(CO)3,46and their hetemuclear ionic clusters,(47)Fe and with Fe(CO)5andCo2(CO)8,(48)La and Rli with Fe(CO)5,Rh with CoNO(CO)3,49)Sc, Ti, V÷, Cr,Fe, Cot, Nit, Cu, Nb, and Ta with Fe(CO)5,0)Cu with Fe(CO)5,1)H20s3(CO)10,(52)La2 with Fe(CO)5,3)large cluster ions containing up to 40 metalatoms from Fe(CO)5 and Re2(CO) bare metal clusters from Fe(CO)5 andRe2(CO) Mn with Cr(CO)6,56)CpV(CO)4,57Fe(13CO) with Fe(CO)5,(58)Cr(6-arene)(CO)3,59)andFe(r-HO)(CO)3. )Gas-phase ion-molecule condensation chemistry of transition metal nitrosyls is alargely undeveloped field, although organometallic nitrosyl chemistry already has become aspecial subject in inorganic chemistry.(61)Cleavage of the N-O bond was not observed inion-molecule condensation studies of CoNO(CO)3,4’46,49) as there is a strong Co-NObond in CoNO(CO)3(which is expected since the Co electronic structure is 3d74s2,and thelinear NO ligand is a three electron donor). The elimination of nitrogen from the NO ligandand oxo chromium clusters containing up to four metal atoms have been observed inclustering of CpCrNO(CO)2,(0)but no further study was attempted. To make acomparison with the clustering of CpCrNO(CO)2,some extensions were made in thiswork. We used a chromium mtrosyl compound containing no carbonyl, CpCr(NO)2H3and because the mass of NO is almost equal to the mass sum of two CH3 groups, thedeuterated compound CpCr(NO)2D3was used to confirm our results. It should be notedthat we have to study a compound containing nitrosyls with other ligands, since only twothermally stable binary metal nitrosyls, Cr(NO)4and Co(NO)3,are known at present.(62)—24—Reaction pathways were determined by the multiple resonance technique of FT-ICR. Thekinetics were studied by monitoring temporal variations of the reactions for up to twentyseconds, and by calibration of nominal pressure. Cluster structural information wasproposed, even though the structure-reactivity rule(29)for gas-phase metal carbonyl ions,which was based on the 18-electron rule, is not suitable to metal nitrosyls.CpCr(NO)2H3was first synthesized by Piper and Wilkinson in 1955.(63) Theirmethod involved reaction of a Grignard reagent, CH3MgI, with the iodo-complexCpCr(NO)21. Another method for preparation of CpCr(NO)2H in which anorganoaluminum reagent, (CH3)A1, was used to react with the chioro-complexCpCr(NO)21, was reported by Hoyato, Legzdins, and Malito in 1975.(64)CpCr(NO)H3possesses a “three-legged piano stool” molecular structure.(63b) Thenitrosyls in CpCr(NO)2H3are linear so that the complex has an 18-electron structure.C?c’4 ‘ND3N0(?-2,4-Cyclopentadien- l-yl) (i-2,4-Cyclopentadien- 1-yl)methyldinitrosylchromium methyl-d3-dinitrosylchromiumThe bond energy of N—0 in metal complexes is generally weaker than that ofC_O.(61) This is consistent with their respective vibration frequencies (vco(65): 1850 —2125 cm1 and VNO(66): 1520 — 1950 cm1) in metal complexes. The N—O bond can bebroken in ion-molecule reactions to form oxygenated complexes, especially for theoxophilic early transition metals. Thus, the condensation chemistry of CpCr(NO)2H3ismuch richer than that of metal carbonyl complexes.H30—25—2.2 Experimental Section2.2.1 Experimental HardwareAll experiments were performed in a home-built FT-ICR spectrometer equipped witha Nicolet FT-MS 1000 console. The size of the cubic ion-trapped cell is 2.54 cm x 2.54 cmx 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 maintainedbelow 5 x i0 Torr and 5 x 10 Torr, respectively. A schematic diagram of the vacuumsystem of the FT-ICR mass spectrometer is shown in Figure 2.1. The basic operatingFigure 2.1 The vacuum system of the FT-ICR mass spectrometer. The sample isintroduced from a sampling tube through a roughing valve, through a leakvalve to the high vacuum region containing the cubic cell. Sample pressureis held constant by continuous pumping by the ion pump.RoughingValvesCubicCellMagnet—26—principle of FT-ICR has been described in Chapter 1. Compounds CpCr(NO)2H3andCpCr(NO)2D3are dark green crystals, which were kindly provided by the research groupof Professor P. Legzdins. No impurity was detected in the mass spectra of these twocompounds. The samples were evaporated into the FT-ICR spectrometer at ambienttemperature through variable leak valves. Pressure was controlled in the nominal pressurerange 6.2— 6.8 x i0- Torr as measured by a VARIAN Dual Range Ionization GaugeModel 971-1008. Other important parameters employed in the FT-ICR experiment arelisted in Table 2.1. The rate constants were measured on different days, using differentemission currents, and different excitation levels in order to increase the reliability. Themeasured 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 parametersParameter name Parameter value RemarkIonization voltage—25.0 V for positive ions,—1.2 V for negative ionsTrapping voltage +1 V for positive ions,—1 V for negative ionsNumber of data points 64 k for a good resolution,for FT (m/Em)5o%>30O.Zero filling for FT 1Ionization time 20 ms for good ion signals.2.2.2 Multiple Resonance TechniqueThrough the triple resonance experiment described in Chapter 1, all ion masses exceptone can be ejected. Figure 2.2 is a positive ion mass spectrum of CpCr(NO)2D3under—27—25 eV electron ionization, which corresponds to a normal mass spectrum obtained fromconventional mass spectrometers without ion-molecule reactions. If only (mlz 52) is tobe studied, the multiple resonance procedure is run as follows. The ions whose masses arebelow m/z 51 are ejected by a if sweep within their resonance frequencies and this processis called ejection 1 in Figure 2.3. The ions whose masses are over mlz 53 are ejected by asecond if sweep within their resonance frequencies and this process is called ejection 2 inFigure 2.3. A third if sweep whose amplitude is weaker than those of the prior two ifsweeps 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 sameprocedure is run again except that a delay time, e. g., 100 ms, is added to permit reactionsbetween the Cr ions and the neutral molecules. All mass peaks containing more than twochromium atoms shown in Figure 2.4 (b) can be assigned to be ion-molecule condensationproducts of Cr with CpCr(NO)2D3.Comparing with Fig. 2.2, many “primary” ions arealso found in Fig. 2.4 (b), especially CpCrNOCD and CpCr(NO). These ions areformed from the reactions of Cr ions with the parent molecules.by charge transferfollowed by fragmentation.2.2.3 Pressure CalibrationThe reactivity of a metal (or metal cluster) ion is a reflection of its coordinativeunsaturation.(29)Therefore, determination of the rate constants of gas-phase metalion-molecule reactions is very important. Ion-molecule reactions in mass spectrometryusually are pseudo-first order, since [molecule] >> [ions]. For example, the ion density inFr-ICR is generally 106 ions/cm3. According to the ideal-gas law, number of molecules Nin a volume V and at a pressure P isN=k’ , (2.1)—28—100 -___________ _______________—CpCrNOCDCpCr11— 50- CpCr(NO)+7/i /r_moIUuIa ion0— _..tI1i 11111 II100 200 300 400 500 600MASS In A.M.U.Figure 2.2 A positive ion mass spectrum of CpCr(NO)2D3.The operating parametersare listed in Table 2.1. Because of a 20 ms experimental beam time, someion-molecule reactions are observed.Ion Ion Ion Ionformation excitation detection quench(rf sweep)_J]aabledelay time [j [j RepeatNormal procedureIon Ion Ion Ion Ion Ionformation ejection 1 ejection 2 excitation detection quench____(if sweep) (rf sweep) (rf sweep)UULfU1ZLTriple resonanceVariable delay timeFigure 2.3 The triple resonance procedure of FT-ICR mass spectrometry. Two ionejections are simply added between “Ion fonnation” and “Ion excitation”.—29—where the Boltzmann’s constant kB = 1.380658x 10-23 3 K-1, and T is temperature inKelvin. At a pressure of 1 x i0 Torr (1.3 x iO—5 Pa) and a temperature of 300 K, thenumber of molecules in a volume of 1 cm3 is( —5’ ( —6’N = = 3x10 (molecules).(1.380658x10—23)x 300That is, the density of molecules is generally 3 x iO times the ion density in FT-ICR. Ifthe true pressure of the neutral gas is known, the second order rate constants of thesereactions then can be calculated. The true pressure is not the nominal value from thevacuum 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 varioussmall inorganic and organic molecules.(67)The true pressure of a sample x is given by theformula— nominal pressure of x 2 2x Rwhere R is relative sensitivity of the ion gauge with respect to the sample x with. Theycompared several empirical methods for calculating the sensitivities and gave the bestgeneral equationR = Sx/SN = (0.36/A3)a+ 0.30 (2.3)where S, is the chemical sensitivity of a sample x, relative to the chemical sensitivity ofnitrogen SN = 1.00, 1. e., the nominal pressure of nitrogen is defined as a standard, and ais the molecular polarizability of x in A3. The molecular polarizability of a molecule is not asimple sum of its atomic polarizabilities. Nevertheless, Aroney, Le Fèvre, andSomasundaram in 1960 found that the polarizability and molar Kerr constant of ferrocene isequivalent to that of a krypton atom sandwiched between two Cp planes.(68) Since only afew data on polarizabiities of metal complexes are available, we assumed:—30—100•1200 300 400 500MASS IN A.M.U.1(a)600100IC,,zLUIzLU‘II-4-JUi600(b)Figure 2.4 (a) A pure Cr spectrum by using the triple resonance technique; (b) Thespectrum under the same conditions after a reaction delay time of 100 ms(nominal pressure = 6.2x107Torr). CpCrNOCD and CpCr(NO) camefrom collision induced dissociation of Cr with CpCr(NO)2D3.100 200 300 400 500MASS in A.M.U.—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 theircorresponding molecules benzene(69)and ethane, respectively. Because terminal nitrosyland carbonyl both contain it-back-bonding to a metal and there is only one electrondifference in their numbers of electrons donated, the a of nitrosyl is assumed to beapproximately 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 CpCr(NO)2H3iscalculated as follows.The polarizabilities of benzene and ethane are known: a(C6H)= 10.32 A, anda(C2H6)= 4.45 A3.C70) The a of carbonyl in Cr(CO)6 and-C)Cr(CO)3can bederived to be 2.82A3.(7’) Thus, according to hypothesis (2)a(C5H)= 10.32 A3 x = 8.60 A3,a(.CH3)= 4.45 A3 x = 2.23 A3,a(nitrosyl) 2.82A3where a(CH3)is quite close to the theoretical value 2.22 A3 obtained from an empiricalmethod suggested by Miller and Savchik.(72)a(Kr) = 2.4844 A3.(70) The polarizabiity ofCpCr(NO)2H3is calculated according to hypothesis (1) to bea(CpCr(NO)2H3)= cz(C5H)+2 x a(nitrosyl) + a(•CH3)+ a(Kr)= 8.60+2x2.82+2.23+2.484418.95 (A3).—32—The chemical sensitivity of CpCr(NO)2H3is then, according to hypothesis (3),RCpcr(NO)CH = 0.36 x 18.95 + 0.3 = 7.1.Hence, for a nominal pressure of 6.3 x i0 Torr of CpCr(NO)2H3,the calibratedpressure is given by“CpCr(NO)2H3= 6.3 x i0 Torr I 7.1 = 8.9 x 10 (Torr).The calculation shows that the nominal pressure of CpCr(NO)2H3as indicated on the iongauge may be very different from the true pressure.For sixteen transition metal complexes whose average molecular polarizabilities havebeen measured experimentally,(68’70-7 1, 73) their theoretical molecular polarizabilities arecalculated on the basis of the three hypotheses and given in Table 2.2. Except TiC14 and0s04,whose structures do not obey the 1 8-electron rule, the theoretical polarizabilities ofall the other carbonyl complexes agree with the literature polarizabilities within an errorrange<5%.2.3 Condensation Chemistry of the Positive IonsTo avoid confusion between the hydrogens of the CH3 group and those of the Cpgroup, the condensation chemistry is presented using CpCr(NO)2D rather thanCpCr(NO)2H3.Under electron ionization of 25 eV, the important primary ions (relativeintensities all over 10%, see Table 2.3) from CpCr(NO)D3are Cr, CpCr, CpCrNO,CpCrNOCD, CpCr(NO),C5HNCD,C6H4D2r, CpCrO, and CpCr(NO)2D. Ofthese primary ions, the methylpyridinium ion,C5HNCD, is very unreactive in gas-phasedue to its aromaticity. Temporal behavior in ion intensity from zero to one second of someions from CpCr(NO)2H3and CpCr(NO)2D3are shown in Figures 2.5 and 2.6,— 33—Table 2.2 A comparison between literature data of average avalues and theoreticalcalculations of a values for some transition metal complexesComplex EXPenmefltal Theoretical error Remark Ref.cx (As) a (A)t (%)TiC14 16.4 17.6 + 7.3 The Ti has 8 valence (70)electrons.Cp2Fe 19.0 19.7 +3.7 (68)0s04 8.17 9.76 + 19.4 The Os has 16 valence (70)electrons.Cr(CO)6 19.4 — — CO in it as a standard. (71)(CH)Cr(CO)3 21.7 21.3 — 1.8 (71)(toluene)Cr(CO) 23.6 23.2 — 1.7 (71)(p-xylene)Cr(CO)3 25.7 25.0 — 2.7 (71)(mesitylene)Cr(CO) 27.6 26.9 — 2.5 (71)(mesitylene)Mo(CO)3 29.8 28.4 — 4.7 (73)(mesitylene)W(CO) 30.1 29.7 — 1.3 (73)(durene)Cr(CO)3 29.0 28.7 — 1.0 (71)(pentamethylbenzene)Cr(CO)3 30.5 30.6 + 0.3 (71)(Me6)Cr(CO) 31.8 32.4 + 1.9 (71)(t-butylbenzene)Cr(CO)3 28.8 28.7 — 0.3 (71)(p-di-t-butylbenzene)Cr(CO)3 35.9 36.1 + 0.6 (71)(1 ,3,5-tri-t-butylbenzene)Cr(CO) 42.4 43.5 + 2.6 (71)¶ The polarizabffities of the ligands are directly taken from reference 70 or calculated fromthe method suggested by reference 72.—34—where relative ion intensity is plotted for each ion intensity I relative to total ion currentintensity El1Relative ion intensity= If E I. (2.4)Reaction times as long as twenty seconds have been utilized and no significant change infinal products was observed (see A1.1 of Appendix Al).Elemental composition assignments of the ions can be determined by accurate massmeasurements. Our FT-ICR mass spectrometer is capable of mass measurements of goodaccuracy to permit elemental composition assignments. Mass measurement errors aregenerally less than 40 ppm as shown in Table 2.4 with every mass peak calibrated by itsleft and right neighboring peaks. CpCrNOCH had a maximum ion intensity, and henceTable 2.3 The primary positive ions from CpCr(NO)2D3(25 eV electronionization and sample nominal pressure = 6.5x10-7Torr)Primary positive ion mlz Relative intensity Electron count(%) for CrFragment ions:Cr 52 16 5CpCr 117 49 10CpCrNO 147 17 13CpCrNOCD 165 100 14CpCr(NO) 177 9 g 16Rearrangement ions:C5HNCD 97 1164D2Cr 132 26 9orllCpCrO 133 14 12Molecular ionCpCr(NO)2D 195 13 17¶ After a short delay time, this ion intensity increases to over 10%.—35 —c;1C190.0 0.2 0.4 0.6 0.8Time (second)1.0Figure 2.5 The temporal behaviors of products from the condensation reactions of Crwith the parent molecule CpCr(NO)2H3.Operating parameters are thosein Table 2.1 and the sample nominal pressure = 6.4x107Torr. Cr is theprimary ion; CpCr2Ois the secondary ion;Cp2r3(NO)O is the tertiaryion; Cp3r4O and Cp3r4(NO)20+ are the quarternary ions;Cp4r5(NO)03is the quinary ion.—36 —c;1C76543210Time (sec)Figure 2.6 The temporal behaviors of products from the condensation reactions of Crwith the parent molecule CpCr(NO)2D3.Operating parameters are thosein Table 2.1 and the sample nominal pressure = 6.5x107Torr. Cr is theprimary ion; CpCr2O4is the secondary ion;Cp2r3(NO)O is the tertiaryion; Cp3r4O and Cp3r4(NO)2O are the quarternary ions;Cp4r5(NO)O is the quinary ion. The relative ion intensity of CpCrO(m/z 185) has been calibrated for the isotopic peakC10H7D3r(m/z 185).— 37 —there was a non-identical unlike-ion interaction for CpCrNOCH ions relative to the otherions. This interaction will be discussed in Chapters 4—6. The resolution of the FT-ICRmass spectrometry was not very good for ion mass range over 400 Daltons. Therefore, thelarge errors (over 40 ppm) in mass measurements of ions CpCrNOCH,Cp3r(NO)O,Cp3r4O,Cp3r4(NO)O, andCp4r5(NO)0 can be expected.The ion-molecule condensation products are deduced from precision massmeasurements for the compositions of the product ions; the strong Cp-Cr bond (Cp-metalbond is generally inert to both nucleophilic and electrophilic reagents.(74)). The neutralproducts in the following reaction equations are suggested because of their stabilities, andtheir simple compositions.2.3.1. CrReactions2.3.1.1 Charger transfer and electrophiic additions of CrCharge transfer between ion and molecule followed by subsequent dissociation of themolecule is very common in MS-MS experiments.(75)Using triple resonance, chargetransfer reactions between Cr and the parent molecule are confirmed as in Eq. (2.4).(C5HN D + Cr + Cr0 + NOCP + CpCr(NO)2D3 CpCrNOCD + Cr + NO (2.4)CpCr(NO) + Cr + CD3177The mass-to-charge ratio (mlz) of every ion is indicated just below its formula. The Cr inCpCrNOCD 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 themetals in them are close to the 18-electron structure. Their subsequent reactions arediscussed in the individual sections for each of these two primary ions. Charge transferwas found in the reactions of CpCr, CpCrN0,C6H4D2r, and CpCr0 with the parent—38—Table 2.4 The identification for the ions from CpCr(NO)2H3Stoichiometry of ion Calculated miz Measured mlz Error (ppm)C6H8N 94.065126 94.064138 — 10.50CpCr 116.979086 116.981384 + 19.65C6Hr 129.986911 129.990027 +23.98CpCiO 132.974001 132.971305 —20.27CpCrNO 146.977075 146.982775 + 38.78CpCrNOCH 162.000550 161.987292 —81.84CpCr(NO) 176.975064 176.979096 + 22.78CpCr(NO)2H 191.998539 191.996018 — 13.13Cp2r 182.018212 182.014131 —22.42CPCr2O 184.914511 184.919162 +25.15CpCr2NO 198.917585 198.910078 —37.74Cp2rO 249.953636 249.959373 + 22.95Cp2rOH 250.961461 250.954970 —25.86Cp2rO 265.948551 265.941574 —26.23CprNOCH 278.980185 278.970193 —35.81Cp2rNO(CH3) 294.003660 294.002970 — 2.35Cpr(NO)H 308.978174 308.981741 + 11.55Cp2r(NO)CH3) 324.001649 323.998438 — 9.91Cpr(NO)O 331.892134 331.891092 —3.14Cp3r(NO)O 396.93 1259 396.901698 — 74.47Cp3r4O 450.863610 450.836560 —59.99Cpr(NO)02 464.866684 464.824477 — 90.79Cp4r5(NO)03 597.841233 597.894157 + 88.53¶ Calculated ion mass has been calibrated for electron mass.—39—molecules as well.In this chapter, we are concerned with the condensation chemistry such aselectrophilic additions by shown in Eq. (2.5), where the dominant reaction is markedbyastar*.CPCr2O + N20 + .033 *+ + CpCr(NO)2D3— cpCr2NO + NO + •Q33 (2.5)199The generation of CpCr2O and CpCr2NO showed that a linear NO ligand inCpCr(NO)2D3became bent under the attack of Cr. The reason that the yield of theO(oxygen)-bridge product, CpCr2O, predominates over the yield of the N(nitrogen)bridge product, CpCr2NO, could be attributed to the extra lone pair of electrons on theoxygen of bent NO ligand (see Figure 2.7). This process is comparable to the reaction ofFe with Fe(CO)5in which the CO served as a 4-electron bridge ligand.(45)2.3.1.2 CpCr2O ReactionsCpCr2O + CpCr(NO)2D3—* Cp2r3(NO)O + NO + •CD3. (2.6)185 332f CpCrO3+ NO + N2 + CD3451I Cp3r4NO) O + NO + CD3Cp2r3(NO) O + CpCr(NO)2D3 465 (27)332 prO+N2+DCN+D467Cp3r(NO)2O + NO +479Cp3r4O+ CpCr(NO)2D4-, Cp4r5(NO)O + NO + CD3. (2.8)451 598Cp3r4(NO)02+ CpCr(NO)2D3—, Cp4r5(NO)O + N2O + CD3. (2.9)465 598Subsequent reaction ofCp3r4O(m/z 467, close to mlz 465) cannot be determined by the-40--Cr(a) 0-bridging(b) N-bridgingFigure 2.7 Two proposed mechanisms for the bent nitrosyl ligand in the Cr-moleculecondensation reaction. (a) 0-bridging to Cr; (b) N-bridging to Cr.multiple resonance at present due to insufficient mass resolution. Because Cp3r4Oshows a lower reactivity thanCp3r4O(mlz 451) andCp3r4(N0)0 (mlz 465) and thereis an ion speciesCp3r40D(m/z 483) whose intensity grows more slowly than that of—41—Cp3r4(NO)2O(m/z 479) as shown in Figure 2.8, the product ofCp3r4O-moleculereaction could beCp3r4OD2.3.02.5b1.00.50.00.0Figure 2.80.2 0.4 0.6 0.8 1.0Time (second)The temporal behaviors (solid lines) of Cp3r4OandCp3r4OD.Operating parameters are those in Table 2.1 and the nominal samplepressure = 6.5x107Torr. The chemical reactivity ofCp3r4O is less thanthose of Cp3r4O andCp3r4(NO)O. Ion intensity ofCp3r4ODgrows more slowly than that ofCp3r4(NO)2O.2.3.1.3 CpCr2NOReactionsCpCr2N0 + CpCr(NO)2D3-Cp2r3(NO)O + N2O + •CD3199 332Eq. (2.7)(2.10)—42—The condensation of CpCr2NO includes addition of an oxygen ligand. Conversely, thecondensation of CpCr2Oincludes addition of a nitrosyl ligand. Cp3r4O andCp3r4(NO)O have the same tendency.The products in Eqs. (2.6)—(2.1O) indicate that only when the positive charge movesto the Cr unconnected to Cp, can the ion-molecule condensation proceed. This isapparently due to spatial freedom of the Cr unconnected to Cp. Without exception, thelargest cluster always comes from the reaction of a bare metal ion with molecules becauseof its largest electron deficiency and unsaturated coordination number. The largest clusterhere is the quinary ion,Cp4r5(NO)O. Typically, the clusters from the reactions of Crwith the parent molecule, CpCr(NO)2D3are of the form CpCr(NO)O, wherem=x+y,andn=m+ 1.2.3.2 CpCr Reactions2.3.2.1 Charger transfer and electrophilic additions of CpCrCharge transfer from CpCr to the parent molecule produces CpCrNOCD andCpCr(NO). The condensation processes of CpCr with the parent molecule are quitesimilar to those of the Cr-molecule reactions in previous section:(Cp2r0 + N20 + cDD31 250I CpCH4DCrO+ N20 + •CHD2 *CpCr + CpCr(NO)2D3— 251 (2 11)117 Cp2rOD + N2 + CD2O *252Cp2rNO + NO + cD3264In the condensations of CpCr with CpCr(NO)2D3,0-bridging is still favored overN-bridging from [Cp2rO] + [CpC5H4DCrO9 + [Cp2rOD] >> [Cp2rNO].—43—Moreover, a-hydrogen transfer from the methyl to form CpC5H4DCr2O(m/z 251) wasobserved in the CpCr-molecule reaction.2.3.2.2 CprO,CpC5H4DCr2O,andCp2rODReactionsResolution of our multiple resonance experiments is not accurate enough todistinguish these three secondary ions because their masses are very close. Their reactionsare discussed together.Cp2rt CpC5H4DCr *Cp Cr O 182 183250 Cp3r(NO) OCpCH4DCrO+C r(NO)D3— + (2.12)251 Cp25H4DCr(NO) 0+ 398252 Cp3r(N0)O D399The tertiary ion Cp2ris known to be very stable.(76)2.3.2.3 Cp2rNO Reactions(CpCr2O+5HN MCp2rNO + CpCr(NO)2D3— 185 (2.13)264 I Cpr3(NO)OC D + C5HN + NO350where M represents the parent molecule. Reactions of CPCr2) have been given in Section2.3.1.2. The intensity ofCp2r3(NO)OCD is weak and its subsequent reaction pathwaysare uncertain.2.3.3 CpCrNO ReactionsCpCrNO preferred to transfer its charge to CpCr(NO)2D3during coffision withCpCr(NO)2D3to produce CpCr (which may come from a dissociation of CpCrNO),CpCrNOCD and CpCr(NO). It did not show obvious condensation chemistry.-44-2.3.4. CpCrNOCD, CpCr(NO), and CpCr(NO)2DReactionsThese primary ions have similar reaction pathways and are discussed together.CpCrNOCD is the most abundant primary ion. Consequently, bimetal complexes aremajor products in the condensation chemistry of CpCr(NO)2D3fC2rO+ NO + N2 + C2D6I 266CprOD+ NO + N2 + •C2D5I 268I CprN0CD + 2N0 + •cD3 *CpCrNOCD + CpCr(NO)2D3 282 (2.14)165 CprNO(CD3)+ 2N0 *300Cp2r(NO)D+ NO + D3312Cr(NO)CD3)+ NO330These product ions in Eq. (2.14), exceptCp2rNOCD (mlz 282), reacted slowly withthe parent molecules ( three second reaction time under the pressure range of 6.2 —6.8x iO- Torr) to produce more stableCp2rO3(CD)(mlz 318) and a deuterated ion ofm/z 286 (more precisely, m/z = 285.97) whose composition remains to be proved yet.Several possible compositions exist for the latter ion, such as(5-CH4)2CrND(m/z285.964),(-C5H2CrO3D(m/z 285.973), andCp2r3ND(mlz 285.976) etc.The molecular ion CpCr(NO)2Dcan fragment to CpCrNOCD and CpCr(NO)through collisions with the parent molecule. Condensation of the molecular ion with theparent molecule occurs in the same way as does CpCrNOCD, except there is one moreNO in the right-hand of Eq. (2.14). CpCrNOCD and CpCr(NO)2Dreaction pathwayswere also confirmed at ionization threshold (nominal 9.7 eV). At the ionization threshold,the mass spectrum showed only two mass peaks, CpCrNOCD and CpCr(NO)2D, andtheir condensation products did not change with higher electron ionization energy.—45—Since, there is no methyl ligand in CpCr(N0), the condensation reactions ofCpCr(N0) with the parent molecule are deduced as:(CP2r0+ 2N0 + N2 + ‘CD3I 266CpCr0D+ N20 + N2 + CD 20CpCr(N0)+CpCr(N0)2D3 268k (2,15)282Cpr(N0)D+ 2N03122.3.5 CH4D2rReactionsDuring collision ofC5H4D2rwith the parent molecule, CpCr(N0)2D3chargetransfer occurred to produce Cr (which may come from a dissociation ofC5H4D2rj,CpCr, CpCrNOCD, and CpCr(N0). Another more complicated reaction ofC5H4D2rwith the molecule was:C5H4D2r+ CpCr(NO)2D3—C6H4D2rC5+ Cr + 2N0 + D2. (2.16)132 211Similar reactions (both the fragmentation and condensation) were also observed in the ion-molecule chemistry of(r6-benzene)Cr with(r6-benzene)Cr(C0)3.(77)Therefore, thecomposition of this primary ion seems to be(i7-benzene)Cr.2.3.6 CpCr0 ReactionsBecause ion mass ofC5H4D2r(m/z 132) is very close to that of CpCr0 (m/z133), the reactions of CpCr0 with the protium molecule CpCr(NO)2H3were monitoredusing triple resonance. Products from charge transfer between CpCr0 and the parentmolecule, CpCr(N0)2H3are CpCrNOCH and CpCr(N0). Condensation reactions ofCpCiO with the molecule are:—46—2.4 Ion-Molecule Chemistry of the Negative IonsNegative ion chemistry of CpCr(NO)2H3is simpler than its positive ion chemistry.There are only four primary negative ions from CpCr(NO)2D3:CpCrO, CpCrNOCD,CpCr(NO), and CpCr(NO)2D(Table 2.5) formed with 1.2 eV electron ionization. Thenitrosyls in these primary ions are considered to be linear. The negative charge in eitherCpCr(NO) or CpCr(NO)2Dmay delocalize onto the two NO ligands.(78)Table 2.5 The primary negative ions from CpCr(NO)2D3(1.2 eVelectron ionization and nominal pressure = 6.5x107Torr)Primary negative ion mlz Relative intensity Electron(%) count for CrCpCrO (superoxide) 149 2 13CpCrNOCD 165 100 16CpCr(NO) 177 5 17 ‘CpCr(NO)2D 195 24 18(Cp2rOCH + 2N0265+ CpCr(NO)2H3 Cp2rOH+ N2 + CH2O133 267CprOH+ N2 + OH280(2.17)c7H30 0¶ If the negative charge is delocalized onto the dinitrosyl.—47 —CpCrO and CpCr(NO) underwent electron transfer reactions to the parent molecule,CpCr(NO)2D3.The reactions of CpCrNOCD with the molecule gave rise to amononitrosyl chromium complex and a bimetallic condensation product.CpCrO + CpCr(NO)2D3— CpCr(NO)2D+ [CpCrO2] (2.18)149 195CpCr(NO) + CpCr(NO)2D3-* CpCr(NO)2D+ [CpCr(NO)2]. (2.19)177 195(cpcrNO(cD3)+ [CpCr(NO )2]•CpCrNOCD + CpCr(NO)2D3 183 (2.20)165 Cp2r(NO)D+ DD3 *34217-electron radical CpCrNO(CD3)should possess a “three-legged piano stool” molecularstructure, like those of the 17-electron mononitrosyl chromium complexes, CpCrNO(L)I (L= PPh3,P(OPh)3,P(OEt)3and CpCrNO(PPh3) H2SiMe.(9Triple resonance experiment showed that some CpCr(NO)2Dions decomposed toproduce CpCrO and CpCrNOCD. The temporal variation of the CpCr(NO)2Dintensity displayed in Figure 2.9 shows that two kinds of CpCr(NO)2D ion speciesprobably exist: excited state CpCr(NO)D* and ground state CpCr(NO)2D (Theunusual temporal behavior (decrease and then increase) was reproducible and measuredfour times (Table 2.9)). We believe that the excited state negative molecular ion is aCpCrO ion source (Eqs. (2.21)). There are three possible decay pathways for the excitedstate CpCr(NO)2D*ion: radiation decay, unimolecular decomposition, and bimolecularreaction.