{"http:\/\/dx.doi.org\/10.14288\/1.0067239":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Physics and Astronomy, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Vyskocil, Eric","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-05-19T14:27:02Z","type":"literal","lang":"en"},{"value":"2009","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Methods for understanding and generating pulsed beams of translationally cold molecules through electrostatic velocity filtering are reported. Pulsed beams of acetonitrile (CH\u2083CN) and calcium monofluoride (\u2074\u2070Ca\u00b9\u2079F) in particular are theoretically examined and experimentally obtained. CH\u2083CN gas molecules are obtained from the vapour pressure above liquid CH\u2083CN and introduced to the electrostatic guide through a pulsed nozzle. The observed time of flight indicates a longitudinal velocity of 31 m\/s and temperature of 2.37 K. \u2074\u2070Ca\u00b9\u2079F molecules are obtained through laser ablation off a CaF\u2082 disk and cooled via buffer gas cooling by a cold pulsed Helium beam prior to being introduced into the electrostatic guide. Longitudinal velocities of 4.47 m\/s and temperatures of 70.9 mK are obtained. While results for CH\u2083CN are within theoretical expectations, results for \u2074\u2070Ca\u00b9\u2079F are not in agreement with theoretical predictions, indicating either a lack of understanding of the process or a limitation in the current experimental design. Nonetheless, the current experimental apparatus may be used, with minor adjustments to perform depletion spectroscopy on these molecules as well as may be used to generate pulsed beams of other molecules such as barium monofluoride (\u00b9\u00b3\u2078Ba\u00b9\u2079F), which may be used in the determination of the electron's electric dipole moment.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/7920?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"4795020 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"Generation of Cold Pulsed Molecular Beams by Eric Vyskocil B.Sc., University of Windsor, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May, 2009 c\u00a9 Eric Vyskocil 2009 Abstract Methods for understanding and generating pulsed beams of translationally cold molecules through electrostatic velocity filtering are reported. Pulsed beams of acetonitrile (CH3CN) and calcium monofluoride (40Ca19F) in par- ticular are theoretically examined and experimentally obtained. CH3CN gas molecules are obtained from the vapour pressure above liquid CH3CN and introduced to the electrostatic guide through a pulsed nozzle. The observed time of flight indicates a longitudinal velocity of 31 m\/s and temperature of 2.37 K. 40Ca19F molecules are obtained through laser ablation off a CaF2 disk and cooled via buffer gas cooling by a cold pulsed Helium beam prior to being introduced into the electrostatic guide. Longitudinal velocities of 4.47 m\/s and temperatures of 70.9 mK are obtained. While results for CH3CN are within theoretical expectations, results for 40Ca19F are not in agreement with theoretical predictions, indicating either a lack of understanding of the process or a limitation in the current experimental design. Nonetheless, the current experimental apparatus may be used, with minor adjustments to perform depletion spectroscopy on these molecules as well as may be used to generate pulsed beams of other molecules such as barium monofluoride (138Ba19F), which may be used in the determination of the electron\u2019s electric dipole moment. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Principles of Electrostatic Velocity Filtering . . . . . . . . . 4 2.1 Symmetric Top Molecules . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Zero-Field Energy Eigenvalues and Eigenstates . . . . 4 2.1.2 Stark Effect: Approximate and Exact Values . . . . . 9 2.2 Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Zero-Field Energy Eigenvalues and Eigenstates . . . . 12 2.2.2 Stark Effect: Approximate and Exact Values . . . . . 13 2.3 Electrostatic Forces . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Guide Geometry . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Transverse and Longitudinal Velocity Filtering . . . . 17 2.4 Probability Calculations . . . . . . . . . . . . . . . . . . . . . 21 3 Buffer Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Elastic Collisions and Cooling . . . . . . . . . . . . . . . . . 24 3.2 Inelastic Collisions and Cooling . . . . . . . . . . . . . . . . 25 4 Laser Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1 Principles of Laser Ablation . . . . . . . . . . . . . . . . . . 28 4.2 Ablation Plume . . . . . . . . . . . . . . . . . . . . . . . . . 29 iii Table of Contents 5 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . 31 5.1 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Hexapole Details . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 CH3CN Gas Source . . . . . . . . . . . . . . . . . . . . . . . 34 5.4 Pulsed Nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 40Ca19F Ablation Sources . . . . . . . . . . . . . . . . . . . . 37 5.6 Observation Setup . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 40 6.1 CH3CN Velocity Filtering I . . . . . . . . . . . . . . . . . . . 40 6.2 CH3CN Velocity Filtering II . . . . . . . . . . . . . . . . . . 48 6.3 CH3CN Velocity Filtering: \u2019Ablation\u2019 Simulation . . . . . . . 50 6.4 40Ca19F Velocity Filtering . . . . . . . . . . . . . . . . . . . 52 7 Future Considerations . . . . . . . . . . . . . . . . . . . . . . . 55 7.1 Alternate Sample Molecules . . . . . . . . . . . . . . . . . . 55 7.2 Multi-Photon Ionization . . . . . . . . . . . . . . . . . . . . . 56 7.3 Laser Depletion Spectroscopy . . . . . . . . . . . . . . . . . . 57 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Appendices A Stark Effect in Symmetric Top Molecules: C Program . . 63 B Stark Effect in Diatomic Molecules: C Program . . . . . . 67 iv List of Tables 2.1 Constants Data for CH3CN . . . . . . . . . . . . . . . . . . . 5 2.2 Constants Data for CaF and BaF . . . . . . . . . . . . . . . . 13 2.3 Probability Calculation for CH3CN . . . . . . . . . . . . . . . 22 5.1 Chamber Pressure Data . . . . . . . . . . . . . . . . . . . . . 31 5.2 Hexapole Properties . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 CH3CN Chemical Properties . . . . . . . . . . . . . . . . . . 35 6.1 Maximum Trappable Velocities for CH3CN . . . . . . . . . . 48 v List of Figures 2.1 Oblate and Prolate Symmetric Top Molecules . . . . . . . . . 5 2.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Stark effect of J=1 states of CH3CN . . . . . . . . . . . . . . 11 2.4 Stark effect of the lowest states of 40Ca19F . . . . . . . . . . . 15 2.5 The Electric Field of a Hexapole Arrangement . . . . . . . . 16 2.6 Transverse and Longitudinal Velocities in a Hexapole . . . . . 18 2.7 Approximate Maximum Velocity for 40Ca19F . . . . . . . . . 20 2.8 Effusive and Collimated Sources . . . . . . . . . . . . . . . . 23 3.1 Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 Ablation Damage Photo . . . . . . . . . . . . . . . . . . . . . 30 4.2 Ablation Plume Photo . . . . . . . . . . . . . . . . . . . . . . 30 5.1 Simple Diagram of the Vacuum System . . . . . . . . . . . . 32 5.2 Hexapole Radii . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Source CH3CN Gas . . . . . . . . . . . . . . . . . . . . . . . . 34 5.4 Source Arrangment Used for CH3CN . . . . . . . . . . . . . . 35 5.5 Pulsed Nozzles Used in Experiment . . . . . . . . . . . . . . . 36 5.6 Source Arrangment Used for Ablation of 40Ca19F . . . . . . . 37 5.7 Source Arrangment Used for Ablation of 40Ca19F . . . . . . . 38 5.8 Simple Diagram Showing the Observation Setup . . . . . . . 39 6.1 Variation of the Backing Pressure of the CH . . . . . . . . . . 41 6.2 Experiment Diagram Showing the Total Distance Travelled . 42 6.3 Pulse Width Variations on the Time of Flight of CH3CN Di- rectly Injected . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.4 Variation of the Large Hexapole Voltage . . . . . . . . . . . . 45 6.5 Time of Flight Spectrum for CH3CN Directly Injected . . . . 46 6.6 Velocity Distribution of CH3CN Directly Injected . . . . . . . 47 6.7 Time of Flight Spectrum for CH3CN Directly Injected II . . . 49 vi List of Figures 6.8 Velocity Distribution of CH3CN Directly Injected II . . . . . 50 6.9 CH3CN Time of Flight: \u2019Ablation Simulation\u2019 II . . . . . . . 51 6.10 40Ca19F Time of Flight Distribution . . . . . . . . . . . . . . 53 6.11 40Ca19F Translational Velocity Distribution . . . . . . . . . . 54 7.1 SrO and Rotating Motor . . . . . . . . . . . . . . . . . . . . . 55 7.2 Multi-Photon Ionization . . . . . . . . . . . . . . . . . . . . . 56 7.3 Depletion Spectroscopy . . . . . . . . . . . . . . . . . . . . . 57 vii Acknowledgements There are many people to whom I am extremely grateful for their assistance, both direct and indirect, throughout my studies. First of all, I would like to thank my supervisor, Dr. Takamasa Momose, for his continued assistance and patience over the two years that I was his student. I have learned much from him and have appreciated the time I have been under his direction and the opportunity to work on this project. I would also like to thank my fellow labmates, Masaaki Tsubouchi, Susumu Kuma, Yuki Miyamoto and Hiroko Nakahara for the many times they have provided assistance or guidance with my experiment. Whenever problems arose I felt fortunate to be surrounded by intelligent and patient labmates who, even when they did not know the solution, were willing to listen and help brainstorm possible solutions. Indirectly, there are many people who have supported me during my time as a student and even though perhaps they were unable to understand my project, were able to relate to my struggle. My family, despite the great distances apart, have always encouraged me to work honourably and do my best. My friends have helped me relax after difficult days, or celebrate acheivements. To all of you, I am grateful. 