OBSERVATION OF THE INFRARED SPECTRUM OF THE HELIUM-HYDRIDE MOLECULAR ION

Item Type text; Dissertation-Reproduction (electronic)

Authors Tolliver, David Edward

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

Download date 07/10/2021 23:40:49

Link to Item http://hdl.handle.net/10150/282038 INFORMATION TO USERS

This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted.

The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction.

1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity.

2. When an image on the film is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of movement during exposure, or duplicate copy. Unless we meant to delete copyrighted materials that should not have been filmed, you will find a good image of the page in the adjacent frame.

3. When a map, drawing or chart, etc., is part of the material b^ing photo­ graphed the photographer has followed a definite method in "sectioning" the material. It is customary to begin filming at the upper left hand corner of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete.

4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department.

5. Some pages in any document may have indistinct print. In all cases we have filmed the best available copy.

University Microfilms International 300 N. ZEEB ROAD, ANN ARBOR, Ml 48106 18 BEDFORD ROW, LONDON WC1R AEJ, ENGLAND 8017776

TOLLIVER, DAVID EDWARD

OBSERVATION OF THE INFRARED SPECTRUM OF THE HELIUM- HYDRIDE MOLECULAR ION

The University of Arizona PH.D. 1980

University Microfilms International 300 N. Zeeb Road, Ann Arbor, MI 48106 18 Bedford Row, London WCIR 4EJ, England OBSERVATION OF THE INFRARED SPECTRUM OF THE

HELIUM HYDRIDE MOLECULAR ION

by

David Edward Tolliver

A Dissertation Submitted to the

COMMITTEE ON OPTICAL SCIENCES (GRADUATE)

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

198 0 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read the dissertation prepared by David Edward Tolliver entitled Observation of the Infrared Spectrum of the Helium Hydride

Molecular Ion

and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy

/ j n •; •-> —1 j .1 O / ^

17 Date7" /O f &(. j A'r o /2. i /'fi d JDate

- g/^L/l,\A "f . ^ ^ 61 Date ^-C. V V?.l/ 8 b Date

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.

If- 1^ Dissertation Director Date STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrow­ ers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduc­ tion of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of schol­ arship. In all other instances, however, permission must be obtained from the author.

SIGNEDDoa^J F- AU that U> gold doeA not glitten, Not all thoie who wande.fi one loit; The old that. ih A&iong doeA not wltkeA, Veep nootA a/ie not reached by the fanott.

Fn.om the aAhes a fa-Oie ihall be woken, A tight (JKom the t>hadow& ihaLt Ap>u.ng; Renewed t>hall be blade that u)a& broken: The cAownlea again bhaJUL be king.

iii ACKNOWLEDGMENTS

I would like to acknowledge the guidance of my research director,

Dr. William H. Wing, and my academic advisor, Dr. Richard L. Shoemaker, during the course of my graduate study at the Optical Sciences Center.

I would like to thank Professor Willis E. Lamb, Jr. for helpful and stimulating discussions regarding this research and physics in general.

I am grateful to Dr. Joseph J. Spezeski for encouragement, valuable discussions, and a careful reading of the manuscript. I also acknowledge the collaboration on this research of Dr. George A. Kyrala. The contri­ butions of Mr. David R. Wickholm and Mr. August H. Johnson toward the development and construction of the apparatus is appreciated. The excel­ lent typing and editorial skills of Ms. Elaine Segura is much appreciated.

This research was supported in part by the National Science

Foundation; the National Bureau of Standards Precision Measurement Grant

Program; the U.S. Air Force Office of Scientific Research and the U.S.

Army Research Office; the Petroleum Research Fund, administered by the

American Chemical Society; and the University of Arizona Physics

Department.

Lastly, I would like to thank my wife for many years' patience in somewhat trying circumstances.

iv TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS vii

LIST OF TABLES ix

ABSTRACT x

1. INTRODUCTION 1

Interest in Molecular Ions 1 History of and Current Interest in HeH 5 Summary of Chapters 8

2. SUMMARY OF THE THEORY OF THE STRUCTURE OF HeH+ 9

Electronic States in HeH+ 9 Vibrational-Rotational States 15 Hyperfine Structure 24 Application to HeH+ 25 Peripheral Theoretical Treatments of HeH 27

3. REVIEW OF PREVIOUS EXPERIMENTAL WORK ON HeH+ 28

Experiments on Molecular Bond Strength . 29 Unimolecular and Collision-Induced Dissociation of HeH . 30 Dissociation of Fast HeH Ions Traversing Thin+Foils . . 31 Observation of the Infrared Radiation from HeH 32 4. EXPERIMENTAL METHOD 35 The Doppler Tuned Ion Beam Laser Resonance Method ... 36 Ion Source Population Distribution 38 Doppler Tuning 40 Laser-Molecule Interaction 43 Collisional Detection Mechanism 47 Signal-to-Noise Ratio Estimate 52

5. APPARATUS 56

Experiment Framework 56 Ion Beam Vacuum System 59 Ion Source 63 Production Mechanisms of HeH+ 63 Ion Source Design 70 Ion Source Performance 78

v vi

TABLE OF CONTENTS--Continued

Page

Ion Beam Optics 80 Carbon Monoxide Laser 90 Optical Cavity and Mechanical Design 90 Laser Power Supply 92 Laser Cooling and Gas Filling Systems 102 Laser Frequency Control 106 Passive Stabilization 106 Active Stabilization 106 Frequency Calibration 112 Laser Beam Optics 114 Signal Processing, Data Acquisition, and Computer Control 119 Signal Processing 120 Microcomputer System Hardware . . . 121 Microcomputer Software 124 Computer Programs 127

6. EXPERIMENTAL RESULTS AND DISCUSSION 129

Experimental Procedure 129 Method of Data Analysis 132 Analysis of Experimental Linewidths and Uncertainties . . 136 Linewidth Contributions 136 Frequency Measurement Uncertainties 139 Presentation of Results and Comparison with Present Theory 141 Comment on the Implications of the Present Work 143 Comment on Possible Future Work 144

APPENDIX A: COMPREHENSIVE LISTINGS OF THEORETICAL WORK ON HeH+ 145

APPENDIX B: VIBRATIONAL-ROTATIONAL SELECTION RULES AND APPROXIMATE TRANSITION DIPOLE MOMENT CALCULATION 152

APPENDIX C: LISTING OF THE MICROCOMPUTER EXPERIMENT CONTROL PROGRAM 158

LIST OF REFERENCES 167 LIST OF ILLUSTRATIONS

Figure Page

1. Correlation diagram for HeH+ 13

2. Potential energy curves for some low lying states of HeH+ 16

3. Electron density contours for ground state HeH+ 17

4. (^He^H)+ infrared emission spectrum obtained by Raitz Von Frentz et al 34

5. Doppler tuned ion beam laser resonance method concepts . . 37

6. Doppler tuned ion beam laser resonance method schematic . 39

7. Optical table-ion beam framework 58

8. High vacuum system schematic 60

9. Ion-molecule reaction kinetics diagram 66

10. Theoretical population distribution for HeH+ 68

11. Ion source schematic 71

12. Ion source wiring diagram 74

13. Focusing electrode power supply schematic 76

14. Ion source flange 77

15. Ion source performance 79

16. Ion beam optics schematic 83

17. Ion beam electrical wiring schematic 84

18. Gas target detail 86

19. Beam defining aperture detail 88

20. Laser current regulator circuit schematic 94

21. Laser current controller circuit schematic 96

vii viii

LIST OF ILLUSTRATIONS--Continued

Figure Page

22. Laser current controller logic circuit schematic 98

23. PZT/bushing current sensors schematic 99

24. Laser anode current sensors schematic 100

25. Laser gas fill system schematic 105

26. Laser stabilization scheme 108

27. Laser frequency locking loop schematic 110

28. PIT driver/amplifier circuit schematic 113

29. Integrating digital voltmeter schematic 122

30. Stepping motor driver/interface circuit schematic .... 125

31. Chart recorder trace of the (1,ll)«-»-(0,12) resonance . . . 131 LIST OF TABLES

Tabic Page

1. Numerical values for quantities appearing in Chapter 4 . . 54

2. Vacuum system operating parameters 64

3. HeH+ theoretical population distribution 69

4. Ion source operating parameters 81

5. Ion beam line parameters 91.

6. Laser gas mixture . 104

7. Summary of laser parameters 115

8. Extrapolation of data to zero conditions 134

9. Estimated contributions to experimental linewidths and uncertainties 137

10. Summary of the measurements of HeH+ vibrational- rotational transition frequencies 142

11. -Comprehensive listing of ab initio theoretical treatments of He!,+ •.:ith method of treatment (in chronological order) 146

12. Comprehensive listing of approximate and model theoretical treatments of HeH w4th method of treatment (in chronological order) .... 149

ix ABSTRACT

This dissertation describes the first high-precision observation of the infrared spectrum of the helium hydride molecular ion HeH+. The frequencies of five vibrational-rotational transitions in the range

1700-1900 cm"* in the X*£+ ground electronic state of ^HeH+ have been measured to ±0.002 cm"* (±1 ppm). The Doppler tuned ion beam laser spec­ troscopic method was used in making the measurements: In a region of constant electrostatic potential, an HeH+ ion beam of several keV energy is intercepted at a small angle by a beam from a carbon monoxide infra­ red gas laser. The energy of the ion beam is adjusted to Doppler-shift an ion transition into resonance with a nearby laser line. On resonance the laser light stimulates transitions to take place. If the resonating states differ in population, the laser-induced transitions produce a net population transfer. The occurrence of population transfer is detected by monitoring the transmission of the ion beam through a gas target down­ stream from the laser beam interaction region. The transmission through the target is dependent upon the ion beam vibrational-state population distribution and therefore is sensitive to changes in the population distribution, because the cross-section for charge-exchange neutraliza­ tion of an incident ion is dependent upon the vibrational state of the ion.