(80)Further studies are needed.CpCr(NO)2D*— CpCrO + N2 + CD3. (2.21a)195 149.1.Eq. (2.18)—50—d =— k’ [A] (2.24)wherek’ k [MJ. (2.25)Integration of Eq. (2.24) givesin [Ak] =— k’ t + in [A]0, (2.26)where [Aj0is the initial intensity of A. In order to correct for fluctuations in ion intensitymeasurement of FT-ICR, the normalized ion intensities [Aj0 and [Ak] are used to plotagainst time. If [M] is calibrated by the method suggested in Section 2.2.3 TTPressureCalibration’, 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(43)to measure the rateconstants for each ion from the decay portion of each temporal intensity curve. The decayportions of the curves were found to give satisfactorily linear logarithmic plots.The rate constants, averaged from six measurements, of the primary positive ions fromCpCr(NO)2H3,are given in Table 2.6. Since gas-phase reactions can be convenientlystudied by measuring partial pressures of the reactants and products, the units of the rateconstants given here are Torr1•sec.1 Torr’sec = 3.1x1017cm3•moleculessec.In Table 2.6, the uncertainty of the rate constant measurements for highly reactive ions isless than ± 10- 15%. The big errors in the rate constants of CpCrNO and CpCr(NO) aredue to their low ion intensity. It is interesting to note that the reaction rates of the primarypositive ions in which Cr has an even electron structure are not proportional to theirelectron deficiencies: CpCr(NO) (16-electron) is not more active than CpCr(NO)2H(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 ofC6Hr(11-electron).Although a smooth relationship curve between the rate constants of these primary ions andtheir electron deficiencies, plotted in Figure 2.10, also can be found, as proposed byWronka and Ridge,(29)there is at least a 1 electron error in the electron deficiency model.The rate constants of all positive condensation ions from CpCr(NO)2H3shown inEq. (2.5) — Eq. (2.17) are listed in Table 2.7. The measurement errors are calculated at95% probability. The precision of rate constant measurements is less when the reactivitiesof the ions listed in Table 2.7 are less or when there is poor mass resolution in the high ionmass region. For comparison, the rate constants of all positive ions from CpCr(NO)2D3are listed in Table 2.8. There is little difference between the rate constants of ionCpCr(NO)2H3reactions and those of ion-CpCr(NO)2CD3reactions. The relative rateconstants in Table 2.6 are independent of the pressure calibration accuracy.Table 2.6 The experimental rate constants of the primary positive ions fromCpCr(NO)H3(25 eV electron ionization)Positive k x i0 Torr•sec Relativeprimary ions 1 2 3 4 5 6 Average Error rate (no unit)C? 23.4 23.7 23.8 22.7 24.0 21.6 23.2 ±0.9 1.000CpC? 19.3 20.3 16.8 19.5 21.2 19.1 19.4 ±1.5 0.836C6Hr 19.9 20.1 16.3 18.6 19.0 20.1 19.0 ±1.5 0.819CpCrO 16.7 18.4 14.8 18.6 16.0 15.4 16.6 ±1.6 0.716CpCrNO 16.9 20.8 15.8 19.0 22.9 15.9 18.6 ±3.0 0.802CpCrNOCH 11.2 10.2 9.26 9.59 11.3 9.45 10.2 ±0.9 0.440CpCr(NO) 5.00 7.47 6.56 8.32 9.06 4.97 6.9 ±1.8 0.30CpCr(NO)2H 9.71 8.94 7.21 7.43 9.95 8.05 8.5 ±1.2 0.37—52—19.519.218.9Ink18.618.318.017.714Electron DeficiencyFigure 2.10 Plot of in rate constants vs. electron deficiency for the primary ions fromCpCr(NO)2H3.2.5.2 Rate Constants of the Negative IonsThe rate constant measurements of the negative ions produced from CpCr(NO)2D3using the electron ionization, measured four times, are given in Table 2.9. Forcomparison, the rate constants of the corresponding negative ions from CpCr(NO)2H3are given in Table 2.10. The decay rate constant of the excited state negative molecular ion,CpCr(NO)2D*,cannot be determined, since its reaction mechanism is not resolved as yet(please see Eqs. (2.12a), and (2.21b)).0 2 4 6 8 10 12—53—Table 2.7 The experimental rate constants of the positive condensation ions fromCpCr(NO)H3(25 eV electron ionization)m/z Positive k x107 Torr•secproduct ion 1 2 3 Average Error185 CpCr2O 11.8 12.7 12.5 12.3 ±1.2199 CpCr2NO 11.0 11.2 11.4 11.2 ±0.5250 Cp2rO’ 9.28 9.06 9.21 9.2 ±0.2251 Cp2rOH4 1.76 2.06 2.22 2.0 ±0.5264 Cp2r,NO 1.63 1.78 1.90 1.8 ±0.4265 Cp2rOCH 1.82 1.63 1.48 1.6 ±0.4266 Cp2rO 0.40 0.35 0.47 0.4 ±0.1267 Cp2rOW 0.23 0.12 0.37 0.2 ±0.2279 CprNOCH 0280 Cp2rH 0285 0294 Cp2rNO(CH 0.11 0.16 0.53 0.3 ±0.6309 Cpr(NO)H 0.034 0.15 0.23 0.1 ±0.2312 CP2rO3(CH) 0324 Cpr(N0)CH <<0.05332 Cp2r3(NO)O 7.36 8.32 7.23 7.6 ±1.5347 Cpr(NO)OCH 2.35 2.27 1.89 2.2 ±0.6397 Cp3r(NO)O <0.05398 Cpr(NO)OW <<0.05451 Cp3r4O 6.15 6.89 4.08 5.7 ±3.6465 Cpr(NO)02 4.96 3.90 2.33 3.7 ±3.3467 Cp3r4O 3.63 3.66 2.94 3.4 ±1.0479 Cpr(NO)20 0481 Cp3r4OH 0598 Cpr5(NO)0 0—54—Table 2.8 The experimental rate constants of the positive condensation ions fromCpCr(NO)cD3(25 eV electron ionization)m/z Positive kx107 Torr•sec m/z Positive kx107 Torr•secprimary ion product ion52 Cr 23.3 185 CpCr2O 12.1117 CpCr 16.7 199 CpCr2NO 8.83132 C6H4D2r 20.7 250 Cp2rO 10.1133 CpCrO 16.8 251 CpC5H4DCr 10.3147 CpCrNO 16.6 252 Cp2rOD 2.00165 CpCrNOCD 9.75 264 Cp2rNO 1.35177 CpCr(NO) 8.64 268 Cp2rOCD — ¶195 CpCr(NO)2D 8.63 266 Cp2rO 0.21268 Cp2r0D — ¶282 CprNOCD 0282 Cp2rD 0286— 0300 Cp2rNO(CD 0.21312 Cpr(NO)D 0.098318 Cp2rO3(CD 0330 <<0.05332 Cp2r3(NO)O 6.80350 Cpr(NO)OCD 2.64397 Cp3r(NO)O <0.05398 Cp25H4DCr(NO)O <0.05399 Cp3r(NO)OD <<0.05451 Cp3r4O 5.40465 Cpr(NO)02 4.08467 Cp3r4O 2.71479 Cpr(NO)20 0483 Cp3r4OD 0598 Cpr5(NO)0 0¶ 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 fromCpCr(NO)2D3(1.2 eV electron ionization)m/z Negative ions k x107 Torr•sec1 2 3 4 Average Error149 CpCrO 7.95 9.29 8.82 7.67 8.4 ±1.2165 CpCrNOCD 8.73 9.74 10.1 9.95 9.6 ±1.0177 CpCr(NO) 8.86 10.1 10.6 10.4 10.0 ±1.2195 CpCr(NO)2D—195 CpCr(NO)2D 0183 CpCrNO(CD 0342 CP2r(NO)D3 0Table 2.10 The experimental rate constants of the primary negative ions fromCpCr(NO)2H3(1.2 eV electron ionization)m/z Negative kx107 Torr•sec ,,, Negative kx107 Torr•secprimary ions product ions149 CpCrO 8.2 192 CpCr(NO)2H 0162 CpCrNOCH 10.3 177 CpCrNO(CH3) 0177 CpCr(NO) 10.0 339 Cp2r(NO)H 0192 CpCr(NO)2H -—56—2.6 Proposed Ion Structures2.6.1 Cluster Ions from the &-Molecule Condensation ReactionsAll these reactions are written in Eq. (2.5) — (2.10) of Section 2.3.1. CpCr2OandCpCr2NOare the smallest condensation complexes here, whose structures have beenproposed in Fig. 2.7. A series of interesting oxo compounds are produced from these twoions, which are as follows: Cp2r3(NO)Ot Cp3r4O,Cp3r4(NO)O, Cp3r4O,Cp3r4(NO)2O,Cp3r4OH, and Cp4r5(NO)O. The largest known neutralchromium cluster complex is pentanuclear[Cp2r(SCMe3)SJin which four sulfursserve as J13 bridge ligands to connect five chromiums with a “bow-tie” frame.(83) IfCp4r5(NO)03has a similar structure, its precursors Cp2r3(NO)Ot Cp3r4O, andCpr(NO)02can be suggested as those in Figure 2.11. The four Cr atoms in bothCp3r4OandCp3r4(NO)O may have a “i” skeleton like a shovel to which can beadded one more Cr to form a bow-tie frame. In the shovel skeleton, the positive charge islocalized on the central Cr with no Cp ligand as an active center. Successively, the three Cratoms inCp2r3(NO)O could be of a triangular skeleton. With reference to Bottomley’swork on oxo chromium cubanes Cp4rOandCp4r(C5H)03,(84Cp3r4O wasnever found in the mass spectra of these two oxo compounds, but a strong Cp3r4Omasspeak was observed.(8485) Bottomley’s work provides a clue to the structures ofCp3r4Oand Cp3r4Oas found in our FT-ICR mass spectra: the four Cr atoms inCp3r4Owould not form a tetrahedron, but the four Cr atoms in Cp3r4Owould. Thus,Cp3r4OHmay have a cubane-like structure as its possible precursor, Cp3r4O.The non-reactivity ofCp3r4(NO)2Oshows that it may have a different structurefrom the shovel skeleton ofCp3r4OandCp3r4(NO)O. Cp3r4(NO)2Ois assumedto be tetrahedral. There may exist double metal-metal bonds in those clusters shown inFig. 2.11. Before qualitative analysis of their molecular orbitals, the metal-metal bonds areindicated by single bold lines.—57 —Cp3r4(NO)20Proposed structures of the cluster ions from the Cr+mo1ecule condensationreactions. The metal-metal bonds are indicated by bold lines. Forsimplification, the metal-metal bonds in clusters Cp3r4O andCp3r4OHare not shown.CpCr.”CrCp2r3(NO)OLCp3r4O0Cp3r4(NO)O;rCpCp4r5(NO)Or=CH2CpCp3r4OCpCp3r4OHFigure 2.11—58—2.6.2 Cluster Ions from the CpCr-Molecule Condensation ReactionReactions involved with CpCr are listed in Eq. (2.11) — (2.13) of Section 2.3.1 andtheir proposed structures are drawn in Figure 2.12. Cp2rOandCp2rNO may havestructures similar to those of CpCr2Oand CpCr2NO, shown in Fig. 2.7. Because therate constants ofCp2rOH andCp2rNOare lower than that ofCp2rOby a factorof 5 (Tables 2.7 and 2.8), the 0 and H in Cp2rOW are assumed to bep2-bridgingligands (rather than a OH ligand) between the two chromiums in order to explain the lowerreactivity of Cp2rOH. Then, the proposed structures of tertiary cluster ionsCp3r(NO)O andCp3r(NO)OH are presumed on the basis of those ofCp2rOandCp2rOH.2.6.3 Other Bimetallic Cluster IonsThe dimer chromium clusters are the major product ions in reactions described in Eq.(2.14) — Eq. (2.17). The oxo bimetallic Cp2rOis relatively inert toward condensation(its rate constant is -. 0.3 x Torr/sec at nominal pressure 6.5x107Torr). Thisinteresting ion has been found in the mass spectra of [CpCr(NO)(OMe)]279a andCp4rO.(85)Bottomley et a!.(86)predicted that any cluster of form CpMetL, where Lis a ligand other than Cp, could exist, when the cluster satisfies the structural geometry ofEuler’s polyhedron theorem:(87)for every convex polyhedron the number V of its vertices plus the number F of itsfacesminusthenumberEofitsedgesisequalto2,i.e.,Vi-F—E = 2.CP2rOmay be the smallest cluster, in which Cr20forms a tetrahedron (Figure2.13) similar to the non-planar four-membered ring Cr20inCp2r(O-t-Bu) synthesizedby Chishoim et a!. in 1979.(88) Furthermore, the oxidation states of the chromium atoms in2Cr0are +2 and +3. Although such a new i-i72-dioxygen ligand has not yet been—59—reported, the u-n2- ligands of other non-metallic elements (S, Se, and Te) in group VIBhave been synthesized.(8990)The most abundant metallic dimer in this work is the extremely stable ion,Cp2rNOCH. When two chromiums in a metallic dimer are Cr(II) and theircoordination numbers are five or six, a quadruple Cr-Cr bond is possible.(91)Cp2rNOCH probably possesses a quadruple metal-metal bond.CPCr\/CrcPcr<>:Cp2rO Cp2rOH0CpCr\,/Crl+Cp3r(NO)OCp3r(NO)OH Cp2r3(NO)OCHFigure 2.12 Proposed structures of the cluster ions from the CpCr-molecule-Cpcondensation reactions.-60--pccpCP2rOcPcK: Cft><;:pC C flT.1+_p2ri.i.iCp2rNOCH Cp2rOH0cPCr,4PCPH3 H3Cp2rNO(CH) CP2r(N0)3 Cp2r(NO)CH3)orCpCrZ NrcpCp2rO3(CH)0H3-rH20 0Figure 2.13 Proposed structures of all the other positive bimetallic cluster ions.—61—Two plausible structures forCp2rO3(CH)are one containing a non-planar fourmembered ring Cr20 as inCpr(O-t-Bu),(88and one containing a planar fourmembered ring Cr20as in[5-CMeCrO2].(92)The two proposed structures of thisbimetallic ion are drawn at the bottom of Figure 2.13.2.6.4 The Negative IonsAlthough the negative ion CpCrO is a fragment ion from CpCr(NO)2H3,itsstructure is of interest. It can be an oxide, peroxide, or superoxide. The structure of theanionic dioxide Cr(CO)3Owas proved to be either a peroxide or a superoxide by theCollision-Induced Dissociation (CD) method of FT-ICR.(93) The authors, Bricker andRussell, preferred the superoxide structure. Therefore, the CpCrO probably is asuperoxide, too, for the reason that mononuclear superoxo complexes are formed almostexclusively by metals of the first transition series because of formal one-electron oxidationrequirement of the metal,(94)and the CpCrO-molecule reaction is an one-electron transferreaction (Eq. (2.18)). The proposed structure of the CpCrO is drawn in Figure 2.14 withCp2r(NO)H, the only bimetallic anion, in which every Cr has 16 electrons(assuming a Cr-Cr single bond).CpCrO Cp2r(NO)H00—i-H3Figure 2.14 Proposed structures of CpCr0 andCp2r(NO)H.—62—2.7 DiscussionCluster research is a very active field. Cluster structure has been recognized as a linkbetween gas and solid phases.(95)The enormous effort being focused on metal carbonylclusters 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 metalvapor synthesis in organometallic chemistry.(12’98) Compared with carbonyl, nitrosylligands show many unusual bonding features (Table 2.11, also see Appendix A1.2, TableA1.1 Coordination Modes of Carbon Monoxide and Table A1.2 Coordination Modes ofNitric Oxide). In contrast with the positive ion chemistry of transition metal carbonyls, theNO ligand can be broken up in the ion-molecule reactions of CpCr(NO)2H3.Typically,oxygen is retained in the ionic product and nitrogen is lost as a part of the neutral (evendinitrosyl 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 fortransition metal carbonyls, the structure-reactivity relationship (electron deficiency model)suggested by Wronka and Ridge(29)on the basis of the 18-electron rule provides only anindication 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 ligandsof these complexes vary greatly, their rate constants are very difficult to relate to electrondeficiencies. No plot of rate constant vs. electron deficiency for the condensationcomplexes is attempted here.The ion-molecule condensation chemistry of CpCr(CO)2N0()is not as complicatedas that of CpCr(NO)2H3.Since there is one more nitrosyl in CpCr(NO)2H3than inCpCr(CO)2N0, it is expected that more oxo compounds can be produced from the ionmolecule condensation chemistry of CpCr(NO)2H3.For example, oxo compounds—63—Cp2rO,Cp2rO,Cp3r4O,Cp3r4(NO)2O,andCp4r5(NO)O (Section 2.3),which are produced from the ion-molecule condensation reactions of CpCr(NO)2H3,were not found in those of CpCr(CO)2N0.() In the ion-molecule chemistry ofCpCr(NO)2H3,many unreacive bimetallic complexes were produced from the reactionsof primary ions CpCrNOCH, CpCr(NO), and molecular ion CpCr(NO)2H with theparent molecules (Section 2.3.4). In contrast, in the ion-molecule chemistry ofCpCr(CO)2N0,()its molecular ion is unreactive, and only one unreactive bimetallicproduct,Cp2r(NO),was observed.Advantages of gas-phase studies arise from the spatial dispersion of theorganometallic species under high vacuum conditions employed. In gas-phase ion-molecule reactions, there is no solvent-shell interaction and associations by ion pairs.C5HNCH,(101) Cp2r,(766-CH)2Cr,(102) and CpCr(NO)2H,(81) which havebeen formed in solid or liquid state, were all found to be unreactive in the gas state withoutcounter-ions or solvent-shell effect. Moreover, (r-C6H)2Crproduced from the reactionof(r6-CH) r with CpCr(NO)2H3(Eq. (2.16)) shows that the Cp-ring may beextended to form a benzene ring through the ion-molecule reaction.If possible, binary nitrosyls Cr(NO)4and Co(NO)3,and cyclopentadienyl nitrosyls[CpCr(NO)2],[CpFeNO]2,andCp3Mn(NO)4are recommended for further study of theion-molecule condensation chemistry of nitrosyl transition metal complexes. Gas-phaseion chemistry of these nitrosyl complexes can be made a comparison with that ofCpCr(NO)2H3.The intrinsic properties of coordination of NO ligand to transition metalatoms will be further revealed, since Cr(NO)4,Co(NO)3,[CpCr(NO)2],[CpFeNOJ2,andCp3Mn(NO)4do not contain methyl or other ligands (except the inert Cp ligand in laterthree nitrosyl complexes).Nitrosyl behavior in transition metal complexes has been also studied recently by theCD method of FT-ICR(1031)in which a triangular structure ofCo3(j1-NXJtOY’ wasproposed, and chemical ionization method in which carbonyl-nitrosyl condensation metal-64-clusters up toCo7(CO)8N )aandFe8(CO)3N )j and pure nitrosyl metal clusters up toCo6(NO)j andFe7(NO)j were reported.(105)Table 2.11 shows a summary of the important characteristics of carbonyl transitionmetal cluster ions and nitrosyl transition metal cluster ions. 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Acta 1991, 185, 103-117.—72—CHAPTER 3REACTIONS OF THE FRAGMENT IONS OFCpCr(NO)2H3WITH SMALL MOLECULESN2, 112, H20, D20, NH3, AND CH4 IN THE GAS-PHASE—73—3.1 IntroductionStudy of the reactivity of gas phase ionic transition metal complexes with inorganicand organic small molecules is another important application of FT-ICR massspectrometry. When a second species such as H2 is introduced with a transition metalcomplex into the FT-ICR mass spectrometer, reactions which differ from the condensationchemistiy occur (e.g., ligand substitution, oxidation-addition of atomic metal ions etc.). Ofparticular interest to chemists are the reactions between the second species and ions derivedfrom the transition metal complex. The bare metal or bare metal cluster products fromtransition metal carbonyls reacting with small molecules have been studied frequently inorder to understand metal-ligand and metal-metal binding energetics. Many studies havebeen summarized in several recent reviews,(14)especially in a new review presented byEller and Schwarz in 1991,() where the literature has been covered up to early 1991. Atrend 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 thesecond-row transition metal ion Ag with tribenzocyclothyne,(5)Fe, Mn, and ligated Cr,Fe, Co, Ni ions with porphine,(6)and Fe, Cow, Ni, Cu, Rh, La, and VO withbuckminsterfullerene (C),(79) Fe with thpeptide val-pro-leu.(10)In this work, reactions of ions derived from the metal nitrosyl CpCr(NO)2H3(orCpCr(NO)2D3)with small molecules were examined. Small molecules H2,H20, NH3,and CH4 have been chosen in the present work as the reactants, because all of these smallmolecules contain hydrogen atoms (from 2 to 4). N2 was also used. In our experiments,H2,H20, D20, NH3, CH4, and N2 were found not to react with the negative ionsproduced from CpCr(NO)H or CpCr(NO)D3.Therefore, only the positive ionchemistry of CpCr(NO)2H3and CpCr(NO)2Dwith the six small molecules arediscussed in this chapter.So far only a few studies have been carried out on gas-phase chemical behavior ofcobalt nitrosyl ions with small molecules.(1114)When the cobalt nitrosyl ions reacted with—74—small molecules CO or 02,(12_14) the NO ligand in the cobalt nitrosyl ions was completelyreplaced by CO or 02, or the N-O bond was broken. In the latter case, the nitrogen wasretained by the cobalt atom, but the oxygen was lost. Obviously, gas-phase ion chemistryof metal nitrosyls with small molecules is a developing research area.3.2 Experimental SectionThe instrumentation of the FT-ICR mass spectrometer has been described in Section2.2.1 of Chapter 2. The ions were produced by electron impact on CpCr(NO)2H3at 25eV energy. N2,H2,NH3, and CH4 were commercial pure gases obtained from Mathesonof Canada, Ltd. H20 was obtained directly from distilled water and 99.9% purity heavywater, D20, was from Merck & Co., Ltd. Chromium nitrosyls CpCr(NO)2H3andCpCr(NO)2D3were provided by the research group of Professor P. Legzdins.Chromium nitrosyl CpCr(NO)2H3,or CpCr(NO)2D3,was first introduced intothe analyzer cell to a nominal pressure of 3.0 x iO- Torr. Then, a second molecular gasH2,H20, D20, NH3, CH4, or N2 was added into the analyzer cell at a total nominalpressure 3.0 x 10 Torr. A high pressure of the second molecular species was used toobtain a high reaction efficiency and to avoid confusion with the condensation reactionsdiscussed in Chapter 2. The ion-molecule reactions of each small molecule withCpCr(NO)2H3were monitored from 0 to 5 seconds. In order to eliminate all reactionsbetween ions produced from H2,H20, D2O, NH3, CH4, or N2 with the molecules, adouble resonance was performed during the beam time (ionization time) in which theresonance frequency for ejection was set to the corresponding resonance frequency ofH2O (mlz 18), or D2O (mlz 20), or NH (mlz 17), or CH (m/z 16), or N (m/z 28).Because the resonance frequency of H (mlz 2), 14.6 MHz, can not be reached by theequipment at 1.9 Tesla magnetic field, no ejection was applied for H. However, it wasfound that there was no signfficant difference between the spectra with and without the—75—double resonance because productions of these primary ions from H2,H20, D20, NH3,CH4,and N2 are limited at 25 eV ionization energy.3.3 Calibrations of the Nominal PressuresIn Section 2.2.3 of Chapter 2, the chemical sensitivity of an ion gauge was shown tovary with different gases, and so the true pressure of a gas is not the nominal pressuretaken from a vacuum meter. According to Bartmess and Georgiadis,(15)the true pressureof a sample x— nominal pressure of x 3 1x R (.)where R is the relative sensitivity of the ion gauge with respect to the sample x. Thechemical sensitivity of CpCr(NO)2H3has been estimated, in Section 2.2.3 of Chapter 2,tobeRCpcr(NO)Cp = 7.1.The molecular polarizabilities of N2,H2,H20, NH3,and CH4 have been reported as 1.00,0.44, 0.97, 1.12, and 1.62 A3, respectively.(15)As the partial pressures of N2,H2,H20(or D20), NH3, and CH4 were much larger than the partial pressure of CpCr(NO)2H3inour experiments, the total sampling pressures in the analyzer cell are taken as the pressuresof N2,H2,H20, NH3, and CH4. The calibrated pressures of N2,H2,H20, NH3,CH4,and CpCr(NO)H3are calculated from Eq. (3.1) and shown in Table 3.1.—76—Table 3.1 Calibrations of the sampling pressures of N2, H2,H20, NH3,CH4, and CpCr(NO)2H3Sample Nominal pressure Chemical sensitivity Calibrated pressureN2 3.0 x 10Toff 1.00 3.0 x 10ToffH2 3.0 x 10 Torr 0.44 6.8 x 10 TorrH20 3.0 x 10Toff 0.97 3.1 x 10ToffNH3 3.0 x 10Torr 1.12 2.7 x 10ToffCR4 3.0x10Toff 1.62 1.8x10ToffCpCr(NO)2H3 3.0 x i0 Torr 7.1 4.2 x iO Torr3.4 Ion-Molecule Chemistry of CpCr(NO)2H3with Hydrogen, H23.4.1 Collisional Quenching and Charge-Transfer ReactionsAll small molecules used in this work can quench the primary ion Cr from its excitedstate to its ground state. As a result, high nuclear metal condensation products whichmainly came from the successive condensations of Cr with CpCr(NO)2H3(as discussedin Section 2.3.1 of Chapter 2) almost disappeared. The spectrum shown in Figure 3.1 (a)was obtained under the nominal pressures of CpCr(NO)2H3and H2 given in Table 3.1with no delay time. After a 100 ms delay time, none of the high nuclear condensationproducts such as Cp3r4O,Cp3r4(NO)O,Cp3r4(NO)2O,Cp3r4O, andCp4r5(NO)O 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.CFb9UIzU>‘-4FaIUFUFzILiWI-’FaIUFigure 3.1 (a) Spectrum of pure Cr obtained by the triple resonance. (b) After a 100ms delay time, large amounts of product ions from charge transfer of Cr toCpCr(NO)2H3appeared.CU90ri iiiiijiIIIjTI1I iiiiCC 2CC11P53 ]NC‘-4I I I I I I I I I I li’ ICC 5CC 6CCP.M.U,(a)CpCrNOCH(Product of thecharge transfer)(Product of thecharge transfer)Delay time =100 msI i iC1 1111.M55 ]N4CIt-I.p U5CC(b)—78—to the parent molecule, CpCr(NO)2H3followed by dissociation, appeared in largeamounts. Nevertheless, some condensations products, such as CpCr2O(mlz 185) andCpCr2NO (m/z 199), can be still observed in the spectrum of Fig. 3.1 (b). A comparisonwith Fig. 2.4 (b) in Chapter 2, after the same 100 ms delay time, showed that the ion-molecule reactions of Cr with pure CpCr(NO)2H3were predominantly condensations.The reactions of Cr with H2 and CpCr(NO)Hare summarized as follows,collisional quenching: Cr + H2 (i + H, (3.2a)other reactions*k2of Cr+* with H2 : Cr + H2 products, (3.2b)kCpCrNOCH+NO+Crcharge transfer: Cr + CpCr(NO)2CH3 3 162 (3.2c)52 CpCr(NO) + CH3 + Cr177k CpCr2O+ N20 + •CH3condensation: Cr+* + CpCr(NO)2H3 185 (3.24)52 CpCrNO + NO + •CH3‘ 199where the superscript” “ indicates the ion or the molecule in an excited state, k1, k2, k3,and k4 are the respective rate constants of the reactions in Eqs. (3.2), and mass-to-chargeratio of each ion is given just under the ion formula. The collisional quenching and otherreactions of Cr+* with H2 (Eqs. (3.2a) and (3.2b)) have been studied by Elkind andArmentrout.(16) These secondary ions in Eqs. (3.2c) and (3.2d), CpCrNOCH,CpCr(NO), CpCr2O, and CpCr2NO, reacted further with CpCr(NO)H3according tothe reactions described in Section 2.3.4 of Chapter 2. Temporal behavior of the five ionsin Eqs. (3.2) is shown in Figure 3.2. The temporal variation of Cr did not show apseudo-first order kinetics as in the reactions of Cr with pure CpCr(NO)2H3—79—C(Fig. 2.5 of Chapter 2). The reactions of Cr with all of the other small molecules, H20,NH3, CH4, and N2, showed the same kinetics (Appendix A2). Since [H2] and[CpCr(NO)2H]are much larger than the total ion intensity, they can be considered to beconstant concentrations in the reactions. Then, the reactions in Eqs. (3.2) can be reducedto pseudo-first order reactions. The rate laws would bed [Dr] =— k1 [112] [Cr+*] — k2 [H2] [Cr+*] — k4 [CpCr(NO)2H31[Cr+*]—1-3-5-7-91.0Figure 3.2 Temporal behavious of Cr, CpCrNOCH, CpCr(NO), CpCr2O, andCpCr2NO in H2 medium. The temporal variation of Cr obviously doesnot obey pseudo-first order kinetics.0.0 0.2 0.4 0.6 0.8Time (second)= _(kj+kj+k)[Cr+*]. (3.3)—80—andd Dr]= k1 [112] [Cr+*] — 1c3 [CpCr(NO)2H][&]= kj [Cr+*] — kj [Cr1 (3.4)where= k1 [H2], (3.5)= k2 [H2], (3.6)k = k3 [CpCr(NO)2H], (3.7)= k4 [CpCr(NO)2H3}. (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 + k3 [Cr] d t = k’1 [Cr+*]0e — “ + + k ) d t. (3.10)After multiplying both sides of Eq. (3.10) by ee d [Cr] + ic’3 [Cr] e d t = k [Cr+*jo e — — — k td t. (3.11)—81—Integration of Eq. (3.11) gives[Cr1=e’ +Ce’. (3.12)where C1 is the integration constant. Since at t =0, [Cr1 = [Cr10,initial ion intensity ofCr in the ground state by electron ionization, the integration constantC= [Cr10— k —k—k— k (3.13)Thus, the total rate equation of Cr4 is the sum of Eq. (3.9) and Eq. (3.12)[Cr+*] + [Cr1— k — k — k (÷* —(k+k÷k)t— b i,., [r ]0e— “1 — —/ k’ [Cr+*]+ (3.14)The right-hand of Eq. (3.14) is a double exponential function of time t which has thefollowing formfQ) = b1e_b2 + b3e1’. (3.15)wherefQ) = [Cr+*] + [Cr], (3.16a)—82—b1= k—k—k—k[Cr+*]0, (3.16b)b2 = + k + (3. 16c)b3= [Cr]o— kk[Cr+*]ok (3.16d)b4 = k. (3.16e)Such a function was designed and input into the Macintosh “Igor” program for curve fittingof the ion intensity ([Cr+*1 + [Cr1) vs. time t. The curve fitting equation was[Cr*] + [Cr] = 0.00612 e 46.43t + 0.01832 e 9.130t (3.17)where t = 0 — 0.4 s. The curve of ([Cr+*] + [Cr1) vs. t fitted very well (sum of squarederror = 1.2 x 10 only), as shown in Figure 3.3. The rate constants are then found to be= 9.130 sec1 and k + k + k = 46.43 sec1. The calibrated pressure ofCpCr(NO)2H3is 4.2 x 1O Torr (Table 3.1).k3 = k / [CpCr(NO)2H3]= 9.130 sec1 / 4.2 x 108 Torr= 2.2 x 108 Torf•sec = 6.7 x i0cm3•molecules1s cThe k3 value is very close to the decay rate constant of Cr in last chapter (Table 2.6).Theoretically, if the initial ion intensities [Cr+*]o and [Cr]0 are known, k1, k2, and k4 canbe evaluated. Accuracy of the rate constants solved from the curve fitting method wasestimated at ±30%.