1 Chapter 1 Introduction Research into the field of cold and ultracold molecules continues to be one that attracts a great deal of attention with numerous applications in both chemistry and physics, including but not limited to high resolution spec- troscopy and the search for the electron\u2019s electric dipole moment (EDM) [1]. As research into the field of cold and ultracold molecules increases, the number of experimentally realized methods and our understanding of these methods increases. Certain ultracold dimers may be created through laser trapping and subsequent photoassociation [2] or by using Feshbach resonances [3] of atoms. Stark deceleration of polar molecules using time- varying electric fields has been acheived and modified by numerous groups [4\u20137] for various purposes. Although similar to Stark deceleration, Zeeman deceleration may decelerate practically any molecules having a having a non- zero spin state and has only very recently been successfully implemented [8] and its appeal has led many others to pursue it as a useful method for ob- taining cold molecules. Buffer-gas cooling, also applicable to virually any molecule, has been experimentally realized [9] and is in widespread use. An- other innovative method involves the use of a rapidly rotating source whose rotational motion slows molecules leaving the source [10]. While the previously mentioned methods are considered to be \u2019active\u2019 cooling methods because energy is removed from the molecules, there also exists passive cooling methods which rely on the fact that even at room temperature there are molecules that possess low kinetic energy. One such method is Stark velocity filtering, which relies on the geometry of electro- static fields to filter slower moving molecules from faster moving ones [11]. Combinations of different cooling methods for molecules are possible, such as buffer-gas cooling and Stark velocity filtering [12]. The experiments reported on in this thesis have applications in high res- olution spectroscopy of the target molecules CH3CN and 40Ca19F and are an important stepping stone to further experiments. Spectroscopic analysis of cold molecules is more accurate than that of molecules at room temperature because of a reduction in the Doppler width of transitions and an increase in the interaction time between the molecules and the probe [1]. The molecule 2 Chapter 1. Introduction 138Ba19F, which may be obtained in a similar method to that for 40Ca19F is a useful molecule in the search for the electron\u2019s EDM as its structure is similar to YbF, a molecule already being used in such experiments [13]. In addition, the future addition of a laser IR cavity to trap molecules will allow for experimentation on coherent quantum control [14]. From a spectroscopic standpoint, 40Ca19F is a very convenient molecule to work with. As a diatomic molecule, it consists of two closed shells with a single electron orbiting it, which can be considered similar to atomic al- kali metals. In its ground electronic state, X2\u03a3+ has no spin-spin coupling and orbital angular momentum and relatively small hyperfine interactions and spin-rotation coupling. Obtaining it has proven to be easily obtained through laser ablation off a solid CaF2 target [15, 16]. 3 Chapter 2 Principles of Electrostatic Velocity Filtering Electrostatic velocity filtering, otherwise known as Stark velocity filtering uses the geometry of static electric fields inside a device, such as a quadrupole or hexapole, to passively select molecules with low longitudinal and trans- verse velocities. The principles needed to understand this process are de- scribed in this chapter beginning with an explanation of the Stark effect in symmetric top and diatomic molecules, through a discussion on the elec- trostatic forces on such molecules traversing the apparatus and concluding by numerically calculating the theoretical net probability that a molecule entering the electrostatic guide comes out at the end of the guide. 2.1 Symmetric Top Molecules 2.1.1 Zero-Field Energy Eigenvalues and Eigenstates Rigid symmetric top molecules have well known and understood zero-field rotational energy levels derived in numerous sources through semiclassica l and quantum mechanical methods [17\u201319]. The semiclassical derivation begins by considering a rigid molecule whose rotational energy can be found by calculating the motion about three principle axes (A,B,C) each with its own moment of inertia (IA,IB,IC). In particular, C is defined as the molecular axis and has the largest moment of inertia. Erot = 1 2 IA\u03c9 2 A + 1 2 IB\u03c9 2 B + 1 2 IC\u03c9 2 C (2.1) Rewriting this equation in term of angular momentum Ji = Ii\u03c9i and using the relation J2 = J2A + J 2 B + J 2 C we find an equation relating the rotational energy to the angular momentum operators. Erot = 1 2 ( J2 IB \u2212 J 2 C IB + J2C IC ) (2.2) 4 2.1. Symmetric Top Molecules A (cm\u22121) B (cm\u22121) CH3CN 5.280 0.307 Table 2.1: Constants Data for CH3CN [20] Applying quantum mechanics, we know that J2 and J2C are quantized in units of h\u03042J (J + 1) and h\u03042K2. Introducing the constants C = h\u0304\/4picIC and B = h\u0304\/4picIB we obtain our final, most general equation for the zero- field eigenvalues for a rigid symmetric top molecule in Equation 2.3 [17]. Values for the relevant constants are given in Table 2.1 for the symmetric top molecule CH3CN. The quantum number K is the projection of the orbital angular momentum on the molecular axis. Erot = BJ (J + 1) + (C \u2212B)K2 (2.3) At this point, it should be noted that there are two kinds of symmetric top molecules, prolate and oblate. The difference between these two kinds of molecules relates to their moments of inertia. If IA = IB < IC then the molecule is referred to as oblate and examples include benzene, cyclobutane or boron trifluoride and Equation 2.3 is correct. Otherwise, if IA < IB = IC then the molecule is prolate and in Equation 2.3 C is replaced by A. Examples of these prolate molecules would include ammonia, methyl bromide and acetonitrile [17]. I B I A I C I B I A I C (a) (b) Figure 2.1: These are the principle axes of rotation and simple diagram of (a) oblate symmetric top and (b) prolate symmetric top molecules [17] 5 2.1. Symmetric Top Molecules The more rigorous quantum mechanical derivation of the zero field eigen- values is not particularly illuminating and results in the same energy eigen- values as the semi-classical derivation. However, knowledge of the eigen- states for a symmetric top molecule is necessary for later calculations involv- ing the Stark effect so the rotational eigenstates are derived below starting from the most general solution for the Hamiltonian [18]. To begin, the an- gles of rotation \u03c6, \u03b8, and \u03c7 are defined as being about the space-fixed Z axis and molecule-fixed y and z axes respectively as shown in Figure 2.2. X Y Z x y z \u00f8 \u03b8 \u03c7 Figure 2.2: This diagram shows the Euler angles as defined for the following calculations. Then we begin by expanding the Hamiltonian in this basis as below. H\u03c8 = EJMJK\u03c8 (2.4) 0 = EJMJK\u03c8 + 1 sin\u03b8 \u2202 \u2202\u03b8 ( sin\u03b8 \u2202\u03c8 \u2202\u03b8 ) + 1 sin2\u03b8 \u22022\u03c8 \u22022\u03c6 + ( cos2\u03b8 sin2\u03b8 + IB IA ) \u22022\u03c8 \u22022\u03c7 \u2212 2cos\u03b8 sin2\u03b8 \u22022\u03c8 \u2202\u03c7\u2202\u03c6 (2.5) 6 2.1. Symmetric Top Molecules Supposing that there is a solution of the form \u03c8 = \u0398(\u03b8)\u03a6 (\u03c6)X (\u03c7), then it may be easily solved that for rotations about the space fixed z axis and molecule fixed z axis the eigenfunctions are simple exponentials where MJ and K are integers. \u03a6 (\u03c6) = e\u2212iMJ\u03c6 X (\u03c7) = e\u2212iK\u03c7 This leaves a complicated relation for the solution for \u0398 (\u03b8). 0 = 1 sin\u03b8 \u2202 \u2202\u03b8 ( sin\u03b8 \u2202\u0398 \u2202\u03b8 ) \u2212 M 2 sin2\u03b8 \u0398\u2212 ( cos2\u03b8 sin2\u03b8 + IB IA ) K2\u0398 + 2cos\u03b8 sin2\u03b8 KM\u0398+ EJMJK\u0398 (2.6) But it may be solved by defining the following relationships. x = 1 2 (1\u2212 cos\u03b8) \u03b1 = | K \u2212M | +1 \u03b2 = | K +M | + | K \u2212M | +2 \u03b3 = EJMJK \u2212 IBK 2 IA +K2 \u2212 ( 1 2 | K +M | +1 2 | K \u2212M | ) \u00d7 ( 1 2 | K +M | +1 2 | K \u2212M | +1 ) \u0398(\u03b8) = x|K+M |\/2 (1\u2212 x)|K\u2212M |\/2 F (x) (2.7) The function F(x) is then solved from the following differential equation to be the summation with constants an as defined. x (1\u2212 x) \u2202 2F \u2202x2 + (\u03b1\u2212 \u03b2x) \u2202F \u2202x + \u03b3F = 0 (2.8) F (x) = \u221e\u2211 n=0 anx n (2.9) an+1 = n (n\u2212 1) + \u03b2n\u2212 \u03b3 (n+ 1) (n+ \u03b1) an (2.10) 7 2.1. Symmetric Top Molecules The result can be simplified by defining the rotational eigenfunctions of symmetric top molecules as DJMK where \u0398 (\u03b8) = d J MK (\u03b8) It is useful to know that the \u2019d\u2019 function cited reduces to the Legendre polynomials in the case where M=0,K=0 (dJ00 (\u03b8) = PJ (\u03b8)) [21] as should be expected since K=0 would reduce the case to that of a simple linear molecule. \u03c8rot (\u03b8, \u03c6, \u03c7) = DJMK (\u03b8, \u03c6, \u03c7) = e\u2212iM\u03c6e\u2212iK\u03c7dJMK (\u03b8) (2.11) 8 2.1. Symmetric Top Molecules 2.1.2 Stark Effect: Approximate and Exact Values Applying an electric field to a molecule peturbs the energy of each state by introducing a \u2212~\u00b5\u00b7E term into the Hamiltonian of the system as a Stark effect (HStark). To the first order approximation we may calculate the Stark effect by solving for the changes in only the diagonal elements of the Hamiltonian matrix. By applying Equation 2.11 and using a relation involving the DJMK function we may then derive the following summation [21]. H \u2032Stark = < J K MJ | \u2212~\u00b5 \u00b7E | J K MJ > = \u2212\u00b5E \u222b 2pi 0 d\u03c6 \u222b 2pi 0 d\u03c7 \u222b pi 0 d\u03b8sin\u03b8cos\u03b8DJ\u2217MKD J MK = \u2211 J \u2032 (\u22121)M\u2212K < J \u2212K,J K | J \u2032 0 >< J \u2212M,J M | J \u2032 0 > \u00d7 \u222b 2pi 0 d\u03c6e\u2212i(M\u2212M)\u03c6 \u222b 2pi 0 d\u03c7e\u2212i(K\u2212K)\u03c7 \u222b pi 0 sin\u03b8cos\u03b8d\u03b8dJ \u2032 M\u2212M K\u2212K = \u2211 J \u2032 (\u22121)M\u2212K < J \u2212K,J K | J \u2032 0 >< J \u2212M,J M | J \u2032 0 > \u00d74pi2 \u222b 1 0 xdxPJ \u2032 (x) (2.12) At this point the summation is greatly simplified by the fact that the final integral over the Legendre polynomial is zero for all but one case. Then, we may easily convert from the current notation to Wigner 3j notation and solve the Wigner 3j symbols. = 8pi2 (\u22121)M\u2212K ( J J 1 \u2212M M 0 )( J J 1 \u2212K K 0 ) = 8pi2 MK J (J + 1) (2J + 1) (2.13) When we normalize the results, using Equation 2.14, we obtain our final value for the first order approximation of the Stark effect (H\u2019Stark) from the diagonal elements of the Stark shifted Hamiltonian matrix. < JMJK | JMJK >= 8pi2\/2J + 1 (2.14) H \u2032Stark = < J MJ K | HStark | J MJ K > = \u00b5E MJK J (J + 1) (2.15) 9 2.1. Symmetric Top Molecules While useful, the first order approximation is insufficient on its own since it incorrectly predicts that there is no Stark effect on states having values of K=0 orMJ = 0. The second order approximation can be obtained in a similar manner as the first order approximation and the results are summarized below. H \u2032\u2032Stark = \u2211 All States < J MJ K | HStark | J \u2032 M \u2032J K \u2032 > EJ \u2032M \u2032JK\u2032 \u2212 EJMJK = \u00b52E2 2hB ( J2 \u2212M2J ) ( J2 \u2212K2) J3 (2J \u2212 1) (2J + 1) (2.16) \u2212\u00b5 2E2 2hB ( (J + 1)2 \u2212K2 ) ( (J + 1)2 \u2212M2J ) (J + 1)3 (2J + 1) (2J + 3) While the second order approximation is an improvement over the first order it is possible to calculate the Stark Effect nearly exactly by first cal- culating and then diagonalizing and solving the Hamiltonian matrix for the system. The degree of accuracy of this method depends on the size of the matrix used; a smaller matrix, while easier to compute is less accurate. To calculate the matrix requires the knowledge of Stark effect on every matrix element. In this we are aided by two useful principles of symmetric top molecules. Firstly, the Stark effect is diagonal with respect to both MJ and K, there are no off-diagonal elements with respect to these quantum numbers. Secondly, the only off-diagonal elements with respect to J are J \u00b1 1, creating a tridiagonal Hamiltonian matrix which is easily solved. The diagonal and off diagonal elements are summarized below[22]. < JM \u2032JK \u2032 | HStark | JMJK > = \u00b5E MJK J (J + 1) \u03b4KK\u2032\u03b4MJM \u2032J < J \u2212 1MJK | HStark | JMJK > = \u00b5E \u221a (J2 \u2212M2) (J2 \u2212K2) J2 (2J + 1) (2J \u2212 1) < J + 1MJK | HStark | JMJK > = \u00b5E \u221a\u221a\u221a\u221a((J + 1)2 \u2212M2) (J)2 (2J + 1) \u00d7 \u221a\u221a\u221a\u221a((J + 1)2 \u2212K2) (2J + 3) 10 2.1. Symmetric Top Molecules -6 -4 -2 0 2 4 6 8 0 2 4 6 8 10 \u03bcE (cm-1) E n e rg y ( c m -1 ) J=1 K=1 M=-1 J=1 K=0 M=0 J=1 K=0 M=-1 J=1 K=1 M=1 J=1 K=1 M=0 Figure 2.3: Stark effect of J=1 states of CH3CN Using these values and a simple computer program that generates and di- agonalizes an appropriately sized Hamiltonian matrix for CH3CN the Stark effect for individual states was solved and plotted with respect to the applied electric field, some of these results are shown in Figure 2.3. The computer program itself was written in the C programming language and is provided in Appendix A. It was created mostly by the author although some parts were taken from or inspired by other sources, specifically Numerical Recipes in C and C How to Program, and modified to meet the needs of the program. 