The current interest in molecular ions in general, and in HeH+ in particular, is explained. The existing theory of the structure of

HeH+ is summarized and a comprehensive listing of theoretical treatments of the structure of HeH+ is given. The meager previous experimental

x The current interest in molecular ions in general, and in HeH+ in particular, is explained. The existing theory of the structure of

HeH+ is summarized and a comprehensive listing of theoretical treatments of the structure of HeH+ is given. The meager previous experimental work on HeH+ is reviewed. The principles of the Doppler tuned ion beam laser resonance method are discussed and the experimental apparatus used is described in detail. The acquisition and analysis of the data is described and the results are compared with the best existing theoreti­ cal predictions of the transition frequencies. The present experimental values (given by D. E. Tolliver, G. A. Kyrala, and W. H. Wing, Phys. Rev.

Lett. 43, 1719) for the measured transitions are (with the corresponding values calculated by D. L. Bishop and L. M. Cheung, J. Mol. Spectrosc.

75, 462, given in parentheses): (v,J) = (1, ll)-<-»-(0, 12), 1855.905 cm 1

(1856.152 cm"1); (1,12)^.(0, 13), 1751.971 cm"1 (1752.198 cm"1); (2,8)+-+

(1,9), 1896.992 cm"1 (1897.139 cm"1); (2,9)<->-(l,10), 1802.349 cm"1

(1802.492 cm"1); and (2,10)+->(1,11), 1705.543 cm"1 (1705.684 cm"1). It is seen that the present experimental values deviate from the theory by typically 0.2 cm 1, and are two orders of magnitude more precise than the theoretical values. CHAPTER 1

INTRODUCTION

This dissertation describes the first high precision infrared

spectroscopy of the helium hydride ion HeH+. HeH+ is a member of the

large set of molecular species that are not electrically neutral, but

are charged; that is, the set of molecular ions. There are probably

more kinds of molecular ions than there are neutral molecules. Nearly

every neutral molecule has a stable singly-charged positive ion; multiply-

charged ions also exist. In addition, there are negative molecular ions

and ions for which there correspond no stable neutral molecules.

Interest in Molecular Ions

Many important physical and chemical processes involve the par­

ticipation of molecular ions. This is partly because molecular ions are

very reactive chemically - even more so than free radicals. Therefore

they tend to dominate the chemistry of media in which they occur in sig­ nificant quantities. Areas where molecular ions might be expected to

play an important role include the earth's upper atmosphere, extrater­

restrial regions, flames, combustors, chemical reactors, and perhaps controlled thermonuclear fusion devices.

That molecular ions are currently the subject of intense inter­ est is illustrated by the recent appearance of two reports. In an in­ vited paper given at the Joint Symposium of the American Physical Society

1 2 and the American Association of Physics Teachers on the Frontiers of

Optical Physics during the annual meeting of the American Physical

Society in Chicago, January 1980, Gerhard Herzberg (1) reported on re­ cent laboratory work involving astronomically important molecular ions.

Also, in Physics News in 1979 (2), a publication designed to call atten­ tion to newsworthy developments in physics in 1979, the American

Institute of Physics reports on recent laboratory work in spectroscopy of molecular ions.

The internal structure (and spectroscopy) of molecular ions, being differentiated from their reaction and chemical properties, is of interest mainly to three groups of researchers: the astrophysicists, the quantum chemists, and the physicists.

Since a large part of the matter in the universe is in the plas­ ma state, it is likely that there are regions where molecular ions exist in significant quantities. Such regions probably include stellar atmos­ pheres, interstellar matter such as planetary nebulae, upper atmospheres of planets, tails of comets, etc. Current theories (3,4) predict that the chemical dynamics occurring in media such as interstellar gas clouds are largely governed by ion-molecule reactions. Because of their impor­ tant role in the chemistry of such media, the detection of molecular ions is of special interest to astrophysicists. Unfortunately, astro­ nomical detection of molecular ions has been hampered by a severe lack of laboratory spectroscopy of such species. In fact, only a handful of molecular ions have been unambiguously identified in extraterrestrial regions. As a result, many details of chemical reactions, which affect 3 the observed abundances of the elements and hence affect hypotheses of the origin of the universe, must be inferred from indirect evidence.

Theoretical quantum chemistry involves the calculation of molec­ ular properties from first principles. This approach is tractable only for the simplest molecules. For this reason, the hydrogen molecular ion

H* and its isotopes may become the "hydrogen atom" of molecular physics.

Similarly, the two-electron systems, H^ (the simplest homopolar two- electron system) and HeH+ (the simplest hete.ropolar two-electron system) may serve as the "helium atoms" of molecular physics, in which electron correlation effects, etc., may be studied. Again, the lack of experi­ mental results has hampered progress in this area. Theoretical chem­ istry is also concerned with calculation of collision cross sections and rate coefficients. Such calculations require a precise knowledge of the potential curves for the various interactions that occur (5); evaluation of these would be greatly aided by experimentally derived energy levels.

Molecular ions are of interest to physicists first of all be­ cause their study involves natural extensions of the experimental and theoretical techniques of atomic physics. Some of the same experimental and theoretical tools that are applied to atomic systems may also be applied to molecular ions. Also, the simplest molecular ions, because of their simplicity, offer the possibility of making new measurements of some of the fundamental constants of nature. In addition, the simple molecular ions can also serve as proving grounds for application of ac­ cepted physical ideas to new areas, for instance, the calculation of the 4

Lamb shift in molecular systems. Again, experimental results would be of great value in these efforts.

As mentioned above, lack of experimental results on molecular ions has hindered progress in several areas. This experimental defi­ ciency is primarily due to the high reactivity of the ions: ion-molecule reactions are so rapid that concentrations of the ions of interest are usually very low. This makes conventional absorption and emission spec­ troscopy very difficult, if not impossible.

Most of the limited success in molecular ion spectroscopy has been in the observation of electronic emission spectra. Work in this field has been reviewed by Herzberg (6). Using a beam technique that is very similar to the one used in the present experiment, Carrington and co-workers have obtained rotationally-resolved electronic spectra in

0* (see Ref. 7 ), C0+ (see Ref. 8 ), and f^O* (see Ref. 9 ). Only a handful of purely rotational spectra have been obtained. The first ro­ tational spectra in molecular ions were detected and identified by radio-astronomers (10-12). Subsequently, a few laboratory spectra have been obtained (13-16). The hyperfine spectrum of the H* ion has been studied by Dehmelt and Jefferts (17) and by Jefferts (18), but this does not seem to have been done for any other species. Direct study of the vibrational spectra of molecular ions has also had very limited suc­ cess. Only one completely resolved vibrational-rotational spectrum has been reported (19-22) prior to the present work. Molecular ions have also been studied by the techniques of photo-electron spectroscopy and photodissociation spectroscopy (see, for example, the work of Moseley et al. (23) and McGilvary and Morrison (24)). The resolution in these 5 types of experiments is usually low by normal spectroscopic standards; however, these spectra are usually the only source of information and are valuable for that reason. In contrast, electronic, vibrational, rotational, and hyperfine transitions in neutral molecules have all been extensively studied for many years, and numerous ultra-high resolution spectra have been obtained.

History of and Current Interest in HeH*

The particular species under study here is the helium hydride molecular ion HeH+. Ions with mass-to-charge ratio 5 were discovered in ion sources containing mixtures of helium and hydrogen and identified to be HeH+ by Hogness and Lunn (25) in 1925. Apparently this was the first observation and preceded any by the pioneer mass-spectrometrist, F. W.

Aston. Aston's 1933 book on mass-spectrometry [26) discusses HeH+, but his 1923 book (27) does not mention it. Since this early discovery, a considerable amount of theoretical work has been done on the HeH+ sys­ tem. Also, ion-molecule reactions involving the production, destruction, or other participation of HeH+ have been extensively studied, both theo­ retically and experimentally. There has been, however, very little ex­ perimental work done on the structure of the molecule. Theoretical and experimental work relating to the structure of HeH+ are summarized in later chapters. Ion-molecule reactions involving HeH+ are also dis­ cussed later, but only to the extent necessary to discuss the operation of the ion source used in the present experiment.