(1718)— 83 —0.0250.0200.0150.0100.0050.0000.5Figure 3.3 Curve fitting for ion intensity ([cr+*J + [CrJ) vs. time t. ExperimentalCpCrNOCH3NH+ C5HN + Cr0179CpCrN + CpCr(N0)2CH3Cr(NO)2NH+C5HH3+ Cr1341940.0 0.1 0.2 0.3 0.4Time (second)data were obtained in 112 medium.3.4.2 Ligand SubstitutionsH2 reacted to replace oxygen of the ligand NO in both CpCrNOCH (m/z 162) andCpCr(NO) (mlz 177) to form a NH3 ligand.CpCrNOCH +2 112 - CpCrNH + 0C112+ 112. (3.18)162 134CpCr(N0) +2 H2 —> CpCrNH + NO + •OH. (3.19)177 134(3.20)—84—In Eq. (3.20), the methyl ligand, CH3 or the nitrosyl ligand, NO, of CpCr(NO)2H3wasreplaced by the ammonia ligand. Temporal variations of CpCrNOCH3NH(m/z 179) andCpCr(NO)2NH (m/z 194) are shown in Figure 3.4. Both the product ionsCpCrNOCH3NHand CpCr(NO)2NHare unreactive. This behavior can be explained bytheir electron structures. The metal in CpCrNOCH3NHhas a 16-electron structure andthe metal in CpCr(NO)2NH has a 18-electron structure. Using deuteratedCpCr(NO)2D3,the methyl ligand was found to be a hydrogen source for ammonialigand, too.CpCrNOCD+2H2 - CpCrNH+OCD2+HD. (3.21a)165 134CpCrNOCD+ H2 -* CpCrNDH + OCD2. (3.21b)165 135(CpCrNOCD3NH+ C5HN + Cr0CpCrNH + CpCr(NO)2D3 182 (3.22a)134 1 CpCr(NO)2NH +C5HD3+ Cr194(cpcrNOCD3NDH+ C5HN + Cr0CpCrNDH + CpCr(NO)2D3— 183 (3.22b)135 CpCr(NO)2NDH +C5HD3+ Cr1953.5 Ion-Molecule Chemistry of CpCr(NO)2H3with Water and Heavy Water,H20 and D20H2° is also a good hydrogen donor in the ion-molecule reactions of CpCr(N0)2H3with H20. All these ion-molecule reactions are exactly equivalent to those ofCpCr(N0)2H3with H2 including the collision quenching, the ligand substitutions, and— 85 —1.0Figure 3.4 Temporal variations of the chromium ammine complexes produced fromthe ion-molecule reactions of CpCr(NO)2H3with H2.additions. These equations are not duplicated here. D20 was used to verify that thesubstituted ligand was ND3, not OD, because masses of NH3 and OH are both 17 Daltonsand there is a 2 Dalton mass difference between ND3 and OD. CpCrND (m/z 137),CpCrNOCD3ND (m/z 183), and CpCr(NO)2ND (m/z 197) were products in the ion-molecule reactions of CpCr(NO)2Dwith D20.3.6 Ion-Molecule Chemistry of CpCr(NO)2H3with Ammonia, NH3From the above analysis, we expected that ammonia would react with CpCrNOCHand CpCr(NO) directly to form CpCrNOCH3NHand CpCr(NO)2NH. The experimentfulfilled this expectation exactly. No intense mass peak of the intermediate CpCrNH (m/z-3-4-5-6-70.0 0.2 0.4 0.6 0.8Time (second)—86—134) was observed. Therefore, the reactions between ammonia and CpCrNOCH andCpCr(NO) are additions, not substitutions:CpCrNOCH + NH3-, CpCrNOCH3NH, (3.23)162 179CpCr(NO ) + NH3 —* CpCr(NO)2NH. (3.24)177 1943.7 Ion-Molecule Chemistry of CpCr(NO)2H3with Methane, CH4CH4 was a good quencher of Cr in the excited state, produced from CpCr(NO)2H3by electron ionization. However, CH4is not a good hydrogen donor for formation of theammonia ligand. Although the chromium ammine complexes CpCrNH,CpCrNOCH3NH, and CpCr(NO)2NHwere all observed from ion-molecule reactions ofCpCr(NO)2H with CH4, their relative intensities as shown in Figure 3.5 were weakerthan those from the ion-molecule reactions of H2 or H20 with CpCr(NO)2H3(Fig. 3.2).Reactions of Cr produced from chromium carbonyl with methane have already beenstudied in detail.(1920)It had been observed that Cr could react with CH4 to give CrCHand the carbonyl ligand of CrCO could be substituted by CH4.(19) Because the yield ofthe CrCH is very low, it will not be considered here. In our experiment, no ligandsubstitution by direct addition of CH4 was observed.3.8 Ion-Molecule Chemistry of CpCr(NO)2H3with Nitrogen, N2In all six small molecules used here, nitrogen was the only molecule without ahydrogen atom. No ligand substitution occurred between N2 and CpCr(NO)2H3.Therefore, nitrogen is a poor quencher of the excited Cr ions. Consecutive condensationsof Cr with CpCr(NO)2H3were still observed.— 87 —-4-5- -6C-7-81.0Figure 3.5 Temporal variations of the chromium ammine complexes produced fromthe ion-molecule reactions of CpCr(NO)2H3with CH4. The measuredrelative ion intensities of CpCrNH, CpCrNOCH3NH, andCpCr(NO)2NH are obviously weaker than those from ion-moleculereactions of CpCr(NO)2H3with H2 (Fig. 3.4).3.9 Discussion3.9.1 The Electron Transfer ReactionAt least two states of Cr ions produced by electron ionization, the excited state andground state, were already discovered by Ridge et al. from the ion-molecule chemistry ofCr with Cr(CO)6.19) Using FT-ICR mass spectrometry they found that variation of the0.0 0.2 0.4 0.6 0.8Time (second)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)6were consideredto be condensations:(19)Cr + Cr(CO)6-, Cr2(CO)L + m CO (3.25)where m = 1— 3. This difference from our ion-molecule chemistry of Cr withCpCr(NO)2H3can be understood by noting that the chromiums in the metal nitrosyl ionsCpCrNOCH 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, 6S, of Cr (3d5) has a half-filled 3d shell. An important resultfrom our experiments on the ion-molecule chemistry of CpCr(NO)2H is that groundstate Cr can undergo charge transfer reactions. In Section 3.4.1, the mechanism of Crreactions were shown in Eqs. (3.2)—(3.17). Because it was verified experimentally herethat the ion-molecule reaction of Cr ions in the ground state produced fromCpCr(NO)2H3was charge transfer and the ion-molecule reaction of the & ions in theexcited state produced from CpCr(NO)2H3was condensation, the fractions of [Cr] and[Cr+*] could be estimated from yields of their product ions. For example, according to thedata shown in Fig. 2.4 of Chapter 4, 40% of Cr ions were in the ground state and 60% ofCr ions were in the excited state by 25 eV electron ionization. Comparing with 25% ofCr ions in the ground state and 75% of Cr ions in the excited state by 70 eV electronionization,(19)populations of Cr ions in the ground and excited states are obviously afunction 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 orderkinetics). 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 notdistinguished as clearly as those of Cr ions in the excited state and the ground state.3.9.2 The Condensation ReactionsSince the ion-molecule condensation chemistry of CpCr(NO)2H3has beendiscussed in Chapter 2, the collisional quenching and the ligand substitution reactions arefocus points in this chapter. However, due to coffisions between large amount of the sinallmolecules and small numbers of ions, the condensation rates of the ions were different indifferent media. As an example, rate constants, k, of the reactions of CpCrN0CH andCpCr(N0) in H2,H20, NH3,CH4, and N2 media, given in Table 3.2, show that thesetwo ions are more reactive in the media of polar molecules 1120 and NH3 than in the mediaof nonpolar molecules H2, CH4 and N2.Table 3.2 Rate constants of CpCrN0CH and CpCr(NO) in H2,H20,NH3,CH4, and N2 mediaMedium CpCrNOCH CpCr(NO)k x107 Torr•sec k xlO7 Torr•secH2 12.3 12.31120 38.5 46.8NH3 33.2 40.8CR4 29.6 30.0N2 26.5 24.1¶ The method for calculating rate constants has been given in Section 2.5.1 of Chapter 2.Calibrated pressure of CpCr(N0)2H3was 4.2 x 10 Torr (Table 3.1).—90—Figure 3.6 Temporal variations of CpCr and the secondary ions Cp2rCp2rO,Cp3r(NO)O in I12 medium. Cp2r,Cp2rO, andCp3r(NO)Owere produced from the ion-molecule reactions of CpCr withCpCr(NO)2H3.For simplification, the variations of condensation ionsCp2rOW,Cp2rNO,Cp3r(NO)OH, which were produced alsofrom CpCr with CpCr(NO)2H3are not shown here. The plot oflogarithmic relative ion intensity of CpCr against reaction time isnon-linear.Il / I0.010.0010.0 0.2 0.4 0.6 0.8Time (second)1.0—91—3.9.3 Further WorkThe ion-molecule chemistry of the metal nitrosyl complex CpCr(NO)2H3with thesmall molecules discussed in this chapter has demonstrated that it is an interesting researcharea. On the basis of this work, many studies should be done. The collision quenchingand charge transfer of Cr has been verified experimentally. The mechanism of thesereactions can be identified from the kinetic data of Cr. In order to get good ion intensitiesto compare with the results given in Chapter 1, a 20 ms beam time was used as in theexperiments of Chapter 1. Because some ion-molecule reactions already occurred duringthe 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 Crreactions (Section 3.4.1), a shorter ionization beam time (for example, 5 ms) isadvantageous to increase accuracy of the determinations. However, this will sacrifice theion intensity. One important result of our research is that the ground state Cr ions prefercharge transfer to the metal nitrosyl CpCr(NO)2H3,and result in different products fromthose of the condensation reactions of the excited state Cr ions. Populations of Cr ions inthe ground state and the excited state, therefore, can be determined according to respectiveyields of the charge transfer products and condensation products for various ionizationenergies.The pressures of the small molecules were all controlled at a constant nominal 3.0 x10 Torr. The collision quenching effects of the small molecules could be changed withdifferent partial pressure ratios of the small molecules to the metal nitrosyl. The effects ofthe media H2,H20, NH3,CH4, and N2 to the condensation rate constants are also worthstudying further.Reduction of metal nitrosyl to metal ammine in acidic solution has been known since1973.(21) In the present study, it was found that the nitrosyl ligand in CpCr(NO)2H3canbe 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 smallmolecules containing hydrogen. It is possible that many new discoveries will come withmore work in this area.—93—REFERENCES1. 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”; AmericanChemical 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 ASMSConference on Mass Spectrometry and Allied Topics, Nashville, TN, 1991, pp 455-456.11. 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 4SIMPLE PHYSICAL POINT AND LINE CHARGE MODELSFOR COULOMB-INDUCED FREQUENCY SHIFTAND INHOMOGENEOUS BROADENINGIN FT-ICR MASS SPECTROMETRY—96—4.1 Space Charge Effects in FT-ICR Mass Spectrometry4.1.1 Space Charge Effects and Mass MeasurementThe simplest theoretical treatment of ion cyclotron resonance (ICR) presumes that theion motion is controlled by exclusively applied forces, such as those from theradiofrequency field, the static electric field, and the static magnetic field. In ahomogeneous static magnetic fieldB=Bk (4.1)where k is the unit vector in the z direction, and the Lorentz force on an accelerated ion isF=qvXB (4.2)where q is the charge of the ion and v is the ion velocity. The solution to the motionequation (4.2) predicts that the ion of mass m will travel in a circular orbit of radius r with acyclotron frequency(1)°o =q, (4.3)and the instantaneous position of the ion related to the Cartesian coordinates, x and y, isgiven byx=rcos (a0t+) (4.4x)y=—rsin(w0t+•) (4.4y)where 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 isindependent 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 numberdensity. That is, the resolution and mass measurement of FT-ICR will become degradedwith growth of the ion number density. This experimental phenomenon apparently comesfrom space charge effects in ICR and FT-ICR. Therefore, a Coulomb interaction termshould 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 asunlike ions. The space charge effects should include: Coulomb interaction of the ions withthe external electhc fields, Coulomb interaction between like ions, and Coulomb interactionbetween unlike ions.4.1.2 Prior Research on the Space Charge EffectsDuring the past decade, a number of authors have studied the space charge effects inFT-ICR mass spectrometry. Ledford et al. experimentally measured the frequency shiftwith ion number, for like ions in the cubic ion-trapped cell of FT-ICR, and first offered anempirical equation to calibrate the frequency shift due to the trapping electrostatic field andion number dependence.(2)Their procedure gave mass measurement errors averaging 3ppm. Jeffries, Barlow, and Dunn first gave a theoretical quantitative study on the Coulombinteraction of ions in ICR and FT-ICR mass spectrometry for various ICR cells.(3) Franciet al. used the space charge theory of Jeffries et al. to calibrate mass measurements of bothscanning ICR spectrometry and FT-ICR spectrometry and gave accurate massdeterminations with errors less than 1 ppm.(4) The basis of Francl et al.’ s procedure is thata 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 ionsare formed along the z-axis in the center of ICR cell and then the ICR motion of these ionsis sequentially excited as the spectrum is scanned. This is inappropriate for the FT-ICRexperiment where all the ions in the cell are undergoing excited cyclotron motion during theFT-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 referenceions).(5) Wang and Marshall studied spectral lineshape broadening due to ion-ion Coulombinteraction in Fr-ICR by using numerical analysis.(6)On the basis of the model derived byLedford et al.,(5)Meek et al. gave an average mass error of less than 0.5 ppm for m/z rangefrom 119 to 556 Daltons in a 7 Tesla magnetic field.(7) Yang, Rempel, and Grossexamined 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 ownmodel(5)to include the time averaged interaction of ions, but no detail was given.(9) Yang,Rempel, and Gross incorporated relative ion intensity into the mass calibration coefficientsaccording to Jefferis et al.’s model.(10) Herold and Kouzes simply used an infinitely longline charge model to calibrate the space charge effects(11) and their result was in keepingwith the experiment of Franci et al.(4) Smith has noticed that ion-ion Coulomb interactionsfrom different m/q ions which are called unlike ions can be averaged and their intensitiesshould be included in the calibration of space charge effect.(12)There is, therefore, an ionintensity term in Smith’s calibration formula based on the models of Jeifries et al.(3) andLedford et al.(5) After ten reference mass peaks were used, Smith shown that calibratingFT-ICR mass spectra by inclusion of unlike ion intensities gives superior calibrationaccuracy.Marshall and Verdun concluded in 1990:(13) “Unfortunately, there is no simpleanalytical treatment for either the static or dynamic effect of multiple-ion Coulombinteractions, and one must resort to computer-intensive trajectory calculations of each of10,000 or more ions in a trap of given geometry.” Grosshans, Shields, and Marshall haverealized that any accurate calibration of space charge effect in FT-ICR mass spectrometrymust employ the interaction potential between ions rather than the simpler Coulombpotential model usually used previously.(14) Until now, no theoretical quantitative—99—discussion has been developed on Coulomb interactions between different ion masses inFT-ICR.Recently, we have proposed simple physical point and line charge models whichmore closely correspond to physical reality than prior models.(15) Coulomb interactionbetween two or more different m!q ions can be calculated by averaging the separationbetween two charges (either points or lines).4.2 The Point Charge Model — Coulomb Shifting of Unlike IonsFor a simple two-dimensional picture as shown in Figure 4.1, two excited ions withdifferent mass m1 and m2 are undergoing cyclotron motion with common orbit centers in ahomogeneous static magnetic field B and their cyclotron radii are r1 and r2. If each ion isinfluenced only by the Lorentz force, the ions will undergo cyclotron motion at their naturalcyclotron frequencies, c for m1 and c002 for m2. Now the instantaneous magnitude, D,of the spatial separation between the ions, D, is given byID I = D = (x1 — x2)+ (y1—y2) (4.5)where x1 and y1 (x2 and y2) are the instantaneous Cartesian coordinates of m1(m2).Substituting Equations (4.4) into Equation (4.5) givesD = ‘\Jr12 + — 2 r1 r2 cos[(co0—2)t+ (1—2)1 • (4.6)If r2 r1, Eq. (4.6) then reduces toD = r1 \J2 —2 cosP = 2 r I sin (cIi /2) I (4.7)wherec1 = (c001—w2) + (—)• (4.8)-100-Figure 4.1 Point model for Coulomb effects. Two positive ions, m1 and m2, areundergoing excited cyclotron motion at their respective cyclotronfrequencies, (Ooi andEqs. (4.7) and (4.8) give the instantaneous spatial separation between two ions m1 and m2,as a function of their equal cyclotron radii, r1, the two cyclotron frequencies, c- and %2’the two initial phase angles, and 2’ and the time t. This separation varies sinusoidallyas indicated by Equation (4.7). The mean value, Dme, of the separation between the ionsis given by averaging D for ‘ =0— it (or by averaging D for CP = It — 2it)—101—Dme =- J 2 r1 sin (/2) d1 (I)=O4r== 1.2732 r1. (4.9)ItBecause the Coulomb force is inversely proportional to the square of the separation, theroot mean square value, of the separation between the ions is given by0.5Dims= ( r?(2—2cos) thP)= = 1.4142 r1. (4.10)Thus, from the point of view of the first ion mass, the second ion mass will appear onaverage a distance 1.2732 r1 away, as indicated in Figure 4.2, if we chose the “average”distance to be the mean distance, Dmean or a distance 1.4 14 r1 away if we choose the rootmean square distance, D5. If we choose Dm to be the “average” distance, the Coulombforce between the ions is given by Equation (4.11),kq1q2 k q12Frad= D2 = 1 6211 2 (4.11)mean Tiwhere k is the Coulomb constant, 8.9876x10 N m2/C,q1 and q2 are the respectivecharges of m1 and m2, and Frad is a radial Coulomb force on m1 which subtracts from theLorentz force of Eq. (4.2). In terms of the cyclotron frequency w and radius r1 of m1,the Lorentz force on m1 from Eq. (4.2) can be writtenF1 =q1o.01rB (4.12)—102—The average distance approach. The figure gives the location of ion in2 asseen by in1 by viewing the system in a coordinate frame which rotates at thefrequency co. The “average” distance between m and in1 is taken to bethe mean distance Dmej and is 1.2732 r1 if r2 = r1. This distance is usedto calculate the average radial Coulomb force on m1 due to in2. If thedistance r1 is used, corresponding to the average laboratory-frame positionof in2 as in scanning ICR mass spectrometry, the radial Coulomb force is1.27322= 1.6211 times as great.FradFigure 4.2— 103 —If any change in the net radial force is assumed to be small with respect to the Lorentzforce, then from Eq. (4.12), we can write for m1,AF1 = E%1qrB. (4.13)Eq. (4.13) relates the change in cyclotron frequency, with the change in radial force,AF1, which caused that change in frequency. Substituting Eq. (4.11) into Eq. (4.13) andrearranging, gives= k q2 = k, (4.14a)or equivalently,Af01 = 1.4137 x i0 2 (4.14b)r1BEq. (4. 14b) gives Af(fl, the change in experimental cyclotron frequency in Hz, of an ionm1, due to the presence of an ion m2 of equivalent cyclotron radius, as a function of q2, thecharge on m2 in units of the electronic charge; r1, the ion radius in cm; and B, the magneticfield strength in Tesla. Eq. (4.14a) is the equivalent equation in SI units.The easiest way to visualize the Coulomb effect of m2 on m1 is in a coordinate framewhich rotates at the frequency ofm1.(6) In this frame, both the position of m1 and theaverage position of m2 are static and the average position of m2 appears as shown in Figure4.2.It follows from Eqs. (4.5) — (4.7) that both the spacing, D, and the frequency shift,Af01, caused by ions of mass identical to the ion mass being monitored, are zero. Thisresult has been previously noted from both theoretical(1718)and numerical analysis.(6)For purposes of illustration, the Coulomb shift due to a particle with 106 charges onthe cyclotron frequency of an ion of cyclotron radius 1 cm in a magnetic field of 2 Tesla—104-would correspond to 106 ions of m1 which collectively are taken to be a point charge.Substitution of these parameters into Eq. (4. 14b) gives a negative frequency shift of 70.7Hz.4.3 The Line Charge Model — Coulomb Shifting and Broadening ofUnlike IonsIn conventional electron ionization experiments, the ions are formed in a longcylindrical shape at the center of FT-ICR cubic cell. When the radius of this cylindrical rodis small relative to its length and excited cyclotron radius, the ion cylindrical rod can betreated as a line charge. The cyclotron radius r of an ion during excitation is only a functionof magnetic field strength B, rf electric magnitude E0, and excitation time t, withoutdependence of its mass and charge(1920)Etroc. (4.15)No matter what the mlq ratios of the ions are, their cyclotron radii are all the same when auniform sweep rate of excitation frequency is used.A physical model which better approximates the actual ICR experiment can be derivedfrom Figure 4.2 by dispersing the ion charges into the z-dimension to create a rotating linecharge of length 1, for each excited ion mass. Figure 4.3 shows this model. Consider theeffect of line charge 2 upon the cyclotron frequency of line charge 1. As for the model ofFigure 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 thisdistance. Now the radial electric field at point P of line charge 1 from a differentialelement, dz of line charge 2, j(2l)—105——zline charge 2line charge 1Figure 4.3 Line model for Coulomb effects. This model is derived from the model ofFig. 4.2 by uniformly distributing the charges in the z direction. The lengthof the charges is 1 for both m1 and m2 ions. The mean distance of linecharge 2 from line charge 1 is Dme = 1.2732 r1. The radial Coulombelectric field on point P of line charge 1 from the differential element dz ofline charge 2 is rad = dE x cos 8.z = 1/2!—106--rad = dExcosO= k N2 q2 Drnean dz3/2 (4.16)(z—z1) + meanand so the total radial electric field created by line charge 2 at point P is(22)z=l/2E— kN2q DmeandZ— z=—l/2 [(z_z1)2 + Dan ]3/2= k N2 q2 [ 1/2 — z1 + 1/2 + Z1 1(4.17)mean J(l/2 — z1)2 + Dan J(l/2 + z1)2 + DeanEq. (4.17) gives the radial electric field at position P in an ICR experiment modelled byFigure 4.3, as a function ofN2q, the total charge on line charge 2; 1, the length of linecharge 2; Dmean, the mean distance between the two line charges. Each element, dz, of linecharge 1 (not shown) will have a mass N1m/l dz and charge N1q/ldz, and the radialCoulomb force from line charge 2 on this element is given by the productFj = Erad N1q/l dz (4.18)where Erad is given by Eq. (4.17). Now Eq. (4.13), which was derived for the pointmodel in Figure 4.2 and which relates the change in frequency and the radial Coulombforce, is also valid for a differential element of line charge 1 in Figure 4.3. Substituting thecharge of the differential element, N1q/ldz, and the radial force in Eq. (4.18), into Eq.(4.13) and solving for the frequency change gives&Ooi— kN2q [ l/2—z1 l/2+z1 ]r1 mean J(l/2 — z1)2 + Dean sJ(l/2 + z1)2 + Dean—107—or equivalently,2.2918x10Nq[ 1/2—zAfoi = r1B I Dmean (1/2 — z1)2 + mean1/2 + z1+__________________g (1/2 + z1)2 + 2 (4.19b)DmeanFor z1 = 0, Eq. (4.19) reduces tokN2q [ 11 r1B Dmean J(l/2)2 2 (4.20a)+ Dmeanor equivalently,2.2918x1ONq[ 1 (4.20b)‘V01= r1B Dmean l/22 + D 2meanIn the limit (i. e., an infinite long line charge),112— 00, (4.21)DmeanEq. (20) reduces to2 k N2 q2 (4.22a)= r1B 1 Dmeanor equivalently,4.5836 x 10N2q2 (4.22b)‘vol= r1 1 DmeanEquation (4. 19b) gives the frequency shift in Hz for a differential element of line charge 1at position z1 in Figure 4.3 as a function of B, the magnetic field in Tesla; N2q2, the totalcharge on line charge 2 in units of the electronic charge; r1, the cyclotron radius of linecharge 1 in cm; 1, the lengths of line charges 1 and 2 in cm; Dmean, the average distancebetween the lines (1 .2732 r1) in cm; and z1, the position in cm of the differential element.—108--Eq. (4. 19a) is the equivalent equation in SI units. Note that the shift is a function of theposition, z1, in line charge 1. This model predicts that different ions of mass 1 wifi havedifferent cyclotron frequencies because of their different positions in line charge 1. In otherwords, the experimental spectral peak will be broadened as well as shifted by the Coulombinteraction with line charge 2. Eq. (4.20a) gives the Coulomb shift in SI units for ions atthe center of line charge 1. This is also the maximum Coulomb frequency shift for thepeak. 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 Dmeanreplaced byr1.01)Figures 4.4, 4.5, and 4.6 show the effects of Coulomb shifting and broadening dueto the model illustrated in Fig. 4.3. Figs. 4.4, 4.5, and 4.6 give the Coulomb shiftcalculated from Eq. (4. 19a) as a function of z1, the z-axis position in line charge 1, and 1,the length of the line charges, forB = 2 Tesla, Dmean = 1.2732 cm, = cm, r1 = 1.0cm, and N2, the number of m2 ions. Figure 4.4 gives the shift for 1=2.4 cm which wouldcorrespond 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 cmand 15 cm, respectively. The charge density in line charge 2 is constant at 4.167 x iOions/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 orCoulomb-induced peak broadening for like ions.4.4 Discussion4.4.1 Choice of ModelOne prior physical model for Coulomb shifts in ICR was developed by Jeffries,Barlow, and Dunn.(3) Their model had a charge cloud on the z axis of Fig. 4.3 with an ionundergoing cyclotron motion in the x-y plane. This model closely corresponds to the—109—8565‘vol(Hz)45251.2Figure 4.4 Coulomb shifting and broadening for the line model of Fig. 4.3. Thefrequency shift for m1 ions, calculated from Eq. (4. 19b), is plotted as afunction of z1, the position of the m1 ion. The parameters used for thesecalculations are: average ion separations Dme = 1.2732 cm and Drmscm, corresponding to a cyclotron radius, r1 = 1.0 cm; the magnetic field,B =2 Tesla; and the z-axis length, 1=2.4 cm. N2, the number of m2 ions,is adjusted so that the ion number density is constant at 4.167 x 10ions/cm. The upper curve gives the Coulomb shifts for the “averageposition” model, and was calculated from Eq. (4. 19b) by replacing Dmewith r1 = 1.0 cm. The lower curve gives the Coulomb shifts for the“average distance” model, using Drms (Eq. (4.10)) as the “averagedistance”. The Coulomb shifts arising from using Dme (Eq. (4.9)) as the“average distance” lie between the “position” and “D” curves.-1.2 -0.8 -0.4 0.0 0.4 0.8z1 (cm)—110—Lf01(Hz)11090503011090705030-7.5(Hz)7°Figure 4.5-4 -3 -2 -1 0 1 2 3 4z1 (cm)Same graph as that in Fig. 4.4 for 1= 8 cm and N2 = 3.33x 106 ions. (N2]!is constant at 4.167x105ions/cm).Figure 4.6-5.0 -2.5 0.0 2.5 5.0 7.5z1 (cm)Same graph as that in Fig. 4.4 for 1= 15 cm and N2 = 6.25x10 ions. (N2J1is constant at 4.167x105ions/cm). Note that although the magnitude of theCoulomb shift is greater for longer 1 , because of the greater number ofions, the dispersion in cyclotron frequencies for most m1 ions is less.