11 2.2. Diatomic Molecules 2.2 Diatomic Molecules 2.2.1 Zero-Field Energy Eigenvalues and Eigenstates Unlike in symmetric top molecules, the calculation for the zero field energy levels in diatomic molecules is rather complex but may be calculated through the solution of the Hamiltonian, given in its most general form below in terms of N, the rotational quantum number, S the spin quantum number and I the nuclear spin quantum number [23] with constants for 40Ca19F and 138Ba19F are listed in Table 2.2. H = \u03b2N2 + \u03b3S \u00b7N+ bI \u00b7 S+ cIzSz + CII \u00b7N (2.17) In general, the hyperfine splitting constants (b,c), the spin-rotational constant (\u03b3), and the nuclear spin-rotation constant (CI) are all extremely small compared to the rotational constant \u03b2 and can be ignored to a good approximation in calculating the eigenvalues of diatomic molecules. While it is possible to calculate the eigenvalues given a large matrix using N, MN , S, MS , I, MI quantum numbers (the eigenstate |N,MN ,S,MS ,I,MI >) it is more convenient to convert to the quantum numbers N, J, F, MF and the eigenstates | N J F MF > before calculating the eigenvalues of the Hamiltonian. Since S = 1\/2 and I = 1\/2, conversion between the two sets of eigenstates is possible through Equations 2.18 and the Clebsch-Gordon coefficients. J = N+ S (2.18) J = N + S ... | N\u2212 S | F = J+ I F = J + I ... | J\u2212 I | (2.19) 12 2.2. Diatomic Molecules \u03b200 \u03b300 b00 c00 40Ca19F X2\u03a3[23] 0.3437044 0.0013228 0.0036419 0.0013381 138Ba19F X2\u03a3[24] 0.2158493 0.0027013 Table 2.2: Constants data for CaF and BaF, all units are in cm\u22121. The spin-rotation constant (CI) is omitted since it is extremely small. Another easy way to calculate each state is to begin with the MF = F state and then to use the lowering operator to drop to every other state. For example, we can start from Equation 2.20 and use F\u2212 = N\u2212 + S\u2212 + I\u2212 to obtain Equation 2.21. This is the method used by the computer program in Appendix B. | N = 0 J = 0.5 F = 1 MF = 1 >= (2.20) | N = 0 MN = 0 S = 0.5 MS = 0.5 I = 0.5 MI = 0.5 > | N = 0, J = 0.5, F = 1,MF = 0 >= (2.21) 1\u221a 2 | N = 0 MN = 0 S = 0.5 MS = 0.5 I = 0.5 MI = \u22120.5 > + 1\u221a 2 | N = 0 MN = 0 S = 0.5 MS = \u22120.5 I = 0.5 MI = 0.5 > 2.2.2 Stark Effect: Approximate and Exact Values As stated in the section on symmetric top molecules, applying an electric field has the effect of adding aHStark = \u2212\u00b5\u00b7E term to the total Hamiltonian. The equations governing the Stark effect in a symmetric top molecule may be used as an approximation to diatomic molecules if we ignore the hyperfine splitting constants (b,c), the spin-rotational constant (\u03b3), and the nuclear spin-rotation constant (CI) and set K = 0. These approximations reduce the eigenvectors from | N J F MF > to | N MN > as the spin (S) and nuclear spin (I) are no longer important. In which case, we know that to the first approximation the Stark effect is zero (Equation 2.22), but to the second order it is not (Equation 2.23)[17]. 13 2.2. Diatomic Molecules H \u2032Stark = \u2212 < N MN | HStark | N MN > = 0 (2.22) H \u2032\u2032Stark = \u00b52E2 2hcBN N (N + 1)\u2212 3M2N (N + 1) (2N \u2212 1) (2N + 3) (2.23) As an approximation, this is fairly good however it is important to be able to calculate the Stark shifted energy eigenvalues more exactly. In order to accomplish this, it is necessary to calculate all matrix elements affected by the Stark effect and to then diagonalize the matrix to find the eigenvalues. This may be accomplished rather easily with a computer program once a general equation is known. Fortunately, a general formula is available and is presented here in terms of Wigner 3j ( () ) and 6j symbols ( ) [25]. < N J F MF | HStark | N \u2032 J \u2032 F \u2032 M \u2032F > (2.24) = (\u22121)I+S+Nmax+J+J \u2032+F+F \u2032\u2212MF \u00d7 \u221a (2F + 1) (2F \u2032 + 1) (2J + 1) (2J \u2032 + 1)Nmax \u00d7 ( F 1 F \u2032 M \u2032F 0 M \u2032 F ){ J F I \u2032 J \u2032 F \u2032 1 }{ N J S J \u2032 N \u2032 1 } A computer program, given in Appendix B, was used to calculate the zero-field Hamiltonian matrix along with the Stark effect elements and then diagonalizes the matrix in order to solve for the eigenvalues given a partic- ular applied electric field. Figure 2.4 was calculated using this program and shows the Stark effect for 40Ca19F and how individual states evolve as the applied electric field is increased from zero. It is expected that because the nuclear spin - rotation constant (CI) was ignored in the calculations that applying Equation 2.24 will cause there to be a degeneracy of 2 for every state. Moreover, because of the small values for other constants it would be expected that many states would share a similar Stark effect. These expectations were fulfilled by the simulation presented. Unfortunately, one limitation of this program is that it is unable to identify exactly the quantum numbers for every observed state. The N quantum numbers are easily esti- mated but the others cannot be reliably identified because they are nearly degenerate for many cases. 14 2.2. Diatomic Molecules -1 -0.5 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Applied Electric Field (\u03bcE) (cm-1) E n e r g y ( c m -1 ) N=1 N=2 N=0 Figure 2.4: Stark effect of the lowest states of 40Ca19F 15 2.3. Electrostatic Forces 2.3 Electrostatic Forces 2.3.1 Guide Geometry Being able to calculate the Stark shifted eigenvalues is only the first step in understanding the dynamics of molecules traversing the electrostatic guide. The eigenvalues depend strongly on the strength of the applied electric field, as seen in Figures 2.3 and 2.4 for both symmetric top and diatomic molecules. As a molecule travels inside the guide the electric field acting on it will change, causing the eigenvalues of the different states to also change. These changes create a force on the molecule, influencing its movement in- side the guide. It is therefore necessary to know the geometry of the static electric field generated by the guide in order to proceed. Since a hexapole arrangement is used in the experiments described in this thesis, that is what will be discussed although quadrupole guides have also been successfully used in velocity filtering [11]. While more complicated and accurate expressions exist for the electric field [26] for the purposes of this discussion it is sufficient to use the approx- imation given in Equation 2.25 where the value for R0 is the radius of the hexapole and is shown graphically in Figure 2.5 [27]. The hexapole arrange- ment is such that, the electric field increases from zero-field in the center (r=0) to a maximum value equal to 3V0 \/ R0 at the edge of the hexapole (r=R0). E = 3V0r2 R30 (2.25) + + + _ _ _ R 0 Figure 2.5: A simple diagram showing the cross section of the hexapole arrangement and the electric field [27] 16 2.3. Electrostatic Forces 2.3.2 Transverse and Longitudinal Velocity Filtering A molecule that begins in the center of the hexapole and moves radially out- ward experiences a force, FE , given in Equation 2.26 due to changes in the Stark shifted energies (HStark). States whose internal energy increases mov- ing from low to high field regions are known as low field seeking states, while states whose energy decreases are called high field seeking states. Since the total energy must be conserved, while the internal energy increases, the ki- netic energy decreases and if Stark shift is greater than the transverse kinetic energy of the molecule, it will be unable to leave the hexapole. This pro- cess is known as transverse velocity filtering, where the transverse velocity is defined as the velocity perpendicular to the local axis of the hexapole. FE = \u2202HStark \u2202r (2.26) A straight hexapole would funnel low field seeking molecules with low transverse velocities along the path of the hexapole, thus differentiating be- tween low field seeking molecules with high and low transverse velocities. High field seeking molecules are drawn away from the center of the hexapole and lost. However, a straight hexapole has no mechanism for differentiating between low field seeking states with high or low longitudinal velocities, that is, velocities along the path of the hexapole. Therefore, the hexapole is bent with a specific radii (\u03c1), which acts to filter molecules with low longitudinal velocities. A molecule travelling through the hexapole experiences an op- posing set of forces at the bend in the hexapole where the centripetal force on the molecule must be less than the force in Equation 2.26 or the molecule will be unable to follow the path of the hexapole and escapes the hexapole and is lost [26]. mv2 \u03c1 \u2264 \u2202HStark \u2202r (2.27) It is then necessary to find the relation between the Stark effect and the position, r, inside the hexapole. Using the approximations for the Stark effects in Equations 2.15 and 2.23 for symmetric top and diatomic molecules, respectively, and Equation 2.25 for the electric field inside the hexapole we may solve Equation 2.27 for the symmetric top molecules (Equation 2.28) and diatomic molecules (Equation 2.29). 17 2.3. Electrostatic Forces Transverse Velocity Longitudinal Velocity Hexapole Figure 2.6: A simple diagram showing definition of and difference between longitudinal and transverse velocities. \u2202HStark \u2202r = \u2202E \u2202r \u00b5MJK J (J + 1) = 6V0r R30 \u00b5MJK J (J + 1) (2.28) \u2202HStark \u2202r = 18V 20 r 3 R60 \u00b52 hcBN N (N + 1)\u2212 3M2N (N + 1) (2N \u2212 1) (2N + 3) (2.29) At this point it is easy to see that to maximize either relation with respect to the radial position we simply must set r to be its maximum value, R0. With this knowledge, the final equations, Equations 2.30 and 2.31 for symmetric top and diatomic molecules respectively, for the approximate maximum trappable velocities are obtained. vmax,symm = \u221a \u03c1 m 6V0 R20 \u00b5MJK J (J + 1) (2.30) vmax,dia = \u221a \u03c1 m 18V 20 R30 \u00b52 hcBN N (N + 1)\u2212 3M2N (N + 1) (2N \u2212 1) (2N + 3) (2.31) 18 2.3. Electrostatic Forces Knowing from Figures 2.3 and 2.4 the exact Stark shifted eigenvalue relative to the applied electric field, and the approximation of the electric field with respect to the position inside the hexapole, the partial differential in Equation 2.27 may be calculated. The three steps are shown in Figure 2.7 for the lowest low field seeking states for 40Ca19F. These states correspond to approximately the N=1 MN=0 state. It is through the final graph (c) in this figure that we may observe the limitations in our previous approximations. Equation 2.29 predicts that for a diatomic molecule the Stark effect increases with the radial position, and hence the maximum trappable velocity would occur at the maximum value for the radial position. With the more accurate calculation based on the exact solution of the Hamiltonian matrix it can observed that this is not true for all states. It must be accepted that the approximations used are only good for small applied electric fields, and that for larger applied electric fields there are more correction terms that must be considered. The breakdown of this approximation may be seen in Figure 2.7, while the prediction shows that the for diatomic molecules the Stark Effect should increase proportional to the square of \u00b5E, in reality this is only true for fields where \u00b5E \u00a1 0.1 cm\u22121. Some states may even change from low field seeking to high field seeking if the applied electric field is strong enough as may be seen in Figure 2.7. 