In recent years, several unrelated studies have renewed interest in the internal structure of HeH+. One problem is concerned with 6

3-decay of tritium in TH and molecules. This problem has been stud­ ied experimentally by Snell et al. (28) and Wexler (29). They found that approximately 90% of the positively charged fragments were the bound daughter molecule ^HeH+ or ^HeT+. More recently, Raitz Von Frentz et al. (30) have detected radiation from ^HeT+ formed in the decay of

T^. The B-decay process can be described theoretically by a "sudden perturbation" calculation in which the daughter molecule is populated according to the Franck-Condon principle. The effects of recoil from the B-particle must also be accounted for. A complete understanding of these processes requires knowledge of all the bound and unbound states of the daughter molecule.

The most interest in HeH+ has been stimulated by astrophysical considerations. Hydrogen and helium are thought to be the two most abundant elements in the universe. In view of this abundance, it seems likely that HeH+ exists in some regions of stellar atmospheres or inter­ stellar regions (31). Merrill et al. (32) have observed fairly strong unidentified features at 2.43, 3.09, 3.27, and 3.4 ym in the infrared spectrum of the planetary nebula NGC 7027. Dabrowski and Herzberg (31) have tentatively identified the 3.09 and 3.27 ym features as being due to vibrational-rotational transitions in HeH+, but agree that this infer­ ence is not conclusive. More recently, Scrimger et al. (33) have searched the range 0.86-1.65 ym in NGC 7027. They failed to find any lines attributable to HeH+, thus casting doubt on the identifications of

Dabrowski and Herzberg.

Stecher and Milligan (34) have observed "broad-band flux defi­ ciencies" (interpreted to be absorption bands) in the 160-260 nm range 7 in the ultraviolet spectra of several early-type stars. They suggest that this might be due to absorption by electronic transitions out of

the ground electronic state occurring in HeH+ in the outer atmospheres

of the stars. Werner (35), however, has argued that this wavelength re­

gion did not match transitions from the ground to the first excited state

of HeH+, but his analysis does not rule out transitions which might occur

between higher excited molecular states (36).

On the theoretical side, Black (37) has calculated that molecu­

lar ions such as HeH+ can exist in detectable quantities in transition zones of ionized nebulae. His calculations were based on a simple model of the transition zone and support the identification of the 3.09 and

3.27 ym features observed by Merrill as being due to HeH+. Flower and

Roueff (38) have done similar, perhaps more detailed, calculations, coming to the opposite conclusion. They suggest that if HeH+ is detec­ table under some favorable circumstances, it would be by observation of the (v,J)= (0,l)-*(v,J)= (0,0) rotational transition at about 149 ym.

It has also been suggested (39) that HeH+ might be important in determining the ionization balance of the ionosphere of Jupiter. The reaction rates involved have been studied in detail by Johnsen and

Biondi (40). Similarly, HeH+ may be important in the problems of ther­ mal stability and convection phenomena in stellar atmospheres. Edmonds

(41) and Fallon et al. (42) have theoretically examined the relative importance of several reactions which might dominate such transport processes.

In view of such scattered and incompatible observations and cal­ culations, it seems safe to say that HeH+ has not yet been identified in 8

extraterrestrial sources. It is hoped that the results of the present

work will help to stimulate searches for HeH+ in extraterrestrial sources

and will aid in resolution of the uncertainties in the identification of

the already observed spectral features.

It is also hoped that the present results will stimulate further

theoretical calculations on the structure of this simple molecular sys­

tem and will aid in the understanding of the details of the various phe- + nomena in which HeH plays a part.

Summary of Chapters

Chapter 2 presents a review of the theoretical description of diatomic molecules and also gives a summary of the theoretical work done on HeH+ from the beginning in 1933 through the present. Chapter 3 recounts the limited experimental studies that have been carried out on

HeH+. The principles of the Doppler tuned ion beam laser spectroscopic method used in the present experiment are described in Chapter 4. Chap­ ter 5 describes the considerable amount of apparatus involved in the present experiment. The experimental results are discussed and compared to existing theory in Chapter 6. Appendix A presents comprehensive list­ ings of theoretical work on the structure of HeH+. A derivation of the vibrational-rotational selection rules and an approximate calculation of the transition electric dipole moment are given in Appendix B. Appen­ dix C consists of a listing of the microcomputer progam which controls the experiment. CHAPTER 2

SUMMARY OF THE THEORY OF THE STRUCTURE OF HeH+

The helium hydride ion is among the simplest heteronuclearmolec­ ular systems. This fact has stimulated considerable theoretical inves­ tigation beginning in 1933 and continuing to the present. Much of this work has been ab initio calculations. By ab initio, I mean a calcula­ tion which attempts to predict properties of the molecule from first principles rather than by some sort of empirical, semi-empirical, or approximate approach. Since the publication of the fairly accurate re­ sults of Wolniewicz in 1965, several authors have used HeH+ as a model for testing various approximate methods of calculation designed for ap­ plication to more complex molecules. The purpose of this chapter is to summarize the ab initio investigations of HeH+ and to indicate what cal­ culations remain to be done in order to make a meaningful comparison with the present experimental results. Since the electronic and vibrational-rotational motions in the molecule occur at very different frequencies, they are discussed separately.

Electronic States in HeH+

In contemplating the structure of molecules, it is appealing to attempt in some way to "build up" a structure from the known constituent components just as is done in the case of atoms. In any such attempt, an important part is played by the fact that the masses of the atomic nuclei are very large compared to the mass of the electrons. Since all

9 10 the particles in the molecule are subject to similar forces, the elec­ trons move very much faster than the nuclei; to some degree of approxi­ mation, the nuclei may then be regarded as fixed centers of potentials in which the electrons move. Thus, a molecule may be hypothetically constructed by fixing a configuration of nuclei and then bringing in electrons and considering how they can be arranged into stable wave functions or orbitals. The different possible arrangements are then the different possible electronic states.

Certain characteristics of the states can be derived from con­ sideration of symmetry. In a diatomic molecule, in contrast to an atom, there is no longer spherical symmetry, so the orbital angular momentum is no longer conserved. However, there is axial symmetry about the line between the nuclei, so that the component of orbital angular momentum in that direction is conserved. Therefore, the orbitals can be labeled according to the value of A, the projection of the orbital angular momen­ tum on the internuclear axis. In analogy with the labeling of atomic states, states with A=0, 1, 2, ..., are designated by I, II, A, ..., re­ spectively, using Greek instead of Latin letters.

There is also symmetry of reflection in any plane containing the internuclear axis. Although the energy is not changed by such a reflection, the state obtained is not identical to the initial state since the sign of the angular momentum (which is an axial vector) about the axis is changed. Thus, states with A/0 are doubly degenerate. For

A=0, the state of the molecule is not changed at all on reflection.

Since the probability density must be invariant, a reflection operation can only multiply the wave function for a £ state by a constant. Two such reflections must result in the original state so the constant is

±1. Thus, we have two possible kinds of E states: those for which the

wave functions are unaltered by reflections and designated by E+, and

those for which the wave functions are inverted by reflection and desig­

nated by £ .

If the molecule consists of two identical atoms, there is addi­

tional symmetry with respect to exchange of nuclear coordinates. Thus,

the states are additionally labeled by the subscripts g (for even) or

u (for odd) according to whether the state is even or odd under this

exchange.

Each electron state is also characterized by the total spin S of

all the electrons in the molecule. As in atoms, the number 2S+1 is

called the multiplicity of the state and is written as a superscript be- 13 fore the Greek letter for the state. Thus, we have Z and II states

for example.

It is also customary (see Ref. 43 ) to precede the term symbol

with an additional, empirical designation. The ground state is labeled

X. Successive states with the same multiplicity as the ground state are

labeled with A, B, C, ..., and successive states with the other multi­

plicity are labeled with a, b, c, ... .

It is useful to consider two limiting cases of the diatomic mole­

cule structure. First, since the molecule is formed from and dissoci­

ates into atoms (including ions), it is useful to consider how the

molecular states go over into the states of the constituent atoms as the

internuclear distance R goes to infinity. This is the so-called

"separated-atom limit." Wigner and Witmer (44) have derived very general rules for determining these correspondences using the methods of group theory. Many of these rules can be derived simply by considering how the orbital and spin angular momenta and parity characteristics of the atoms can combine to give molecular states of given character (45; pp. 315-322). The other limit is taken as R goes to zero, the so-called

"united-atom limit." In this limit the wave functions must approach the wave functions for a single atom (or ion). Again, there are obviously connections between the characteristics of the molecular state and the united-atom state. Carrying out these studies leads to "correlation diagrams" for the molecular states.