—111—scanning ICR experiment, in which ions are formed along the z axis and then the ICRmotion of these ions is sequentially excited as the spectrum is scanned. The experimentallyobserved negative direction of the Coulomb frequency shift was correctly predicted by thismodel. This model is inappropriate, however, for the Fr-ICR experiment where all theions in the sample are undergoing excited cyclotron motion during the FT-ICR detectionperiod, as indicated in Figs. 4.1 and 4.2. Condensation of the charge cloud into a chargedline gives a static line charge on the z-axis(11)which again, is applicable only in thescanning ICR experiment. Why then does this model work for FT-ICR? An examinationof Fig. 4.2 shows the reason. A two-dimensional Jeffries model can be created bycondensing 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 negativefrequency 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 otherions can be qualitatively explained as being due to an average radial force on the particularion caused by the other ions. Attempts to quantitatively derive the magnitude of thisfrequency shift within the framework of the simple physical models of this work give riseto three possible approaches. What is desired is the average Coulomb radialforce on theparticular ion, which we assume gives the frequency shift via Eqs. (4.13) and (4.14).Unfortunately, the average radial force is infinite for r1 = r2 (Fig. 4.1), due to themomentary superposition of the two ions during one complete cyclotron orbit.A second approach, which is developed here, is to derive the average distance of theperturbing ion from the particular ion and then assume that the “frequency-shifting radialforce” is equal to the radial force from the “average distance”. The average distance can beeither 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 forcefrom the average position. This, in effect, is the approach which others(4’5,7—12) haveused. For point charges, the “average distance” model predicts lower shifts by a factor of2.0 for and 1.6211 for Dme from the “average position” model. The predictions ofthe average-position model can be derived from the average-distance equations in thischapter by replacing in any of Eqs. (4.16) — (4.22) by r1.As discussed above, for a specific case, the two-particle model in Fig. 4.2, predicts aDme Coulomb shift of 70.7 Hz for 106 m2 ions. When the 106 ions are spread out over adistance of 2.4 cm (Fig. 4.4), to create the line model, the corresponding Coulomb shift isreduced to a maximum (Eq. (4.20b)) of 51.4 Hz. This reduction is not surprising, sincethe perturbing ions m2 are on average, farther away from m1 ions in the line model than inthe point model. The average distance models are not only for two line charges, but alsocan be extended to arbitrary numbers of line charges. When the ion-ion Coulomb effectson line charges m1 are studied, all other line charges can be averaged to the distance Dmenor Drms provided their separations vary sinusoidally.In another example, where ion masses are very similar, let m1 = 1000 Daltons and m2= 1001 Daltons in a 2 Tesla magnetic field, in which their resonance frequencies arev01 =3.07120x104Hz and v = 3.06813x104Hz. In a detection time T = 8.398 ms (fairlyshort), they oscillate for 257.92 cycles and 257.66 cycles, respectively. Their difference ismuch less than 1 cycle, so that their separation can not be averaged from 0 to 2ir.However, the theoretical resolution of the magnitude mode(23)ism — gBT — 1.60x109 x 2 x 0.008398— 213Am50% 7.582 m 7.582 x 1000 x l.66x1027 —It is impossible to detect the Coulomb interaction between these two ion species at such apoor mass resolution. Therefore, this case can be ignored in our models.—113—It is common in the development of physical models for certain approximations orsimplifications to be made.() The point model of Figs. 4.1 and 4.2 treats the problem asa two-dimensional one, with no radial electric field. Inclusion of a z-axis distribution ofcharge, as in Fig. 4.3, gives rise to a position-dependent frequency shift. In addition, themodels in Figs. 4.4 — 4.6, while including axis position, omit an electrostatic trappingfield, and omit any z-axis motion.4.4.2 Inhomogeneous Broadening in FT-ICRICR spectral peaks can be broadened by two different types of processes; calledhomogeneous and inhomogeneous broadening.(25)The corresponding processes in thetime domain are called homogeneous and inhomogeneous relaxation. Homogeneousrelaxation and broadening result from any process which limits the lifetime of theoscillation. Inhomogeneous relaxation and broadening result from a dispersion in theresonant frequency of the individual components of a macroscopic sample.(25)From thesedefinitions, it follows that the position-dependent Coulomb-induced frequency shiftingprocess of Fig. 4.3 and Eq. (4.18) gives rise to an inhomogeneous broadening/relaxationmechanism.Wang and Marshall(6)have discussed Coulomb broadening in FT-ICR and concludedthat a two-dimensional model, such as those in Fig. 4.1 or 4.2, cannot account forCoulomb broadening. These authors concluded that z-axis oscillations could giveCoulomb broadening if the Coulomb force was comparable to the Lorentz force. Thisconclusion is supported here. Fig. 4.3 and Eq. (4.19) show that it is the dispersion in zaxis position, not z-axis oscillations, which causes Coulomb broadening; and that smallCoulomb forces, which are much less than the Lorentz force, can create Coulombbroadening. 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 IonsAs noted above, our model as well as the work of others(6’ 17-18) predicts nofrequency shift for ions of identical mass. Yet, such shifts have been observed when theICR experiment is conducted on just a single ion mass.(4’6) Prior treatments make thequadrupolar approximation in which it is assumed that all ions of a particular mass aresubjected to an identical radial component of the trapping field. However, it is well knownthat in practice, the observed cyclotron frequency depends upon the z-coordinate, as theexperimental field is quadrupolar only at z = 0. We believe that these actual nonquadrupolar electrostatic fields may give rise to a Coulomb-induced frequency shift asfollows: consider m1 ions located at z = 0 which have been excited to some radius, r1;consider next ions of identical mass located at ± z whose cyclotron motion has also beenexcited. These ions will have a different cyclotron frequency from the ions at z =0, due tothe non-quadrupolar radial component of the trapping electric field. Compared tom1ionsat z = 0, they will appear to be ions of differing mass, and can be treated as m2 ions in thepresent model, and will produce a Coulomb-induced frequency shift and a Coulomb-induced broadening. Consequently, a Fl’-ICR experiment conducted with a single ionmass can give an ICR lineshape which is both Coulomb-shifted and inhomogeneouslyCoulomb-broadened.4.4.4 ICR Cell Design and Coulomb EffectsScreened,(26)shaped,(27)segmented-trap-electrode,(2829)and elongated(30)ICR cellshave been developed to create what approaches zero trapping fields in much of the interiorof 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 highermass resolution than is present in the cubic ICR cell.(3132) The analysis in this chaptersuggests that there is afurther advantage to these cells, namely that the lower trapping fielddispersion gives rise to lesser Coulomb effects, both shifting and broadening, because aparticular ion will “see” the Coulomb effects from other ions, but only to a lesser extentfrom ions of the same mass. Hence, the like-ion contamination can be minimized in theseICR cells with negligible trapping fields.For either the point charge model or the line-charge model, the average radialCoulomb force is infinite for r1 = r2, due to the momentary superposition of the two ionsduring 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 acylinder charge model (33) which can overcome this limitation of the point charge modeland the line charge model.—116—References1. 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. Jeffries, 3. B.; Barlow, S. E.; Dunn, G. H. mt. J. Mass Spectrom. Ion Proc. 1983,54, 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. mt. J. Mass Spectrom. Ion Proc. 1986, 68, 287-301.7. 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 onMass Spectrometry and Allied Topics, 1988, 586-587.9. Rempel, D. L.; Gross, M. L. Proceeding of 37th Conference on Mass Spectrometryand Allied Topics, 1989, 1222—1223.10. Yang, S. S.; Rempel, D. L.; Gross, M. L. Proceeding of the 37th ASMS Conferenceon Mass Spectrometry and Allied Topics, 1989, 1224-1225.11. Herold, L. K.; Kouzes, R. T. 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 Trappingin Mass Spectromerry, 1990.13. Marshall, A. 0.; Verdun, F. R. “Fourier Transforms in NMR, Optical, and MassSpectrometry”; Elsevier: Amsterdam, 1990; p 243.—117—14. Grosshans, P. B.; Shields, P. 3.; Marshall, A. G. J. Chem. Phys. 1991, 94, 5341-5352.15. Chen, S.-P.; Comisarow, M. B. Rapid Commun. Mass Spectrom. 1991,5, 450-455.16. 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. 1978,26, 369-37 8.20. Grosshans, P. B.; Marshall, A. G. mt. 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 CyclotronResonance 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. mt. J. Mass Spectrom. Ion Proc. 1990,100, 38 1-395.—118—29. Naito, Y.; lnoue, M. Proceeding of 36th ASMS Conference on Mass Spectrometryand Allied Topics, San Francisco, CA, 1988, 608—609.30. Hunter, R. L.; Sherman, M. G.; Mclver Jr., R. T. mt. J. Mass Spectrom. Ion.Phys. 1983,50, 259—274.31. 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 5SIMPLE CHARGED-DISK MODELAND SIMPLE CHARGED-CYLINDER MODELFOR COULOMB SHIFTING AND COULOMB BROADENINGIN FT-ICR MASS SPECTROMETRY—120—5.1 IntroductionIn Chapter 4, we developed two simple physical models (the point and line chargemodels)(1)which explained Coulomb-induced frequency shifting and Coulomb-inducedinhomogeneous line broadening in FT-ICR mass spectrometry. The first model consistedof two point ion masses, in1 and m2, which were undergoing excited cyclotron motion.When viewed in a coordinate frame which rotated at the frequency of m1, the modelshowed that the motion of in2 gave rise to a radial Coulomb force of repulsion at ml whichsubtracted from the Lorentz force on m1 (because of its opposite direction) and lowered thecyclotron frequency of m1. Thus, the Coulomb interaction between in1 and m2 shifted thefrequency of m1.We derived the frequency shift, given by the simple SI formulakN qO1 D2rB’ (5.1)where k is the Coulomb constant, q2 is the charge on m2,N2 is the number of in2 ions, B isthe magnetic field strength, r is cyclotron radius of m1 and in2, and D is the “apparentCoulomb distance” of in2 from in1. By extending the point model into the magnetic-field(z) direction, a charged-line model(1)for Coulomb shifting in FT-ICR was developed,whose Coulomb-induced frequency shift was given by the SI formula— k N2q ( 1/2 — z1 1/2 + ) 5 2(l/2_z)2+D2 + (l/2÷zi)2+D2 ‘ . )where z1 is the z-axis position of a particular m1 ion, 1 is the length of the line charge andthe other parameters are as in Eq. (5.1). Note that for a macroscopic sample of ionsdisthbuted along the z direction, Eq. (5.2) predicts a Coulomb-induced line broadening.For z1 = 0, Eq. (5.2) reduces to—121—— kN2q5301— rB q (112)2+ D2which 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 theperturbing m2 ions gave rise to an average force, AF, on in1 which gave rise to a frequencyshift, &o , given by= &o q1 rB. (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 Coulombinteraction is infinite because of the momentary superposition of the ions during onecomplete cyclotron cycle of the frequency difference°kn — (002. This, of course, is aninherent flaw in the model. So, instead of attempting to calculate an average Coulombforce for the point-charge or line-charge models, we calculated an apparent distance of m2as seen by m1 and then modeled the apparent Coulomb force as the force from the“apparent Coulomb distance”. If we took the average location of m1 in the laboratoryframe as the appropriate distance model, the “apparent Coulomb distance” was r, thecyclotron radius of m1 and m2, and if we took the mean distance between m1 and in2 as theappropriate distance model, the “apparent Coulomb distance” was 4 nit (or r for theroot mean square distance). These models gave rise to simple analytical formulae for bothCoulomb shifts and Coulomb broadenings.O) Moreover, the simplicity of these modelsprovided the physical basis for both Coulomb shifting and Coulomb broadening. For eachof the point charge and line charge models, quantitative prediction of the magnitude of theshifts 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 benecessary 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 chapterhave appeared in communication form.(2)5.2 The Charge-Disk Model5.2.1 Two Charged Disks in a Rotating Laboratoiy-FrameFigure 5.1 shows the charged-disk model for Coulomb frequency shifting in FTICR. This 2-dimensional model takes the distribution of m1 ions to be a uniformly chargeddisk of radius r’, which is undergoing excited cyclotron motion at a laboratory-framefrequency with a cyclotron radius of r. A second ion mass, m2, also uniformlydistributed over a disk of radius r’, is also undergoing excited cyclotron motion withcyclotron radius r, but at a laboratory-frame frequency %2. The distribution of m1 and m2ions in a coordinate frame is shown in Fig. 5.1, which is rotating(3)at the laboratory-framefrequency co01. In this coordinate frame, the m1 disk is stationary and the m2 disk rotatesat a frequency —At any instant of time the relative positions of the disks are characterized by an anglecP (Fig. 5.1), which varies linearly with time. Disk 2 will exert on Disk 1 a Coulombforce, the radial component of which is of interest here because the averaging over a cycleof its tangential component wifi be zero as shown Figure 5.2. Calculation of this radialCoulomb force depends upon whether or not the disks are overlapping. uch’ the anglewhen the two disks just touch, is given bytouch = 2arcsin(---) (5.5)and separates the overlapping from the non-overlapping values of Cl).—123—Figure 5.1 Uniformly-charged-disk model for FT-ICR. The figure gives theinstantaneous location of Disk 1 and Disk 2 in a rotating frame whichrotates in the x-y plane at the cyclotron frequency of Disk 1. In this rotatingframe, the location of Disk 1 is constant and Disk 2 rotates on the circularpath shown, at a frequency (O— %• The cyclotron radius of both disksis r. The disk radius is r’ for both disks. ‘ is the instantaneous angularcoordinate 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-pointmodel of reference 1. Dispersal of the charges of Disk 1 and Disk 2 into thez-direction creates the charged-cylinder model of the text.F—124—Therefore, the average ofF andFis zero, that is, the averaging over acycle ofF is zero. However, their radial Coulomb forces on are:ForF’dF F’Figure 5.2 The averaging of tangential components of the Coulomb interaction betweentwo ions m1 and m2. Whether their cyclotron radii, r1 and r2, are the sameor different in the rotating laboratory-frame of Fig. 5.1, when m2 movesfrom the position (r2, )) to its mirror position m, (r2,—), their tangentialcomponents of Coulomb forces F and F’ on m1 are:F= _F’tan•= F.— 125 —5.2.2 Radial Coulomb Force on Disk 1 for Non-Overlapping Disks (0 >0uch)For values of the angle 0, (Fig. 5.1) for which the disks do not overlap, theinstantaneous radial force can be evaluated with aid of Figure 5.3 as follows: Consider thedifferential element th2xdy of Disk 2 located at (x2,y2). This element will exert aCoulomb force dF on the differential element dx1xdy of Disk 1 located at (x1,y1). Thisdifferential force is given by— U2 2 dy2 c71 dy11 2 Y12where k is the Coulomb constant, o is the charge density of Disk 1 and U2 is the chargedensity of Disk 2. The charge densities are given by— N1(2.)q1(2)O’l(2)— i r’2where N1 and q1 (N2 and q2) are the number of and charge of the ions which make up Disk1 (Disk 2). The radial component, dF, of the total differential force dF is given byUdxdy UdxdydF d = k2 222 1 12 sin a, (5.8)ra ( ‘ ( e— X2j +—where ae, the angle of elevation (Fig. 5.3), which gives the direction of the totaldifferential force, is given byY12ae = arcsm ( i (5.9)/—126—Differential elements of the charged-disk model for Coulomb shifting inF1’-ICR. The differential element at P2 of Disk 2 exerts a radial Coulombforce on a differential element of Disk 1 located at P1. This force is givenby Eq. (5.8). Disk 1 is static in the rotating frame of the figure and islocated at 01. Disk 2 can rotate on the circular cyclotron path shown. 02 isthe instantaneous location of Disk 2. ) is the instantaneous angularcoordinate of Disk 2. ae is the angle of elevation of the total differentialCoulomb force from P2 to P1.yx12+(y1—r)2=r’Disk 1(x2—r sin 1)2+ r cos)2=r’2Disk 20Figure 5.3x— 127 —Now for Disk 1,+ (y1 — r = r’2, (5.10)and for Disk 2,(x2—rsin) + (y2—rcoscl.’) = r2. (5.11)The total radial Coulomb force on Disk 1 is then given by the integral of Eq. (5.8) over allrelevant values of the coordinates x1,y1,x2, andy2fYimx fYzxn fX fXFj = kirirJ2J J j J sin a dx21dy21 (5.12)Yi y x1 (x — x2) + (y1 — y2)Substituting Eq. (5.9) into Eq. (5.12) givesYi Y X1 XF=ku1 i Lm L [(x1 —x2)÷(y1_y2)2j312dX2 dx1 dy2 dy1, > (5.13)The integrals overx1 andx2 in Eq. (5.13) can be analytically evaluated,(4)reducing Eq.(5.13) to(Yi (Y2Fr =ku1cY2J J I dy21, (5.14)Yi Yi—128—where= Y 1 Y2(q (Ximin — X2max) + (y1— Y2)2+ (Y1 —y2) + •\I(Xlmax_X2min)2 + ()‘i Y2)_g(X1j_2)2+ b’ _y2)2); (5.15)Ylma r + ?; (5.16)y1 = r — (5.17)= rcos+r’; (5.18)Ym= rcoscl—r’; (5.19)Xlmax = Ir’ 2_ (y — r)2 ; (5.20)Xlmjn = — 4r’ 2 — (y1 — r)2 ; (5.21)x= r02_2_rcosø)2 + rsin; (5.22)andx2= — r’ 2— (Y2 — r cos )2 + r sin ci’. (5.23)Eq. (5.14) gives the radial Coulomb force on Disk 1 arising from Disk 2, for valuesof D> eLouch.—129—5.2.3 Radial Coulomb Force on Disk 1 for Overlapping Disks (ci) < touch)Figure 5.4 shows the two disks when 0 <Otouch• Because each of the twooverlapping disks has the same cyclotron radius r, both crossing points, P and P_, willmake an angle 0i2 with y-axis. Let a be the distance between P÷ and the origin, 0.P+O = a. (5.24)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_ r2 sin2 (0/2). (5.26)Eq. (5.26) with the positive term gives the polar coordinate, a÷, for the point, (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 — r2 sin2 (0/2) 1 sin (0/2), (5.27x)= [r cos (0/2) + r’ 2 — r2 sin2 (0/2) 1 cos (0/2). (5.27y)—130—xFigure 5.4 Overlapping charged disks. When 1 is less than ouch (Eq. (5.5)), thedisks overlap, creating the hatched region shown, which does not contributeto the Coulomb force between the disks.y(x, .v+)Disk 1Disk 20—131—Similarly, the Cartesian coordinates (x_, y_) of P_are given by Eqs. (5.28).x = [r cos (0/2) — r’ 2 — r2 sin2 (0/2) ] sin (0/2), (5.28x)y_ = [r cos (0/2) — r’ 2 — r2 sin2 (0/2) 1 cos (0/2). (5.28y)Evaluation of the integral for the Coulomb force between two overlapping disks isfacilitated by dividing each disk into three sections based upon the crossing points, Pf andP_, as shown in Figure 5.5. Disk 1 is divided into the sections A11,A12, and A13. Disk 2is divided into the sections A, A22, and A23. The overlapping area (shadowed sectionsin Figs. 5.4 and 5.5) does not contribute to the Coulomb interaction. Therefore, the totalradial Coulomb force on Disk 1, arising from an overlapping Disk 2, is given by Eq.(5.29), as a sum of six integrals.F1 = F1 + F2 + F3 + F4 + F5 + F6, 0 <Ouch (5.29)where F1 — F6 are given by Eqs. (5.30), (5.35), (5.38), (5.40), (5.42), and (5.43),respectively.F1 is the Coulomb force between regions A11 and A21 of Fig. 5.5.Yi=Y+ y2=y x1=X2( I I I__________________F1 = k u2J1= iy2t . JX1WiD JXzX [(x1 — x2) + (y1 —y2)2]312X dx21y. (5.30)—132—Disk 1A13fY1Y+ (Y2=Y+F1 = k a1 a2 J J ‘2 dy2 dy1YiY- YzYP (x÷, y..)A22Disk 212 y1 Y2((X1min_X2max)2 + (Y1 —Y2)—(X12— X2max) + (y1 — y2) + (X12 — X22) + (y1 —P_ (x_, yJFigure 5.5 Sections of overlapping disks used for piecewise evaluation of the totalradial 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).where(5.31)\I(Ximin —X22) + (y1 —Y2) ) (5.32)—133—andX12 = — r’ 2_— r cos + r sin ; (5.33)X22 = 2_ (y2 — . (5.34)F2 is the Coulomb force between regions A11 and A22 and between regions A13 andA21 of Fig. 5.5.py=y+ y x1=X2 x2,,,Yi— Y2F2 = 2k 2J Limb I kxi — + (y1 —YiYX dx21y. (5.35)PYiY+ fYm,F2 = 2kcT1cTJ J I3dy21 (5.36)“YiY- YzY.where113= —12 — X2max)2 + (y1—y2)— (Ximin— X2max )2 + — y2) + (X1 — X2min) + ( —— J(Ximin— X2min) + (Yi — Y2) ) (5.37)—134—F3 is the Coulomb force between regions A11 and A and between regions A12 andA21 of Fig. 5.5.YiY+ YY- x1X2 x2,F3=2ku1a21L 1 [(x_)2+(yy]312X dx dx1 dy2 dy1. (5.38)(YiY+ (Y2Y_F3=2ku1a2)- ) 13 dy21. (5.39)Yi—Y-F4 is the Coulomb force between regions A13 and A23 and between regions A12 andA22 of Fig. 5.5.fYi fY fXi fXF —2k I I I I__________________4 — U1 U2j1 Jy2= + Jx1 Jx2 [(x1 — x2) + (y1 —y2)2]3/X dx2th1y. (5.40)(Yi (YF4 = 2ka1a) J I1dy2. (5.41)YiY. YzY.F5 is the Coulomb force between regions A13 and A22 of Fig. 5.5.—135—Y1Y- Y2max X1 X2F—k I I I________________O°2J, Jy2= , Jxj, [(x1 — + (y1 —y2)2]3/X dx21y. (5.42)(Ya=Y (YZmF5 = k o J Ii dy2 dy1. (5.43)Yimin Yz=Y+F6 is the Coulomb force between regions A12 and A23 of Fig. 5.5.(Yi fyy- fX1F—k I I I I______6 — i °2J1=JX111 Jx21,, [(x1 — x2) + (y1 —y2)2]312X dx21y. (5.44)(Yi (YzYF6=ku1Y2) J I dydy1. (5.45)YiY+5.2.4 Instantaneous Radial Coulomb ForceThe instantaneous radial Coulomb force is given by Eq. (5.13) for (1)> ouch andEq. (5.29) for CX) < touch The integrals in these equations were numerically evaluatedusing a multiple Gaussian integration algorithm(5-7)on an IBM 3081 mainframe computer(the corresponding Fortran program is given in Appendix A3.1). Several of the—136—calculations were duplicated using a Romberg integration algorithm(5)on a Nicolet 1180minicomputer (the corresponding Fortran program is given in Appendix A3.2), withagreement 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 functionof the position angle, ‘1 (Fig. 5.1). The force curve in Fig. 5.6 is the plot of the value ofEq. (5.13) or Eq. (5.29) for a total of 291 different values of 0. The ordinate shows theforce in Newtons for r = 1 cm, r’ = 5 mm, for which the ratio of cyclotron radius to thedisk radius is 2. When the calculations were repeated for other values of r and r holdingthe ratio r/r’ constant, the shape of the force curve remained the same if the ordinate scalewas adjusted by a factor of hr2. For example, if r = 2 cm and r’ = 10 mm, rir’ still equals2, and the force curve superimposes with that of Fig. 5.6, if the ordinate scale extendsfrom 0 — 0.5x1016 Newtons, rather than 0 — 2.0x1016 Newtons as in Fig. 5.6. Theoverall 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 =0touch For touch’ the overlap area (Fig. 5.3) increases as 0 decreases and theradial Coulomb force becomes zero when the disks completely overlap (0 = 0).Figure 5.7 shows the instantaneous radial Coulomb force over the range 0 < 0< it,for rir’ = 5. The general features of Fig. 5.7 are similar to those of Fig. 5.6, but themaximum in the instantaneous radial Coulomb force occurs at the smaller touching angleappropriate 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—‘ad(10_lB N)2.01.61.20.80.40.0Figure 5.6 Radial Coulomb force on Disk 1, a uniformly charged disk of m1 ions, dueto a uniformly charged disk of m2 ions, as a function of the position of Disk2. This position is characterized by the angle 0 (Fig. 5.1). The force iscalculated from Eq. (5.13) if 0 < < 0uch and from Eq. (5.29) if<0< it, where ouch is the touching angle, the value of 0 where the twodisks 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 ratioequals 2. The ordinate gives the force in Newtons for r, the cyclotronradius = 1 cm, r’, the disk radius = 5 mm, and N2, the number of m2 ions,= i0. The average radial Coulomb force, <Fave >, was calculated fromEq. (5.46). Figs. 5.7 — 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 — 5.10.0 it/4 lt/2 32t/4 itcJ (radian)—138—Frad(1O_16 N)Figure 5.7ad(1O_16 N)(I) (radian)Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forr/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>, arelarger than their counterparts in Fig. 5.6, but smaller than their counterpartsin Figs. 5.8 — 5.10.10864200 (radian)Figure 5.8 Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forrir’ = 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>, arelarger than their counterparts in Fig. 5.6 and 5.7, but smaller than theircounterparts in Figs. 5.9 and 5.10.0 it/4 lt/2 3it/4 it0 7t12 37t/4 it:—139—2016ad 12(1O_16 N)840‘ (radian)Figure 5.9 Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forrir’ = 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> arelarger than their counterparts in of Figs. 5.6 — 5.8, but smaller than theircounterparts in Fig. 5.10.10080Frad 60(1016N)40200cJ (radian)Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forr/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>, arelarger than their counterparts in any of Figs. 5.6—5.9.0 7t/4 1t12 31t/4 itouch = 0.0200003r/r’= 100r = 1 cmr’ = 0.1 mm<‘ad> =2.1750 it/4Figure 5.10it/2 3n/4—140—Comparing Figs. 5.6—5.10 shows that the maximum in the radial force appears at aprogressively smaller touching angle, 0touch’ as r/r’ increases and further, that themagnitude 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(1)with its infmite radial Coulombforce at 0=0.5.2.5 Average Radial Coulomb ForceEqs. (5.13) and (5.29) give Frad the instantaneous radial Coulomb force, as afunction 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 integralfusi f7<Fraj> = - (I Eq. (5.13) dO+ J Eq. (5.29) dO ). (5.46)77.; JoThe value of < Frad > for the force curve in each of Figs. 5.6 — 5.10 is indicated in thefigure. Note that in Figs. 5.6 — 5.10, the only parameter which was varied was the diskradius, r’ and therefore the ratio r/r’. Note that as this ratio increases, <Frad> the averageradial force, increases and in the limit r/r’ — oo, where the disk model becomes the pointmodel, the average radial Coulomb force becomes infmite.5.2.6 Apparent Coulomb Distance, DOf greater interest than <F>, which is characteristic of N1, the number of m1ions, and N2, the number of m2 ions, is the “apparent Coulomb distance”, D. Thisdistance, which would be a function only of the model, can be obtained as follows.Combining Eqs. (5.1) and (5.4) gives—141—AF= kN2q1 (5.47)Equation (5.47) gives the average radial Coulomb force on a single perturbed ion. For N1ions of mass m1, the force would beAF= k N2 q2 N1 q1 (5.48)Now if we assume that the average Coulomb force (Eq. (5.46)) for the charged-disk modelof 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 giveD/kN2qN149c ‘4 <Frad>Eq. (5.49) gives the “apparent Coulomb distance”, D, as a function of N1 q1, the totalcharge on Disk 1, N2 q2, the total charge on Disk 2, and <Frad > (Eq. (5.46)), the averageradial Coulomb force for the uniformly charge-disk model. D is the distance at which thecharges of Disk 2 appear to be seen by Disk 1. Hence, it is termed “apparent Coulombdistance”.Table 5.1 lists values of D for many possible situations. D is proportional to thecyclotron radius, r; and the most insightful way of tabulating values of D is as a functionr/rt, as is done in Table 5.1. When tabulated this way, D varies modestly, from 1.792 rwhen rlr’ =2 to 1.030 r when rir’ = 100. The dependence of D upon r/r’ was confirmedfor 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/2mm, 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 CoulombBroadening in FT-ICRModel Apparent Coulomb Distance ReferenceDc*Average Position 1Mean Distance 1.273 (4/it) r 1Rms Distance 1.414 (‘J) r 1Charged disk and charged cylinder; rir’ = 2# 1.792 r this workCharged disk and charged cylinder; r/r’ = 3@ 1.629 r this workCharged disk and charged cylinder; r/r’ = 4 1.539 r this workCharged disk and charged cylinder; rir’ = 5# 1.478 r this workCharged disk and charged cylinder; rir’ = 8 1.371 r this workCharged disk and charged cylinder; rir’ = 10 1.327 r this workCharged disk and charged cylinder; r/r’ = 15@ 1.257 r this workCharged disk and charged cylinder; r/r’ = 20# 1.2 15 r this workCharged disk and charged cylinder; r/r’ = 25 1.183 r this workCharged disk and charged cylinder; rir’ = 50# 1.102 r this workCharged disk and charged cylinder; rir’ =100# 1.030 r this work*) Calculated from Eq. (5.49). D gives the Coulomb-induced frequency shift when usedwith 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, and10 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 ofD from Table 5.1 vs. rir’. We were ableto fit these values to a double-exponential equation of logarithmic rir’= [0.5504 + 1.128_3bOl08 + 0.5338e2511 losfr/r’)] r. (5.50)Several other curve fitting equations are also available (Appendix V. 4), For example,= { 2.172 — 1.535 log(rIr) + 1.014 [log(r/r’)]2— 0.3827 [log(r/r’)]3+ 0.05836 [log(r/r’)]4} r. (5.51)Eq. (5.50) is the best one among those curve fitting equations for calculating values of Dwith any r/r’ ratio between 2 and 100 (Appendix A3.4). There is no physical meaning tobe associated with Eqs. (5.50) and (5.51).5.2.7 Validity of the Charged-Disk Model5.2.7.1 Assumption of small like-ion interactionsThere are several approximations which have been made in the above modelling inaddition to those which already have been identified above as assumptions. The firstadditional assumption is that each of Disk 1 and Disk 2, the distributions of perturbed ionsand perturbing ions respectively, maintain their integrity as each traverses its cyclotronorbit. For Disk 1, this integrity will be maintained if the Coulomb forces within the diskare 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 whosecharge density is o, disk radius is?, and cyclotron radius is r. The mass and number of-144-(cm)2.Or1.81.61.41.21.0Figure 5.11 Normalized apparent Coulomb distance parameter, D, as a function ofTI?, the ratio of the cyclotron radius to the disk radius, for the charged-diskmodel. 