19 2.3. Electrostatic Forces (a) (b) (c) 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 Applied Electric Field E (cm-1) E ne rg y (c m -1 ) 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 Radial Position (mm) E ne rg y (c m -1 ) 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 Radial Position (mm) M ax V el oc ity (m \/s ) Figure 2.7: (a) The Stark effect on the lowest energetic (N=1,MN=0) low field seeking states with respect to the applied field, (b) is the Stark effect on the same states with respect to the radial position inside the hexapole instead, using the dimensions of the hexapole and an applied high voltage of 4kV. From the slope of this graph we may obtain (c) the maximum trappable velocity with respect to the radial position inside the hexapole arrangement. 20 2.4. Probability Calculations 2.4 Probability Calculations Through the previous sections, most of the details needed in order to solve for the probability that a molecule entering the electrostatic guide exits the electrostatic guide at its end have been discussed. The energy eigenvalues of low field seeking states for both symmetric top and diatomic molecules are known and the maximum trappable velocity for individual states may be calculated. It is known that the probability that a molecule entering the hexapole aparatus leaves it at the end is given by the summation in Equation 2.32 where Q is the partition function [26] and with the knowledge discussed previously and knowledge of the intial velocity probability distribution of the molecular beam this equation may be solved. Pnet = \u2211 lfsstates e\u2212\u03b2Elfs Q \u222b vmax 0 P (v)dv (2.32) Q = \u2211 allstates e\u2212\u03b2Es (2.33) \u03b2 = 1 kBT The initial velocity distribution depends on the method in which the target molecules are obtained. For a typical pulsed source there can be two possible distributions depending on the pressure of the gas behind the valve, known as the backing pressure. The two possibilities are effusive and collimated distributions. For collimated sources, the backing pressure is high, creating a velocity distribution with a higher percent of molecules having high longitudinal velocities. For effusive sources, the opposite is true, the backing pressure is low and the velocity distribution is weighted more strongly towards longitudinally slower molecules and has a larger percent of high transverse velocities. Equations 2.34 and 2.35 are the equations repre- senting the velocity probability distribution for collimated [26] and effusive [28] sources respectively and Figure 2.8 graphically represents them. P (v)dv = m2 2 (kBT ) 2 v 3e \u2212 mv2 2kBT dv (2.34) P (v)dv = mv kBT e \u2212mv2 kBT dv (2.35) 21 2.4. Probability Calculations Using the approximation given in Equation 2.30, the values for the Stark effect calculated by the program mentioned previously and whose results were shown in Figure 2.3 and the knowledge of the velocity distribution of the intial molecular beam as Equation 2.35 at an initial temperature of 300 K, we may calculate the probability that a molecule entering the hexapole leaves it at the end as shown in Table 2.3. There are two sources of degeneracies that need to be considered in the probability calculations for the symmetric top states. Firstly, the energy eigenvalues and maximum trappable velocity for the states J=j, MJ=mj, K=-k and J=j, MJ=-mj, K=k are the same. Secondly, there is a degeneracy equal to 2 for states where K is not divisible by 3. This method of calculating the probability of traversing the hexapole, while applicable to the the case where we are using CH3CN as a target molecule, is not as useful in calculating the probability for 40Ca19F molecules because in that case the target molecules are obtained through ablation rather than from a pulsed nozzle source. J K MJ D x e\u2212\u03b2EStark\/Q \u222b Ek,max 0 P (E) dE Probability\/State 1 1 -1 3.0843 x 10\u22124 3.4207 x 10\u22125 1.0551 x 10\u22128 2 1 -1 3.0684 x 10\u22124 2.2224 x 10\u22126 6.8192 x 10\u221210 3 1 -1 3.0432 x 10\u22124 3.9411 x 10\u22127 1.1994 x 10\u221210 4 1 -1 3.0079 x 10\u22124 1.1004 x 10\u22127 3.3098 x 10\u221211 5 1 -1 2.9641 x 10\u22124 3.9957 x 10\u22128 1.1844 x 10\u221211 ... 56 1 -1 2.8185 x 10\u22126 3.4268 x 10\u221213 9.6585 x 10\u221219 2 2 -1 2.8551 x 10\u22124 1.2492 x 10\u22125 3.5666 x 10\u22129 3 2 -1 2.8329 x 10\u22124 2.2224 x 10\u22126 6.2956 x 10\u221210 ... 22 13 -1 2.6753 x 10\u22126 2.0845 x 10\u22128 5.5768 x 10\u221213 ... Pnet = 1.3658 x 10\u22126 Table 2.3: The relevant quantities needed for the calculation of the total probability (Pnet) are given for the symmetric top molecule CH3CN where all values are as defined in Equation 2.32 and D is given as the degeneracy for each state. 22 2.4. Probability Calculations (a) 0.003 300 0.002 0.001 200 0 1000 Speed (m\/s) 0.004 500400 P(v) (b) 3002001000 0.007 0.006 0.005 0.002 0.001 Speed (m\/s) 400 P(v) 0.004 0.003 0 500 (c) 300200 0 1000 P(v) 0.002 Speed (m\/s) 0.01 0.008 0.004 400 0.006 500 Figure 2.8: The longitudinal velocity probability distributions for effusive (blue) and collimated (red) sources at (a) 300K, (b) 100K and (c) 40K. 23 Chapter 3 Buffer Gas Cooling From the discussions in the previous chapter it is evident that the intial ve- locity distribution of the target molecules is extremely important in attaining a high probability that initial molecules will be guided by the electrostatic guide. For this purpose, buffer gas cooling is an obvious and easily imple- mented method to reduce the translational velocity of the target molecules. 3.1 Elastic Collisions and Cooling Buffer gas cooling, in contrast to velocity filtering, is an \u2019active\u2019 cooling mechanism since it removes energy from target molecules. It relies on the thermalization of the target molecules via elastic collisions with the cold buffer gas molecules, often He because of its low freezing point. It is useful in many situations because of its generality, it can work with any molecule or atom regardless of energy structures and can cool not only translational but also rotational degrees of freedom. M m p 1 , E 1 p 2 , E 2 m M p 2 \u2019 , E 2 \u2019 p 1 \u2019 , E 1 \u2019 p 1 +p 2 =p 1 \u2019+p 2 \u2019 E 1 +E 2 =E 1 \u2019+E 2 \u2019 |J K M J > |J K M J > Figure 3.1: This simple diagram shows the principles and equations used to illustrate how translational cooling by collisions occurs. 24 3.2. Inelastic Collisions and Cooling Supposing that the molecules behave as hard spheres and momentum and energy are conserved during collisions we may derive an approximate model for the cooling mechanism after one, and after n collisions. The temperature drop after a single collision between a target molecule of initial temperature T\u2019 and mass M and a buffer gas molecule having temperature T and mass m is given below where \u03ba is also defined [29]. \u2206T = T \u2032 \u2212 T \u03ba (3.1) \u03ba = (M +m)2 2Mm (3.2) Generalizing this equation into differential form and solving we may obtain a general equation for the final temperature after n collisions [29]. Tn = ( T \u2032 \u2212 T ) e\u2212n\/\u03ba + T (3.3) Together with the knowledge of the mean free path of the target molecules through the buffer gas, it can be estimated the number of collisions and hence, the final temperature of one of the target molecules. In reality the situation is more complex since the experiment discussed in this thesis con- sists of introducing the target molecules in a perpendicular direction to the flow of a cold pulsed beam of Helium gas, but this approximation aids in our understanding of the process. 3.2 Inelastic Collisions and Cooling While elastic collisions may cause translational cooling, inelastic collisions will change the energy of internal states and cause relaxation to lower energy states. In general, inelastic collisions can take on many forms and most of the time in experiments dealing with cold molecules a focus is placed on reducing inelastic collisions, however inelastic collisions also produce the ro- tational cooling effect that is desireable for most cold molecule experiments. Methods that cool molecules translationally do not necessarily cause rota- tional cooling which can reduce the efficiency of methods, as is the case with velocity filtering as described in the previous chapter. 25 3.2. Inelastic Collisions and Cooling M m p 1 , E 1 p 2 , E 2 m M p 2 \u2019 , E 2 \u2019 p 1 \u2019 , E 1 \u2019 p 1 +p 2 =p 1 \u2019+p 2 \u2019 E 1 +E 2 \u2260E 1 \u2019+E 2 \u2019 |J K M J > |J\u2019 K M J > Figure 3.2: This simple diagram, together with Figure 3.1 shows the differ- ences between elastic and inelastic collisions of molecules. A semiclassical explanation of rotational relaxation experienced by sym- metric top molecules has proven satisfactory in the case of H2CO [30] The semiclassical explanation assumes a rigid molecule with atoms positioned at fixed angles and distances from other atoms inside the molecule spinning like a top. If the rotational energy is less than the kinetic energy of the He atom then the collision excites the molecule to a higher rotational state from which it may relax back to a lower state. Otherwise, if the rotational energy is less than the kinetic energy of the colliding buffer gas atom then rotational cooling occurs. From this semi-classical understanding it is cor- rectly understood that the onset of rotational cooling is strongly dependent on the density and temperature of the buffer gas as has been experimentally shown [12]. Besides rotational relaxation, other forms of relaxations caused by in- elastic collisions between buffer gas atoms and target molecules includes electronic and vibrational relaxation. For 40Ca19F molecules in particular, it is known that electronic relaxation rates are practically zero and vibra- tional relaxation rates are small relative to the rotational relaxation rate [15]. The rotational relaxation rates in buffer gases are, for many molecules including 40Ca19F, too fast to be measured accurately and only a lower limit is available [15, 31]. This knowledge however, means that once the onset of rotational cooling is reached we can be relatively sure that it will be ex- tremely efficient in reducing the populations of higher rotational states to lower rotational states as has also been observed experimentally [12]. Zeeman relaxation, the relaxation from higher spin states to lower ones is 26 3.2. Inelastic Collisions and Cooling an important consideration in experiments involving buffer gas cooling and magnetic trapping of molecules. These relaxations, as suggested before, are considered detrimental to the trapping process since the relaxations trans- fer the population from trappable to non-trappable states. However, these relaxations are not considered important in this experiment and so will not be discussed further [15]. 