The application of these ideas to HeH+ is shown in Fig. 1. The united-atom limit for HeH+ is the two-electron ion Li+. The configura­ tions of several low-lying states of Li+ are given on the left-hand side of the diagram. There are several possible separated-atom limits cor­ responding to dissociation into helium (ground state or excited) and a proton, or into a helium ion (ground state or excited) and a hydrogen atom (ground state or excited). Several of these are given on the right- hand side of the diagram. The considerations outlined above have been applied to arrive at the correlations to the molecular states as indi­ cated by the connecting lines.

The considerations discussed so far can only give information on the number and types of states possible in a molecule. No mention has been made as to which states lead to stable molecules and which lead to repulsive interactions of the two atoms. To address these questions, it is necessary to replace the schematic correlation lines in Fig. 1 by adiabatic potential energy curves; that is, by curves giving the 13

Hels2p3P + H+ -2.133 3 Li * IS3S S Hels2s's + H+ -4.752 -2.146

Hels2s3S + H+ + Li IsZp 'P -2.175 -4.993 Li+ Is2p 3P -5.028 Li+ Is2s 'S -5.041 Li + Is2s 3S -5.111

He+ ls2S + Hls2S -2.500

Hels2 'S + H+ -2.904 X'L+ Li+ ls2lS -7.280

FIG. 1. Correlation diagram for HeH . -- The left-hand side repre­ sents the united-atom limit and the right-hand side the separated-atom limit. The energy coordinates have been arbitrarily scaled for conve­ nience. The indicated energies are the known exact values in hartrees (1 hartree=2.19x10^ cm"*). The correlations are arrived at by applica­ tion of the considerations given in the text. After Ref. (36). potential energy of the configuration as a function of R as the inter-

nuclear distance is changed very slowly. A minimum in the curve means

that a stable state of the molecule is possible, whereas a purely repul­

sive curve means the configuration is unstable.

Historically, two loosely-defined schools of thought have devel­

oped in approaching this problem. Both make extensive use of the varia­

tional principle. One method grew out of the approach first applied by

Heitler and London (46) to the hydrogen molecule and can be termed

"valence-bond" calculation. This method has since been extended in quite

a variety of ways. Generally, some sort of plausible electronic wave- function, with adjustable parameters, is invented to model the various bonding mechanisms thought to occur in the molecule. The variational principle is then used to optimize the adjustable parameters at every value of the internuclear distance R. The second method, initiated by

Hund (47) and Mulliken (48), and generally termed the "method of molec­ ular orbitals" (MO), proceeds similarly but uses linear combinations of atomic orbitals (LCAO) with adjustable coefficients as the trial wave functions.

A large body of ab initio theoretical work on HeH+ has grown up beginning in 1933 with the short letter by Glockler and Fuller (49), which was an early valence-bond type of calculation. Since their work, both of the above methods have been applied to HeH+ in a wide variety of ways. Table 11, containing a comprehensive list of references to ab initio work in HeH+, together with a brief description of the method of treatment in each case, is given in Appendix A. These various attempts have all been more or less successful in obtaining realistic potential 15 energy curves for several low-lying electronic states. Some of the curves of Michels (36) are reproduced in Fig. 2 as an example. From

Fig. 2 we see that the ground electronic state is the X*I+ state and is stable. This is the most important state and the only one of interest here.

The wave functions determined from the above calculations can be used to obtain other properties of the molecule such as the electronic charge distribution. The charge-density contours calculated by

Peyerimhoff (50) are reproduced in Fig. 3. Notice that as the inter- nuclear distance gets large the electrons tend to leave the hydrogen nucleus and concentrate around the helium nucleus. This confirms that the ground state correlates to a helium atom and a proton in the separa­ ted limit as indicated in Fig. 1. Knowing the charge distribution al­ lows calculation of the electric dipole moment, which is necessary for predicting optical activity. (An estimate of the transition dipole mo­ ment is derived in Appendix B.) However, accurate calculation of vibra­ tional and rotational energy levels of the ground electronic state in which we are interested here depends on the development of a much more detailed theory which is discussed in the next section.

Vibrational-Rotational States

In the preceding discussions, the potential energy of an elec­ tronic state was found as a function of the internuclear separation.

Because the nuclei move much more slowly than the electrons, this energy can be regarded as an average potential in which the nuclei move. These nuclear motions can best be considered by going back to the complete 16

Hels2s 'S + H 2.146 He|s2s 3S -I- H' -2.20 -2.175

ai -2.40

-2.500

^ -2.60

-2.80

-2.904

20 oo INTERNUCLEAR SEPARATION (bohrs)

FIG. 2. Potential energy curves for some low-lying states of HeH . - These curves are only qualitatively correct. After Ref. (36). FIG. 3. Electron density contours for ground state HeH+. -- (a) Contours for internuclear separation R=1.0 bohr on the repulsive part of the potential curve, (b) contours for R=1.455 bohr at the equilibrium separation, (c) contours for R=2.0 bohr on the attractive part of the potential curve, (d) contours for R=3.5 bohr near the dissociation limit. After Ref. (50). 1 1 He H He H (a) 1.0 bohr (b) 1.455 bohr

-+- He H~ He H (c) 2.0 bohr (d) 3.5 bohr

FIG. 3. F.lectron density contours for ground state HeH+. 18

Schr&dinger equation for the system. The nonrelativistic, spinless,

time-independent SchrBdinger equation for a two-electron, diatomic mole­

cule is

3CV (R^.R^) = E V , (1)

where the Hamiltonian is given by

h2 _ 2 h2 _ 2 ft2 _ 2 h2 2 V V V V + V ^ " 2Ma Ra " 21^ Rb " 2m Rj " 2m R2 ' ^

in which the potential energy is

2 2 2 Z I e 2 Z e Z e V = a b + e a a |R-i| iR.-iU |R-1| |R-L| 'an' '12' 'a 11 'a 21 C3) Z,7 e 2 Z,7 e 2 b b Mil IV*2

Here the subscripts a and b refer to the two nuclei, and the subscripts

1 and 2 refer to the electrons. Z and M are the atomic number and a a mass of the ath nucleus, respectively, and the R's are the positions of the various particles. Equation (1) is an exact equation for the com­ plete molecular wave function including electrons and nuclei, in the nonrelativistic limit. An analytic solution to this four-body problem has not been given, so we must proceed with approximate calculations.

Following Kolos and Wolniewicz (51), it is convenient to sepa­ rate off the translational motion of the center of mass and transform the problem to a set of internal coordinates in which R=^a~^ is the 19 internuclear distance and r .=R .-^(R +R, ), for •£=1,2, is the position of Z- U 3.D the ith electron measured from the geometrical center of the molecule.

In these coordinates the kinetic energy portion of the Hamiltonian is written (neglecting the center-of-mass motion):

T " - II 'R - 55 - §iJ t'l*V2 - 2T (V'W)- <4) where

p " M + M, ' y M. " M 1 J 3D a. D a.

The first two terms are the kinetic energies of the nuclei and the elec­ trons, respectively. The two cross-terms represent the so-called mass polarization. They will lead to coupling of motions of the two elec­ trons and of the electrons and the nuclei.

To proceed with the theory as developed by Born (52,53), the

Hamiltonian is divided into two parts. Then

X = K + JC , (6) o where ( v 7 *0' - H 'iV) • • < > and

/ h2 2 ft2 „ ,2 ft2 x '5? 'R - 8? < VV - 2jr(V a.

3C is the Hamiltonian for the "clamped nuclei," that is, the Hamiltonian I for the electrons in the field of two fixed nuclei; and 3C describes the kinetic energy of the relative motion of the nuclei and the coupling be­ tween the electronic and nuclear motions. 20

We now suppose that the "clamped nuclei" SchrBdinger equation

for the electrons,

3C ijj (r.;R) = U (R) \p (r.;R) (9) oni n n t,

has been solved for all electronic states, perhaps by one of the methods

discussed in the preceding section. The electronic wave function t/i is

a function of the electronic coordinates and also depends parametrically

on the nuclear separation R. The subscript n denotes all the quantum

numbers necessary to specify a state. Since the wave functions depend

parametrically on R, the eigenvalues also depend on R.

We are striving for a solution of the complete SchrBdinger

equation

XV = E f (?.,£) . (10)

Since the eigenfunctions form a complete set, a solution of Eq. (10)

can be assumed in the form of the expansion

' (11) where the $£$)'s are nuclear wave functions to be determined. Here the

sum denotes summation over the discrete spectrum and integration over

the continuum.

Upon substituting Eq. (11) into Eq. (10), multiplying through

from the left by and integrating over the electronic coordinates, one gets a set of equations for the nuclear functions ^(R): 21

c (it) e S) Vib tl2) f S 7R * VR> * » " K< " "S '

Here the so-called dynamical correction operators are

C „$) = 5f' (— VD + 5— (V.+VjW dr.dJ.Lv_ . (13) ml ml |J m\ p R 2\i 1 2J/Yl 1 2j R v 1

Equation (12) represents a rigorous set of equations, fully equivalent to the exact SchrBdinger Eq. (10). Complete solution is not tractable and at this point approximations must, be made.