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 exponentiallogarithmic equation shown. This equation can be used to predict values ofD for any rir’ ratio between 2 and 100. No physical meaning can beattached to this equation.r / r’—145—ions in Disk 1 are m1 and N1. Consider next an identical differential element which isadjacent and radially aligned with the first differential element. Since r’ >> dr’, the twodifferential circular elements can be approximated to two point charges. The radialCoulomb force, dFr&d on the first element, arising from the second element would bek 01 dA 1 dA = k 1 (dr’)2 01 it (dr’)2r&d 4 (dr’)2 4 (dr’)2= k o it2 (dr’)2 (5.52)The differential Lorentz force on the first differential element would bedFi..orentz = o.j it (dr’) w01 r B. (5.53)If we assume that the criterion for maintaining the integrity of the charged disks isdifferential like-ion Coulomb force << differential Lorentz force, (5.54)then, substituting Eqs. (5.52) and (5.53) into Inequality (5.54) yieldsk o it4 << rB, (5.55)or4w01rB 4q1rB2=, (5.56a)or equivalently,—146—<< 8.52 x 108 q1rB2 (5.56b)Since Disk 1 is uniformly charged,01=(5.57)Substituting Eq. (5.57) into Inequality (5.56a)4 r B2 r’2N1<< km (5.58a)or equivalently,B2 ‘2N1 << 2.68 x iO T miT (5.58b)Inequalities (5.56b) and (5.58b) describe the criterion for validity of the assumption ofnegligible like-ion Coulomb interactions, contained in the models of this chapter, in termsof o, the charge density of Disk 1 in electronic charges per square mm; N1, the ionnumber in Disk 1; r’, the disk radius in mm; m1, the ion mass in Dalton; r, the cyclotronradius 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 N1 = 10 ions, r’ = 1 mm,r = 1 cm, B = 1 Tesla and m1 = 100 Daltons, the left hand side of Inequality (5.58b) equalsiO, whereas the right hand side of Inequality (5.58b) equals 2.68 x and theinequality 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 modelor any line-charge model, where by defmition, r’ =0. This violation is a characteristic ofalmost all prior treatments of FT-ICR motion. In these prior treatments, the violation isimplicitly made and then implicitly ignored. Inequalities (5.58) provide the criterion forwhen it is safe to ignore like-ion Coulomb interactions.5.2.7.1 Assumption of small unlike-ion interactionsThe derivation of Eq. (5.4) from the cyclotron equation implicitly requires that theperturbation of the motion of Disk 1 by Disk 2 is small. That isI 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 Coulombforce 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. SubstitutingEq. (5.47) into Eq. (5.59) givesk q1 N2 q2<< q1 c rB. (5.60)Solving for N2 givesD2w rB (D’)2q r3B2N2 << k q—k q21m (5.61 a)or equivalently,— 148 —(D’)2q r3B2N2 << 6.70 x 1010 , (5.61b)q2 1where D’ is the numerical component of D (Table 5.1). Inequality (5.61b) gives thecriterion for validity of our uniformly charged-disk model as a function ofN2, the numberof m2 ions, q1 and q2, the charges of m1 and m2 in units of the electronic charge, m1, theion mass of Disk 1 in Dalton, r, the cyclotron radius of the disks in cm, Do’, the numericalcomponent 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 inour model that the disks are rigid and will maintain their integrity in the presence of theCoulomb perturbation. For example, for N2 = 10 ions, D’ = 1.327 (r/? = 10, Table 5.1),q1 = q2 = 1 electronic charge, B = 1 Tesla and m1 = 100 Daltons, the left hand side ofInequality (5.61b) is 10, whereas the right hand side is 1.18 x iO, and the Inequality issatisfied.5.2.7.3 Assumption of high frequency perturbationsAnother assumption implicit in our treatment is that the perturbation frequency,—k2’ is large with respect to the perturbation, A0 That is,I%1—C002 >> (5.62)Substituting Eq. (5.1) into Inequality (5.62) giveskN2q>>D2B ‘C— 149 —or alternatively,1m2-m I kNm21 >> (D’) r3B2 (5.64a)or equivalently,Im-mi Nm2 m1 >> 1.49 xlO(D’)2 r3 B2 (5.64b)f ‘ = q2. Inequality (5.64b) gives the “high frequency criterion” for the validity of thedisk model as a function of m1 and m2, the masses in Dalton of the perturbed andperturbing ions, Do’, the numerical component of D in Table 5.1, N2, the number of m2ions, B, the magnetic field strength in Tesla, and r, the cyclotron radius in cm of Disk 1and Disk 2. For example, for B = 1 Tesla, r = 1 cm and N2 = 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 massdifference of 1 Dalton, the left hand side of Inequality (5.64) would be 10” at 100Daltons, 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 barelysatisfied. At masses above 1,000 Dakons, we would expect that greater mass separations,higher magnetic fields or lower numbers of perturbing m2 ions would be needed for themodels 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. Forexample, Inequality (5.62) has its counterpart in NMR where magnetic field gradientswhich would broaden the signal from a macroscopic sample are averaged to zero byspinning the sample, if the sample-spinning frequency exceeds the inhomogeneouslybroadened linewidth. The same process applies to FT-ICR, where the cyclotron motion—150—itself averages x, y magnetic inhomogeneities to zero.(9) The physical basis for theseaveragings of oscillatory perturbations is the uncertainty principle.5.3 The Charge-Cylinder Model5.3.1 Apparent Coulomb Distance and Charged-Cylinder ModelIn many Fr-ICR experiments, ions are formed in a cylindrical volume whose axiscoincides with the z axis of the ICR cell. The cylinder has a radius r’ and extends from —112to +1/2, where l is the z-axis length of the cell. After Fr-ICR excitation, which we willassume is uniform throughout the spectrum, the dynamic disthbution of ions in the cellwould be a sum of cylinders, one for each ion mass, each cylinder rotating around the zaxis with cyclotron radius r. For purposes of calculating Coulomb effects between ions ofdifferent masses, we can examine the system in a rotating frame, just as was done for thecharged-point, charged-line and charged-disk models. This charged-cylinder model can becreated 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.Derivation of the Coulomb-induced frequency shift for the charged-line model(1)gaveEq. (5.2) which was a function of the same “apparent Coulomb distance”, D, as waspresent in Eq. (5.1), the frequency shift equation for the charged-point model. Wetherefore assume that our “apparent Coulomb distance” D for the charged-disk model ofthis 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 calculateCoulomb-induced frequency shifts and Coulomb-induced line broadenings for the charged-cylinder model. Rigorous conformation of the legitimacy of this assumption would have tocome from the evaluation of 6-dimensional integral equations, analogous to the 4-dimensional integral Eqs. (5.13) and (5.29).—151—t4%00ISSSFigure 5.12 The Charge-cylinder model. The cyclotron radius r, cylinder radius r’, andz direction length 1 of Cylinder 1 and Cylinder 2 both are the same. Thedirections of applied magnetic field B and their cyclotron motions areFCylinder 1Cylinder 22r’0xIIIIIIISSSS SS SS SS SS SSSSSS544SSzshown as indicated.—152—5.3.2 Validity of the Charged-Cylinder ModelThe charged-line model predicts smaller Coulomb-induced frequency shifts than doesthe charged-point model.(1) That is, Eq. (5.2) (Eq. (4.19) in Chapter 4) gives smallerfrequency shifts than does Eq. (5.1) (Eq. (4.14) in Chapter 4). This is not surprising sincedispersing the m2 ions into the z direction will reduce the Coulomb force seen by any singlem1 ion in line 1. We would similarly expect that the Coulomb-induced frequency shifts forthe 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 charged-disk model, will also apply to the charged-cylinder model. However, for any examplecharacterized by a particular N2, r/r’, and r, etc., the criteria will be more easily satisfied forthe 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 thecharged line and charged-cylinder models. The corresponding charged-disk expressionswere derived using Eq. (5.1).5.3.2.1 Assumption of small like-ion interactionsConsider a differential disk element of radius dr’ and thickness cii at coordinate z1 = 0in Cylinder 1, whose cylinder radius is r’, z-length is 1, and cyclotron radius is r. Themass, charge, and number of ions in Cylinder 1 are m1, q1 and N1, respectively. Consideranother differential cylinder element of radius dr’ and z-length 1 which is adjacent andradially aligned with the first differential disk element. Since Cylinder 1 is uniformlycharged, the ion numbers in the differential disk element, dNdl, and the ion number in thedifferential cylinder element, dNa, aredN = N (dr’)2 dl = N (dr’)2 dl (5.65a)itr 1 r 1—153—and= N1 d = N1 (d?/r’)2. (5.65b)As ci,’ <<r’, the line-charge model (Eq. (4.17)) presented in Chapter 4 can be applied tocalculate the radial Coulomb force, dFr on the differential disk element at =0 due tothe differential cylinder element. Then,=— k dNdl dN1q12 1— 2 dr l 4(l/2)2 + (2 dr’)2k (N1 q1)2 (dr’)3 dlr’4l2The Lorentz force on the differential disk element would be‘‘Lorentz = N1(dr2dlq1 oi r B. (5.67)If we assume that the criterion for maintaining the integrity of the charged disks isI like-ion Coulomb force I << Lorentz force, (5.68)then, substituting Eqs. (5.66) and (5.67) into Inequality (5.68) yieldsk N q dr’<< 0o rB. (5.69a)Since dr’ <<r’, Inequality (5.69a) can be simplified by setting 10 dr’ r’ in order to get—154—the desired result,kNq10 ‘ << rB, (5.69b)or alternatively10 rB2 r’ 1N1<< k (5.70a)or equivalentlyN1 << 6.70 x 1010 r Br 1 (570b)Inequality (5.70b) gives the criterion for validity of the assumption of negligiblelike-ion Coulomb interactions, contained in the charged-cylinder model of this chapter, interms of N1, the ion number in the cylinder; m1, the ion mass in Dalton; r’, the cylinderradius 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 correspondingexpression in SI units. For example, for N1 = 10 ions, r’ = 1 mm, 1= 2.4 cm, r = 1 cm,B = 1 Tesla and m1 = 100 Daltons, the left-hand side of Inequality (5.70b) equals iowhereas the right-hand side of Inequality (5.70b) equals 1.61 x iO, and the inequality issatisfied. Analogous inequalities, of course, apply for Cylinder 2. Note that for theexample quoted, the ratio between the right and left hand sides, 1.61 x iO: iOn, ofInequality (5.70b) is 1.61 x i05, whereas in the corresponding example for the diskmodel, 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 chargedcylinder 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 maximumvalue 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 interactionsInequalities (5.61) gave the criterion that the Coulomb interaction between the disksshould be small with respect to the Lorentz force on either disk. The correspondingexpression for the charge-cylinder model from Eqs. (5.3) (the maximum Coulomb shift)and (5.4) isD J(l/2)2+D2 q rB2N2 << C C 1 , (5.71a)q2 m1or equivalently,D .\/(112)2+D2 q rB2N2 << 6.70 x 1010 C C 1 • (5.71b)q2 m1Inequality (5.71b) gives the criterion of small intercylinder Coulomb interactions for ouruniformly charged-cylinder model as a function of N2, the number of m2 ions, q1 and q2,the charges of m1 and in2 in units of the electronic charge, m1, the ion mass of cylinder 1 inDalton, 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 correspondingequation in SI units. Note that unlike Inequalities (5.61) which is a function of Do’, thenumerical 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 maintaintheir integrity in the presence of the Coulomb perturbation. For example, for N2 = iO—156—ions, D = 1.327 r (r/r’ = 10, Table 5.1), q1 = q2 = 1 electronic charge, r = 1 cm, 1 = 10cm, B = 1 Tesla and m1 = 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 theexample quoted, the right hand side of Inequality (5.7 lb) was 4.60 x 10’°, whereas in thecorresponding example for the disk model, the right hand side of Inequality (5.61b) was1.18 x iO. These examples illustrate the lesser sensitivity of the charged-cylinder modelto violations of the “small unlike-ion interaction” assumption.5.3.2.3 Assumption of high frequency perturbationsAs for the charge-disk model we assume in our treatment that the perturbationfrequency, % — is large with respect to the perturbation, tco on the chargedcylinders. Substituting Eq. (5.3) (the maximum Coulomb shift) into Inequality (5.62)giveskN qI W — (002 I >> 2 2 , (5.72)rB D q(l/2)2+Dor alternatively,Im-mi kN2 1 2 (5.73a)m21rB2Dc4(l/2)2+Dc2or equivalently,I m2—m1 I>> 1.49x 10-11N2, (5.73b)m21rB2Dc4(l/2)2+D=“2 Inequality (5.73b) gives the “high frequency criterion” for the validity of the—157--charge-cylinder model as a function of m1 and m2, the masses in Dalton of the perturbedand perturbing ions, D, the apparent Coulomb distance in cm, N2, the number of m2ions, r, the cyclotron radius in cm of cylinder 1 and cylinder 2, 1, the length of the cylindersin cm, and B, the magnetic field strength in Tesla. For example, for B = 1 Tesla, r = 1 cm,1 = 10 cm, N2 = iO, and D = 1.327 cm (r/r’ = 10, Table 5.1), the right hand side ofInequality (5.73b) is 2.17 x For a mass difference of 1 Dalton, the left hand side ofInequality (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 highfrequency perturbations is only barely satisfied. At masses above 1,000 Daltons, wewould expect that greater mass separations, higher magnetic fields or lower numbers ofperturbing m2 ions would be needed for the models of this work to be valid. Note that forthe example quoted, the right hand side of Inequality (5.73b) was 2.17 x i0-, whereas inthe 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-cylindermodel to violations of the “high-frequency perturbation” assumption.5.4 The Coulomb-Induced Frequency Shift and InhomogeneousBroadening of the Charged-Cylinder ModelOn the basis of the above discussion in Section 5.3, we can predict the Coulomb-induced frequency shift and inhomogeneous broadening of the charged-cylinder modeldirectly from Eq. (5.2) if this cylinder model is valid. For a certain rir’ ratio, the apparentCoulomb 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 = 1Tesla, r = 1 cm, 1 = 2.4 cm, N1 = N2 = m1 = 250 Daltons, m2 = 251 Daltons, and D= 1.327 cm and 1.215 cm (r/r’ = 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—1513tf011(Hz)975141210(Hz)864z1 (cm)Figure 5.13 Coulomb shifting and broadening for the charged-cylinder model ofFig. 5.12. The frequency shift for m1 ions, calculated from Eq. (5.2b), isplotted as a function of the position of of the m1 ion. The parameters usedfor these calculations are listed in Table 5.2 and z-axis length of the cylinderas indicated. The upper curve in each figure gives the Coulomb shifts forrir’ = 20 (D = 1.2 15 cm), and the lower curve in each figure gives theCoulomb shifts for rIr’ = 10 (D = 1.327 cm).-1.2 -0.8 -0.4 0.0 0.4 0.8 1.2z1 (cm)-4 -2 0 2 4—159—Coulomb-induced cyclotron frequency shifts and inhomogeneous broadenings calculatedfrom Eq. (5.2) are shown in Figure 5.13(a) for these parameters. As a comparison, for thesame parameters, except that 1 is changed to 8 cm, the same calculationsaxe processed and shown in Figure 5.13(b). The longer length of the cylinders willeffectively reduce both the Coulomb-induced cyclotron frequency shift and inhomogeneousBroadening. 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 = 1 Tesla,r = 1 cm, 1 = 2.4 cm, N1 = N2 = q1 = q2 = 1 electron charge,= 250 Daltons, n.2= 251 Daltons, andD = 1.327 rand 1.215 rD Eq. (5.70b) Eq. (5.71b) Eq. (5.73b)N1 right-hand N1 right-hand left-hand right-hand1.327 r 6.43x108 6.36x108 1.59x10 6.28x1071.215 r 3.22x108 i05 5.49x108 l.59x10- 7.18x105.5 Experimental Tests of The Charged-Cylinder Model5.5.1 The Unlike-ion Coulomb-Induced Frequency Shifting from Experiment ofFranci et al.(10)Francl et al.(lO) in 1983 measured the experimental ICR frequency vs. ion number ina pulsed (no Fl’) ICR mass spectrometer with an elongated cell (3.18 cmx3.18 cmxl5.34cm) 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. Thedependences of ICR effective frequencies of these four compounds on ion numbers wereexperimentally all linear. The slope of the ICR effective frequency vs. ion number forbenzene, which has only one mass peak, was measured to be —58 HzJV, where V isvoltage of the ion Coulomb potential in the analyzer cell. According to the data given byFranci et al., we deduce1 million positive ions = + 0.186 V (ion Coulomb potential). (5.74)Since there is no mass dependence in the ion Coulomb potentia1,(111) the slope of benzenecurve 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 wassolved by the perturbation method (Eqs. (4.13) and (5.4)). It has been proved that theexperimental ICR frequency approximately was equal to the ion natural cyclotron frequencyminus the frequency shifts induced by the trapping electric field and the ion Coulombpotential.(’0)Therefore, these frequency shifts are superimposed on the ion cyclotronmotion. When there exist two ion species simultaneously in the analyzer cell, the unlike-ion Coulomb-induced frequency shifts can be calculated by Eq. (5.75),fi41ei4t= [slopexN—(—58Hz/V)x(N+N2)]xO.186V (5.75)where bf1 is the unlike-ion Coulomb-induced frequency shift component in Hz, of an ionspecies m1, due to the presence of an ion species m2; Afei is the total change inexperimental cyclotron frequency in Hz, of m1; 4f is the frequency shift componentinduced by the trapping electric field in unit of Hz, ; N1 is the ion number of m1 in unit of— 161 —million ions; and N2 is the ion number of m2 in unit of million ions.The slopes of bromobenzene curves of two mass peaks m/z 156 (CH79Br) andm/z 158 (C6H81Brj were measured to be —131 Hz/V and —146 Hz/V, respectively.Although Francl er at. did not give the relative ion intensities of these two ion peaks, theintensity ratio of rnlz 156 to mlz 158 should be 100 : 98 from isotopic abundances of 79Brand 81Br. 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 Coulombinduced 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 mlz156 is=— (146 Hz/V x 0.99 — 58 Hz/V x 2) x 0.186 V = — 5.3 Hz. (5.76b)The parameters in the experiment of Franci et at. are: r = 1.5 cm, 1 = 15 cm (for anelongated cell 3.18 cmx3.18 cmxl5.34 cm), B = 3.4 Tesla, q1 = 2 = 1 electron charge,and D = r (for scanning ICR), the maximum unlike-ion Coulomb-induced frequency shiftsexpected from the charged-cylinder model (Eq. (4.20b), or Eq. (5.3)) are:=— 2.2918x10 x 0.99x106 =— 3.9 Hz, (5.77a)1.52x 3.4 xJ752 + 1.52=— 2.2918x10x 1.01x106= —4.0 Hz. (5.77b)11.52x 3.4 xJ752 + 1.52—162—Comparing Eqs. (5.76) with Eqs. (5.77) shows that the charged-cylinder model agreeswell with the experimental results. Even though the error range is ±25 — 30%, for such asmall frequency shift and without calibration for the Coulomb effects from like-ioncontamination, the agreement seems acceptable.5.5.2 Unlike-ion Coulomb-Induced Frequency Shifting in FT-ICRIn our experiment, CpMn(CO)3(Cp = ?75-CH)was chosen as the parent moleculeto study the unlike-ion Coulomb interactions, because only four primary ions, Mn (mlz55), CpMn (m/z 120), CpMnCO (mlz 148), and CpMn(CO) (m/z 204), were producedfrom CpMn(CO)3by electron ionization, and these mass peaks are separated well in themass spectrum. The experiments were run in a 2.54 cm cubic trapped-ion cell of a home-built FT-ICR mass spectrometer combined with a Nicolet FTMS 1000 console which hasbeen 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 points64 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 ineach cylindrical ion cloud. A method for measuring ion current (ion number then can becalculated) was described by Hunter, Sherman, and Mclver.(12) The total ion number inthe analyzer cell was measured using an electrometer (e. g., Keithley, Model 616) and asmall modification of the trapping bias was made to separate the ion current from theelectron current.(12) We studied the unlike-ion Coulomb interactions by monitoringelectron emission current, because for a pulsed electron beam such as employed in FTICR, the number of positive ions, N, produced during the beam pulse or duration tb isproportional to electron emission current‘e from the filament used for ionization(13)QI P1 tN qk Tb (5.78)—163--where Q1 is the total ionization cross section of the gaseous molecules at a specifiedionizing potential, 1e is the path length that ionizing electron travel, P is the pressure of themolecules, kB is the Boltzmann’s constant (kB = 1.380658x 10—23 J K—1), and T is thetemperature. The total ionization cross sections, Q1, have been measured for a veiy limitednumber of compounds. There is no available Q. value of CpMn(CO)3.Nevertheless, inSection 2.2.3 of Chapter 2, the three hypotheses have been suggested, and can be appliedto calculate the theoretical molecular polarizabilities of transition metal complexes following18-electron rule and to calibrate the nominal pressure. Hence, the molecular polarizabilityof CpMn(CO)3would bea(CpMn(CO)3) = a(C5H)+3 x a(carbonyl) + a(Kr)= 8.60 + 3 x 2.82 + 2.4844= 19.54 (A3) (5.79)where a(C5H),a(carbonyl), and a(Kr) are the polarizabilities of Cp ligand, carbonylligand, and Kr atom, respectively; and their values have been given in Section 2.2.3 ofChapter 2. According to Eq. (2.2) in Chapter 2, the calibrated pressure of CpMn(CO)3isthe nominal pressure of CpMn(CO)3“CpMn(CO)3— 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 molecularpolarizabilities.(’4)For general cases, the empirical relationship between total ionization-164-cross sections and molecular polarizabilities can be estimated as(14)Q1 =2.2 a (5.81)where Q is in A2 and a in A. Total ionization cross sections generally are constant atelectron energy around 75— 100 eV.(15) Assuming that this relationship in Eq. (5.81) canbe extended to the organometallic complexes whose molecular polarizabilities are around 20at electron energy 90 eV, total ionization cross section of CpMn(CO)3will beQ(CpMn(CO)3)= 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 longitudinalmagnetic field.(15)le can be assumed to be slightly less than twice the length of the ICRcell(12) (le = 4.8 cm for the 2.54 cm cubic analyzer cell). When the pressure P = 5.9 x10-8 Torr (7.8 x 10-6 Pa) and the temperature T = 300 Kelvin, for a 100 nA electroncurrent in aS ms beam time, the number of the positive ions produced from CpMn(CO)3at90 eV electron ionization, according to Eq. (5.78), isN— (43x1020)x (4.8x10-2)x (7.8x106)x (lOOxlO-9)x (5.OxlO-3)+— (1.6022x109 x (1.380658x1023x 300= 1.2 x i0 (ions). (5.82)After the FT-ICR mass spectrometer with a water cooling system was turned on forpreheating for 30 minutes, drift of the magnetic field was monitored at the cyclotronfrequency of m/z 204 for 40 minutes until the drift continuously increased as indicated inFigure 5.14. The single resonance and the triple resonance, described in Section 2.2.2 ofChapter 2, were used to study the frequency shifts in the FT-ICR mass spectrometer.—165—143.62fen :o(kHz)143.591 43.580 5 10 15 20 25 30 35 40Time (mm)Figure 5.14 Drift in the 1.9 Tesla magnetic field used in the experiment after preheating30 mm. The cyclotron frequency of CpMn(CO) (m/z 204) was monitoredwith time by using emission current = 335±20 nA.Using the triple resonance, CpMn was retained in the cubic analyzer cell, but the otherthree ions, Mn, CpMnCO, and CpMn(CO), were ejected. The effective cyclotronfrequency of CpMn was monitored as the nominal electron emission current was raisedfrom 100 nA to 1 pA. Then, the experimental procedure was transferred from the thpleresonance to the single resonance. The three ion species Mn, CpMnCO, andCpMn(CO), which were ejected before by the triple resonance, were added in the cubicanalyzer cell. Their ion intensity sum was about 1.5 times the ion intensity of CpMn. Theion intensity of CpMn remained constant with or without the ejection pulses. Theeffective cyclotron frequency of CpMn was monitored again while the nominal electronemission current was scanned from 100 nA to 1 iA. The measurements are shown inFigure 5.15 and are listed in Table 5.3. The unlike-ion Coulomb induced frequency shift isgreater in the 2.54 cm cubic cell and 1.9 Tesla magnetic field than in the 3.18 cmx3. 18cmx 15.34 cm elongated cell and 3.4 Tesla magnetic field used by Francl et al.(10) The—166—kHz/million ions) of the plot of the effective cyclotron frequency of CpMn vs. electronemission current with unlike-ion Coulomb interaction is about 3.5 times the slope (0.0555kHzlmilhion ions) of that of single ion species CpMn vs. electron emission current withoutunlike-ion Coulomb interaction, as shown in Fig. 5.5 (rather than 2.5 times, if thefrequency shifts were caused only by the ion Coulomb potential in the trapping electricfield). Since the magnetic field steadily increased (Fig. 5.14), the frequency shifts (towardlower frequencies) in Fig. 5.15 were caused undoubtedly by ion Coulomb interactionsincluding the ion Coulomb potentials in the trapping field, the unlike-ion Coulombinteractions, 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 1million ions. According to experimental slopes in Fig. 5.15, the unlike-ion induced-Coulomb frequency shift from Eq. (5.75) isfcpM = — 0.194 x 1 — (— 0.0555 x 2.5) = — 0.055 (kHz)= —55 Hz. (5.83)The unlike-ion induced-Coulomb frequency shift according to the charged-cylinder modelforD = 1.327 r (Eq. (4.20b) or Eq. (5.3)) is2.2918x1Ox 1.5x106AfcpMfl+ = —___________= —43 Hz. (5.84)1.327 x 1 x 1.9 x q 1.22 + 1.3272Comparing Eq. (5.83) with Eq. (5.84), the charged-cylinder model produced the desiredvalue.