27 Chapter 4 Laser Ablation The process of laser ablation of materials is a commonly used one to obtain radicals and other sample molecules that are not readily available in either gas or liquid form otherwise. It also has other applications such as in thin film deposition techniques. The focus of this chapter is to introduce the principles of laser ablation and how they apply to the current experiments. 4.1 Principles of Laser Ablation The difference between laser ablation and desorption is mostly a question of scale. Desorption involves the removal of a sample without mesoscopic dam- age to the solid, whereas ablation involves large scale removal of material in which the surface of the solid is structurally modified at a mesoscopic scale. For example, examining the principles of laser desorption would require a detailed knowledge of the bond strengths of a single atom or molecule in a lattice structure that would remain relatively unaltered by the process, but laser ablation would provide suffient energy to melt and vapourize large quantities of atoms or molecules from the solid, destroying any lattice struc- ture in the process. The exact method of removal differs for the target substance and is a broad field of continuing research but in general follows the following steps [32]. 1. Energy from the photons of the ablation laser is absorbed by the solid either through the excitation of free electrons in the solid or by exciting electronic or vibrational transitions. For example, the energy may induce quantized lattice vibrations, known as phonons, in the target solid. For target solids that have more than two different ions in a lattice cell, when the ions move out of phase with one another the vibration is known as an optical phonon and these optical phonos be directly excited by an infrared laser. 28 4.2. Ablation Plume 2. Absorbed energy is released through a number of methods by the solid. However, should a single site of the solid retain sufficient energy for a long enough period of time then the bonds which hold it in its position in the solid may break and allow the atom or molecule to move away freely. For laser desorption these processes would only occur at surface sites, however with ablation entire layers of the surface may be removed in a single pulse and an understanding of the bulk properties of the material is required such as the rate of thermal diffusion of the material [32]. 4.2 Ablation Plume One of the main limitations of laser ablation is the variability in the pro- duction of the desired target molecules [33]. Laser ablation creates a low to medium density plume of atoms, ions, molecular fragments and radicals above the point of ablation on the target. The exact numbers of produced molecules varies from laser pulse to pulse and it is not known how many molecules are directly ablated off the target and how many are the result of recombination of atoms above the target [15]. Another limitation is the disk itself, which experiences damage from the ablation process. After a certain amount of time, the laser may ablate a hole in the solid, altering both the quantity of molecules ablated and the plume distribution. Moreover, separating out the molecules of interest requires mixing the ablation plume with a buffer gas of sufficiently high density to prevent re- combinations of atoms and radicals into molecules that are not of interest. For radicals such as 40Ca19F it has been estimated that for each laser pulse up to 1011-1013 molecules may be created per pulse depending on the laser and the strength of the pulse [15]. 29 4.2. Ablation Plume Ablation Damage Figure 4.1: This photo, taken with a standard digital camera, is of the damage done to a CaF2 disk that was used as a source for 40Ca19F radicals. The ablation laser used was a Nd:YAG laser operating at 532nm with a 10ns pulse width. There are two obvious points of damage where laser ablation occured, the top one is a hole straight through the disk. Also obvious is the fact that the disk was split in half, which was also a result of ablation. Ablation Plume CaF 2 Figure 4.2: This photo, taken with a standard digital camera, is of an abla- tion plume formed by ablation off a CaF2 plate with an ArF laser operating at 193nm and having a 2 ns pulse width. The plume is clearly visible. 30 Chapter 5 Experimental Details 5.1 Vacuum System Attaining a high vacuum is important both for reducing the number of collisions between target molecules traversing the guide and background molecules in the chamber and for preventing electrostatic discharges be- tween neighbouring poles in the hexapole. Additionally, it is important to maintain a pressure of less than 1 x 10\u22128 in the region of the mass spectrom- eter. Acheiving high vacuum in the system is acheived by four turbomolec- ular pumps; a TG800 series Osaka Vacuum compound molecular pump, a Turbo-V 550 Varian turbomolecular pump, and two TMU 260 series Pfeiffer compact turbopumps backed by an Edwards 40 two stage rotary pump. The pressure is monitored by four identical Balzer full range compact gauges po- sitioned as shown in Figure 5.1. Typically, the turbomolecular pumps attain a pressure in the range of 10\u22126 in the source and bend chambers, and 10\u22127- 10\u22128 in the QMS chamber. To reduce the pressure of the QMS chamber sufficiently, baking out is performed for between three and four days. Source Bend QMS Valve Closed 4.2 x 10\u22126 3.3 x 10\u22127 9.7 x 10\u22129 Valve Open 6.1 x 10\u22126 3.4 x 10\u22127 9.7 x 10\u22129 Table 5.1: Sample chamber pressures for when the valve was open and closed for a backing pressure of 750 mTorr and a pulse width of 50ms for CH3CN sample molecule. 31 5.1. Vacuum System Figure 5.1: This diagram shows the vacuum chamber and the path of the hexapole apparatus as well as the position of the vacuum pumps and guages. Vacuum gauges are represented in red. The hexapoles and chambers are not to scale. 32 5.2. Hexapole Details 5.2 Hexapole Details Two separate hexapoles were used in the operation of this experiment. The first hexapole, referred to as the \u2019large hexapole\u2019, is 0.225 m in length and used to channel the molecules from the gas source to the second hexapole. The second hexapole, referred to as the \u2019main hexapole\u2019, is used in velocity filtering as shown in Figure 5.1 and extends from the source chamber to the QMS chamber and is 1 m in total length. The important properties of the two hexapoles are summarized in Table 5.2. As discussed previously, the dimensions of the hexapole are important in calculating the velocity filtering performed by the hexapole. The voltages of the hexapoles are controlled by three separate devices, one device constructed by the UBC Chemistry Electronics Shop for the large hexapoles capable of applying voltages of up to 10kV, and two Stanford Re- search Systems, Inc. Model PS350 high voltage power supplies each capable of applying a voltage of 5kV. Location Max Voltage Length Radius Bend Radius Source Chamber 10 kV 0.225 m 12 mm - Bend Chamber 5kV 1 m 6 mm 75 mm Table 5.2: Summary of the properties of the two hexapoles used. Bend Radius Radius Figure 5.2: This figure graphically shows the definitions for the \u2019radius\u2019 and \u2019bend radius\u2019 given in Table 5.2. 33 5.3. CH3CN Gas Source 5.3 CH3CN Gas Source There were four different sets of experiments, however only two different internal arrangments were required. The first three sets of experiments were performed with CH3CN as the target molecule and used the arrangement shown in Figure 5.4. These first two were performed in preparation for the third experiment, which used the target molecule 40Ca19F ablated off a CaF2 target as shown in Figure 5.7. The second experiment was meant to simulate the process of ablation as much as possible in order to optimize the parameters of the experiment. While the Helium gas may be obtained in a straightforward manner through a compressed gas cylinder and controlled by a series of needle nose valves before the pulsed, CH3CN is not readily prepared in such a fashion. In order to obtain it then, a sample of liquid acetonitrile is prepared in a container which is attached to the system externally so that it may be easily refilled. The liquid acetonitrile vapourizes, providing a gas sample that may be used. The pressure from CH3CN behind its gas valve is also controlled by two needle nose valves placed in series with one another and the pulsed nozzle. For the first and second sets of experiments, the acetonitrile is attached directly to the main pulsed nozzle, while in the third, it is attached to the \u2019ablation\u2019 pulsed gas line. Figure 5.3: This apparatus was used to introduce CH3CN into the system by extracting the vapour formed above the liquid acetonitrile from the glass chamber and into the experiment. 34 5.3. CH3CN Gas Source Main Pulsed Nozzle Teon Tubing Hexapoles (\u00b1 8kV) Kapton Tape Main Gas \u2018Ablation\u2019 Gas Input Figure 5.4: The source arrangment used for the first two sets of experiments where CH3CN was the target molecule. In the first set, the source molecules were fed directly through the main gas line and its pulsed nozzle. In the second set, Helium was fed through the main pulsed nozzle, while CH3CN was pulsed from the \u2019ablation\u2019 gas input line. Vapour Pressure (300 K) Boiling Point Melting Point 128 mbar 354.8 K 228 K Table 5.3: Summary of some important properties of acetonitrile [34]. 35 5.4. Pulsed Nozzles 5.4 Pulsed Nozzles The pulsed gas nozzles used in the experiment, shown in Figure 5.5, were general valves produced by Parker Hannefin. One was placed inside the experiment and could be cooled by a Helium refrigerator to a minimum of 40 K. The second one was positioned externally to the system. Both were controlled by Iota One controllers, also built by Parker Hannefin, which could vary the rate and pulse width that the valves open, from a minimum pulse width of 300\u00b5s. The Iota One which controls the external pulsed valve acts as the master for the other Iota One controller. A pulse generator placed between the two Iota One controllers inserts a delay between the activation of the two valves needed to synchronize the arrival of molecular beams at the same point. Figure 5.5: The two nozzles used in the experiment are shown, (a) the external nozzle used to pulse the CH3CN during the \u2019ablation\u2019 simulation and (b) the pulsed nozzle placed inside the experiment and is used as the main valve. 36 5.5. 40Ca19F Ablation Sources 5.5 40Ca19F Ablation Sources For the fourth experiment,40Ca19F radicals were the target molecules and were generated through laser ablation off a CaF2 positioned inside the ex- periment. The ablation plume generated expands rapidly into the vacuum, a consideration which is important since maximizing the number of molecules that enter the hexapole is paramount. This is acheived by positioning the solid target in a way in which the majority of the molecules in the plume are not lost. Deciding on the positioning of the CaF2 solid relative to the pulsed valve was a challenging task and multiple positions were attempted without success before the position in Figure 5.7 was decided on. Several at- tempts were made by positioning the ablation point closer to the hexapoles, as shown in Figure 5.6 however it was discovered that the ablation plume created during the process causes arcing between the poles. The arcing is understandable given the high density of particles in the ablation plume but disrupts the flight path of the molecules by changing the electric fields in- side the hexapole and damage the hexapoles themselves. Reducing the laser intensity reduces the ablation plume and eliminates the arcing, however it also reduces the likelihood that sufficient numbers of 40Ca19F radicals are produced. Covering the ends of the hexapole with insulating Kapton tape was found to be an insufficient solution. Nd:Yag Laser (Q-Switched) CaF2 Helium Gas Nd:Yag Laser (Q-Switched) CaF2 Helium Gas (a) (b) Figure 5.6: These two ablation setups (a) and (b) were attempted with- out success, both arrangements caused arcing between the hexapoles, even through the protective captum tape coating on the ends of the hexapoles. 