If we neglect all the C j^'s and retain only one term in the ex­ pansion (11), we get the "clamped nuclei" equations for the nuclear mo­ tion. This is equivalent to the famous Born-Oppenheimer approximation

(54) in which the nuclear and electronic motions are not coupled. For light, simple molecules such as HeH+, the Born-Oppenheimer results are accurate to about 5 cm ^ for the vibrational-rotational energy levels and to about 1 cm * for the transition frequencies.

If the right-hand side of Eq. (12) is neglected, that is, if the off-diagonal correction terms are neglected and the diagonal term is re­ tained, and again only one term in the expansion (11) is retained, one gets an equation of the form

M E R) 0 {- tv 'R • - jv = • ("3 in which the diagonal correction term has been absorbed as part of the t effective internuclear potential Um(R). This is the so-called adiabatic approximation and is equivalent to letting the nuclei move in an 22 effective potential obtained by averaging the full Hamiltonian over the electronic motion rather than averaging only Hq, as is the case in the

Born-Oppenheimer approximation. This is the highest level of approxi­ mation for which an interatomic potential is a good model. Adiabatic calculations have been carried out only for one- and two-electron mole­ cules. The differences in the adiabatic and the Born-Oppenheimer results are on the order of 5 cm * in the energy levels and on the order of

1 cm * in the transition frequencies.

Thus far, no mention has been made of the separation of the nu­ clear wave function into vibrational and rotational parts. Assuming the nuclear wave function can be written in the form

= £ fCRD 7(e,

the rotation can immediately be separated off, giving an equation for the vibrational wave function f(R):

T--4- + U'(R) + J(J+j:) - E .if _(R) = 0 , (16) V J V J 2y d2R 2yR ' ( > where v and J are the vibrational and rotational quantum numbers, respec­ tively. This equation is then solved numerically, giving as the eigen­ values the vibrational-rotational energy levels E . It should be V,J T noted that the separation effected in Eq. (15) implies a neglect of various small terms which couple the vibrational and the rotational motion. At this level of approximation, these interactions are usually not important in vibrational-rotational transitions, but could be 23 important in pure-rotational spectra since they are then much larger effects relative to the transition frequency.

Improvement in the adiabatic approximation can be made in two ways. One could apply perturbation theory, treating the previously ne­ glected correction terms as a small perturbation. The result could be termed a nonadiabatic correction. Alternatively, one could solve the complete set of coupled Eqs. (12), or equivalently solve the complete

Schrodinger equation (Eq. 1 ), by a variational method, for instance.

The result of this approach would be exact (in as far as the equations could be solved) and could also be termed a nonadiabatic result. For one- and two-electron molecules, the nonadiabatic corrections are less -1 -1 than about 5 cm in the energy levels and less than about 1 cm in the transition frequencies.

Even given an exact solution of the SchrBdinger equation (1), there are various corrections arising from relativity and quantum elec­ trodynamics that have been neglected and that make significant contribu­ tions to the vibrational-rotational energy levels and transition frequencies. The exact, relativistically and quantum electrodynamically correct equation has not been given. The corrections can be taken into account either by using perturbation theory, or by including appropriate relativistic and radiative terms in the SchrBdinger equation Hamiltonian and obtaining a corrected interatomic potential curve. For the only two-electron system for which calculations have been carried out, both the relativistic and the radiative corrections are on the order of

0.02 cm * in the transition frequencies. However, they are of opposite sign and thus tend to cancel out. 24

Hyperfine Structure

In addition to the spatial coordinates discussed so far, there

are the spin angular momenta coordinates arising from the relativistic

theory. Interactions among the spin and other angular momenta in the system give rise to hyperfine structure. In the ground electronic state

of HeH+ the only unpaired angular momenta are the proton spin and the

molecular rotation. Interaction of the magnetic moments associated

with these angular momenta results in a splitting of the energy levels

into two hyperfine levels with the splitting given by (55; p. 216):

AE = B -^hY* J^N^r . 4, (17) hfs Ii 4nZ-i E - E I cR ( 11 where I is the proton spin, J is the molecular angular momentum, I is the moment of inertia of the molecule, qHe is the net charge of the helium nucleus plus the electron in a closed shell about it, and is the (or­ bital) angular momentum of'the valence electron about the proton. Here the sum over n is over all electronic states (|o) and |n) are the ground

and excited electronic states, with energies Eq and E^, respectively) and

b - h -7 , 1 v f 1 Ol 2 ' aon I < _3 > ' (18) 8HTT I T av where the average is over the nth excited electronic state, and and liQ are the magnetic moments of the proton and the electron, respectively.

It should be noted that Eqs. (17) and (18) are correct only for the ground vibrational state, since I is evaluated at the equilibrium internuclear separation. More generally, Eq. (17) should be vibrationally-averaged with the variation of I with R included in the average.

The splitting given by Eq. (17) is usually very small and has not been calculated for HeH+. In particular, the a 's have not been r on evaluated and I will not attempt it here. It is reasonable to expect that in HeH+ the splitting will be of the same order of magnitude as in

+ the H2 molecule (which is isoelectronic to HeH ) where it is about 100 kHz. This is negligible compared to the instrumental resolution in the present experiment, so hyperfine structure is of no consequence in the present work.

Application to HeH+

I will now discuss the extent to which calculations of the vibrational-rotational states have been applied to HeH+. Very accurate interatomic potential curves were first obtained by Wolniewicz (56) in

1965. This was a valence-bond type of calculation using trial wave functions of the form N

Y (1,2) = £ ^1^(1,2) + (2,1)j , (19) where

•id'2) = exP |"al5l " a252 + 6lnl + 32T12 }

* I™1 x Zi1 x x x • (20)

Here p=2r12/R; a^, o^, Bj, S2> and C^are variational parameters; £., n.are the coordinates of the jth electron in elliptic coordinates; and J 26

and R are the interelectronic and internuclear separations, respec­ tively. Each basis function is prescribed by the (integer) values of the exponents m., n., k., SL., and q.. Wolniewicz used a 64-term wave t' 1* 1* I* Is function of this type. Kolos (57) and Kolos and Peek (58) extended

Wolniewicz's calculations to larger and shorter R, respectively. In

1977 Dabrowski and Herzberg (31), using these potential curves, calcu­ lated, for the first time, vibrational-rotational energy levels and transition frequencies. This was done in the Born-Oppenheimer approxi­ mation, and the transition frequencies were expected to be accurate to

"a few cm

Very recently Bishop and Cheung (59) have recalculated the po­ tential curves, using a 255-term wave function, also of the form (19).

The Born-Oppenheimer transition frequencies derived from their potential curves differ from those of Dabrowski and Herzberg by typically 0.15cm*.

This indicates that the 64-term basis used by Wolniewicz was not suffi­ ciently large. Computational problems prevented Bishop and Cheung from increasing the basis set to beyond 255 terms.

Bishop and Cheung also calculated vibrational-rotational energy levels in the adiabatic approximation. They estimate that the errors in their adiabatic energy levels are less than 1 cm * and that, due to error cancellation, the errors in the transition frequencies are likely to be less than this.

This represents the present limit of the theory of HeH+. To date no nonadiabatic calculations have been done. Neither have any ap­ proximations of nonadiabatic corrections been given. Likewise, no cal­ culations or approximations of relativistic and radiative corrections 27 have been given. A comparison of the best theory available at present

(the adiabatic results of Bishop and Cheung) with the present experimen­ tal results will be given in Chapter 6. There I will also give estimates of the neglected corrections.

Peripheral Theoretical Treatments of HeH+

In addition to the previously discussed ab initio treatments of

HeH+, other theoretical work has been done involving this molecule.

Schopman and co-workers have observed unimolecular dissociation in beams of HeH+ (see Chapter 3). Peek (60) was the first in giving the correct explanation of this phenomenon in terms of quasi-bound, rotationally- predissociating states of the ground electronic state. Further calcula­ tions involving these quasi-bound states have been reported by Bernstein

(61) and by Price (62).

Since the accurate results of Wolniewicz in 1965, several papers have appeared in which various approximate methods designed for use with more complex systems have been applied to HeH+ as a test of the methods.

Since these results have no bearing on the present work, they are not discussed here. However, since the division of variational calculations into ab initio and approximate methods is somewhat arbitrary, the treat­ ments not listed in Table 11 in Appendix A as ab initio calculations are listed in Table 12 in Appendix A as approximate treatments.

There also exists a large body of theoretical work concerning the details of the formation of HeH+ from helium and hydrogen. This topic is beyond the scope of this summary and is mentioned briefly in

Chapter 5 in the discussion of the production of HeH+ in the ion source. CHAPTER 3

REVIEW OF PREVIOUS EXPERIMENTAL WORK ON HeH+

In contrast to the extensive theoretical work on the structure of H

+ HeH + H2 -* He + H* . (1)

As a result, very limited quantities of HeH+ can be obtained for experi­ mentation. Because of this difficulty, only a handful of experimental results dealing with the internal structure of HeH+ exists. All but one of these observations involve some sort of beam experiment, where chemi­ cal reactions such as reaction (1) are minimized. These few previous results are the subject of this chapter.