—167—The effective cyclotron frequency, feff’ of CpMn vs. number of ionswith and without unlike-ion Coulomb interactions. The data indicated by“0” were obtained by ejecting all ions except CpMn, and there was nounlike-ion Coulomb effect on CpMn. The data indicated by “•“ wereobtained without triple resonance, and the effective cyclotron frequency ofCpMn was affected by unlike-ion Coulomb interaction.244.244.50fett 244.46(kHz)244.42244.380.0 0.1 0.2 0.3 0.4 0.5Number of ions (million ions)0.6Figure 5.15—168—Table 5.3 The effective cyclotron frequency feff of CpMn with the change ofelectron emission current, EMC. The experimental parameters aregiven in the text.Without unlike-ion Coulomb interaction With unlike-ion Coulomb interactionEMC (i) feffOZ) EMC (jiA) feffOZ)0.085 244.492016 0.095 244.5059970.170 244.491344 0.160 244.5016260.330 244.486645 0.270 244.4931180.490 244.481521 0.400 244.4789880.595 244.479028 0.515 244.4682980.665 244.478110 0.615 244.4602040.760 244.474919 0.765 244.4439970.905 244.467348 0.850 244.4264881.045 244.467348 0.925 244.4112891.135 244.464552 1.100 244.3962465.5.3 Experimental Inhomogeneous Broadening in Fr-ICRThe mass peak width at 50% height for CpCr (mlz 117) as a function of electronemission current was monitored, and the experimental data are listed in Table 5.4. Theother four unlike-ion species in the analyzer cell were:C6Hr (m/z 130), CpCrO (mlz133), CpCrNO1 (m/z 147), and CpCrNOCH (mlz 162). The sum of their ion intensitieswas double that of Cp&. The experiment was run after the FT-ICR mass spectrometerwas turned on for 2 hours, so that the drift of the magnetic field was minimized. The masspeak 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 beamduration 20 ms, beam voltage —25 V. data points 64 k, zero filling 1, trapping voltage +1V, and strength of the magnetic field 1.9 Tesla.Table 5.4 The peak width at 50% height of CpCr with the change of electronemission current, EMC. The experimental parameters are given inthe text.EMC (uA)Left frequency* Right frequency* Peak widthat 50 % height (kHz) at 50% height (kHz) at 50% height (Hz)0.125 250.6785 250.5368 141.70.230 250.6605 250.5197 140.80.330 250.6423 250.5001 142.20.435 250.6327 250.4900 142.70.565 250.6251 250.4819 143.20.665 250.6155 250.4724 143.20.805 250.6073 250.4631 144.20.965 250.6059 250.4610 144.91.125 250.5958 250.4518 144.01.300 250.5901 250.4459 144.21.555 250.5831 250.4385 144.6* 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 sixdigits after the decimal point as given by the Nicolet computer in Table 5.3.— 170 —In the charged-line model(1) and the charged-cylinder model,(2) the maximumfrequency shift is at z1 = 0 and minimum frequency shift is at z1 =1/2. The broadening,however, is not a simple subtraction of the minimum frequency shift from the maximumfrequency shift, because the frequency spectrum of FT-ICR is given by the Fouriertransform of the time domain signal. Therefore, the inhomogeneous broadening (the timedomain signal) induced by unlike-ions can be expected from Eq. (5.2), or Fig. 5.13, butthe inhomogeneously broadened FT-ICR peak shape, such as the experimental data givenin Table 5.14, can not be evaluated directly from Eq. (5.2). Further work is needed toformulate the inhomogeneously broadened FT-ICR peak shape for Eq. (5.2), or Fig. 5.13.5.6 DiscussionIn this chapter, a two-dimensional charged-disk model is proposed to explainCoulomb-induced frequency shifting in FT-ICR mass spectrometry. This modelcorresponds more closely to the actual FT-ICR experiment than does the charged-pointmodel,(1)which has recently been developed. The model consists of a uniformly chargeddisk of ions of mass m1, whose excited cyclotron motion is perturbed by a seconduniformly charged disk of ions of mass m2, whose cyclotron motion is also excited.Apparently, the second disk creates a radial force on the first disk which lowers thecyclotron frequency of the first disk. This is most easily seen in a rotating frame whichrotates at the cyclotron frequency of the first disk. This radial force is numericallyevaluated 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,(1)which has aninfinite average radial Coulomb force, the average radial Coulomb force for the chargeddisk model is finite. This average Coulomb force allows use of formulae which permitcharacterization 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 rotatingframe would give the same frequency shift. It is argued that the same “apparent Coulombdistance” would apply in the case of a charged-cylinder model, which accounts forCoulomb-induced line broadening in addition to Coulomb-induced frequency shifting. The“apparent Coulomb distances” are tabulated for convenient use with simple analyticalexpressions which quantitatively give the Coulomb-induced frequency shifts and theCoulomb-induced line broadenings in any experimental situation covered by the charged-disk or charged-cylinder models. The assumptions of the disk model are examined andanalytically defined by formulae in terms of like-ion Coulomb interactions, unlike-ionCoulomb interactions and the magnitude of the Coulomb interaction relative to thefrequency difference between the perturbed and perturbing ion. From our numericalanalysis, the “apparent Coulomb distances” is found to be a function of ratio of the ioncyclotron radius to the cylindrical radius of the ion cloud, rir’.The experimental tests for unlike-ion Coulomb-induced frequency shifting showedthat the charged-cylinder model worked fairly well for either the elongated analyzer cell orthe cubic analyzer cell. In our experiment, the number of ions was estimated theoreticallywith respect to the electron emission current (Eq. (5.78)). We assumed: (1) the methodsuggested in Section 2.2.3 of Chapter 2 can be used to evaluate the molecular polarizabiityof CpMn(CO)3,and then, to calibrate the sample pressure; (2) the total ionization crosssection of CpMn(CO)3was linearly related to the molecular polarizabiity of CpMn(CO)3.These assumptions has not been proved experimentally yet. Further study onpolarizabilities 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 andMarshall in 1986.(’) Theoretically, through numerical analysis of the ion motion in FrICR, and experimentally, through examining positive benzene ions, they proved that as theions 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 chaptercan be used to explain the unlike-ion Coulomb-induced broadening in FT-ICR. Theunlike-ion Coulomb-induced broadening effect on CpCr due to the presence of fourunlike-ions CpCrO,C6Hr, CpCrNO, and CpCrNOCH has been studied here.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), theinhomogeneous broadenings caused by the unlike-ion Coulomb interactions in FT-ICRwere apparently not serious. As a matter of further interest, it should be noted that whenthe 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 ionclouds could be expanded because of the ion-ion Coulomb repulsion; that is, the rir’ ratiowas decreased, and the D value was increased (Table 5.1). From Eq. (5.2), this change inD should lessen the Coulomb-induced frequency shifting.Compared to the superconducting magnetic field, the magnetic field drift in thisexperiment was serious as indicated in Fig. 5.14, because a water cooling system wasused. Even after the magnets had been turned on for 2 hours, the ICR frequency shift dueto the magnetic field drift can be almost 10 Hz within 30 mins. For a superconductingmagnetic field of 1.9 Tesla, ICR frequency shift due to the magnetic field drift was reportedless than 1 Hz per dáy(14) Consequently, the FT-ICR mass spectrometer in our laboratoryshould be equipped with superconducting magnets in the future.The procedures for calibrating Coulomb-induced frequency shifts, proposed by eitherJeffries, Balow, and Dunn,(11)or Ledford, Rempel, and Gross,(16)all make use ofcalibrant masses to correct the trapping field effect, geometrical shapes of ion clouds, andion-ion Coulomb interactions. In the charged-line model (Chapter 4) and charged-cylindermodel (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 thiswork could allow absolute mass calibration of FT-ICR spectra, independent of any knownmasses, only requiring a knowledge of the magnetic field strength and a calibration of theFT-ICR signal strength vs. number of ions.— 174 —REFERENCES1. 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 CyclotronResonance 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.; MclverJr., 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, 169-187.12. Hunter, R. L.; Sherman, M. 0.; Mclver, R. T. Jr. mt. J. Mass Spectrom. Ion.Phys. 1983, 50, 259—274.13. 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, 6129-6132.15. Wang, T.-C. L.; Marshall, A. G. In:. J. Mass Spectrom. Ion Proc. 1986, 68, 287-301.16. 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 AlliedTopics, San Francisco, CA, 1988, 608—609.— 176 —CHAPTER 6TAYLOR’S EXPANSION APPROXIMATION OFION-ION STRONG COUPLING COULOMB INTERACTIONIN FT-ICR MASS SPECTROMETRY—177--6.1 IntroductionThe charged-line model and charged-cylinder model developed recently (1-2) explainion-ion Coulomb-induced frequency shifting and ion-ion Coulomb-induced broadening inFT-ICR mass spectrometry. The charged-cylinder model(2)corresponds more closely tothe actual FT-ICR experiment than does the charged-line model.(1)However, there arethree essential prerequisites in the charged-cylinder model, which have been discussed inChapter 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 >> A%1(), perturbation frequency shift.The second prerequisite is related to the third prerequisite, because they both depend on thenumber of unlike ions, N2 (Eqs. (5.61) and (5.64)). Coulomb interaction between two ionspecies will be sharply increased for small separations (non-overlapping or overlapping alittle) as described in Figs. 5.6 — 5.10 of Chapter 5. It is desirable to know what willhappen if the unlike-ion Coulomb interactions for small separations are comparable to theLorentz force on the ions, i. e., the second prerequisite does not hold; with increase in thenumber of unlike-ions, N2, the unlike-ion Coulomb interactions will become strongenough to overcome the Lorentz force. Also, the perturbation frequency shift will becomelarge enough to nullify the relationship in the third prerequisite.Obviously, it is impossible to solve the effects of strong unlike-ion Coulombinteractions for small separations by using the averaging method in Chapter 5. In thischapter, a Taylor’s expansion approximation is used to study the strong Coulombinteractions between two ion species within small separations in FT-ICR. A preliminary—178—I 112 1/2 10 zFigure 6.1 A cylinder in cylindrical coordinates (p. q,, z). Radius and z-length of thecylinder are? and 1, respectively. Its center is at (r, £2), off the origin of thecylindrical coordinates.report of this study was presented in 1989,() in which it was showed that the strongunlike-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. Integrationmust be used to develop equations for the ion-ion Coulomb interactions of a large numberof ions. The ion cloud in the FT-ICR experiment assumes the shape of an approximatelycylindrical rod. Assume that this charged-cylinder has a radius r’ and length 1 and ismoving circularly with a cyclotron radius r. The analytical equations for a cylinder incylindrical coordinates (q, p, z) of Figure 6.1 arepr0—179—p2 = r’ — + 2 p r cos(q — £2), (6. la)—112 z 112 (6.lb)where £2 is the angular coordinate of the center of the cylinder. When the cylinder ismoving in a circular orbit, the angular position £2 of the cylinder is given by£2 =co0t+ (6.2)where w- is natural resonance frequency of the ions, t is time, and is initial phase angle ofthe cylinder at t = 0. Coulomb interaction between two charged cylinders will be a sixdimensional integral equation. Evaluation of such a six dimensional integral requiresapplication of a numerical method. However, it is most desirable to obtain the analyticalanswers with certain physical meanings, which can be compared to numerical solutions.Therefore, a simpler case will be considered: two uniformly charged tetragonal ion cloudsin the analyzer cell of FT-ICR, since integration is easier relative to the interaction betweentwo uniformly charged cylinders, and will be similar to the latter. The edge lengths of bothsquare cross sections both are 2 a0 and both z-axis lengths are I as shown in Figure 6.2.The ion densities for N1 ions in Tetragon 1 and N2 ions in Tetragon 2 in Cartesiancoordinates areNfl1X)Z A 2i ‘a0Nn2xYz A 2a0Applying Taylor’s expansion approximation, the Coulomb interaction between two—180—Figure 6.2 The charged-tetragon model. Both tetragons have the same cyclotronradius, R; the same edge width and height, 2 a0; and the same z-axislength, 1. The directions of applied magnetic field B and their cyclotronmotions are shown as indicated.Tetragon 1Tetragon 2I2a00xSSSzSS I—Il_IlSIIIIV—181—tetragonal ion clouds within a range of small separations is integrable. In order todistinguish the cyclotron motion of an individual ion and the collective cyclotron motion ofan ion cloud, coordinates of an individual ion in an ion cloud will be symbolized as r, andcyclotron radius of the ion cloud, as R. Note that these two symbols are different fromthose used in Chapter 4 and 5.Potential energy is a scalar quantity and force is a vector quantity. Thus, evaluationsof 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 CloudWhen an excited tetragonal ion cloud, Tetragon 1, in which each ion has a mass m1and a charge q1 is moving circularly in a cyclotron radius R1 = (xRl, YR1’ 0), its potentialenergy will be the sum of the external potential where the ion cloud is located and itsinternal Coulomb potential,(4)(xRI + ao (YR1 + 1o (112 q NU(R1) ) ) 1 2 1’ dx dy dzXRI — a0 YRI — a0 —1/2 4 a0 I+ (a0 (a0 faa (O (1/2 (1/2 k q12N dx dx’ dy dy’ d zdz’ 6 4i—a0 i-a0 i—a0 J-a0 J—l/2 J-l/2 16 2 12 2 I r — r’ I ‘where f’is the external potential; Coulomb constant k = 8.98756 x i0 Nm2/C;rand r’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 iscounted 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) isV= V’—A.v (6.5)where iy, A, and v are electric potential, magnetic vector potential, and ion velocity,respectively.(5)The electric potential, i; which is the trapping potential of the analyzercell, is another perturbation with respect to the Lorentz potential. For the sake ofcalculation, 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 samevector in Cartesian coordinates (x, y, z) byA = Aq, e + A ep + A c = i + A,, j + k (6.7)where e, es,, and e are unit vectors in cylindrical coordinates, and e = k. The magneticvector potential of an uniform static magnetic field in cylindrical coordinates(6)Aq, = Bp, (6.8a)A= A = 0 (6.8b)where p = Ix2 + y2. Hence,—183—A,e,= Ai + Ai. (6.9)The cylindrical unit vector e, is related to the Cartesian unit vectors i and j bye = — i sin q’ + j cos q’, (6.10)so, the three components of the magnetic vector potential in Cartesian coordinates areA = —- By, (6.lla)= Bx, (6.llb)A = 0. (6.llc)The elementary motion equation of an ion in ICR mass spectrometry, which was given inSection 4.1 of Chapter 4 isx=Rcos (w0t+), (6.12x)y =—R sin (w0t+ ). (6.12y)The derivatives of Eqs. (6.12) aredx=w0y=v, (6.13x)= —w0x = v,. (6.13y)Substituting Eqs.(6.1 1) and (6.13) into Eq. (6.6) gives the external potential—184—= BwO(22) (6.14)Since the ions in Tetragon 1 are taken as a collection, x = XR1,v YR1’ and co0 = c• Thefirst integral in Eq. (6.4) is solved to berBeo 2 1U(R1) = q1N[ (6.15)Eq. (6.15) gives the the potential energy of Tetragon 1 as a function of the ion number inTetragon 1, N1, the strength of the magnetic field, B, the cyclotron frequency of Tetragon1, the edge width and height of Tetragon 1, 2a0, and central coordinates of Tetragonl,XR1 and YR16.3 Potential Energies of Two Tetragonal Ion Clouds6.3.1 Taylor’s Expansion Approximation of the Coulomb Potential EnergyIf there are two tetragonal ion clouds of masses m1 and m2 whose cyclotron radii areR1 and R2 in the analyzer cell of F1’-ICR, the potential energy of each tetragon should beits own potential energy plus the Coulomb potential energy due to another tetragon. ForTetragon 1,U1(R,R2) = U01(R1) + U12(IR1— R2 I) (6.16)where U01 is the potential energy of Tetragon 1 given by Eq. (6.15) and U12 is theCoulomb potential energy due to Tetragon 2. If Tetragon 1 contains N1 ions of charge q1,Tetragon 2 contains N2 ions of charge q2, and their geometric sizes are as indicated in Fig.—185—6.2, the ion densities of the two tetragons are those of Eq. (6.3). The Coulomb potentialenergy of Tetragon 1 due to Tetragon 2 isU12(IR1— R2 I) = kq1q2N d3r1d3r2Tetragon 1 r1 — F2Tetragon 2— kq1q2N— 16a042 X(XR2+aO fXRI+ck (YRZ+ao (YRI+ao ( 1/2 (1/2 dz12yx)XR2—ao JXR1_CO JYR22O )yRI—ao J—l/2 )—1/2 ‘V’(x+(y-- z (6.17)U12 is a six-dimensional integral. Because two particles cannot occupy the same positionin space, i. e., for x1 = X2, Yi = Y2’ and z1 = z2,1= 0. (6.18).%J (x1—x2)+ (Y1—Y2)2 + (z1—z2)Thus, the singular points (x1 = x2, Yi = Y2’ and z1 = z2) must be deleted from Eq. (6.17).The following substitutions are used to simplify the upper and lower integral limits of Eq.(6.17):= X_Xj+XR1—X2, (6.19x)Y1—Y2 = Y—YjYR1--YR2 (6.l9y)where x =x1—XR1,x2 =X2—XR2,yj =Y1YR1’ andy =Y2—YR2’ such that there is novariable in the upper and lower limits of Eq. (6.17).—186—u12(IR1— R2 I)— kq1q2N— 16a042 X(xR2+ao (x+a0 (YR2+ao (YRI+ao (1/2 (1/2 dz12yxJXp2 JXRrao JYiwao Jyp.i—ao .11/2 .L112 .v’(x_)2+(y+(z— kq1q2N (ao p0 p0 p0 (1/2 (1/2— 16 a j2 J J J J. J_i/2 J_i/2dz1 dz2 dy1 dy2 dx1 dx2 (6.20)(x—Xj+xR1—xR2)2+ 7i_y2+yR1_yR2) + (z1—z2)The magnitude of Coulomb interaction for a small separation AR R1 — R2 between twoion 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 = 0, i. e., (xRl — xR2)o = 0 and (YR1 —YR2)O =0. Taylor’s expansion of the integrand in Eq. (6.20) to second-order terms is1.‘.J (xj—xj+XR1--xR2)2+ (yj-y+yR1yR2) + (z1—z2)1+ (y—y)2+ (zl_z2)2x—xj— (xRl XR2) [ (xj—xj)2+ (Yj—y)2+ (z1—z2)j3/2y —y—(YR1 YR2) [ (xj—xj)2+ (y—y)2+ (z1—z2)j3/2( )2 (yj—y)2+ (zl_z2)2 — 22 XR1 XR2 [ (xj—xj)2 + (yj—yj)2+ (z1—z2)j52—187—‘ ‘1 2 l21 ‘12’ — ‘2 Ri YR2 [ (x—4)2+ (y—y)2+ (z1—z2)]5l2(xj—4)(y—y)+ (Xj1— XR2)(YR1— YR2) r , 2 ‘ 2 2 15/2I (x1—x2) + (y1—y2) + (z1—z2) jSince the expansions were made at (xRl—xR2)o =0 and (y1—Y2) =0, x = x, yj = y,and z1 = z2 are still singular points and must be deleted. As shown in Figure 6.3, thesingular points in the area (x, 4) are composed of a straight line xj = 4, and also theareas (y, y) and (z1,z2). All these singular points are composed of a parallelepipedsingular subspace in six-dimension space.6.3.2 Integration of the Coulomb Potential EnergyAt 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 completeoverlapping of the two ion clouds. Because two completely overlapped ion clouds areequivalent 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-orderlinear differential equation of ion motion in FT-ICR,(7)any constant can be ignored in thelinear differential equation (or simultaneous linear differential equations). The first-orderterms and the cross term in Eq. (6.21) are odd functions of (—4) and (y—yj). After thesingular points in their integral areas (Fig. 6.3) are deleted, the integrals of the first-orderterms and the cross term should be zero. Only the second-order terms should beconsidered for a small separation R. The expansion coefficient of (XR1—XR2)2 in Eq.(6.21) actually has the same form as the expansion coefficient of (YR1—YR2) Therefore,—188—1/2z2Figure 6.3 The coordinate expression of singular points for x = x, y = y, andz1 = z2. Three coordinates x vs. x, y vs. y, and z1 vs. z2 are used.The integral limits of x, 4, yj, and yj are from — a0 to a0. The integrallimits of z1 and z2 are from —1/2 to 1/2.x,1aY2z1—1/2—112 I—189—the Coulomb potential energy of Tetragon 1 due to Tetragon 2 is predicted to beU12(IR1— R2 I) U12(0) + N N2 c [ (xRl — + (YR1 — YR2)2] (6.22)where U12(0) is the Coulomb potential energy when the two ion clouds are overlappingcompletely and the coefficient C can be obtained from the integrations of the second-orderterms. We define C as the Coulomb interaction coefficient between two ion clouds. FromEqs. (6.20) and (6.21),— kq1q—— 16 a04(ao (ao fao p0 (112 (1/2 (yj — y)2+(z1— z2)— 2 (x — x)2J—a, i-a0 J.ao i—a0 J112 p1/2 [ (x — x)2 + (yj — y)2 + (z1 — z2) j512x dz1 dz2yl dy dx dxj. (6.23)It is important in evaluating C that singular points are deleted by talcing the left and rightlimits for xj = x, y = y, and z1 = z2. The details of the integrations, in which severalparticular integral formulae must be used are given in Appendix A4. After integrations,C = 41hi72 { 5.9464a03_2al+8alth[l+Jl2+ 8 a021_8al1n[l+gl2+ 402]_6a121n[2a+q12÷ 8 a02]+6aol21n[2a0il2+ 4]_a121n(12+4a2)+ a1—8aln [2a0+l2 + 8 a02]+4a(l2+4a)+4agl2 + 8 a02— 190 —_4a02Il2 + 4 a02 I (12+8a2)m+ (12+4a2)3” 13 }• (6.24)The results of Eq. (6.24) have been checked by numerical evaluation for a given a0 and 1(Appendix A4). The first two terms in brackets of Eq. (6.24) arise from the Coulombinteraction in x-y plane, and all the other terms arise from the Coulomb interaction forseparation 1 (Appendix A4). Of course, the Coulomb interaction in the x-y plane ispredominant. The coefficient C can be approximated askq1q (5.94Mao3_21ra021)=(2.9732_ i). (6.25)For a0 = 1 mm and a 2.54 cm x 2.54 cm x 2.54 cm cubic cell,(8)the error in C given byEq. (6.25), i. e., the difference from the exact value of C given by Eq. (6.24), is less than0.2%. Furthermore, since 1>> 2a0 generally, Eq. (6.24) reduces to—itlcq1q2626—— 2a01 (.The Coulomb potential energy of Tetragon 2 due to Tetragon 1 is the same as U12 butof opposite signU21( IR1—R2I) = — U12( IR1—R2I). (6.27)The potential energies of Tetragons 1 and 2 in the magnetic field from Eq. (6.15)—191—U01(R1) = q1 N1 B w (4 + y1 + a02), (6.28a)U02(R2) = q N2 B w02(+y2 + (6.28b)Total potential energies of Tetragons 1 and 2 areU1(R,2)= q1NB (q+ya02)+U12O)+ N1 N2 C [(xRl — XR2)2 + (YR1 — YR2)l (6.29a)U2(R1,R2) = q2NB 002(2+y2+a02) — U12(O)— N N2 C [(XR1 — XR2)2 + (YR1 — YR2)2]. (6.29b)6.4 Motion Equations and Solutions of Two Tetragonal Ion CloudsAfter the total potential energies of the two ion clouds are known, their FT-ICRequations of motion can be solved. The force field of a velocity-dependent potential (S)aLl aiiFg= g=x,y,z. (6.30)Because the ion velocity is replaced by the angular frequency in Eqs.(6.12), the force fieldin FT-ICR is independent of time and is still conserved. That is,—192—F=—VU (6.31).a .a awhere the operator V i + j + k. For simplicity, settmgN1=N20, (6.32)the simultaneous equations of motion of the two tetragonal ion clouds are -F1= NOm1idu1 = —VU1(R,2)(6.33)F2 = NOm2ddR2 = —VU2(R1,)Substituting Eqs.(6.29) into Eq. (6.33),2 2 N0CD XR1 + (Ooi XR1 + m1 (xRl — XR2) = 0D2yRl+ayRl+ (—‘) = 0(6.34)2 2 N0C.D XR2 + (002 XR2— m2 (XR1 — XR2) = 0D2yR + °O2YR2— m2 1 —y) = 0where D = d/dt, the time derivative operator. The reduced matrix of Eq. (6.34) is2 2N0C N0C__— m1 1 (6.35)NC 2 NC0 D+w+ 0‘—193—From Eq. (6.35), two determinantal solutions of the operator D are obtained. These twosolutions are the eigenfrequencies, w and c of Eq. (6.34), where1 If N0C\ / N0C= 1.Vo?+ in1 ) + V1)02+ m2ri N0C N0C2 4N02C11/2+ LP)o?+ m1 —CO— 1 + m1 J (6.36a)2 1 1/ 2 N0C 2 N0C2 _2O1+m)+,P)O2+— [(2N0C 2NOC)2 + (6.36b)Many important conclusions can be derived from the frequency solutions a and a, whichare discussed in the following section.6.5 Strong Coupling Coulomb Interaction between Two Ion Clouds6.5.1 Strong Coupling ConditionThe relationshipO2 = (6.37)can be used to rewrite the terms in the square root in Eq. (6.36a) (or Eq. (6.36b))f NC NC\2 4N2C0_2_ 0 + 001 m1 02 m12— 194 —4/rn1 + rn2 \2=a01 rn1nz2{ m) ]2 + 2T0 m rn2) rn — rn2 + N02 C2 }• (6.38)Making a comparison among the three terms in Eq. (6.38), a strong coupling conditionbetween the two ion clouds, Tetragon 1 and Tetragon 2, for a small separation is defined asN0C rn1(rn—rn2) (6.39)rn‘0ior alternatively,N0C (rn—mi_____1 2 (6.40)qB2 m126.5.2 Two Oscillations under the Strong Coupling ConditionUnder the strong coupling condition of Inequality (6.39), two oscillations in F1’-ICRdue to the Coulomb interaction between the two ion clouds are obtained from Eqs. (6.36):1, rn+m\(02 = o?÷+2NoC m’12) (6.41)(+ ). (6.42)Note that w is smaller than co2, since the coefficient C in Eq. (6.41) is negative(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.