37 5.5. 40Ca19F Ablation Sources Narrow Slits Focusing Lens Mirrors Laser Beam Inside Chamber Outside Chamber CaF 2 Target Glass Window He Gas Pulsed Nozzle (>40K)Large Hexapole (\u00b18kV) Figure 5.7: The source arrangment used in those trials where 40Ca19F was the target molecule. Ablation of the CaF2 disk by a Nd:YAG laser vapour- ized target molecules which are then cooled by collisions with the cold pulsed He gas beam, which acted as a buffer gas. The strength of the ablation is controlled by adjustable slits placed outside the vacuum chamber. 38 5.6. Observation Setup 5.6 Observation Setup Observing the molecules exiting the hexapole is an RGA 200 mass spec- trometer that has been modified to feed its output to a current amplifier and voltage amplifier in series before displaying the data on an oscilloscope, from which it may be exported onto a USB drive. The setup is illustrated in Figure 5.8 Knowing the sensitivity of the mass spectrometer and the am- plification settings on the current and voltage pre-amplifiers it is possible to derive the number of molecules being guided through the hexapoles. Figure 5.8: This shows the setup for the observation of molecules arriving at the mass spectrometer. The output from the mass spectrometer is fed into a current and voltage pre-amplifier in series before being recorded on an oscilloscope. 39 Chapter 6 Results and Discussion 6.1 CH3CN Velocity Filtering I This section details the results of the first and by far the most straight- forward of the four types of experiments performed. The target molecules were introduced into the system through the main pulsed nozzle as explained previously with pulse width, backing pressure and pulse frequency all con- trollable. Moreover, the voltages on the two hexapoles were controllable and were varied in order to observe the effect on the observed signal. The results from varying the backing pressure are indicated in Figure 6.1. It is impor- tant to remember that the properties of the pulsed molecular beam depend on the backing pressure as mentioned before, and at a backing pressure higher than a few hundred milliTorr the pulsed beam becomes collimated as opposed to effusive. So beyond a certain point, while increasing the backing pressure may increase the signal by increasing the density of the CH3CN molecular beam, the transition from an effusive beam to a collimated beam may nullify this signal increase. The experimental results support this as it is evident that while increasing the backing pressure from 400 mTorr to 500 mTorr increases the signal significantly, increasing it further to 700 mTorr has proportionally a smaller effect on the signal strength. 40 6.1. CH3CN Velocity Filtering I (d) (c) (b) (a) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) Figure 6.1: This diagram shows how the signal varies with the backing pressure behind the pulsed valve: (a) 0.70 Torr, (b) 0.60 Torr, (c) 0.50 Torr and (d) 0.40 Torr. The width of the CH3CN pulse can be varied and increased, this is shown in Figure 6.3. With pulses greater than 5 ms it is necessary to consider the width of the injected pulse in calculating the velocity distribution. With pulses of 5 ms, it is only necessary to use the simple v = d \/ t relationship to find the velocity distribution, where d is the total distance travelled by the molecules is the distance from the pulsed nozzle to the mass spectrometer, 1.254 m. From the most probable velocity it is possible to calculate the longitudinal temperature of the molecules as below. T = mv2 2kB (6.1) 41 6.1. CH3CN Velocity Filtering I Large Hexapole (+ - 9kV) Main Hexapole (+ - 4.5kV) RGA Mass Spectrometer Cold Head d = 1.254 m Large Hexapole (+ - 9kV) Main Hexapole (+ - 4.5kV) RGA Mass Spectrometer d = 1.225 m (a) (b) Figure 6.2: This diagram shows how the total length, used in calculating the velocity distribution, is calculated for the first two experiment arrangements, (a) being the first with results described in this section where the pulsed nozzle is directly in front of the large hexapole, and (b) being that for the experiments described in the next section where the nozzle is fixed to the stand which supports the ablation target. 42 6.1. CH3CN Velocity Filtering I The two hexapoles, as seen in Figure 6.2, have separately controllable applied voltages. It was found after some experimentation with directly injected CH3CN that until the large hexapole applied voltage dropped be- low the applied voltage of the main hexapole there was little difference in the time of flight signal. For example, as shown in Figure 6.4 when the main hexapole had an applied voltage of 4kV applied, unless the applied voltage large hexapole dropped below 4kV the variations between the scans was much larger than the difference between two different applied voltages. This fact should not be surprising considering the discussion on transverse velocity filtering. Both the main and the large hexapoles filter transverse velocities and this filtering is dependent on the amplitude of the Stark effect rather than the gradient of the electric field. So molecules with a transverse velocity that is trappable by the small hexapoles at 4kV is similarly trap- pable by the large hexapole. However, a molecule that is only trappable by the large hexapoles when their applied voltage is greater than 4kV will be lost when it reaches the main hexapole. What these trials show however, is that there is efficient transfer from the large to the main hexapoles, other- wise we would expect there to be a significant difference in the signal as we vary the large hexapole voltage. Transverse Filtering \u221d EStark (6.2) Longitudinal Filtering \u221d \u2202EStark \u2202r (6.3) Understanding now how the different parameters affect the signal we may optimize them and obtain a time of flight scan that we may convert to a velocity distribution. Two figures, Figure 6.5 and Figure 6.6 show the time of flight and velocity distribution measurements taken from it. The most likely velocity is 43.75 m\/s, and from this we may obtain our longitudinal temperature as 4.72 K. v = \u221a 2kT m = 43.75 m\/s (6.4) T = 6.81 x 10\u221226 kg 43.752 m2 s\u22122 2 x 1.38 x 10\u221223 kg m2 s\u22122 T\u22121 = 4.72K (6.5) 43 6.1. CH3CN Velocity Filtering I (a) (b) (c) (d) (e) (f ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) -0.03 -0.02 -0.01 0 0.01 0 50 100 150 200 250 300 Time (ms) S ig n a l (a rb . u n it s ) Figure 6.3: This diagram shows how the pulse width of the directly injected CH3CN can affect the time of flight distribution: (a) 5ms, (b) 10ms, (c) 20ms, (d) 30ms, (e) 40ms and (f) 50ms. 44 6.1. CH3CN Velocity Filtering I (a) (b) (c) (d) (e) -0.04 -0.03 -0.02 -0.01 0 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.04 -0.03 -0.02 -0.01 0 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.04 -0.03 -0.02 -0.01 0 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.04 -0.03 -0.02 -0.01 0 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) -0.04 -0.03 -0.02 -0.01 0 0 50 100 150 200 Time (ms) S ig n a l (a rb . u n it s ) Figure 6.4: This diagram shows how the pulse depth varies with the applied voltage on the large hexapole, the small hexapole voltage was held constant at 4kV by the large hexapole varied from (a) 3kV, (b) 4kV, (c) 5kV, (d) 7kV to (e) 9kV. 45 6.1. CH3CN Velocity Filtering I (a) (b) (c) -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) S ig na l ( ar b. u ni ts ) -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) S ig na l ( ar b. u ni ts ) -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 0.05 0.1 0.15 0.2 0.25 0.3 Time (s) S ig na l ( ar b. u ni ts ) Figure 6.5: This diagram shows two diferent scans taken, (a) one where the main hexapole was turned off (b) one with the main hexapoles turned on at 4.5kV. Both were injected starting at time zero with a 5ms pulse of CH3CN. (c) The difference between the two scans is shown and represents the time of flight signal of CH3CN through the hexapoles. 46 6.1. CH3CN Velocity Filtering I -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0 50 100 150 200 Velocity (m\/s) S ig n a l (a rb . u n it s ) Figure 6.6: This diagram shows the velocity distribution drawn from the time of flight signal taken in Figure 6.5. To understand if this value is reasonable or not, we must consider the predicted velocities from our simulations of the hexapole field and Stark effect. With an applied voltage of 4.5kV and the properties of our main hexapole in Table 5.2 that for a symmetric top molecule the approximate maximum velocity can be derived numerically for the state | J K MJ > through Equation 2.30. While these may provide a general idea that the results are reasonable, without a more exact knowledge of the population densities of individual states its impossible to predict more accurately what the velocity distribution should be. vmax = 108.04 m \/ s \u221a |MJ K | J (J + 1) (6.6) From Table 6.1 it is clear that our observed time of flight would be consistent with an average molecule being in the | J = 2, K = 1, MJ = -1 > eigenstate. This is a not unreasonable expectation since there has been no rotational cooling applied to the molecular beam prior to electrostatic filtering by the hexapole. It is in line with experimental results obtained by other groups performing velocity filtering [12]. 47 6.2. CH3CN Velocity Filtering II J K MJ vmax (m\/s) 1 1 -1 76.40 2 1 -1 44.11 3 1 -1 31.19 4 1 -1 24.16 ... Table 6.1: Numerical calculation of the predicted maximum trappable ve- locities for CH3CN. 6.2 CH3CN Velocity Filtering II To check that the addition of an apparatus to support the ablation process would not significantly affect the results after adding additional parts as shown in Figure 6.2 (b) the tests performed in the previous section were repeated. The results are similar however show an increased level of noise. As before we may obtain a time of flight scan that we may convert to a velocity distribution. Two figures, Figure 6.7 and Figure 6.8 show the time of flight and velocity distribution measurements taken from it. The most likely velocity is 31 m\/s, and from this we may obtain our longitudinal temperature as 2.37 K. v = \u221a 2kT m = 31 m\/s (6.7) T = 6.81 x 10\u221226 kg 312 m2 s\u22122 2 x 1.38 x 10\u221223 kg m2 s\u22122 T\u22121 = 2.37K (6.8) 48 6.2. CH3CN Velocity Filtering II (a) (b) (c) -0.006 -0.004 -0.002 0 0.002 0.004 0 100 200 300 400 500 Time (ms) S ig na l ( ar b. u ni ts ) -0.006 -0.004 -0.002 0 0.002 0.004 0 100 200 300 400 500 Time (ms) S ig na l ( ar b. u ni ts ) -0.006 -0.004 -0.002 0 0.002 0.004 0 100 200 300 400 500 Time (ms) S ig na l ( ar b. u ni ts ) Figure 6.7: This diagram shows two diferent scans taken, (a) one where the main hexapole was turned off (b) one with the main hexapoles turned on at 4.5kV. Both were injected starting at time zero with a 5ms pulse of CH3CN. (c) The difference between the two scans is shown and represents the time of flight signal of CH3CN through the hexapoles. 