The area of ion-molecule chemical reactions involving HeH+, as opposed to the area of the internal structure of the molecule, has been studied quite extensively experimentally. This field of study is not within the scope of the present experimental program, and such work is

28 not mentioned in this dissertation except peripherally as needed in the discussion of the ion source in Chapter 5.

Experiments on Molecular Bond Strength

Perhaps the most fundamental measurement one can make on a dia­ tomic molecule, after one knows that it exists as a stable molecule and what its constituents are, is that of the molecular bond strength. This is usually done by measuring in some way the dissociation energy of the molecule. The usual methods are photodissociation or electron-impact dissociation. An indirect method of obtaining the same information is by studying collisional interactions of the particles making up the molecule.

In the case of HeH+, the appropriate collision partners are helium atoms and protons (H+). The most reliable experiments of this type have been done by Weise et al. (63). They studied the low energy scattering of protons off helium. From analysis of their data, the parameters of a model interatomic potential (the so-called Morse poten­ tial) were determined. Then a value of the dissociation energy Dg, which is the difference between the minimum energy of the potential well and the infinite separation energy, was obtained. Huber and Herzberg

(64) have critically reviewed the data of Weise et al., and they con­ clude that the best experimentally derived dissociation energy for HeH+ is De=2.00±0.1 eV (=16000±1000 cm *). This compares with the theoret­ ical value of 2.0402 eV given by Dabrowski and Herzberg (31) (who used the potential curves of Kolos (57) and Kolos and Peek (58)). The

±1000 cm * uncertainty (1 part in 16) in this experiment should be compared to the 0.002 cm * uncertainty CI part in 10^) in the present transition frequency measurements (65).

Unimolecular and Collision-Induced Dissociation of HeH+

When a molecule of several keV energy collides with a target t atom, it can be dissociated through several distinct mechanisms. By analyzing the momentum distributions of the various fragments, the ef­ fects of the various dissociation mechanisms can be sorted out. The two principal mechanisms for dissociation are (i) a vibrational-rotational excitation of the molecule into the continuum of the ground electronic state due to a hard collision of the target atom with one or both nuclei of the molecule, and (ii) an electronic excitation of the molecule to an excited state which may be repulsive or predissociative. In both cases, a broad distribution of momenta of the fragments would be expected.

In a series of experiments, Schopman and associates (66-69) have analyzed the momentum distributions of proton and He+ fragments from

10 keV HeH+ colliding with rare gas atoms. Instead of broad,relatively featureless distributions, they obtained distributions with several sharp peaks. Some of the peaks persisted even when the collision cham­ ber pressure was reduced to a level which should give an essentially zero background signal. They termed the mechanism giving rise to the zero pressure signal unimolecular dissociation. The narrow distribu­ tions of the proton fragments observed without a target gas were ini­ tially attributed to the decay of quasi-resoncoit states in the vibrational-rotational continuum of the electronic ground state (67,68), which were populated by mechanisms occurring in the ion source. The additional feature observed with a target gas were presumed to be due to the collisional excitation of these and additional quasi-resonant states.

Additional data (69) made this conclusion difficult to support. Probably the correct explanation was given by Peek (60). He calculated the ener­ gy and widths of all quasi-bound, rotationally-predissociating states and obtained reasonably good agreement with the experimental results.

Subsequently, Bernstein (61) used Peek's identifications and the data of

Schopman et al. (69) to calculate experimentally-derived spectroscopic constants (Dunham coefficients), which in turn can give information about the shape of the molecular potential well. More recently Fournier et al. (70) have repeated Schopman's experiments and have obtained even better agreement with Peek's theory.

Dissociation of Fast HeH+ Ions Traversing Thin Foils

If a molecular ion with a few MeV energy strikes a thin solid target, all its binding electrons are stripped off within a few angstroms after penetrating the front surface. There remains a closely-spaced cluster of ions or nuclei which continues to propagate through the tar­ get foil. These positively charged particles repel each other and move apart in a "Coulomb explosion," changing their initial potential energy into center-of-mass kinetic energy. Since the "snipping of the molecu­ lar bond" (that is, the stripping off of the electrons) occurs very quickly, analysis of the trajectories of the fragments emerging from the foil reveals information about the geometrical structure of the molecular ion (71). The trajectory analysis must take into account the phenomenon known as a "polarization wake." The polarization wake is an electron density oscillation induced in the target foil by the passage

of the charged projectile particles (72). If the scattering angle and

energy of the emerging fragments are analyzed and a joint distribution

in angle and energy is plotted, for a diatomic ion a ring-shaped distri­

bution pattern results. The diameter of the ring is determined by the

bond length in the molecule, and the "thickness" of the ring reflects

the variations in internuclear separations due to vibrational motion of

the molecule.

Gemmell and co-workers (73,74) have carried out this program for

3 to 4 MeV HeH+ ions impinging upon approximately 100-A thick carbon

foils. They have determined the vibrational ground-state equilibrium

+ internuclear separation in HeH to be RQ=0.79 A. The corresponding cal­

culated value (50) is 0.77 A. Further analysis of the observed distri­

bution (75) of internuclear separations reveals that the ions in the

incident beam are mostly in the ground vibrational state with a rela­

tively small fraction in the first or second vibrationally excited states. This result is consistent with the population distribution to

be expected for HeH+ formed from He and H* as discussed in the section on the ion source in Chapter 5.

Observation of Infrared Radiation from HeH*

+ One problem of interest involving HeH is that of 3-decay of tritium in TH or T^ molecules (see discussion in Chapter 1). Raitz

Von Frentz et al. (30) have observed 4.6 ym radiation emitted by 3 3 + 3 3 + ( He H) undergoing transitions in the v=l-*v=0 band. The ( He H) was formed in the B-decay of tritium in T£. The radiation was detected by by incoherent up-conversion of IR photons to visible photons, which were detected using standard photon counting techniques. A tunable IR interference filter swept out the spectrum. The spectrum they obtained is reproduced in Fig. 4. Also shown in Fig. 4 are the particular vibrational-rotational transitions thought to contribute to the emission band they observed. While these authors claim this to be the first ob­ servation of a vibrational-rotational transition in an isotope of HeH+

(76), it should be pointed out that in their experiment particular vibrational-rotational transitions were not resolved, but rather a band, made up of several vibrational-rotational transitions was observed.

Their instrumental resolution of 0.1 ym (20,000 ppm of the transition frequency) and transition frequency uncertainty of 0.03 ym (6000 ppm of the transition frequency) should be compared to the observed linewidths of 5-8 MHz (0.1 ppm) and transition frequency uncertainty of 60 MHz

(1 ppm) in the present experiment (65). 34

Spectral resolution

o> !0.5 o-r

3.5 4.0 4.5 5.0 Wavelength (/*)

4 3 + FIG. 4. ( He H) infrared emission spectrum obtained by Raitz Von Frentz et al. -- (a) Photon counting rate as a function of IR filter setting. (b) Emission spectrum obtained after correction for background radiation. (c) Expected vibrational-rotational spectrum in the v=l->-v=0 band, labeled by values of J, J'; the absolute band position is fitted to the curve in (b). From Ref. (30). CHAPTER 4

EXPERIMENTAL METHOD

As pointed out in Chapter 1 and reiterated in Chapter 3, the chief obstacle in doing spectroscopy of molecular ions is the very re­ active character of the ionic species. The high rate of chemical reac­ tions makes the ion concentrations invariably very low. An additional hinderance for vibrational-rotational spectroscopy is the relatively low performance of radiation detectors in the infrared region of interest.

A means of overcoming these problems is provided by the experimental method described in this chapter.

Forming the ions into a beam has several major advantages, as discussed by Carrington (77). First and most importantly, ions that are present in the beam experience a collision-free environment. Therefore, once molecular ions are established in a beam, chemical reactions which would destroy the species of interest are no longer a problem. Second, many molecular ions may be produced by ion-molecule reactions in a source when it is operated at an intermediate-to-high pressure. These ions may be extracted and accelerated to form a beam very quickly, so that they may retain the internal excitation they acquired in the source.

Third, mass spectroscopic techniques made available through the use of beams enable positive identification and specific selection of a partic­ ular ionic species for study. Fourth, light molecular ions (mass num­ bers less than about 20) that are accelerated to potentials of a few

35 kilovolts travel at velocities of about 105 m/sec. If a laser beam is colinear with the ion beam, the ions will experience a substantially

Doppler-shifted laser frequency. The shift can be varied easily by changing the accelerating voltage, making spectroscopy possible with fixed wavelength lasers.

A potential disadvantage of ion beam methods is the low particle densities in the beam. High quality ion beams with currents of up to a few microamps, corresponding to particle fluxes of about 10*^ ions/sec, 2 with beam cross-sections on the order of 1 cm are readily obtainable. 6 3 This gives ion densities in the beam of about 10 ions/cm . For such a small number of ions, absorption or emission of radiation is likely to be undetectable. However, the high degree of spatial control over the beam can be utilized to provide very efficient indirect methods for de­ tecting when spectroscopic transitions have been made.