(9) Theoretically, two oscillatory motions were alsofound 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 intheir experiment. Jungmann et a!. thought that in the 01 mode, the distance between thetwo ion clouds was stretched and compressed about an unchanging center-of-massposition, and hence this collective oscillation cannot give rise to the electric signal. If thisexplanation is correct, the mode will not be observed in Fr-ICR, either.The co2mode in Eq. (6.42) is a cyclotron motion of the two ion clouds with a fixeddistance. The Lorentz forces on two ion clouds are balanced by the ion-ion Coulombinteraction between them. Consequently, only one mass peak will be observed,corresponding to a root-mean-square value of the natural cyclotron frequencies of the twoion clouds, under a strong coupling condition of Inequality (6.40).6.5.3 Strong Coupling Critical CurveSubstituting the value of C (Eq. (6.26)) into Inequality (6.40), and q1 = q2 = unitcharge (If q1 and q2 are not unit charges, they can be set to unity by changing ion massesm1 and m2 to their mass-charge ratios.)ickN Im-mi2a01B >> (6.43)In a first-order approximation, when a physical quantity is ten times greater thananother, this quantity can be defmed as “much larger”. The strong coupling condition isredefined asitkN Im-mI2 a02 1 82x 0.1m1 m2 (6.44)—196—Solving for N0 gives20a21B21m—m IN0 0 1 2 , (6.45a)it km1 m2or equivalently,2lB2Im-m IN0 4.27x109 U 1 2 (6.45b)m1 m2Inequality (6.45b) gives a criterion of the strong coupling Coulomb interaction betweentwo (tetragonal) ion clouds in terms of N0, ion number in each ion clouds; a0, half edgelength of the square cross section of the tetragonal ion cloud in mm (approximately as theradius of a cylindrical ion clouds); 1, the z-axis length of the ion clouds in cm; m1, ion massof an individual ion in Tetragon 1 in Dalton; m2, ion mass of an individual ion in Tetragon2 in Dalton; and B, strength of the magnetic field in Tesla. Inequality (6.45a) is thecorresponding expression in SI units. Resolution of any mass spectrometer can beexpressed as the ability to distinguish two adjacent mass peaks,m2 = m1 + 1 (Dalton). (6.46)Substituting Eq. (6.46) into Inequality (6.45b)N0 4.27x109 (6.47)Typical parameters in a 2.54 cm cubic analyzer cell are as follows: 1= 2 cm, a0 = 1 mm,and B = 2 Tesla. N0 as a function of m1 (m2 = m1 + 1 (Dalton)) for a mass range from100 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 couplingcondition, Eq. (6.47) will be independent of the size of the ion cloud.N D2n = 1.07x1O 2 (6.48)4a01 m1where the unit ofa0 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 m1 = 100 Daltons and m2 = 101 Daltons in a 2 Tesla magnetic field and aconventional 2.54 cm cubic analyzer cell, if their ion numbers N1 and N2 are both greaterthan 3.4 million ions, the strong coupling Coulomb interaction between ions m1 and m2can occur, that is, only one mass peak will be observed. When ion masses are increased to= 300 Daltons and m2 = 301 Daltons, the ion numbers are reduced to 3.8x10 ions.Therefore, no matter how long the acquisition time of FT-ICR is, when the Lorentz force isunable to overcome the strong coupling Coulomb interaction between two ion clouds, theresolution of FT-ICR will deteriorate.6.5.4 Strong Coupling Condition for N1 N2When the ion numbers N1 N2, the solutions in Eq. (6.36) are changed to2 1 1/ 2 N2C\ ( 2 N1C(01 oi+ m1 )+i%+ m2rr N2C 2 N1C2 4N1N2C1/2+ I VOo? + — 02— m2 ) + m1 m2 J J (6.49a)1 11 N2C\ I N1Cm1 )+O)o+ in2— [( + — — + I ‘ }. (6.49b)—198—Figure 6.4 Strong coupling critical curve. When the ion number N0 or ion density n,as a function of the ion mass m1 (m2 = m1 + 1 (Dalton)), is in the shadowedarea, the strong coupling region, the FT-ICR resolution deteriorates.The left ordinate is for the ion number N0 in a 2 Tesla magnetic field and aconventional 2.54 cm cubic cell (a0 = 1 mm and 1 = 2 cm). The rightordinate is for ion density in a 2 Tesla magnetic field without dependence ofgeometrical size of an ion cloud. The strong coupling critical curve iscalculated from Eqs. (6.47) and (6.48).43N0(million)2103n(107/cm3)2100 200 300 400 500 600m1 (Dalton)—199—The terms in the square roots of Eqs. (6.49a) and (6.49b) can be rewritten as/ N2C N C\2 4N1N2C_2_ 1 +ma-’ m1nz2= w { [ml(m_)C(mlNl_mm2+C2(m1N+)} (6.50)The strong coupling condition for N1 N2 is defined asICI (m1N+2)>> Imj2_m2I. (6.51)Substituting Eq. (6.26) and c = q1B 1m into Inequality (6.51) and setting q1 = q2, thestrong coupling condition becomes2 a02 1 B2 I m m 21m1 N1 + m2N >> 1 2 (6.52)t km1 m2Form2= m1 + 1 Dalton, and rn1>> 1 Dalton, Inequality (6.52) reduces toa 2 B2N1 + N2 >> 8.53 x 108 2 (6.53a)m1or alternatively using the ion densities of Eqs. (6.3),B2nl + n2 >> 2.13x lOb (6.53b)where N1 and N2 are the ion numbers in Tetragon 1 and Tetragon 2, respectively; a0 is half— 200 —edge length of the square cross section of the tetragonal ion cloud in mm; 1 is the z-axislength of the tetragonal ion cloud in cm; B, strength of the magnetic field in Tesla; m1 ismass of the ions in Tetragon 1 in Dalton; and n1 and n2 are the ion densities of Tetragon 1and 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, B2N1 + N2 8.53x iO 2 (6.54a)m1or alternatively,fl1 + 2.13x1011 (6.54b)When N1 = N2, i. e., n1 = n2, Inequalities (6.54a) and (6.54b) reduce to Inequalities (6.47)and (6.48), respectively.6.6 Weak Coupling Coulomb Interaction6.6.1 Weak Coupling ConditionWhen the inequality sign of Inequality (6.39) is reversedN0C m1(m—m2) (6.55)in2or alternatively,Im1—m2 2tkN0m12 2a01BInequality (6.56) is the weak coupling condition for the Coulomb interaction between the—201—two ion clouds. The frequency solutions of Eqs. (6.36) becomeN0C m +m1 01 2 12’N0C m1+m2= + 2 m12 • (6.57b)The squares of the natural resonance frequencies, w and w, of the two ion clouds areboth shifted by a term N0 C (m1 + m2) I 2 m1 m2, i. e., shifted to lower frequencies,because C is negative (Eq. (6.26)). It has been known that Coulomb-induced frequencyshifts for two weak-coupling ion clouds in an electric quadrupole trap are both shifted tohigher frequencies,(9)wlich is exactly opposite to the Coulomb-induced frequency shift inFT-ICR (Chapters 4, 5, and 6). This difference is understandable. In an electricquadrupole trap, the radial force of Coulomb interaction between two ion clouds has thesame direction as that of the quadrupole electric field. In FT-ICR, the radial force ofCoulomb interaction between two ion clouds has an exactly opposite direction to theLorentz force.Because the Lorentz force can overcome the Coulomb weak coupling interactionbetween two ion clouds, the distance between two ion clouds will change periodically.Then, the average models(12)discussed in Chapters 4 and 5 can be applied to explain andpredict the Coulomb-induced frequency shifts in FT-ICR.6.6.2 Validity of the Weak Coupling ConditionIn Section 5.3.2.3 of Chapter 5, we gave a criterion for high frequency perturbationsIm1—m2 kN2 (6.58)m12rB2Dc4s,I(l/2)2+Dc2— 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 forweak Coulomb interaction between two ion clouds? In a conventional 2.54 cm cubic cellwhere N1 = N2 = N0, 1=2 cm, r = 1 cm, D = 1.327 r, and a0 = 1 mm, the ratio of the weakcoupling 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 be7rkN kN22= 173 : 1. (6.59)2a0 lB rB2D.\I(l/2)2+D2We 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 theaverage Coulomb interaction between two ion clouds.Eq. (6.57a) and Eq. (6.57b) can be rewritten asNCm +m=(w1—a01)(o,+ = , (6.60a)andNCm +m= ((02 ‘°& 02 + %2) = m1 m2 (6.60b)Noting that w o and C02 c, the frequency shifts from Eqs. (6.60) are=2’ (6.61a)and— 203 —N0C m1+m2=. (6.61b)o2m12Eqs. (6.61a) and (6.61b) are only valid at the instant when the separation between two ionclouds is small, from zero (overlapping disks) to 2 a0 (just touching). After the two ionclouds are separated further, Eqs. (6.61a) and (6.61b) become invalid. Therefore, thecharged-cylinder model presented in Chapter 5 is still correct for calculation and predictionof the Coulomb-induced frequency shifts in FT-ICR. The Coulomb-induced frequencyshift predicted from Eqs. (6.61a) and (6.61b) would be about one hundred times thefrequency 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’sexpansion approximation, not just the radial component of the Coulomb force as was thecase in the charged-cylinder model.6.7 Electric Trapping Potential and Frequency ShiftsThe electric trapping potential of the analyzer cell of FT-ICR, whose radialcomponent induces a negative resonance frequency shift to the ions.(7 1041) also can beincluded in the Taylor’s expansion approximation. For example, if the electric trappingpotential, r, of a cubic ion-trapped cell shown in Figure 6.5 is taken as a quadrupoleapproximation (12)= (VT+V0)+(VT—Vo){y— _(x2+y2_2z2)] (6.62)where y, iç and 1 are geometric factors of the cubic ion-trapped cell, VT is the trapping— 204 —p = x,oryplateFigure 6.5 The cubic ion-trapped cell configuration employed in the F1’-ICR massspectrometer, where the trapping voltage, VT, is applied across the twotrapping plates, and the voltages of the other four plates are set at zero. Thisapplied electric field produces a quadrupole trapping field, Eq. (6.62), nearthe center, 0, of the cubic cell. The magnetic field, B, is along the z-axis.voltage across the two trapping plates, V0 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.(12) Conventionally, the trapping voltage VT = 1 V,and 1/ =0 V. Substituting these values into Eq. (6.62)Receiver plate orTransmitter plateTrapping plateBz— 205 —i’ =5.166667—2.15x103(x2+y—2z). (6.63)Substituting Eqs. (6.63) and (6.14) into Eq. (6.5), the external potential of an ion in thecubic ion-trapped cell is= 5.166667 —2.15 x103(x2+y—2z)—A. v= 5.166667 — 2.15 x i& (x2 + y2 —2 z2)+ Bo.0 (x2 + y2). (6.64)The simultaneous ion motion equations of two tetragonal ion clouds, Eq. (6.34), ischanged toD2xR1 + W XR1 — 4.310 XR1 (x1 XR2) = 0D2yR1 + W — 4.3Q<10 iyR1 + (YR1 —YR2) = 0(6.65)D2xR + (OXR2 — 4.3Ox10 2xR2_’, (xRl xR2) = 0D2yR + WOYR2 — 4.3O10 YR2—,,2 (YR1 —YR2) = 0The reduced matrix of Eq. (6.65) is___________N2C1 1 (6.66)02 m2— 206 —The eigenfrequencies of the determinant solutions areco2 = [+ [(2 + N2 C — 4.30x iO q1— 2_ N1 C — 4.30x q2)2m2________1 (6.67a)m12=1 { (2 + N2 C — 4.30x10 1) + (2 + N1 C — 4.30x10 q2)m22 NlC_4.3Ox103)24N1N2C12 }. (6.67b)+ m1 m2Defining4.30x10 q1= — (6.68a)m1and4.30x q2=2 — (6.68b)the eigenfrequency solutions in Eqs. (6.67a) and (6.67b) reduce to— 207 —= { + N2 C) + ((w12 + N1C)m1+ [((W’1)2 + N2C— (co2)2— N1C)2 + 4N1N2C11121 (6.69a)m1 m2 m12 - ‘and2_’ {((w1)+C)+(( NC(02)2 +m2— [((&)2 + N2 C — (WI)2 — N1 C)2 + 4 N1 N2 C2 ] 112 } (6.69b)m12The following relationship can be found from Eqs. (6.68a) and (6.68b)m1= X — ‘i (6.70)where the coefficient—J2B2_4.3ox1o3m2— q12B — 4.30x10 m1 q1=4qq2(q2B4•30x’om2)(6.71a)q1 (q1 2 — 4.30x 10 m1)Since generally q2 B2 >> 4.30xi0 m2 and q1 B2 >> 4.30x 103 m1,z 1. (6.71b)— 208 —Then,(02 w’. (6.72)Analogous to Eq. (6.50), the strong coupling condition with the quadrupole trappingpotential correction for the Coulomb interaction between two ion clouds is found to be()2 (in1N1 + in2N) >> I m12 — 2 (6.73)Inequality (6.73) is comparable to Inequality (6.51).6.8 Discussion6.8.1 The Charged-Cylinder Model and Taylor’s Expansion ApproximationIn Chapters 4 and 5, only the radial component of the Coulomb interaction betweentwo ion clouds was taken into account in the average models.(12) In the Taylor’sexpansion approximation, the whole Coulomb interaction between two ion clouds isaccounted for. The different treatments arise from different sources. When the separationbetween two ion clouds in FT-ICR varies sinusoidally, as is conventional, the frequencyshifts induced by their Coulomb interaction can be predicted by the charged-cylinder modelon the basis of the average Coulomb effect. Thus, since the average of tangential Coulombinteraction is zero, only the radial Coulomb interaction is involved in the averagingprocedure. However, the charged cylinder model does not predict the strong couplingCoulomb interaction, i. e., two mass peaks would virtually be merged into one when theion 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’sexpansion approximation, Inequality (6.56), overrides that of the weak coupling conditionfrom the charged-cylinder model, Inequality (6.58). In the Taylor’s expansionapproximation, the magnitude of the Lorentz force was compared to the magnitude of thetotal Coulomb force within small separations, especially in strongest interactions, ratherthan averaging the interactions. When the Lorentz force cannot overcome the strongcoupling Coulomb interaction, the charged-cylinder model will become ineffective.However, the Taylor’s expansion approximation cannot be used to evaluate the unlike-ionCoulomb-induced frequency shifts (Section 6.6.2) and the inhomogeneous Coulomb-induced broadening. Therefore, the charged-cylinder model and the Taylor’s expansionapproximation compensate each other.6.8.2 The Electric Quadrupole Trap and Analyzer Cell of FT-ICRThere are many similarities between an electric quadrupole trap and FT-ICR. Bothexperiments 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 andmagnetron 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 ion-trapped cell of FT-ICR both have quadrupolar trapping fields.The Coulomb-induced frequency shifts of stored ions in a radiofrequency electricquadrupole trap have been investigated by Jungmann et al. for both the strong couplingcondition and weak coupling condition.(9)These authors presented a two-particle modeland a two-cloud model to explain the strong coupling Coulomb interaction, which forcedtwo mass peaks of Ho (mlz 165) and Er (mlz 167) to merge into one; and the weakcoupling 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 separationsbetween two ion clouds. Although cyclotron motion is detected in Fr-ICR rather than axialoscillation as in the quadrupole trap, their Taylor’s expansion approximations have asimilar form except that the direction of the Lorentz force applied in FT-ICR is opposite tothe direction of the trapping force in the quadrupole trap (over a time average). For theirspecial purpose (to maintain a small separation between ion clouds), Jungmann et a!. usedan electric quadrupole trap in which the narrowest space between the electrodes was orily5mm. The hole diameter for transmitting an electron beam was not stated. If the holediameter is estimated to be 2 mm, the largest separation between two ion clouds was closeto that of touching in Jungmann’s quadrupole trap. Therefore, the strong couplingcondition is easily attained in Jungmann’s quadrupole trap. On the other hand, theconventional analyzer cell used in Fr-ICR is the 2.54 cm cubic ion-trapped cell, and larger-sized 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.(9) The physical reason for thenegative 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 ConditionObviously, a transition exists between the strong coupling regime and the weakcoupling regime — intermediate coupling. Jungmann et a!. defmed this transition as the“minimum resolvable condition”.(9)That is, when the intermediate coupling condition is ineffect, 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, theintermediate coupling condition is defined as(6.74)From Eq. (6.74), and setting m2 >N0 C — m2— m1 cEoo?. (6.75)Substituting Eq. (6.75) into Eqs. (6.36),2= (Oo2+W02((O01_COo)’\/1 +‘a , (6.76a)and22= %O)o2(COo1(Oo2)’\J1 + . (6.76b)If m2 m1, the frequency solutions of Eqs. (6.76a) and (6.76b) are reduced toWi2 = c+Jo02(w1—%), (6.77a)and(02 = o2 — %2 (ooi — %2) (6.77b)— 212 —The solutions in Eqs. (6.77a) and (6.77b) indicate that one ion cloud will be acceleratedand 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 FunctionsAt the beginning of this chapter, the ions in the analyzer cell of FT-ICR wereassumed to have uniform distributions, i. e., the ions were assumed to have zero initialthermal velocity. The cylindrical symmetric pseudopotential of a if electric quadrupole trapcorresponds to a static quadrupole potential averaging over time.(13) It has been provedthat ion spatial distribution in such a potential is nearly uniform at —300 K for non-zero ionthermal velocities.(14)Because a stronger static magnetic field is applied in FT-ICR alongthe z-direction, the ions in an analyzer cell of the ICR could have a Maxwellian distributionalong the x- and y-axes.(’5)For a cylindrical ion cloud containing N1 ions, in which eachion has a mass m1 and charge q1, its Maxwellian distribution in the center of the analyzercell(15)will ben1 (x,yz) = “‘‘ exp[_131(x2+y)J (6.78)where 1 is the z-axis length of the cylinder and fl = m1 co / 2 kB T, kB being theBoltzmann constant, 1.380658x10 J•K, and T being the absolute temperature. Afterthis ion cylinder is excited to its cyclotron radius R1 = (xRl, YR1)’ Eq. (6.78) becomesn1 (r1,R1) = i” ‘‘ exp { — f3 [(x1 —XR1)2+(yl —YR1)i }. (6.79)—213—Analogously, a second cylindrical ion cloud containing N2 ions, in which each ion has amass in2 and charge q2, has a Maxwellian distribution after excitationn2 (r2,R2) = N2 I2 exp— 132 kx2 — XR2)2+ (Y2 — YR2)] } (6.80)where f2 = m2 co /2kB T. The Coulomb potential energy of Cylinder 1 due to Cylinder2 will then beU12 (1R—1)= kqjq2xf f f n1 (r,R) n2 (r2,R)d3r12Cylinder 1 r1 —Cylinder 2poo pee l/2 pee fOe p1/2— kq1q2Nf3 I I I I I I— 2 j2 J_eo J—l/2 J- J_oe J—l/2— Pi [(Xl—XR1)2+ (Y1—YR1)J — P2 [(x—xp) + (Y2YRZ)Je e dx1 dy1zdx2yz. (6.81)(x — x2) + (y1— y2) + (z1 — z2)The transformsxj =X1—XR1,4 =X2XR,Yj =Y1—YR1’ andy =Y2—YR2 can beused, and the integrand of Eq. (6.81) can be expanded into Taylor’s series. After themasses of two particular ion species are preset, numerical integration can be used to solvethe integral in Eq. (6.81). Because the integral of Eq. (6.81) has the same form as thatderived by Jungmann et al.(9), the solution of Eq. (6.81) is expected to beU12(IR1— R2 I) U12(0) + N1 N2 C’ [ (xR1 — XR2)2 + (YR1 — YR2)2 1 (6.82)—214--where the Coulomb coefficient C’ should be negative. Then, the strong coupling, weakcoupling, 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’sexpansion approximation.6.8.5 Further workThe most important conclusions in this chapter are those obtained first for the strongcoupling 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 interactionbetween ion clouds in FT-ICR, using the analysis in Section 6.4.3, two ion species ofmasses 300 Daltons and 301 Daltons are a suitable ion pair. For the 2.54 cm cubic cell, ionnumbers must exceed 5x10 from Fig. 6.4. Conventional electron ionization can producethis 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 inthe Taylor’s expansion approximation. If a model of the cylindrical ion clouds can beapplied in the Taylor’s expansion approximation, the results obtained here would be moreprecise.The eigenfrequency solution co in Eq. (6.41), which is the oscillation about anunchanging center-of-mass position, is interesting. Verification of its physical regime isrequired, both theoretically and experimentally. Undoubtedly, Taylor’s expansionapproximation 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 ionCoulomb potential in the cubic analyzer cell. Jeifries, Barlow, and Dunn have examinedthe ion Coulomb potential of a prolate ion cloud in the center of ICR analyzer cell, and—215—found it to be harmonic.(10)More recently, the ion Coulomb potential of a spherical ioncloud at cyclotron radius r in the hyperboloidal Penning cell of FT-ICR was solved usingan image charge model developed by Vogel, Kluge, and Schweikhard.(16)In the imagecharge model, the ion Coulomb potential of the ion cloud was increased not only as thenumber of ions increased, but also as its cyclotron radius r increased, which agreed withthe experimental results measured by Vogel et al.(16) In reality, ion clouds in the analyzercells of FT-ICR are cylindrical or prolate, and revolve in the circular orbits. Therefore,these two models, developed by Jeffries et al.(10)and Vogel et al.(16), cannot be applied tothe cubic analyzer cell of FT-ICR. The application of these two models with respect to thecubic analyzer cell remains to be solved in the future.—216—References1. 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 ofPacific 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.; MclverJr., 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. IonPhys. 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 2A1.1 The “Twenty-Second” Ion-Molecule Reaction Mass SpectraCUFJuI—ciIUCI—LII—11LflI-aIUn(b) Parent molecule CpCr(NO)2D3Delay time =20 secL.___C ‘ I I I I I I 1 I I I I 1 I [ 1 I I l[l I1TlFT1F1 I100 200 600 ‘-iOO 500 600F1P55 ]N P.M.U1— 220 —A 1.2 Coordination Modes of Metal Carbonyls and Metal NitrosylsTable A1.1 Coordination Modes of Carbon MonoxideMode Formal ElectronCharge Donated Structure Character Ref.p2(C, 0)-iL3(7)-(0) 4(C,Oj0 202MM02 IIM ‘M02 IIM Mr(M—C): —1.85ALMCO: 165—180°1.2, 1.31.2, 1.3M—COTerminal 0 2Symmethcal 0 2#201’)-Asymmetrical 0Semibridging 00(no M-M bond)Linear 01.1cc77-90°;/3=y 1.113C0±M—M 1.113cC0not±M—M 1.1r(M....M): 3.1—3.4A 1.2a: 104—120°13: 165—175° 1.2, 1.3ZCOM’: 135—180°r(O—M’): 2.05—2.15A4/C\M M(orM)M—C0—M’0M-)M(or Mp3(2or)- 0 4 1.2, 1.3—221 —M odeTable A 1.1 ContinuedFormal ElectronCharge Donatedp3(i’,’i- 0 6 1.2, 1.3p3(2C, 0)- 0 1’j:;’co__r’ ZCOM’: 165_1800 1.2,1.3j.t4(3C, 0)- 0 4 r(0—M): 1.80—2.05A 1.2, 1.3M\ /CO_MM14(371,772) 0 /_7ç\(orM)Mç M(OrM)M(or M)it-Carbene with 1 Mtwo carbonyls’MO”1.4¶ The hypotheticalp4(1r’)CO,p5(’)-CO,p6(’)-C0 are not included in this table. M is used to representa 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 12-47 15.1.2, 1.3Structure Character Ref.— 222 —Table Al .2 Coordination Modes of Nitric OxideMode Formal ElectronCharge Donated Structure Character Ref.Linear terminal1113 (M N=O) r(M—N): —1.76±0.1ALMNO: —180°2.1Bent terminal7111 M—N r(M—N): —1.83±0.15ALMNO: 120—170°2.1SymmetricalAsymmetrical(no M-M bond)30VMM0MMr(M....M): —3.2 Aa: —102°2.1#2(1’ ,ij2) 2.2Isonitrosylbridge# M—NO-4MZWNO = 169°ZNOMg = 135.6°2.33 fl=-Yr(M—N): —1.85 A 2.13M M2.1113(1?’)-#4(112,172)-37—10M’’M2.12.1#4(311 ,i72)-2.1— 223 —Table A1.2 ContinuedFormal Electron Structure Character Ref.Mode Charge DonatedTerminal 2 N°\ 2.4II Mcis-N20’ N0/Hyponitrite M/N\.M 2.5bridge’b¶ The hypothetical ji(’)-NO linked to heteronuclear metals, j.i3(a/ic)-NO,i4(’)-NO are not included inthis 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.— 224 —Al.3 Known Oxo Chromium ComplexesTable A1.3 Known oxo chromium complexes(in order of oxidation state of chromium)Compound Oxidation state Electron Coordinationcounting number Ref.Monomer:[Ci021 +3 11 4 3.1Cr02(CO) +4 16 6 3.2Cr02 +4 10 4 3.1CrOF2 +4 10 6 3.1CrO(Porphrins) +4 14 6 3.1CrO(Pc) +4 14 6 3.3[Cr04] +4 18 8 3.1[Cr05] +4 3.1[Cr06j +4 3.1[(CrTPP)OJ +5 13 6 3.1Cr0[O2(CF] +5 13 6 3.1Cr0[OOR] +5 13 6 3.1, 3.3[Cr0(sa1en)j +5 13 6 3.1CrO(MEC) +5 13 6 3.3CrOX3 (X=F, Cl) +5 11 5 3.1CrOC13L(L=bipy, phen) +5 15 7 3.3[CrOX4] (X=F, Cl, Br) +5 13 6 3.1[Cr0X5](X=F,C1) +5 15 7 3.3Cp*CrOBr +5 15 7 3.4CrOF4 +6 12 6 3.1[CrOF5J +6 14 7 3.1CiO2X (X=F,Cl) +6 12 6 3.102(N03) +6 16 or 20 6 or 8 3.1, 3.5Cr0(OCF +6 16 8 3.1Ci0(O2),py +6 14 7 3.1CrO(02)(1, 10-phen) +6 16 8 3.1Cr03 +6 12 6 3.1[CrOXJ(X=Halogens, OH) +6 14 7 3.1— 225 —Ci03(py)2[Cr04]2 and its derivativesDimer:[(H2O)5CzOCr(H][(NH3)CaOCr(NH4[Cr(NCSXTPyEA)120[(2-picetam)4rO(OH)J3[(bipy)Cr O)(OH)Cr(bipy)[(ph(O)(OH)Cr(phen)j[(bipy)2Cr(Or(bipy][(phen)Cr(Or(phen(CrOC1)2CHPh(CrC1TPP)0(CrPcO)2[(CpCr)O(OCMe3)](C1)HO)OCrOCrOCHPh(C1)[(C1)2O r]CHPh[Sfp*CrcJ1[OCrO2C(cF3)4-p-O][(C1)(HO)OCrJCHPh[Cr2O?Trimer:Cr30(O2CFH)6pyr0(OCCF)(4-cynopy)[Cr30(O2C H)6(H)1[(CpCr)O(OCMeTetramer:[Cr4O(OH)5(H2)10](CpCrO)4(Cp’CrO)4(Cp’Cr)S)2SCuBr)(PhP3[Cr(ji-O)(SO1JMeCN+4, +4 15k, i4+4, +6 10, 12+5, +5+5, +5+5, +5+6, +6+6, +6222+2, +2, +2-222+2, +2, +2+3, +3 15, 15, 15 6, 6, 617, 17, 17 6, 6, 6average +3÷3, +3, +3, +3+3, +3, +3, +3+3, +3, +3, +3+3, +3, +3, +33.183.14Table A1.3 continuedCompound oxidation Electron Coordinationcounting number Ref.8 3.64 3.1÷6 16+6 10+3, +3 17, 17+3, +3 17, 17+3, +3 15, 15+3, +3 15, 15+3, +3 15, 15÷3, +3 15, 15+3, +3 16, 16+3, +3 16, 16+41+46, 66, 66, 66, 66, 66, 66, 66, 64, 46, 66, 66, 64, 65, 56, 66, 66, 67, 73.73.73.83.93.103.103.103.103.113.123.133.143.153.113.163.173.113.116, 1613, 1314, 146, 6, 6 3.186, 6, 6 3.1818, 18, 18, 1818, 18, 18, 186,6,6, 66, 6, 6, 66,6,6, 66, 6, 6, 66,6,6, 63.193.203.203.203.21— 226 —Table A1.3 continuedElectmn Coordination Ref.Compound Oxidation statecounting number(OPh)10Cr4p-O)NaTJv1EDA) +3, +3, +3, +3 6, 6, 6, 6 3.22[(Cp’Cr)(S)3O] +3, +3, +3, +4 6, 6, 6, 6 3.20CP4rr(T15-2H) +3, +3k, 18, i4, 16, 18 6, 6, 6, 7 3.20+4 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, 141-150.3.10 (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. mt. Ed.EngI., 1985, 24, 601-602.— 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, 1380-1384.— 228 —-3-4-5-6-7-8-9A2 Appendixes of Chapter 3Kinetic Behavior of Cr Produced from CpCr(NO)2H3in H20, NH3,CH4, and N2 MediaThe experimental parameters are given in Section 3.2 of Chapter 3. The kineticbehavior of Cr produced from CpCr(NO)2H3in H20, NH3,CH4, and N2 Media all arenot pseudo-first order, like the behavior of Cr in H2 medium shown in Fig. 3.2.0.00 0.05 0.10Time (second)0.15N-3-4-5-6-70.00 0.02 0.04 0.06 0.08 0.10Time (second)0.12— 229 —-6-80.00 0.150.100-3-4-5-70.05 0.10Time (second)-3-4-5-6-70.000 0.025 0.050 0.075Time (second)— 230 —A3 Appendixes of Chapter 5A3. 1 The FORTRAN Program for Calculating Ion-Ion Coulomb Interaction betweenTwo Charged Disks Based on Double Gaussian Numerical ComputationThis 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 forcalculating overlapping disks and increment from point to point. The program will listtouch angle of two disks, the instantaneous distance, angle, radial Coulomb force betweenthe two disks, average radial Coulomb force, and apparent Coulomb distance.1 C: THIS PROGRAM FOR CALCULATION OF RADIAL COULOMB FORCE2 C BETWEEN TWO DISK ION CLOUDS WITH OVERLAP AND WITHOUT3 C OVERLAP BASED ON DDBLGS (DOUBLE GAUSSIAN INTEGRATION)4 C OFMTS.5 C r(or A0) radius of the disk6 C R(or RC) = ion cyclotron radius7 C N0(orDN) =ion number in each disk8 C DIO = initial separation between two disks with overlap9 C DIN initial separation between two disks without overlap10 C.. N =..data..points...for..ca1culatin. f overlap11 C M = data points for calculation of non-overlap12 C Z1,22 = increments in N and M do loopC.. ANG = angle between two disks ifl.. a..yloton....(radi.an)14 C ANGTOUCH = “touching angle” (radian)15 C FORC radial Coulomb force for each calculation16 C FAV = average Coulomb force17 C Dc = Apparent Coulomb distance18 DOUBLE PRECISION A0,DAI,BA1 ,BA2,BA3,UA1,UA2,UA3,TOT1,19 ...DAI.M.1.,PAIMEN2.,PAIMEN3.PAIMEN4,.PAIN.ENS., SUM,TQT.,DN,D.I20 DOUBLE PRECISION Z1,Z2,RC,Q,COUL,API,S0,FAVE,TOD,TEV,21 + ANG,ANGTOUCH,DAIMEN6,DAIMEN7COMMON .,IFLAG/CCf,BA.1. BA3,UA1.,U 3LPD/.AQ.,.ANG,, RC23 DIMENSION SUM(512)24 PARAMETER (Q=1.6021892D—19,COUL=8.987551787D+09,.5 ..A!’I=3.1415.92 535897.932.4.P.O)26 1 WRITE(6,*) ‘r =27 READ(5,*) A028 WR.IT.E.61* R =29 READ(5,*) RC30 WRITE(6,*) ‘NO31 READ (.5., “.) DN32 DIN=2.D0*AO33 WRITE(6,*) ‘DIN = ‘,DIN34 DIO = O.D035 WRITE(6,*) ‘DIO = ‘,DIO36 WRITE(6,*) ‘M = (odd).7 READ(5,*) H—231—38 ANGTOUCH2 .DO*DASIN (AO /RC)39 Z2 (API-ANGTOUCH) /FLOAT (H-i)40 WR.I.T.E(6,..*) N (vcn)41 READ(5,*) N42 3 WRITE(6,*) ‘Zi.4.3 RE?.D.(.5 .) Zi44 IF (Z1*FLOAT(N)_ANGTOUCH) 6,4,445 4 WRITE(6,*) ‘Zi SHOULD BE SMALLER !‘46 GOTO 347 6 WRITE(7,8) AO,RC,DN,A.NGTOUCH48 8 FORNAT(1X,’r ‘,1PG7.1,’, R ‘,1PG7.1,’, NO.9 1.PG7.1., ANGTOUCH 1..1PG 12 6)50 C: CALCULATION OF THE COEFFICIENT OUTSIDE OF INTEGRAL.51 C COUL THE COULOMB CONSTANT.5.2 C Q ION ARGE..53 SO=COUL*Q*Q*DN*DN/ (API*API*AO*AO*AO*AO)54 DO 10 I=1,N5.5 ANG.?.i*FLQAT )56 DS1=2.DO*RC*DSIN(O.5D0*ANG)57 UA1=RC+A058 BA1RC-A0.5 D.COS(ANG)A60 EA2=RC*DCOS fANG) -A061 UA3=(RC*DCOS(O.5D0*ANG) +DSQRT(AO*A0_RC*RC*DSIN(O.5D0*ANG).62 + *D.SI.N(0....5D0*ANG).).).*.DCOS.(O.,5D0ANG)63 BA3=(RC*DCOS(0.5D0*ANG)_DSQRT(A0*A0_RC*RC*DSIN(0.5D0*ANG)64 + *DSIN(0.5DD*ANG)))*DCOS(0.5D0*ANG)65 21 IFLAG=166 DAI=DDBLGS(BA3,UA3,48,48)67 DAIMEN1=DAI68 22 IFLAG=269 DAI=DDBLGS(UA3,UA2,48,48)70 DAIMEN2=DAI*2.D071 23 IFLAG=372 DAI=DDBLGS(BA2,BA3,48,48)73 DAIMEN3=DAI*2.D074 24 IFLAG=475 DAI=DDBLGS(UA3,UA2,48,48)76 DAIMEN4=DAI*2.D077 25 IFLAG=578 DAI=DDBLGS(UA3,UA2,48,48)79 DAIMEN5=DAI80 26 IFLAG=681 DAI=DDBLGS(BA2,BA3,48,48)82 DAIMEN6=DAI.83 SUM (I).=S0*. (DAIMEN1.DAH,N PAl N34DAIMEN .PIMEN.5.+.PAIMEN6)84 WRITE(7,30) DS1,ANG,SUM(I)85 30 FORMAT(1X,’D = ‘,1PG1O.4,.’ ANG = ‘,1PG12.6,.86 FQRC ,2.PG12 6)87 10 CONTINUE88 C: SIMPSON’S RULE WILL BE PERFORMED FOR OVERLAP CASE.89 TOD=0.D090 DO 32 I=1,N—1,291 TOD=TODSUM(I)92 32 CONTINUE93 TEVtO.D094 DO 33 I=2,N—2,25 TEV=TEV.SUM (.1.)96 33 CONTINUE97 TOTtZ1*(SUM(N)+4.DO*TOD+2.D0*TEV) /3.D0.9.8 WM.TEfl,..34) N,TQT99 34 FORMAT(1X,’N ‘,13,’ TOT = ‘,1PG12.6)100 27 IFLAG=711 DC 17 K1,,.M— 232 —102 ANGLANGTOUCH+Z2*FLOAT(K_1)103 DS62.D0*RC*DSIN(0.5D0*ANG).1.0.4 C.+A0105 BA1RC-A0106 +A01.0.7 O.S..(AN A0108 DAI=DDBLG5(BA2,UA2,48,48)109 St.IM(N+K) =DAI’SO.1.1 WRITE (.7.35.) PS6.,.AN,SUM (N+K)111 35 FORMAT(1X,’D ‘,1PG1O.4,.’ ANG ‘,1PG12.6,112 + ‘ FORC ‘,1PG12.6)113 17 CONTINUE114 C: SIMPSON’S RULE WILL BE PERFORMED FOR NON-OVERLAP CASE.115 TEVO.D0116 DO 36 I=N+3,M+N—2,2.117 TEY=TEV.SUM (.1)118 36 CONTINUE119 TOD=0.D0.120 DO 37 I.2.,M.N —1,,2121 TOD=TOD4SUM(I)122 37 CONTINUE.123 TOTI=Z2*(SUM(N+1),SUM(N+M)+2.D0*TEV+4.D0*TOD)/ DO124 FAVE (TOT+TOT1+0.5D0* (SUM(N) SUM(N+1))125 + *(ANGTOUCN_Z1*FLOAT(N)))/API.126 WRITE(7.,38).M,TO ,FAVE127 38 FORMAT(1X,’M ‘,I3,’ TOT1 ‘,1PG12.6,/1X,128 • ‘FAVE ‘,1PG12.6).129 WRITE(7139) p.