49 6.3. CH3CN Velocity Filtering: \u2019Ablation\u2019 Simulation 0 0.001 0.002 0.003 0.004 0.005 0 20 40 60 80 100 Velocity (m\/s) S ig n a l (a rb . u n it s ) Figure 6.8: This diagram shows the velocity distribution drawn from the time of flight signal taken in Figure 6.7. 6.3 CH3CN Velocity Filtering: \u2019Ablation\u2019 Simulation Before proceeding to the trials involving ablation of 40Ca19F it was desired to test the parameters to find the best conditions under which to perform ablation. As shown in Figure 5.4 a pulsed gas source was placed where ablation would occur and cold Helim gas was pulsed through the main valve. The timing between the Helium pulse and CH3CN pulse could be controlled and varied in order to determine the best set of conditions. While initial results were promising, it is obvious that there exists some dependences that are not fully understood. The best time of flight distribution with possible signal is shown in Figure 6.9. Calculations to determine the longitudinal velocity and temperature were not performed as this simply was to test experiment parameters and the signal was not of sufficient quality. 50 6.3. CH3CN Velocity Filtering: \u2019Ablation\u2019 Simulation (a) (b) (c) -0.01 -0.006 -0.002 0.002 0.006 -50 50 150 250 350 450 Time (ms) S ig na l ( ar b. u ni ts ) -0.02 0.06 0.14 -0.01 -0.006 -0.002 0.002 0.006 -50 50 150 250 350 450 Time (ms) S ig na l ( ar b. u ni ts ) -0.02 0.06 0.14 -0.01 -0.006 -0.002 0.002 0.006 -50 50 150 250 350 450 Time (ms) S ig na l ( ar b. u ni ts ) -0.02 0.06 0.14 Figure 6.9: This diagram shows the time of flight distributions where the CH3CN gas pressure was 1.04 Torr, and the Helium gas pressure was 120 mTorr with the (a) main hexapole was off (b) main hexapole was on at 4.5kV and (c) the difference between the two representing the time of flight distribution of molecules through the hexapole. The blue square pulse is the Helium buffer gas valve opening and closing, while the red square pulse comes from the CH3CN pulse opening and closing. 51 6.4. 40Ca19F Velocity Filtering 6.4 40Ca19F Velocity Filtering Velocity filtering of 40Ca19F proved more difficult than anticipated as ex- plained in the previous chapters. The laser ablation would burn a hole through the CaF2 disc if it was left ablating on the same point, although the exact time depended on the intensity of the laser beam. This was avoided by using slits to decrease the intensity of the ablating laser and taking short scans of several minutes. After each scan the position of abla- tion was changed by adjusting the mirrors controlling the laser\u2019s path. Not only was the positioning of the ablation point the source of much trouble, but also interference between the 10 Hz Nd:YAG laser and the electronics induced a background noise that could not be removed. Fortunately the interference could be removed by subtracting the background with main hexapole off from the signal with the main hexapoles turned on. The signal seen in Figure 6.10 was the summation of seven different scans taken at what was considered the best conditions of backing pressure and hexapole voltage. The optimum signal was observed at roughly 100 mTorr, main hexapole voltage of 4kV, large hexapole voltage of 8kV. The time of flight distribution was converted to the velocity time distri- bution in the same way as was done for the case of CH3CN with v = d \/ t. However in this case the distance travelled by the molecules was slightly different, instead of measuring from the nozzle to the mass spectrometer, the distance taken was from the point of ablation to the mass spectrometer for a total distance of 1.233 m. Moreover, instead of time being measured relative to when the nozzle opens it was measured relative to when the ablation laser was fired. 52 6.4. 40Ca19F Velocity Filtering (a) (b) (c) -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time (s) S ig na l ( ar b. u ni ts ) -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time (s) S ig na l ( ar b. u ni ts ) -0.007 -0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002 0.003 -0.1 0 0.1 0.2 0.3 0.4 0.5 Time (s) S ig na l ( ar b. u ni ts ) Figure 6.10: These scans represent the (a) time of flight distribution with main hexapole turned on at 4kV (b) the time of flight distribution when the main hexapole was turned off and (c) the difference between those two scans, representing the mass signal that was filtered by the main hexapole. 53 6.4. 40Ca19F Velocity Filtering -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 2 4 6 8 10 12 14 16 18 20 22 24 Velocity (m\/s) S ig n a l (a .u .) Figure 6.11: This velocity distribution for 40Ca19F molecules was determined from Figure 6.10 The most probably velocity is approximately 4.47 m\/s. From the previ- ous section it is known how to convert this from the translational velocity to temperature. v = \u221a 2kT m = 4.47 m\/s (6.9) T = 9.7998 x 10\u221226 kg 4.472 m2 s\u22122 2 x 1.38 x 10\u221223 kg m2 s\u22122 T\u22121 = 70.9mK (6.10) This value is not far below what was expected for 40Ca19F from pre- vious calculations shown in Chapter 2 although without knowing the state populations it would be impossible to confirm that to confirm theoretical predictions. The state populations could be determined by depletion spec- troscopy. 54 Chapter 7 Future Considerations 7.1 Alternate Sample Molecules There are numerous alternative sample molecules that could be investigated using this apparatus. Some offer the potential for improvements in the reliability of the process and others are more challenging to implement in the current setup. 138Ba19F was one molecule mentioned previously because of its similar- ity to the current sample molecule 40Ca19F and its usefulness as a target molecule in experiments dealing with the search for the electron\u2019s electric dipole moment. BaF2 is readily available, like CaF2, as a solid often found in optical equipment and so may be used without significant change to the experimental apparatus. 40Ca1H was another molecule that had been con- sidered but was found to be inconvenient. The molecule typically exists as a power and is highly reactive to water, it will react extremely quickly even with water vapour in the air. A third molecule that might generate more reliable ablation plumes is SrO because of an addition that would be enable the ablation target to rotate, eliminating concerns about ablating for too long on a single point. This rotation would be acheived by a 5-phase step- ping motor that is already available. More reliable ablation of CaF2 and BaF2 could be similarly be acheived with a motor that moves up and down rather than rotating since those samples exist in disk form but SrO exists in bar form. Figure 7.1: This diagram shows the (a) 5-phase stepper motor that can rotate a cylindrical sample and (b) SrO sample. 55 7.2. Multi-Photon Ionization 7.2 Multi-Photon Ionization An addition onto the experiment that would enhance detection of molecules at the exit of the main hexapole would be a detector that operates based on multi-photon ionization. A laser positioned at the exit of the hexapole would be used to ionize molecules of a particular state by multi-photon ionization. The molecules would then be accelerated upwards by a series of increasing applied electric fields towards a detector. Unlike resonance enhanced multi- photon ionization however, there is not be an intermediate electronic state between the first and subsequent photon induced excitations [35]. The limitation of this method is that an accurate knowledge of the time of flight of the pulsed molecules would be required in order to synchronize the firing of the laser with the arrival of the molecules at the exit of the hexapole. So, multi-photon ionization would be best used in addition to the RGA mass spectrometer rather than as a replacement. The necessary parts for this have already been assembled and prepared but have not yet been tested with pulsed beams of molecules. +V 0 Laser 0 V +4V 0 \/5 +3V 0 \/5 +2V 0 \/5 +V 0 \/5 Cold Pulsed Beam MCP Detector Multi-Photon Ionization Ions Figure 7.2: This diagram shows how multi photon ionization would work. The cold molecules are ionized by the laser, after which they are accelerated by electric fields upwards to a detector. 56 7.3. Laser Depletion Spectroscopy 7.3 Laser Depletion Spectroscopy The process of depletion spectroscopy is one that has been used before successfully in experiments involving electrostatic velocity filtering. It has proven to be a useful tool in determining the state population densities for a target molecule [12]. Depletion spectroscopy works by exciting molecules to a higher state from which they may either dissociate or relax to a high field seeking state which would then be lost to the trap [36]. Such a method requires the depletion laser be able to excite an appropriate transition in the target molecule, a not irrelevant consideration. Depletion spectroscopy can be easily applied to the current setup by aligning an appropriate laser along the main hexapole between the two bends and lasing during the time when the pulse is travelling through it. Hexapole (+ - 9kV) Hexapole (+ - 4.5kV) Ablation Laser Depletion Laser Figure 7.3: This diagram shows the proposed arrangment of the depletion laser relative to the hexapoles and the ablation laser. 57 Chapter 8 Conclusions The experiments with methodology and results described in previous chap- ters have demonstrated the viability of the use of laser ablation, buffer gas cooling and velocity filtering to create cold pulsed molecular beams. Tests involving CH3CN pulsed directly through the nozzle showed a longitudinal temperature of 2.37 K and most probable velocity of 31 m\/s. Where 40Ca19F molecules were the target, a longitudinal temperature of 70.9 mK and most probable velocity of 4.47 m\/s. Improvements on the time of flight detec- tion method using multiphoton ionization are currently underway and plans for implementing depletion spectroscopy to calculate the state population densities are in progress. 58 Bibliography [1] Hendrick L Bethlem and Gerard Meijer. 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[33] D Egorov, W C Campbell, B Friedrich, S E Maxwell, E Tsikata, L D van Buuren, and J M Doyle. Buffer-gas cooling of NH via the beam loaded buffer-gas method. Eur. Phys. J. D, 31:307\u2013311, 2004. [34] J Dojcansky and J Heinrich. Saturated vapour pressure of acetonitrile. Chem. Zvesti, pages 157\u2013159, 1974. 61 [35] David M Lubman. Lasers and Mass Spectroscopy. Oxford University Press, 1990. [36] M Motsch, M Schenk, L D van Buuren, M Zeppenfeld, P W H Pinkse, and G Rempe. Internal-state thermometry by depletion spectroscopy in a cold guided beam of formaldehyde. Phys. Rev. A, 76:061402(R), 2007. 