The Doppler Tuned Ion Beam Laser Resonance Method

A technique which takes advantage of the properties discussed above has been developed in this laboratory (19-22) and has been called the Doppler-tuned ion beam laser resonance method. (Doppler tuning of laser radiation into resonance with molecules or atoms in beams by vary­ ing the interaction angle had previously been used by others (78,79).)

The basic concepts of the method are illustrated in Fig. 5. Molecular ions are produced in an ion source in such a way that the ions have sub­ stantial inequalities among internal state populations. It will be seen below that population differences are critical if net stimulated transi­ tions are to be observed. Next, the ions are accelerated and formed 37

INFRARED ION COLLISIONAL TRANSITION SOURCE DETECTOR REGION

FIG. 5. Doppler-tuned ion beam laser resonance method concepts. - Ions are produced in the ion source, then proceed through the infrared transition interaction region; transitions are detected by the colli- sional detection mechanism described in the text. into a beam. Then, in a region of constant electrostatic potential, the beam of several keV energy is intercepted at a small angle by a beam from an infrared molecular laser. The energy of the ion beam is adjusted to

Doppler shift an ion transition into resonance with a nearby laser line. On resonance, the laser light stimulates transitions, resulting in a net population transfer between vibrational-rotational states (pro­ vided there was an intial population difference in the resonating states).

After interaction of the ions with the laser beam, some property of the ion beam that is dependent on the internal population distribution is monitored. Thus, stimulated transitions can be detected. A collisional detection mechanism (19) that takes advantage of vibrational-state depen­ dent collision cross sections was used in the present experiment. This mechanism is discussed in more detail below. Other detection mechanisms have been cited by Carrington (77) and include two-photon (or multi- photon) absorption leading to dissociation, predissociation after tran­ sition, and ion-molecule reactions.

A schematic diagram of the physical apparatus involved in the method is given in Fig. 6. The apparatus itself is described in detail in Chapter 5. I will now discuss in some detail more significant aspects of the method.

Ion Source Population Distribution

As indicated above, it is critical that the ions be produced with substantial inequalities in internal state populations. This is usually possible if the ions of interest can be extracted from the source soon after their formation, before they can be thermalized by 39

ION SOURCE VOLTAGE EXTRACTOR LENS SWEEP I CONDENSER LENS CO LASER- DEFLECTION PLATES <^1$ BEAM FOCUSING COLLIMATOR LENS OPTICS

LASER BEAM CHOPPER

INTERACTION REGION MICRO COMPUTER

POWER METER- RECORDER FOCUSING LENS DEFLECTION PLATES ANALYSER MAGNET-' GAS TARGET INTEGRATING VOLTMETER

FARADAY CUP- LOCK-IN VACUUM CHAMBER AMPLIFIER

FIG. 6. Doppler tuned ion beam laser resonance method schematic. -- The laser beam direction can be reversed so that it is nearly anti- parallel to the ion beam. Not shown are two long-coil pairs which can­ cel transverse components of the laboratory magnetic field. collisions. In the case of direct ionization of neutral molecules by

single electron impact, the ions produced are populated according to the

Franck-Condon vibrational and Maxwell-Boltzman rotational factors, and

therefore have substantial population inequalities. It seems to be gen­

erally true that ions produced by ion-molecule reactions also have sub­

stantial population inequalities.

Discussion of the specific population distribution expected for + + HeH ions formed by collision of Hmolecular ions with He atoms will be

deferred to the section on the ion source in Chapter 5. The point that

is important here is that there are substantial inequalities in the

vibrational-rotational states of the HeH+ ions produced by the source.

(Specific numerical values of this and other quantities appearing in

this chapter are given in Table 1 at the end of the chapter.)

Doppler Tuning

Now consider the Doppler-tuning of the laser radiation into res­

onance with the molecule. An observer traveling in a reference frame moving with the molecules in the beam would see a Doppler-shifted laser

frequency given by (80):

a , A-B cose A (J) V Vi-B2 / where 3= and u is the molecular speed, 0' is the (laboratory) angle between the molecular velocity and the light beam direction, and £2' is the (laboratory frame) laser radial frequency (radian/sec). Here, and in the discussions which follow, the primed variables refer to the 41 laboratory reference frame. For a fixed interaction angle 0'=O (laser beam and ion beam parallel, as is nearly the case in practice), the

Doppler shift is approximately

A£2 = FT - ' {« - £21 (~) • (2)

Now

where V is the ion beam voltage and M is the molecular mass. Therefore, the Doppler tuning range is determined by the unshifted laser frequency, the mass of the ion, and the ion beam voltage range.

For HeH+, the molecular mass is approximately 5.0 a.m.u. (in 2 energy units, Mc = 4700 MeV). A typical carbon monoxide laser frequen- n' -1 cy (in wavenumbers) is v' = — = 2000 cm . The present maximum attain­ able beam voltage (limited by high voltage discharges in the ion source vacuum chamber) is about 10 kV. Therefore, the maximum available Doppler shift is approximately -4.1 cm * (=-120,000 MHz). The laser beam can be reversed so that Doppler up-shifts of +4.1 cm * are also possible. There is also a lower limit to the Doppler shift because a certain minimum beam voltage must be maintained to sustain an ion beam of appreciable current. A practical lower limit on the beam voltage is about 1500 V, which corresponds to a Doppler shift of ±1.6 cm *.

The carbon monoxide laser has useful power on about 150 lines

-1 -1 from 1612 cm (6.2 pm) to 1923 cm (5.2 ym), for an average line spac­ ing of about 2 cm-1. Since each line can be tuned by ±(1.6-4.1) cm"1, the percent coverage of the spectral range is about 100%. In fact, most transitions should be observable with two or more laser lines.

However, CO laser lines tend to occur in pairs rather than being equally spaced, so there may be some small gaps which cannot be covered.

A fortuitous result of forming the ions into a beam and acceler­ ating them to several kilovolts energy is a "kinematic compression" ef­ fect. The following description of the effect traces that given by

Kaufman (81). Consider ions in the source with a velocity distribution corresponding to a translational temperature T. Further consider two identical ions, initially having velocity components in the z-direction of u^ =0 and ^^^kT/M)'2. Now suppose both ions are accelerated through a potential difference V. The two ions gain the same amount of kinetic energy and then have final velocities

(2eV/M)^, 'if (4) 2 _ /2eV 2\h ( 2 2\h U2i + U + U U + U w U f 2f ~ \ M 2i ) ~ V lf 2i / l 2ulf

The ratio of the difference in final velocities to the difference in initial velocities is

U U 2f " lf 1 fkT . fn B " u . - u.. " 2 \ iV ' (5) 2i0 li that is, the difference in velocities is reduced during the acceleration by the "velocity bunching" factor B. A physical explanation of this effect is that the ions having large initial velocities spend less time in the accelerating electric field and therefore gain less velocity than the initially slower ions. This velocity bunching effect has been used to advantage for several years in merged-beam experiments. See, for example, Trujillo et al. (82). Since the velocity distribution is com­ pressed by the factor B, the spread in Doppler shifts, and therefore the

Doppler-limited width of the resonance lines, is reduced by the same factor.

Suppose the ion source produces HeH+ ions with an energy spread of about 0.5 eV. This corresponds to a temperature of about 6000 K and an initial velocity spread of about 4x10^ cm/sec. Consider an acceler­ ating potential difference of 5 kV. Then B=0.005 and the velocity 3 spread after acceleration is approximately 2x10 cm/sec. This should be compared to the velocity spread that might be expected in a thermal plasma at 300 K, a situation that might be encountered in a non-beam 4 experiment. In that case, the velocity spread is approximately 7x10 cm/sec, a factor of 35 larger. Thus, the linewidths obtained with the beam method should be considerably less than those obtained by conven­ tional methods.

Laser-molecule Interaction

Consider now the interaction of the beam molecules with the laser radiation. Because of the very narrow linewidth of the laser ra­ diation and the collision-free environment of the ion beam, it is suffi­ cient to consider two isolated energy levels, which are the only ones assumed to be involved in the interaction. For brevity of notation, the upper (v,J) level will be denoted by |a) and the lower (v,J) level by |b). As a molecule traverses the laser beam, it experiences a pulse of radiation, whose frequency is the Doppler-shifted laser frequency. Bccause of the shape of the laser beam, the molecule sees a Gaussian-

shaped pulse amplitude and a slight variation in the phase (due to cur­

vature of the laser beam wavefronts). This pulse will be approximated

by a pulse of length T=L/U that begins at time t=0 and has constant am­

plitude and phase. The treatment given here will be a calculation of

the transition probability between states |a) and |b) for the ensemble

of molecules in the beam.