$Q.{cQUL*.pN.pN.*Q.*.Q/FA ).../.R.C130 39 FORMAT(1X,’Dc = ‘,1PG13.7)131 STOP132 99 END133 C134 FUNCTION BLIM(Y).135 DOUBLE PRECISION... BM.,.BA uAl, UA136 COMMON IFLAG/CC/BA1,BA3,UA1,UA3137 IF (IFLAG.EQ.1) BLIM=BA3138 IF...(IFLAG.EQ.2) BLIM=BA3139 IF (IFLAG.EQ.3) BLIM=BA3140 IF (IFLAG.EQ.4) BLIM=UA3.141 IF..(IFLA.G.EQ.5)....BLIM=BA142 IF (IFLAG.EQ.6) BLIM=UA3143 IF (IFLAG.EQ.7) BLIM=BA1144 RETURN145 END146 C147-148149150FUNCTION ULI$CYDOUBLE PRECISION BA1,BA3,UA1,UA3COMMON IFLAG/CC/BA1 ,BA3,UA1,UA3IF (.IFLAG.EQ.1.) Z.M=UA3151 IF (IFLAG.EQ.2) ULIM=UA3152 IF (IFLAG.ZQ.3) ULIM=UA3.153 IF (IFLAG.EQ...4) UUM=UA1..154 IF (IFLAG.EQ.5) ULIM=BA3155 IF (IFLAG.EQ.6) ULIM=UA1.15 .F (IFLAG.EO.7) ULIM=UA1157 RETURN158 END159160161162163 END164 CI 5 UNCTIOCFUNCTION GFUN(Y)GFUN=1 .ODORETURN— 233 —166 DOUBLE PRECISION A0,X,Y,Y0,Y11,Y12,Y21,Y22,R0,ANG,RC167 COMNON IFLAG/DD/A0,ANG,RCIF EQ..1.) OTO 41169 IF (IFLAG.EQ.2) GOTO 42170 IF (IFLAG.EQ.3) GOTO 42.1.7 IF (.IFI.AG.EQ..4.) OTO 44172 IF (IFLAG.EQ.5) GOTO 44173 IF (IFLPG.EQ.6) GOTO 44174 IF (IFLAG.EO.7) GOTO 44175 41 Y0=X-Y176 Y11=DSQRT (A0*A0_ (X_RC*DCOS (ANG) ) **)177 Y12=DSQRT (A0*A0_ (X—RC) **2)178 G)).2.)179 Y22=DSQRT (A0*A0_ (Y—RC) **2)180 RO=RC*DSIN(ANG)181 IF (Y0 .EQ.O.D0) THEN182 AUXIN0.D0183 ELSE184 AUXIN=..(_DSQRT(.(Y11+Y21).*.*2+Y0.*Y0)+DS.QRT.((RO+Y12+Y21)*.*2185 +yQ*yO)+DSQRT((RO_y11_y22)**2+yO*yQ)_DSQRT((y12+y22)186 + **2+yO*yO))/yO187 ENDIF188 GOTO 88189 42 YO=X—Y1.90 Yll DSQRT (Ao.*A0..(X.pcOS.(ANG)..)..z)191 Y12=DSQRT(A0*AO_ (X—RC4 **2)192 Y21=DSQRT(A0*AO(Y_RC*DCOS(ANG) )**2).193 RO=RC*PSIN(ANG)194 IF (Y0.EQ.0.D0) THEN195 AUXIN=0.DO196 ELSE197 AUXIN=(_DSQRT((Y11+Y21)**2+YO*Y0)+DSQRT((R0+Y12+Y21)**2198 + +yQ*yO)+DSQRT((y11_y21)**2+yQ*yO)_DSQRT((R04y12_y21)199 + **2+Y0*YOfl./YQ200 ENDIF201 GOTO 88202 44 Y0=X—Y203 Y11=DSQRT(A0*A0_(X_RC)**2)204 Y21=DSQRT(A0*A0_(Y_RC*DCOS(ANG))**2)205 RO.=RC*PSIN.(ANG)206 IF (Y0.EQ.0.D0) THEN207 AUXIN=0.D0208 ELSE209 AUXIN=(_DSQRT((R0_Y11.Y21)**2+Y0*Y0).DSQRT((R0+Y11+Y21)**2210 +yQ*yO)+DSQRT((RO_y11_y21)**2.yQ*yO)_DSQRT((RQ+y11_y21)211 *.*2+.Y0YO)..)..,tY Q212 ENDIF213 88 RETURN214 END— 234 —A3.2 The FORTRAN Program for Calculating Ion-Ion Coulomb Interaction betweenTwo Charged Disks Based on Double Romberg Numerical ComputationA3.2.1 Program ‘DISK.ROMB” for Two Charged Disks without OverlappingThis 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 ofiterations. The program will list instantaneous angles between the two disks (from theangle of the two disks just touching to the maximum angle it), and radial component of theinstantaneous Coulomb force.CC * DISK.ROM8 *CC r (Dr AG) = radius of the diskC R (or RC) = ion cwclotron radiusC NO (or DN) = ion number in each diskC ANGTOIJCH = ‘touching angle (radian)C N = number of iteration (maximum 15)C 10 = data points for calculation of non—overlapC Zi = increment in TO do loorC ANG = anle between two disks in a cclotror, (radian)PARAMETER (Q=1,6021892E—19,CDUL=8.987551787E+09,+ API=3.14159265358979324)PRINT*,’r =READ*AOPRINT*p’R =READ*,RCPRINT*,’NO =READ*,DNANGTOUCH=2. *ASIN C AO/RC)PRINT*,’ANGTOUCH = ‘,ANGTOUCHC Special coefficient in Romber numerical interatior.AL=1 .558 PRINT*,’AL = ‘,ALC Tolerance in this roramE1 .E—14PRINT*,’E =PRINT*v’N =READ*rMPRINT*,’IO =READ*, TO— 235 —Z1=(API—ANGTOUCH)/(FLOAT(I0) )**1.6DO 9 J0r10IF (J.EQ,0) THENANG=ANGTOUCHELSEANG=ANGTOUCH+Z1*(FLOAT(J) )**1 .6END I FPRINT*,’ANG = ‘,ANGCALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY)COEFF=COUL*O*Q*DN*DN/ (API*A0*A0 ) **2PRINT 200, FORC*COEFF200 FORMAT(/1X,’Fore ‘,E16.9//)9 CONTINUE99 ENDCSUBROUTINE 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/H=A0/4.KEY=OIF (M,LT.1.OR,AL.LT.1,5.OR,AL.GT.2.) RETURNIF (H.LE,0,,OR.H.GT.1.) RETURNEE=AMAX1 (E,TOL)MM=MINO(M, 15)10 L1CC=2.*A021 U=0.KT=0G=1000.*H/K(L)NN=1 ./G+.522 P=CC*GBA2=RC*COS (ANG )—A0BA1=RC—A031 NN2=NNDO 30 I2=1,NN2X(2)=BA2+P*( 12—.5)NN1=NNDO 30 I1=1,NN1X(1)=BAI+P*(I1—.5)30 U=U+F(X,ANG,A0,RC)DO 40 1=1,240 LJ=U*PPRINT*, ‘U’ ,UV(L)=UIF (L—1) 43,43,4443 AA(1)=V(1)L=L+1GO TO 2144 EN=K(L)DO 45 LL=2,LI=L+1—LL45 V(I)=V(I+1)+(V(I+1.)—V(I) )/( (EN/K(I))**2—1.)RESU=V(1)KEY=1IF (ARS(RESU—AA(L—1)) ,LT.A8S(RESU*EE)) RETURNKEY=—1IF (L.EQ.MM) RETURN— 236 —AA(L)=RESUL=L+1K(L)=AL*K(L—1)GO TO 21ENDCFUNCTION F(XrANG,AOrRC)DIMENSION X(2)Y0=X(1 )—X(2)IF (Y0.E0.0.) THENF=0.ELSEY1=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*yoZ ))/YOENDIFRETURNENDA3.2.2 Program “OVER.ROMB” for Two Charged Disks with OverlappingThis 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 ofiterations. The program will list instantaneous angles between the two disks (from 95%overlapping to 5% overlapping), and radial component of the instantaneous Coulombforce.CC * OVER.ROME *CC All smbo1s used in this roram are the same as thoseC in roram ‘DISK.ROMB’ except:C 10 = data points for calculation of overlapPARAMETER (0=1 .6021892E—19,COUL8,987551787E+09r+ API=3. 14159265358979324)COMMON IFLAGvAO,RC,ANGPRINT*,’r =READ* , A0PRINT*,’RREAD*,RCPRINT*,’NO =READ*,DNANGTOUCH=2 .*ASIN ( A0/RC)PRINT*,’ANGTOUCH = ,ANGTOUCHC Special coefficient in Ronher numerical integrationAL=1 .558 PRINT*,’AL = ‘,AL— 237 —C Tolerance in Romber numerical interatioriE=1 .E—14PRINT*,’E ‘EPRJNT*i’READ*rMPRINT*,’IO =READ*, 10Z1=ANGTGUCH*0.95/(FLOAT( 10) )**1 .6liD 9 J=1,I0ANG=Z1*(FLOAT(J) )**1 .6PRINT*,’ANG ‘,ANGCOEFF=COUL*Q*Q*DN*DN/ (API *A0*A0 ) **2IFLAG=1CALL NEIIMRI (A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY, IFLAG)FO1=RESUIFLAG=2CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,pE,M,RESUpKEY,IFLAG)F02=RESU*2,IFLAG=3CALL NDIMRI (A0,RC,ANG,ANGTDUCHALrEMRESUKEYr IFLAG)F03=RESU*2.IFLAG=4CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY,IFLAG>F04=RESU*2,IFLAG=5CALL NL’IMRI (AOpRCpANGpANGTOUCHpALpE,MpRESU,KEYp IFLAG)FO5RESUI FLAG =6CALL NDIMRI(A0,RC,ANG,ANGTOUCH,AL,E,M,RESU,KEY,IFLAG)FO6=RESUFORC=COEFF* (FOl +F02+F03+F04+F05+F06)PRINT 200, FORC200 FORMAT(/1X,’Force =‘,E16.9//)9 CONTINUE99 ENDCSUEIROUTINE 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 =KEYOIF (M.LT.I..OR.AL.LT.1.5.OR.AL,GT.2,) RETURNIF (H.LE.0,..OR.H,GT,1.) RETURNEE=AMAX1 (ETOL)MH=MINO(M, 15>UA1=RC+A0A1=RC—A0(JA2=RC*COS (ANG ) +A0BA2=RC*CDS(4N0)—A0UA3= (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—IF (IFLAG.LE.3) C(1)=U3—BA3IF (IFLAG.EQ.4.OR.IFLAG.EE.6) Ccl )=UA1—UA3IF (IFLAt3.EO,5) C(1)=EA3—EA1IF (IFLAG.EQ.1) C(2)=C(1)IF (IFLAG.EO,2) C(2)=UA2—UA3IF (IFLAG.EO.3.OR, IFLAG.EO.6) C(2)=E3—12IF (IFLAG.EO.4,OR,IFLAG.ED,5) C(2)=UA2—UA321 U=0.KT=0G=1000.*H/K(L)NN=1 ./G+.5DO 22 1=1,222 F’(I)=C(I)*G31 NN2=NNIF (IFLAO..LE,3)IF (IFLAO,EQ,4,OR. IFLAG.EQ.6) D(1)=UA3IF (IFLAG.EO,5) EI(1)=BA1IF (IFLAG.EO.1) D(2)=EA3IF (IFLAG.EO.2) D(2)=U3IF (IFLAG,EQ,3.OR.IFLAG.EQ.6) D(2)=I’A2IF (IFLO,EQ,4.OR. IFLAG.EQ.5) D(2)=UA3DO 30 12=1,NN2X(2)=DC2)+P(2)*(12—.5)NN1=NNtiD 30 11=1,NN1X( 1 )=tI(1 )+P(1 )*(I1—.5)30 U=U+F(X,ANG,A0,RC,IFLAO)DO 40 1=1,240 U=U*P(I)PRINT*, ‘U’ UV(L)UIF (L—1) 43,43,4443 AA(l)=V(1)L=L+1GO TO 2144 EN=K(L)£10 45 LL=2,LI=L+1—LL45 V(I)=V(I+1)+(V(I+1)—V(I))/((EN/K(I))**2—1,)RESU=V( 1)KEY=1IF <ABS(RESU—A(L—1)).LT.AES(RESU*EE)) RETURNKEY=—1IF (L.EQ,MM) RETURNAA(L)=RESUL=L+1K(L)=AL*K(L—l)GO TO 21E Nt’CFUNCTION F(X,ANG,A0,RC,IFLAG)DIMENSION X(2)IF (IFLAG.EO,1) GOTO 71IF (IFLAG.EQ,2,OR,IFLAG.EO.3) GOTO 72IF (IFLG.GE.4) GOTO 7371 Y0=X(1)—X(2)IF (Y0.EQ.0,> THENF=0.— 239 —ELSEY11=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) )/YOENDIFGOTO 8872 YO=X(1)—X(2)IF (YO.EO,O,) THENF=O.ELSEY11=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) )/YOEND IFGOTO 8873 YOX(1)—X(2>IF (YO.EQ,O.) THENF0.ELSEY12=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))/YOEN III F88 RETURNEN P— 240 —A3.3 Results from the Double Romberg Numerical ComputationThe programs “DISK.ROMB” and “OVER.ROMB were used to calculate theinstantaneous 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 = 1cm, ion number in each disk = i0 ions. Both programs were run on a Nicolet 1180minicomputer. Because its operating speed was not fast enough, only 9 points werecalculated for overlapping disks; and for non-overlapping disks, 20 points for r’ = 1 mm,22 points for r = 0.5 mm, and 26 points for r’ = 0.1 mm. All these data were input theRIgor” program of a Macintosh microcomputer. Then, curve fitting was used to fit thesedata and the average radial Coulomb forces were solved.A3.3. 1 Average Radial Coulomb Force for? = 1 mm and r 1 cmTwo curve equations, Cl and C2, from the “Igor” program, conform well with theplot of the instantaneous radial Coulomb force, Fr vs. the angle between the two disksfor r’ = 1 mm and r = 1 cm, as shown in Figure All.For ‘ (radian) from 0 to 0.200335:Cl = 1O_’ x (0.02833— 3.985 c1 — 97.17 + 9.908x— 1.373x105 Ti÷ 8.516x10 t — 1.867x106 t); (A3.1)for CI (radian) from 0.200335 to it:C2 = 1016 x (0.6161 + 5.758e2302 + 117.5e1725 ). (A3.2)The average-over-one-cycle radial Coulomb force was evaluated from the integral—241 —0.20035<F C1d+I C2d)rad 0 0.200335= 1.311x10’6(Newton) (A3.3)where the answer is very close to <Frad> = 1.310x106 Newton in Fig. 5.8. The lattervalue was calculated from the double Gaussian numerical computation for the sameparameters.A3.3.2 Average Radial Coulomb Force for r’ = 0.5 mm and r = 1 cmThree curve equations, C3, C4, and C5, from the “Igor” program, conform well withthe plot of the instantaneous radial Coulomb force, Frad vs. the angle between the twodisks 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 + 51.46 0 — 3.746x10 02 + 1.926x105— 4.273x1060 + 5.089x107— 2.258x1080); (A3.4)for Cl) (radian) from 0.100042 to 0.406 165:C4 = 1016 x (2.147 + 19.44e7986 + 856.3e51”07 <T); (A3.5)for ci) (radian) from 0.406165 to it:C5 = 1016 x (0.5465 + 2.897e516 + 8.518e5880 ). (A3.6)— 242 —108Frad(1O_16 N)6420Figure A3.1 Radial Coulomb force on Disk 1, a uniformly charged disk of m1 ions, dueto a uniformly charged disk of m2 ions, as a function of the position of Disk2. This position is characterized by the angle 1 (Fig. 5.1). The force iscalculated from Eq. (5.13) if 0 < 1’ < touch and from Eq. (5.29) if touch<c1 <it, where touch is the touching angle (Eq. (5.5)), and is theinstantaneous angle between the two disks. The shape of the curve dependsupon the ratio r/r’, the ratio of the cyclotron radius to the disk radius. Forthis figure this ratio equals 2. The ordinate gives the force in Newtons forr, the cyclotron radius = 1 cm, r’, the disk radius = 1 mm, and N2, thenumber of m2 ions, = i04. The average radial Coulomb force, <Fave>was calculated from Eq. (A3.3).r/r’= 10r =1cmr’=lmmCurve fitting Cl— Curve fitting C2o ad from Rombergnumerical integration0 7t/4 it/2 3it/4 itcJ (radian)— 243 —20 -Curve fitting C316-r/r’= 20 “ Curve fitting C4r — 1 cm — Curve fitting C5r’ = 0.5 mm ad from Rombergnumerical integrationad 12- h(1016N)8- .0 it/4 lt/2 3it/4P (radian)Figure A3.2 Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forr/r’ = 20. This figure was derived the same way as was Fig. A3. 1, exceptthat r’ = 0.5 mm.The average-over-one-cycle radial Coulomb force was evaluated from the integral0.100042 0.406165 ItC3d+I C4d÷J C5d)r 0 0.100042 0.406165= 1.566x106(Newton) (A3.7)— 244 —where the answer is very close to = 1.564x10—6 Newton in Fig. 5.9. The lattervalue was calculated from the double Gaussian numerical computation for the sameparameters.A3.3.3 Average Radial Coulomb Force for r’ = 0.1 mm and r = 1 cmThree curve equations, C6, C7, and C8, from the “Igor” program, conform well withthe plot of the instantaneous radial Coulomb force, Frad vs. the angle between the twodisks 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 — l.788x105 + 5.210x107— l.606x1O9); (A3.8)for ‘i) (radian) from 0.0200003 to 0.257698:C7 = 10_16 x (3.739 + 34.34e1445 + 381.3e9982 (A3.9)for cJ) (radian) from 0.257698 to t:C8 = 1016 x (0.5556 + 3.317 e1635 + 11.44e7270 ). (A3.l0)The average-over-one-cycle radial Coulomb force was evaluated from the integral0.0200003 0.257698C6d÷J d+J C8d)ra 0 0.0200003 0.257698= 2.167x10’ (Newton) (A3.ll)— 245 —where the answer is very close to = 2.175x10—’6Newton in Fig. 5.10. The lattervalue was calculated from the double Gaussian numerical computation for the sameparameters.100 -80 Curve fitting C6rlr’= 100 Curve fitting C7r = 1 cm — Curve fitting C8- r’ = 0.1 mm ad from Rombergnumerical integrationrad 60-(1016N)40-20-0--0 it/4 7t12 37/4 ItcJ (radian)Figure A3.3 Radial Coulomb force on Disk 1 as a function of the position of Disk 2 forr/r’ = 100. This figure was derived the same way as was Fig. A3.l, exceptthat 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’ areobtained from “Igor” program on a Macintosh microcomputer. The following fourequations in which the fitting coefficients are expressed to four significant figures give agood fit forD vs. nT’:D = [0.5504 + 1.128 e 0.4310 1og(rIr + 0.5338 e 2.511 1o(rIr’)] r. (A3.12)= { 2.172 — 1.535 log(r/r’) + 1.014 [log(r/r)]2 — 0.3827 [log(rlr’)13+ 0.05836 [log(r/r)]4} r. (A3.13)= [2.341— 1.171 exp(— 1.54) — 0.2177 exp(_3)]r. (A3.14)r 0 7142 log(rlr’) i= [ + 1.292 e j r. (A3.15)In Table V.4.1, the numerical components of D, Dr’, calculated from the above fourequations, Eqs. (A3. 12) — (A3. 15), are listed together with the theoretical values of D’ inTable 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) = [(D’)1— (Dc’)theo ]2 (A3.17)— 247 —where (DC’)fit is the calculated values from Eqs. (A3.12) — (A3.15) and (Dc’)theo is thetheoretical values in Table 5.1. From their deviations listed in Table A3.1, Eq. (A3.12) isthe best fitting equation.Table A3.1 Comparison of the values of D’ calculated from Eqs. (A3.12) —(A3.15) with theoretical values of Do’.Theoretical D’ from D’ from D’ from fromrir’ D’ Eq. (A3.12) Eq. (A3.13) Eq. (A3.14) Eq. (A3.15)2 1.792 1.792 1.792 1.792 1.7393 1.629 1.630 1.632 1.634 1.6144 1.539 1.538 1.540 1.539 1.5375 1.478 1.477 1.478 1.476 1.4828 1.371 1.370 1.370 1.369 1.37910 1.327 1.327 1.327 1.328 1.33615 1.257 1.258 1.259 1.261 1.26620 1.215 1.215 1.217 1.217 1.22125 1.183 1.184 1.187 1.185 1.18850 1.102 1.100 1.103 1.096 1.100100 1.030 1.030 1.033 1.034 1.026Deviation 1x105 3.3x105 8.5x10 3.3x10(sum of squared errors)— 248 —A4 Appendixes of Chapter 6A4.l Four Particular Integrals Applied in the Taylor’s Expansion Approximation ofIon-Ion Coulomb InteractionMost integral formulae used to calculate the value of the Columb interactioncoefficient, C, have been taken from “Table of Integrals, Series, and Products”(1)and“Integrals and Series”(2). The following four particular integral formulae which will beused 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 b2> a2.Lettinga+J2+x = u,x = ‘I(u—a)2—b.J ln(a+b2÷x2)= J lnudg(u—a)2—b2= — a)2— b2 in— J u2— 2a u+ a2— b2 du= — a)2— b2 in u—u2— 2a u + a2— b2 duu Iu2_ 2a u + a2— b2=(u—a)2—bin I U 4uJ ?— 2a u + a2— b2— 249 —= g(u—a)2—b mu—‘Ju2 2a u + a2— b2—ainu— a + u2_ 2a u + a2— b21+2ainu — a + u2_ 2a u + a2— b212au+2(a b2+ ‘1b2— a2 arcsin ‘ + integral constant.u ‘J4 a2— 4 (a2— b2)= (u—a)2—b mu— J(u—a)2b +ainI u — a + J(u_a)2_ b2+ gb2— a2 arcsin — a u +a2— b2 + integrat constant=xln(a+b2+x2)_x+ainx+Jb2+ x2+ gb2— a2 arcsin — b2 — a ‘J b2 + X + integral constant. (A4. 1)b (a + ‘Jb2+ x2 )The integral formula (A4. 1) is valid for x 0.(2) J in (a + a2+x2) dx, where the constant a >0.Lettinga+ga2÷x = u,x = q(u—a)2—a2.I ln(a÷ga2÷x2)= I inud(u—a)2—a2= g(u — a)2— a2 in u — (u — a)2— a2 + am Iu — a + ‘J (u — a)2— a2+ integral constant= x in (a + Ia2+x2) — x + a in — x + J a + + integral constant. (A4.2)— 250 —(3) J x(a+b2+)dx,wheretheconstantsa>O,b>O.Letting b2-f x2 = u2, x = 4u2— b2.I x(a÷gb2+x2) dx= I ln(a+u)du2= { uin(a+u)—a udu }( 2 2 22 u—a +a=iuin(a+u) a+u du= {u2ln(a+u)—u_a+a du }= { t?ln(a+u) — (u_a)2_a2ln(a+u)} +integralconstant.= { (b2 + x2) in (a + + x2) — (qb2 + — a)2— a2 in (a + + x2) + integral constant= { (b — a2 + x2) in (a + + x2) — (gt2 + x2 — a)2 }+ integral constant. (A4.3)—251—Ib2 + ab2+ X dx,wherea>O,b>O,and b>a.(4) arcsin________b ( a + /b2+ x2 )Ib2+aJbxtharcsin__b ( a + Jb2÷ x2 )• b2+ab2+x2—I•b2+aVb2+x2xd[arcsm= x arcsin__________b (a + Ib2÷ x2 ) b (a + b2+ x2 )______xdx• b2 + a b2+ X+b2— a2gb2+ x2 (a + b+ x2 )= x arcsin_______ _____b ( a + ‘Jb2+ x2 )• b2+aJb2+x2 .Ib2_a2J d(b2+x)= xarcsin_ _b(a+b2+x2) 2 b2+x2(a+b2+x2)• b2+aJb2+x2 Jb2_a2J dgb2+x=xarcsin +_b(a+Jb2÷x ) (a÷b2÷x2 )• b2 + a b2+ X+b2—a in [a+gb2x2 1. (A4.4)= xarcsinb (a + Jb2÷ x2 )The integral formula (A4.4) is valid for x 0.All of these four integral formulae have been verified by two ways: either by takingthe differential of the four antiderivatives or by the Romberg numerical integration of singlevariation.(3)— 252 —A4.2 Integrations of the Coulomb Interaction CoefficientIn Section 6.2.2, the integral form of the Coulomb interaction coefficient C (Eq.(6.23)) was found to be— kqq2——16 a04a0 a0 a0 a0 1/2 1/2 (y—y)2+ (z1—z2)— 2 (xj—xj)i f f f £112 £112 { (xj_4)2+ (y-y)2+ (zl_z2)2 j512J-0xdz12yjdydxjd4. (A4.5)When this multiple integral is solved, the singular points can be deleted by taking the leftand right limits for= 4, y = y, and z1 = z2.A4.2.1 Integration for and 4For convenience, x and 4 are integrated first. The integration order of (A4.5) hasbeen changed toa0 a0 1/2 1/2 ‘a0 X12—C fa0J lim(f +1f Lao £1/2 £1/2 = x2 + £(yj—y)2+ (zl_z2)2_ 2(x4)2[ (x—4) + (y—y)2+ (z1—z2)j512 dxd4dzzydya0 a0 112 1/2 ‘a0= L f £112 L1 J ‘ {___________________________= -2 £ 0 [ (44)2 + (Yj—Y)2+ (z1—z2)]3/2 ax-4+ I[ (x—4) + (yj—y)2+ (z1—z2)j3/2 Xi = X2 + dz12ydy— 253 ——2£112 1/2 a0 a0 a0lim= 11/2 11/2 Lo f £ o [ e2 + (y—y)2+ (z1—z2)]3/22 (a0+x)+ [(a0+4)2 + (YjY)2+ (z1—z2)]3/2 } 4dz1dz2ydy (A4.6)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 secondintegrand is needed in order to integrate further, and singular points 4 = — a0, yj = y,and z1 = z2 must be deleted.ao ao 1/2 1/2urn 2 (a0+xj) d4dz1z2y yf Lo 11/2 £1120J—a+e [(a +4)2 + (yj—y)2+ (z1—z2)]3/2Fa0 a0 1/2 1/2= —2 J f 11/2 £112 { 4a2+ (yjy)2+ (z1—z2)-a01-lim______£ 0 e2 + (YjY)2+ (z1—z2)th1dz2ydy. (A4.7)The singular points y = y, and z1 = z2 must be deleted from the second integrand in Eq.(A4.7) in the following integration for z1, z2, y, and y.A4.2.2 Integration for z1 and z2ao ao 1/2 1/2—2 L f 11/2 11/2 { 4a02+ (yj—y)2+ (z1—z2)— 254 —1(Yj—Y)2+(z1—z2dz12ydy‘a0 1/2____________________________= —2£12 {in (z1—z2)+ 4 a02 + (Yj—Y)2+ (z1—z2)J-a0 -a_________________1/2—1n (z1—z2)+ (y—y)2+ (z1_2)2 J L th2dydypa0 a0 1/2_______= —2 1 f f12 { 1n (1/2 + z2) + 4a02+ (yj—y)2+(1/2 + z2)—1n — (1/2 + z2) + 4 a02+ (yj—y)2+ (l/2 + z2)- in (1/2 + Z2) + (yj-y)2+ (1/2 + z2)2+ in — (1/2 + z2) + (yj—y)2 + (1/2 + z2) } dz2ydya0 1/2__= —21 f £/2 { 2in (1/2 + z2) + 4a02+ (yj—y)2+(1/2 + z2)21+2 in Iy—y I } dz2ydy— 255 —= _2f°fao { 2 (l/2+z)ln [ (l,2+z2)+’!4a02+ (yj—yj)2+(1/2 + z2) I—2 44 a02 + (yj—y)2+ (l/2 + z2) — z2 in [ 4 a02 + (y—y)2 j—2Ufl+z)in (1/2 + z2) + q(y_y)2 + (1/2 + z2)_______________________1/2+2 [_y)2 + (1/2 + z2) +2 z2 in I y—y I } 1/2 dydy= 2fao jao { 2 lin[l+[i2 + 4 a02+ (yj—y)2 ]_2l2 + 4 a0+ (yj—y)2+244 a02+ (yj—y)2 —lln[4a2+ _y)2I_2lin[l+l2 + (yj_y)2]+2 + (yj-y)2 -2 (yj-y)2 +21 in I y-y I } dy dy= 2fao J’7° { 2 un {l+/l2 + 4 a02+ (yj—y)2 ]_2Jl2 + 4 a0+ (yj—y)2_2lin[l+l2 + (y_y)2j+2l2 + (yj—y)2 +24a02+ (y—y)2—lin[4a?+ _y)2J_2(y;_y)2 +2lln Iy—yI ) dydy (A4.8)where eight integrands are developed for integration of y and y. It is worth noting thatthe first four integrands, which contain 1, the z-iength of the tetragonal ion cloud, in theirsquare roots, represent the Coulomb interaction for separation 1 in the z-direction; and the— 256 —last four integrands, which do not contain 1 in their square roots, represent the Coulombinteraction in x-y plane. These eight integrands will be integrated methodically in eightsteps, 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 thefollowing integral.2fo2l1nl+\Il2÷4aQ2+ (y1_y)2dy’dyt= —41 {f° f ln [1 + Jl2 + 4 a02 + (yj—y)2 I dydy4° in [1 + l2 + 4 a02 + (y—yj)2 I dydy+1 f ln[l+l2+4a02 (y_y)2] dydy+ in [1 + J 12 + 4 a2 + (y j—y)2 I dy dy }= —4! { 2L J ln [ 1.f2 + 4 a02+ (yj+y)2]+ in [1 + i 12 + 4 a o2 + (y jy)2] dy{dy }= —8 1 J { f° in [1 + g12 + 4 a02 + (yj+y)2] dyIIUI+00+II+II++)_0000__0::‘..,,-a.___-___+II+0+t.3I+-a-+++I)+I++I--a’4.I++—‘—++-‘t)—-3IS:I_+0+l0•__•I__I+I+.)+a,+°I-+oj.+++o-1‘+-‘+1‘1++1i++++1I+1_a—’+a-’+IIa-.,+I.I•.•0Ii-a-.II’.) It..+IIII•—Ia__I1.31-.+iIoj+I+-iI+1IoI1.31+11.3I4I‘-I++i±-Io+I1.cii+1I+IciII+1III0I0I1’3100i-a-IIIa-SIII1+1a-SI‘—I—a.’I,—II—’’_IICII‘.-—-II+1+I-a-i+I+I+i01+1a,--I-I±iI_—-+—Isf-I-%_-‘-.,_‘-II1.-IIt’.)t’31.301.3I1.3‘IL.Ja,+S....-.-0‘.C—II+I+cLII+II+i+I_)0000_0000Ir—Ia—.9I+++t’)I+t.)+++I_+I+++I_+II+i-_+1.31,++1.,H0__I1.3.II0+1r11.3++13I4I+..++çI+-+.++‘+11.31+—+1i+-+10)..-1.3-I-I-II+1-+.d&iI—iicI9!4I++131I+11..)131I1+1‘•1_+1I-0131+.II_I01+1+1+1J+1i4.i‘—‘131IIi00I1•II•_.I1.—i+III—Icici131oloIi+Ici1.3+‘i+I+;i+IC__II____IIII_+L-I.I,-j__—IIoI131__io‘I1.31+o+_cli’131-+I+IIIII‘-1::)I) III13-II13-I131+I4II’13I13’I1.31131++1-.__01.)I00.irJ+10‘—I13131+1I—i1.3-I1.31‘—‘I13-I 13’I+Ico1%)1’3IU...C .-‘C—0h)t)+)_+•C••+..It.31+±1+1;i+1uI+1-__±1C)Icxio,J+++VI,ICI‘l—t31It31Ii_i4I’-++1•.cciI3ciI.__C)I+IIt’.)+ o++I-I)I’—’•. Ct) +-‘—J.—+1?1-I-CC-a++F-’F-’+C- t’.)IIt%)C IJ+ + t)+0t’)+t•3-+ C + + Ct’)+t’)- t’3IIIIi—iC)c9C++++s’ CC3-0Ct)+%_%—I—J‘•‘E‘-••++o+ 0+t’.)-IF-’0I+C+o + p Cl)I.-.+t’.)++ +-.+C__‘.3-.-+01.31)+C’.)+ 1.3+0‘.3+t)- 1’.).1— 260—12 + 4 a02+ 1 4 j2 + 8 a 2—4aarcsin0_4ao21n[l+Ji2 + 8a02 ]4—;-;:a [1 + J i + 8 a02 J12 + 4 a02+ I Ji + 5 a 2+2aarcsin +4afl[l+’[i+5(bO2l4i2÷4ao2 [j+Jl2+5ao2 ]4a021n[ i+4 5 a]— [+ a02 _112—2aln[ l+Jl2÷ 4 a2]+ [[i- 4 a02 j)2+a01 in[a0+[12+ 5 a02 1—i[+ 5 ao2+h[i+ 4 a02l2+4a2+j)j2+5a20—2 a arcsin—________[i+4ao [l+4ll2+5ao2 I—4a in[i+[i2+ 5a ]+4aç ln[l+\[12+ 4 a02]_aolln(12+4a02)+2ltao }—41 (_6a02+4aol 1n[2a0+8 a ]_ifi+ 8 a+l[+ 4 a0212+4a0+lIl2+8ao2—+4a0In[ l+\f4ao2I—8 a02 arcsinJ+4a [l+Il2÷8ao2]—2 a0 l in (12+4 a) + 4it a } (A4.9)faojao(2) _2) 3 —2 + 4 a02 + (yjY dydy(ao’= 2) 1. 4 a0+ (yj—y)2-a0_____________________lao(yjY)2 ]} dy—261—= 2f { 2 (a0-i-yj) !l2 + 4 a02+(a0+y)2+(124a0)ln[(a0+)+l2+ 4 a(2+(a0+y)2 ]_(l2 + 4a02) in [— (a0-i-yj) + Ii2 + 4 a2+(a0+y)2 j } dy= 2f { 2 (a0+yj) gi2 + 4 a2+(a0+y)2+2 j2 +4 a02) in {(a0+) + + 4 a02+ (a0÷y)2 I—(1+4a0)ln(l } dy12 3/2= 2 t-[l+4a(yj)j+ 2 (j2 + 4a02)(a-t-yj) in {(a0-i-) + 4i2 + 4 a02+(a0÷yj)2 I—2 (l2 + 4 a02) gi2 + 4 a02+ (a0+y)2 — (12+4 a02) in (12+4 a) y }12 3t2 2 3/2= 2 t - (l2 + 8 a02) — (j2 +4a02)+4a0(l2)in{2a0+g12+ 8 ao2]_2(l2+4ao2)Il2 + 8 a02+2(l2+4a02)[l2 + 4 a02 —2a(l+4)in (12 +4a0) }. (A4.1O)(3) The integral formulae (A4.2), (A4.3), and (A4.4) have been used to evaluate thefollowing integral._2J f —2lln 1 + 12 + (yj—y)2 dydy— 262 —= 41f f in[l+Ji2 + (yj—y)2]dydy= if { ;-yin[i+l2+ (yj-yj)2]--+ (yjy)2] } dy= 4lf { 2(a0+y)in[l+4i +(a0+y)2]—2a+ Un [(a0+y) + Ji2 + (a0-t-y)2j — un [— (a0+) + .\1i2 + (a0-f-y)2I } dy= 4 if { 2 (a0+y) in [i + Jl2 + (a0+y)2]— 2a+21 in [(a0+) + Ji2 + (a0+y)21 —21 in i } dyj= 4 i { (a0-i-y)2 in [i + Ji2 + (a0-f.y)2I — [i2 + (a0+y)2 —1 j2 — 2a0y+ 2i (a0+yj) in [(a0+) + i2 + (a0+y)2]_2ll2 + (a0+y)2—2l1n(l) }= 41 [ 4ao21n[l÷.Jl2 + 4a02j_[2i2+4a _2l4i + 4 a02 ]—4a+4a0lin{2+4i2+ 4 a02 ]_2ii2+ 4 a02 +2l—4a11n1 }=4i{4ao2in[l+l2+4ao2]_6a_ll2+4a2 l2+4a0lin[2a0-t-Jl + 4 a02 ]—4a0linl } (A4.11)— 263 —pa0 pa______(4) —2) J 2(l + (yj—y)2 dydypa0__ _ __ _ _____Ilao= —2) { (y’—y’) + (YY)2+ j2 in [(y;—y) + gi2 + (yj—y)2] } dypa0_ ____________ _______________= —2) { 2(a0+))!l2 + 4 a+ (aO+y)2+l2in[(a0)) l2 + (a0-i-y)2j-a0_l2 in [— (a0+) + g2 + (a0+y)2 ] } dy= —2 1 2 (a0+) Il2 + 4 a0+(a0+y)2‘—a0+2 12 in [(a0+) + .Jl2 + (a0-t-y)2 ] —2 j2 in l } dy= —2 { [12 +(a0+yj)2j3+2 j2 (a0+y) in [(a0+y) + gi2 + (a0+y)2 ]___________-= —2 { (l2+4ao2)3/_l3+4aol2in[2ao+/l2 + 4 ao2J_2l2,/l2 + 4 a02+2l3—4a0l2n }. (A4.12)(5)—21 J 2\J4a0+ (Yj—Y)2 dydyv—a0 —a0= —2) { (y—y) \I 4 a(2 + (y j—y)2-a0— 264 —+ 4ao2ln [j-y) + 4 a2 + (yj-y)2] } dy= —2 J { 2 (a0+)4 a2 + (a0+y)2+ 4a2in [ (a0+yj) + J4 a02 + (a0÷y)2I—4a1n [—(a0-l-)) + g4a02+(a0+y)2J } dy= _2J { 2(a0+yj)4a+(a0+y)2+ 8 a02in [(a0+y) + 4 a0 + (a0-fy)2j —8 a02in (2 a0) } dyj= —2 { [4 a+ (a0+yj)2] + 8 a02(a0+yj) in [(a0+) + ‘4a0-- (a0÷y)2I—8a04a+(y)in 2) }=—2 { [16qa3_8] 1o ln(2a0)+1631n(2a0i-2J)—16Ja03+ }= —2 { [16qa3_8]+16 1n(1+1)—16’Ja03+ 6a }. (A4.13)(6) -2 f f -1 in [4 a + (y—y)2 I dydy= 21 f° f In [4 a02 + (yj—y)2]dydy= 21 f { (yj-y) in [4 a + (yj-y)2 j - 2(y-y)— 265 —+4a0cnY2lao2a0 J1dya0+yj= 21 J { 2 (a0+) in [4 a+ (a0÷yj)2]— 4a + 8 a0 arctan ( 2 a0 } dy= 21 { [4a02+(a0+y)2]in [4a02+(a0+y)2j— [4a02+(a0+y)2]a0(ao+Y2)82th[1(ao+Y2)1i—4a0y+8 arctana0= 2l{ 8a 1n(8a)—4a }= 21 { 8ao21n(2a0)_12ao2÷47ao2} (A4.14)pa0 pa0________(7) _2J J _2((yj_y)2 dydy-a0pa0 j’a0= 4 J J I y—y I dydy-a0-aoa0 a0= L0 {-*P2-ao + (yj-y)2 J dya0= 2 f { (y+a0)2+ (yj-a0)2 } dyIa= 2 { (y+a0)3+ - (y—a }32= Tao. (A4.15)— 266 —(8) —2fQ f 2 1 in I y—y I dy dy= —41 f f in I yj—y2 I dy dy=4lf { -y)1nIy-yI -(y-y) } dy= _4lf { 2(a0+yj)in(a)—2 } dy= —41 { (a0+y)2 in (a0+y) — (a0+y)2— 2a0 }= —41 {4a02in(2a—2a —4a02 }=— 16 a 1 in (2 a) +24 a021. (A4. 16)All of these eight integrals are verified by Romberg numerical integration of multiplevariabies.(4)A4.2.4. Value of the Coulomb Interaction Coefficient, CEqs. (A4.9), (A4.1O), (A4.1 1), and (A4.12) represent the Coulomb interaction forseparation 1. Their sum is—2 f J { 2 tin [i + 412 + 4 a02 + (yj—y)2 1—2 .4i2 + 4 a02 + (y—y)2_2lln[l+gl2+(y1y) j } dydy— 267 —= —41 {_6a02+4 ln[2a0+l2+ 8 a02 ]_i4i2+ 8a02+li,il2+ 4 a02—8a02arcsinl2+4a2+l,Il2+8a+4aln[j+fr+4a2]Il2+4a0 [ l+.q12+8a0 ]—2aln(+4a)÷4ia } +2 { (l2+8a)3_(l2+4a02)/+4a0(l24a)ln[2a0+Il2 + 8ao1_2(l2+4a)gl2 + 8 a02+2(l24ao)/l2 + 4 a02—2a(l+4)ln(la) }41{ 4aln{l+/l2 + 4a2]_6a0_lil2 + 4 a02 +12+4a011n [2a0+l2 + 4 a02 ]—4a011n1 } —2 { (l2+4a)3_l3+4al2n[2aIl2 + 4a02]_2li1 + 4 a02 +213—4a1n }= 4 {_2a0l2ln[2a2+l2+ 8 2]+a121n(12+4a)l2+4a02+hfl2+8a+8a2larcsm______________—4ita2lJl2+4a0 [ l+Jl2+8a0 ]+ + 8 a02]_4aln(l2+4a)_4a44l2 + 8 a02 +4a,/l2 + 4 a02+2a0ln[2aJl2 + 4a021_2aol2lnl }. (A4.17)For 1=2.4 cm and a0 = 1 mm, the analytical solution of Eq. (A4.17) is 6.65923x1010m3and the numerical solution of the sum of Eq. (A4.9), Eq. (A4.1O), Eq. (A4.l 1), and Eq.(A4.12) from the Romberg multiple integration(4)is6.65892x1(r’° m3.Eqs. (A4.13), (A4.14), (A4.15), and (A4.16) represent the Coulomb interaction in xy plane. Their sum is— 268 —f jao { 2J4a0+ (yj-y)2 -lin[4a÷+21 in I yj—y I } dy dy= —2 { [16qa03_8a3]+1oa n(1+q)—16qa03÷16a }+ 16 a02 un (2a0)— 24 al+ 87t al+a?— l6alln (2 a0) + 24 a021= 4 { a03[— 16J }= 4 (— 5.9464 a03 + 2it a02 1 (A4.18)For 1=2.4 cm and a0 = 1 mm, the analytical solution of Eq. (A4.18) is 5.79400xi- m3and the numerical solution of the sum of Eq. (A4.13), Eq. (A4.14), Eq. (A4.15), and Eq.(A4.16) from the Romberg multiple integration(4)is5.79079x10 m3.Substituting Eqs. (A4.17) and (A4.18) into Eq. (A4.5),c= 4’”72 { 5.9464ao3_2ta02l+2aP n[+l +8 a0212 2 2—a012n(1+4)—8a02larcsin +4ica021q12+4a0 [ l+sJl2+8a0 ]_.(j28a02)3+(l2+4a2)/_l3_83 n[2÷.\Jl2+ 8 a021+4aln(l4a)+4a\Il + 8 a02 _4astqIl2 + 4 a02—2a012n[2a0+l2 + 4 aj +2a012n1 }. (A4.19)— 269 —Reference1. 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”; Gordonand 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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