62 Appendix A Stark Effect in Symmetric Top Molecules: C Program This appendix contains the code used to calculate the Stark effect for sym- metric top molecules. The code is written in the C language and is largely written by the author although part of the program was taken from the code for tqli.c in Numerical Recipes in C, however it was altered slightly to operate more smoothly in this program. The sorting program is similar to one provided in the text C How to Program. #include #include #include void main() { int ji,mi,ki,cnt,cmax; int jmax=20,mmax=5, kmax=5; \/*d represents the diagonal elements of the matrix and e represents the off-diagonal elements such that the matrix is tridiagonal*\/ float d[10000],e[10000]; \/*the constants A and B for CH3CN are defined*\/ float A=5.28,B=0.307; float muE,itgr,JQ,MQ,KQ,J0; \/*for varying the applied electric fields*\/ int z; \/*there are for the program to find the eigenvalues*\/ 63 Appendix A. Stark Effect in Symmetric Top Molecules: C Program int n,m,l,iter,i,k; float s,r,p,g,f,dd,c,b; \/*the following variables were defined for the sorting mechanism*\/ int pass, count; float hold; FILE *fptr; if ((fptr=fopen(\"StarkKM(-5to5)(-5to5)uE2.dat\", \"w\"))==NULL) printf(\"ERROR! Cannot open file\\n\"); else { \/*loops from K=-5 to K=+5*\/ for (ki=-kmax;ki<=kmax;ki++) { KQ=ki; \/*loops from MJ=-5 to MJ=+5*\/ for (mi=-mmax;mi<=mmax;mi++) { MQ=mi; fprintf(fptr,\"\\n\\n\\nM=%f, K=%f\\n\\n\\n\",MQ,KQ); \/*J0 represents the minimum allowable J value for a particular set of K, MJ values*\/ if (abs(MQ)>=abs(KQ)) J0=abs(MQ); else J0=abs(KQ); \/*this loop varies the electric field applied from muE = 0 to 2 cm-1*\/ for (z=0;z<200;z++) { muE=0.01*z; fprintf(fptr, \"\\n%f \", muE); 64 Appendix A. Stark Effect in Symmetric Top Molecules: C Program cnt=1; \/*if the minimum allowable J value is zero, then one possible eigenvalue is zero, this is manually programmed to prevent errors from occuring by dividing by zero in the computer program. Otherwise the equations are as known.*\/ if (J0==0) d[cnt]=0; else d[cnt]=B*J0*(J0+1)-muE*MQ*KQ\/(J0)\/(J0+1)+(A-B)*KQ*KQ; e[cnt]=-muE*sqrt( ( (1+J0)*(1+J0)-KQ*KQ)*((1+J0)*(1+J0)-MQ*MQ) \/( (1+J0)*(1+J0)*(2*J0+1)*(2*J0+3) ) ); for (ji=1;ji<=jmax;ji++) { \/*starting from J0 the values are found for J quantum numbers above J0, thus filling the matrix*\/ JQ=ji+J0; cnt++; e[cnt]=-muE*sqrt( (JQ*JQ-KQ*KQ)*(JQ*JQ-MQ*MQ) \/(JQ*JQ*(2*JQ-1)*(2*JQ+1)) ); d[cnt]=B*JQ*(JQ+1)-muE*B*MQ*KQ\/JQ\/(JQ+1)+(A-B)*KQ*KQ; } n=jmax+1; \/*this is where the code for the tqli.c program from Numerical Recipes in C would be, it is removed for copyright purposes.*\/ \/*this part of the program sorts the d values*\/ for (pass=1;passd[count+1]) { hold=d[count]; 65 Appendix A. Stark Effect in Symmetric Top Molecules: C Program d[count]=d[count+1]; d[count+1]=hold; } } } \/*outputting our data to a file*\/ for (l=1;l<=n;l++) fprintf(fptr, \"%12.5f\", d[l]); }\/*end loop on z*\/ printf(\".\"); }\/*ends for loop over mi*\/ printf(\".\"); }\/*ends for loop over ki*\/ }\/*end if-fptr*\/ fclose(fptr); }\/*end main*\/ 66 Appendix B Stark Effect in Diatomic Molecules: C Program The following code was used to calculate the exact Stark effect in the di- atomic molecule 40Ca19F and is created by the author except for the Wigner 6j and 3j calculators which uses the programs made available by the GNU Scientific Library and the tred.c and tqli.c programs that were taken and slightly modified from Numerical Recipes in C. Again, the sorting program is similar to one provided in the text C How to Program. #include #include #include #include #include \/*defining the maximum as being 12000*\/ #define smax 12000 \/*a, d and e are used for the diagonalization solution*\/ float d[smax+1]; float e[smax+1]; float a[smax+1][smax+1]; \/*C stores the Clebsch-Gordon coefficients*\/ float C[smax+1][5]; \/*Mn, Ms, and Mi are the magnetic quantum numbers*\/ float Mn[smax+1][5],Ms[smax+1][5],Mi[smax+1][5]; \/*QN stores the good quantum numbers*\/ float QN[smax+1][5]; void main () { 67 Appendix B. Stark Effect in Diatomic Molecules: C Program \/*these are the fundamental constants for 40Ca19F*\/ float gamma=0.0013228, lb=0.0036419, B=0.3437044, lc=0.0013381; \/*these are the quantum numbers*\/ float N,J,Mj,F,Mf,S=0.5,I=0.5,Mn0; \/*Nsize,Nmax,exit,times and top are all used in the calculation of the CG coefficients*\/ int Nsize, Nmax=20; int exit, times; int top; \/*these are used for tred2 program*\/ int l,k,j,i,n=smax; float scale,hh,h,g,f; \/*these are used for tred2 program*\/ \/*these are used for tqli program*\/ int m,iter; float s,r,p,dd,c,b,temp; \/*these are used for the tqli program*\/ \/*pass, hold and count are used in the sorting program*\/ int pass, count; float hold,hold2; \/*these are used for calculations involving the Stark effect*\/ float muE,Nma, SS; int cn; float debugger; \/*this is used for the file pointer*\/ FILE *fptr; if ((fptr=fopen(\"19032009NJFMF2.dat\", \"w\"))==NULL) printf(\"ERROR! Cannot open file\\n\"); else { 68 Appendix B. Stark Effect in Diatomic Molecules: C Program \/*we start at muE=0 and slowly increment by 0.1*\/ for (cn=0;cn<=11;cn++) { muE=(float)(cn)\/10; printf(\"Initializing...\\n\"); \/*initializing the matrices*\/ for (j=0;j<=smax+1;j++) { d[j]=0; e[j]=0; for (k=0;k<=smax+1;k++) a[j][k]=0; } for (j=0;j<=smax+1;j++) { for (k=0;k<=5;k++) { C[j][k]=0; QN[j][k]=0; } } \/*we start the counter i at 1 and increment it for each state*\/ i=1; printf(\"Calculating Clebsch-Gordon coefficients...\\n\"); \/*we loop up to Nmax starting at N=0*\/ for (Nsize=0;Nsize<=Nmax; Nsize++) { \/*switching N quantum number to a float*\/ N=Nsize; \/*re-initializing exit and times counters: the time counter determines which of the sub-states we go to*\/ exit=0; times=0; while (exit!=1) { \/*if it is the first time through we automatically start at 69 Appendix B. Stark Effect in Diatomic Molecules: C Program |N J=N+0.5 F=N+1 Mf=F>*\/ if(times==0) { \/*The following section works for the |NJFMf> state where J=N+0.5,F=N+1, Mf=F*\/ F=N+1.0; J=N+0.5; Mf=F; Mj=Mf-0.5; Mn0=Mj-0.5; \/*saving Quantum number*\/ QN[i][1]=N; QN[i][2]=J; QN[i][3]=F; QN[i][4]=Mf; Mn[i][1]=Mn0; Ms[i][1]=0.5; Mi[i][1]=0.5; C[i][1]=sqrt( (J+Mj+1)\/(2*J+1) )*sqrt( (N+Mn0+1)\/(2*N+1) ); Mn[i][2]=Mn0+1.0; Ms[i][2]=-0.5; Mi[i][2]=0.5; C[i][2]=sqrt( (J+Mj+1)\/(2*J+1) )*sqrt( (N-Mn0) \/ (2*N+1) ); Mn[i][3]=Mn0+1.0; Ms[i][3]=0.5; Mi[i][3]=-0.5; C[i][3]=sqrt( 1\/(2*J+1)\/(J+Mj+1) )*(N+Mn0+2) *sqrt( (N-Mn0)\/(2*N+1) ); Mn[i][4]=Mn0+2.0; Ms[i][4]=-0.5; Mi[i][4]=-0.5; C[i][4]=sqrt( (N+Mn0+2)*(N-Mn0-1)*(N-Mn0)\/(2*N+1) \/(2*J+1)\/(J+Mj+1)); times=1; 70 Appendix B. Stark Effect in Diatomic Molecules: C Program \/*there\u2019s never a problem with the next lower state so we can simply move on to it next loop*\/ } \/*on the next time through, if the top state is already done we move to the |N J=N+0.5 F=N Mf=N> state*\/ else if (times==1) { \/*the following section works for the |NJFMf> state where J=N+0.5, Mj=Mn+0.5, F=J-0.5, Mf=Mj+0.5*\/ F=N; J=N+0.5; Mf=F; Mj=Mf-0.5; Mn0=Mf-1; \/*saving Quantum number*\/ QN[i][1]=N; QN[i][2]=J; QN[i][3]=F; QN[i][4]=Mf; Mn[i][1]=Mn0; Ms[i][1]=0.5; Mi[i][1]=0.5; C[i][1]=sqrt( (J-Mj)\/(2*J+1)*(N+Mn0+1)\/(2*N+1) ); Mn[i][2]=Mn0+1.0; Ms[i][2]=-0.5; Mi[i][2]=0.5; C[i][2]=sqrt( (J-Mj)\/(2*J+1)*(N-Mn0)\/(2*N+1) ); Mn[i][3]=Mn0+1.0; Ms[i][3]=0.5; Mi[i][3]=-0.5; C[i][3]=-sqrt( 1\/(J-Mj)\/(2*J+1)*(N-Mn0)\/(2*N+1) )*(N+Mn0+2); Mn[i][4]=Mn0+2.0; Ms[i][4]=-0.5; Mi[i][4]=-0.5; C[i][4]=-sqrt( 1\/(J-Mj)\/(2*J+1)*(N+Mn0+2)*(N-Mn0-1) 71 Appendix B. Stark Effect in Diatomic Molecules: C Program *(N-Mn0)\/(2*N+1) ); \/*so long as N!=0 we move to the next state on the next loop otherwise the loop exits*\/ if (N>0) times=2; else exit=1; } else if (times==2) { \/*the following section works for the |NJFMf> state where J=N-0.5, Mj=Mn+0.5, F=J+0.5, Mf=Mj+0.5*\/ F=N; J=N-0.5; Mf=F; Mj=Mf-0.5; Mn0=Mf-1; \/*saving Quantum number*\/ QN[i][1]=N; QN[i][2]=J; QN[i][3]=F; QN[i][4]=Mf; Mn[i][1]=Mn0; Ms[i][1]=0.5; Mi[i][1]=0.5; C[i][1]=sqrt( (J+Mj+1)\/(2*J+1)*(N-Mn0)\/(2*N+1) ); Mn[i][2]=Mn0+1.0; Ms[i][2]=-0.5; Mi[i][2]=0.5; C[i][2]=-sqrt( (J+Mj+1)\/(2*J+1)*(N+Mn0+1)\/(2*N+1) ); Mn[i][3]=Mn0+1.0; Ms[i][3]=0.5; Mi[i][3]=-0.5; C[i][3]=sqrt( (J-Mj)\/(2*J+1)*(N+Mn0+1)\/(2*N+1) )*(N-Mn0-1); 72 Appendix B. Stark Effect in Diatomic Molecules: C Program Mn[i][4]=Mn0+2.0; Ms[i][4]=-0.5; Mi[i][4]=-0.5; C[i][4]=-sqrt( (J-Mj)\/(2*J+1)*(N+Mn0+1)*(N+Mn0+2) *(N-Mn0-1)\/(2*N+1) ); times=3; \/*if this state is okay, the final state is also undoubtably okay as well*\/ } else if (times==3) { \/*the following section works for the |NJFMf> state where J=N-0.5, Mj=Mn+0.5, F=J-0.5, Mf=Mj+0.5*\/ F=N-1; J=N-0.5; Mf=F; Mj=Mf-0.5; Mn0=Mf-1; \/*saving Quantum number*\/ QN[i][1]=N; QN[i][2]=J; QN[i][3]=F; QN[i][4]=Mf; Mn[i][1]=Mn0; Ms[i][1]=0.5; Mi[i][1]=0.5; C[i][1]=sqrt( (J-Mj)\/(2*J+1)*(N-Mn0)\/(2*N+1) ); Mn[i][2]=Mn0+1.0; Ms[i][2]=-0.5; Mi[i][2]=0.5; C[i][2]= -sqrt( (J-Mj)\/(2*J+1)*(N+Mn0+1)\/(2*N+1) ); Mn[i][3]=Mn0+1.0; Ms[i][3]=0.5; Mi[i][3]=-0.5; C[i][3]= -sqrt( 1\/(2*J+1)\/(J-Mj) )*sqrt( (N+Mn0+1)\/(2*N+1) ) 73 Appendix B. Stark Effect in Diatomic Molecules: C Program *(N-Mn0-1); Mn[i][4]=Mn0+2.0; Ms[i][4]=-0.5; Mi[i][4]=-0.5; C[i][4]=sqrt( 1\/(2*J+1)\/(J-Mj) )*sqrt( (N+Mn0+1) \/(2*N+1)*(N+Mn0+2)*(N-Mn0-1) ); exit=1; } while (-Mf<=F) { \/*lowering operator*\/ i++; Mn[i][1]=Mn0-1.0; Ms[i][1]=0.5; Mi[i][1]=0.5; if ((N-Mn0+1)*(N+Mn0)>0) C[i][1]=C[i-1][1]*sqrt( (N-Mn0+1)*(N+Mn0) ) \/sqrt( (F-Mf+1)*(F+Mf) ); else C[i][1]=0; Mn[i][2]=Mn0; Ms[i][2]=-0.5; Mi[i][2]=0.5; if ((N-Mn0)*(N+Mn0+1)>0) C[i][2]=(C[i-1][2]*sqrt( (N-Mn0)*(N+Mn0+1) ) + C[i-1][1])\/sqrt( (F-Mf+1)*(F+Mf) ); else C[i][2]=C[i-1][1]\/sqrt( (F-Mf+1)*(F+Mf) ); Mn[i][3]=Mn0; Ms[i][3]=0.5; Mi[i][3]=-0.5; 74 Appendix B. Stark Effect in Diatomic Molecules: C Program if ((N-Mn0)*(N+Mn0+1)>0) C[i][3]=(C[i-1][3]*sqrt( (N-Mn0)*(N+Mn0+1) ) +C[i-1][1] )\/sqrt( (F-Mf+1)*(F+Mf) ); else C[i][3]=C[i-1][1]\/sqrt( (F-Mf+1)*(F+Mf) ); Mn[i][4]=Mn0+1.0; Ms[i][4]=-0.5; Mi[i][4]=-0.5; if ((N-Mn0-1)*(N+Mn0+2)>0) C[i][4]=(C[i-1][4]*sqrt( (N-Mn0-1)*(N+Mn0+2) ) + C[i-1][2] + C[i-1][3] )\/sqrt( (F-Mf+1)*(F+Mf) ); else C[i][4]=(C[i-1][2]+C[i-1][3])\/sqrt( (F-Mf+1)*(F+Mf) ); Mf=Mf-1; Mn0=Mn0-1; \/*saving Quantum number*\/ QN[i][1]=N; QN[i][2]=J; QN[i][3]=F; QN[i][4]=Mf; } } } top=i; printf(\"i=%d\\n\", top); printf(\"Calculating Matrix...\\n\"); \/*now that we have our matrix of clebsch-gordon coefficients we can start comparing terms to calculate our Hamiltonian matrix*\/ for (i=1;i=QN[j][1]) Nma=QN[i][1]; else Nma=QN[j][1]; 76 Appendix B. Stark Effect in Diatomic Molecules: C Program if ((int)(floor(QN[i][3]+QN[j][3]+QN[i][2]+QN[j][2]+1 +Nma-QN[i][4]+0.5)) % 2 == 0) { SS=muE*sqrt( (2*QN[i][3]+1)*(2*QN[j][3]+1) *(2*QN[i][2]+1)*(2*QN[j][2]+1)*Nma ) *gsl_sf_coupling_3j((int)(2*QN[i][3]),2,(int)(2*QN[j][3]), (int)(-2*QN[i][4]),0,(int)(2*QN[j][4])) *gsl_sf_coupling_6j((int)(2*QN[i][2]),(int)(2*QN[i][3]),1, (int)(2*QN[j][3]),(int)(2*QN[j][2]),2) *gsl_sf_coupling_6j((int)(2*QN[i][1]),(int)(2*QN[i][2]),1, (int)(2*QN[j][2]),(int)(2*QN[j][1]),2); a[i][j]+=SS; } else { SS=-muE*sqrt( (2*QN[i][3]+1)*(2*QN[j][3]+1)*(2*QN[i][2]+1) *(2*QN[j][2]+1)*Nma ) *gsl_sf_coupling_3j((int)(2*QN[i][3]),2,(int)(2*QN[j][3]), (int)(-2*QN[i][4]),0,(int)(2*QN[j][4])) *gsl_sf_coupling_6j((int)(2*QN[i][2]),(int)(2*QN[i][3]),1, (int)(2*QN[j][3]),(int)(2*QN[j][2]),2) *gsl_sf_coupling_6j((int)(2*QN[i][1]),(int)(2*QN[i][2]),1, (int)(2*QN[j][2]),(int)(2*QN[j][1]),2); a[i][j]+=SS; } } } n=top; printf(\"\\nStarting tred2 program\\n\"); \/*this is where the tred2 program from Numerical Recipes in C would be, it is removed for copyright purposes*\/ printf(\"\\nBeginning tqli program\\n\"); \/*this is where the tqli program from Numerical Recipes in C would be, it is removed for copyright purposes*\/ 77 Appendix B. Stark Effect in Diatomic Molecules: C Program printf(\"\\nSorting\\n\"); \/*this part of the program sorts the d values*\/ for (pass=1;passd[count+1]) { hold=d[count]; d[count]=d[count+1]; d[count+1]=hold; } } } \/*outputting our solutions to file*\/ for (i=1;i