The interaction Hamiltonian for the beam molecule and the laser

radiation is

V , = -pE cos At, (6) ab o

where is the amplitude of the electric field of the laser radiation,

= s is the laser frequency, and P Pa^ i the transition electric-dipole

matrix element. An approximate evaluation of p ^ and a derivation of

the vibrational-rotational electric-dipole selection rules are given in

Appendix B. For molecules in state ja) ax t=0, the probability for tran­

sition to state |b) is (83; p. 26 and 84; p. 119):

Pab(t) * 2 2 5i"2 |[{2 4 (2«J2] '4 ' l') 6 + (2otJ v where the interaction strength

PE a = 2ff~ »

and 6=f2-u is the detuning from resonance. Here to is the Doppler-shifted

molecule transition. (Note that in this section the laboratory frame of reference is taken. That is, the molecule transition frequency is viewed as being Doppler-shifted into resonance with the fixed laser frequency.

Then, from this viewpoint, there is a spread in resonance frequencies due to the velocity spread in the ion beam.) The same transition prob­ ability given in Eq. (7) obtains for molecules initially in state ]b).

2 2 2 In Eq. (7), the factor vR=[6 +(2a) ] is called the Rabi flopping fre­ quency and is the frequency at which the molecule oscillates between the two states. Notice that this formulation has neglected any decays from the two states. In the present case, such decays are entirely negligible.

Equation (7) gives the transition probability for a molecule with a particular Doppler-shifted resonance frequency. In the actual beam, there is a spread of Doppler-shifted resonance frequencies due to the velocity spread of the ions and the spread in the interaction angles.

There is also a spread in the transit times T through different sections of the laser beam. The proper course of action at this point would be to average the contributions of all the molecules. This would take the form of the convolution of the response given in Eq. (7) with the com­ plicated distribution function implied by the above spreads. This is hardly tractable, so an approximate solution is sought. If the laser field is strong enough so that the Rabi frequency vD is much greater than 1/T where T is the pulse length (transit time), that is, if 2a>>l/x, then the molecule undergoes many oscillations between the two states during its transit through the laser beam. (This condition is satisfied for reasonable values of laser power and ion beam voltages.) In this case the approximate effect of averaging over the transit time varia- 2 tions will be to replace the sin factor in Eq. (7) by its average value

av 2 2 ' 6 + (2a)

This is a power-broadened Lorentzian function with half width at half maximum 2a. Equation (9) must now be convolved with the resonance fre­ quency distribution function, which has a width Aw. If the power- broadened response Eq. (9) is much narrower than Aw, the resulting response curve will approximately replicate the distribution function.

In this case, molecules of each resonance frequency group contribute one-at-a-time as the ion beam voltage is swept through resonance. In the other extreme, if the power-broadening linewidth is much larger than

Ah), the resulting response will be approximately a replication of Eq. (9).

In this case, all the molecules will contribute to the interaction at the same time, but the response amplitude will be diminshed because it is spread over a wide range. If the power-broadened linewidth is approxi­ mately equal to Aw, then the optimum situation is reached, because mole­ cules of all resonance frequencies will participate in the interaction at the same time, and the signal is not overly broadened beyond Aw. The 2 2 necessary laser flux is determined by (2a) = (Aw) . Solving for a, using Eq. (8), and multiplying by appropriate constants, the laser flux required to make the power-broadening equal to the spread in resonance 2 frequencies is (in W/m ): /e~~ i.2 o c I Q n ft. >2 (10) SI = Jp 2 (Au) \ o v

This quantity is' evaluated for the present case in Table 2 at the end of this chapter.

Now consider the effect of transitions on the populations of the two states. Suppose the two states enter the laser beam with frac­ tional populations p and p, . Then, for transition probability P, the ci D populations after interaction are

pa = Pa(1-P) + pb P ' (11)

pb = pb(1"P) + pa P

The change in population of state |a) is

p; - pa = P(pb-pa) • (12)

Taking the limit of Eq. (9) as a approaches infinity gives a maximum transition probability of Thus the maximum change in population is half the initial population difference. By conservation of number of molecules, the change in state |b) is equal but opposite in sign.

Therefore the initially different populations have been equilibrated.

Collisional Detection Mechanism

Now consider detection of the transitions driven by the laser light. After interaction with the laser beam, the ion beam is mass analyzed and a selected species passes through a gas target and is then collected in a Faraday cup (see Fig. 6). The detection mechanism is based on the fact that the transmission of the ion beam through the gas target is dependent upon the vibrational-state population distribution of the ions in the beam. In the following discussion, it is assumed that the collection efficiency of the Faraday cup is 100% for beam ions and 0% for neutrals and target gas ions. These assumptions are reason­ able because of the construction of the gas target and the Faraday cup.

When a fast particle (neutral or charged) strikes a metal surface, sec­ ondary particles (electrons or perhaps ions) may be ejected. If the secondary particles are not collected, an erroneous measurement of the ion beam current will result. If the Faraday cup is constructed so as to collect all the secondary particles as well as the incident particle, beam ions will contribute with 100% efficiency and fast neutrals will make no contribution to the measured current. Ions originating in the gas target cell will also contribute with 100% efficiency, but because of the small solid angle subtended by the exit aperture of the target cell only a small fraction of the target gas ions will make it to the

Faraday cup. The effect of the failure of these assumptions (due to incomplete collection of secondary particles or a small fraction of tar­ get gas ions being detected) would be to reduce the measured beam cur­ rent (and therefore the signal) and increase the noise. In practice, departures from these assumptions should be small.

In passing through the gas target the ion beam is attenuated by a variety of mechanisms. In the case of HeH+, the strongest attenuation mechanism in the several keV energy range is probably charge-exchange neutralization followed by dissociation into neutral fragments. Ions that suffer direct collisional dissociation into a neutral and a charged fragment will not produce an apparent attenuation of the beam because the geometry of the apparatus is such that all heavy fragments (ions) are collected. However, dissociation accompanied by electron ejection would produce observable attenuation, as would large-angle scattering.

The present experiment cannot distinguish among these modes of beam attenuation.

It seems to be generally true that collision cross sections in­ volving a molecular projectile exhibit some dependence on the vibra­ tional state of the incident molecule. This phenomenon has been manifested in several collision experiments involving different molecular ions.

Several workers (85-88) have observed that changes in their ion source operating parameters introduce systematic changes in measurements of cross sections involving projectiles. Similar effects have been noticed with H* projectiles (85-86, 89). Such "ion source effects" are attributed to ions in different states having different collision cross sections. Then changes in the ion source that affect the vibrational population distribution would result in changes in the vibrationally- averaged measured cross section. Measured changes can be substantial;

McClure (85) has observed changes of up to 20%.

Although no similar results involving HeH+ projectiles in the energy range of interest have been reported, the results cited above for different ions, involving different collisional processes, at dif­ ferent energies suggest that the dependence of collisional cross sec­ tions on vibrational state is quite general. 50

The way in which the variation of cross section with vibrational state can be used as a detection mechanism can be seen by the following arguments. The total current (of the desired species) in the ion beam

is

(13) i where p^ is the fractional population of the ith vibrational state

(Zp.=l). Then the current in one particular state is Is

I . = I p. . (14) % 01

Let the total current that survives the gas target, neglecting secondary collisions, be

(15)

where n is the target gas density, t is the length of the gas cell, and a. is the collisional cross section for the ith vibrational state. Here i all collisional processes except for the charge-exchange process of in­ terest are neglected. With a different vibrational distribution, such as that induced by stimulated transitions, the surviving current is

(16)

If the P^'s are identical except for the two levels that are connected in the transition, that is, if 51

i p. = p • , i t a,b,

P>Pa , C17)

pb * pb ' then by conservation of number of molecules,

f fpa " pa} = • (pb ' pb} * C18)

Now the "signal" is the difference in beam current collected when the laser is turned on and off. That is,

C T' T VT ' -ntoj V1T e -nlant- ,in. S = I - I = 2-j^-0Pi e - 2^ 0Pi • (19) i i

Carrying out the algebra, using Eqs. (17) and (18), reveals that

. T -nla -nta, a b S = I (p - P ) e - e (20) o a a

Substituting Eq. (12) into Eq. (20), one gets

T -nta -n^0v, (21) S = J*Vpb-paHe a'e

Consideration of Eq. (21) shows that the signal vanishes if either (p^-pa)=0 (no initial population difference) or (aa-ajj)=0 (no difference in cross sections). Notice also that the sign of the signal

a _a is determined by (Pb"Pa)( a ^)• Equation (21) also shows that if n-0

(that is, if there is no target gas) the signal vanishes; likewise, if

«->•<*> (and no ions survive the target cell passage) the signal also 52

vanishes. Somewhere between these extremes there is a point of opera­

tion which gives maximum signal. Setting the derivative of S with re­

spect to n equal to zero reveals that the target density for maximum

signal is given by

However, since the values of the cross-sections are rarely known, Eq.

(22) is not very useful and the target gas density must be selected by

empirical methods.

Signal-to-Noise Ratio Estimate

The formulae developed in the previous two sections can be used

to calculate an estimated signal-to-noise ratio for the experiment.

The signal given in Eq. (21) can be written approximately:

(23) where

(24) v is the average cross-section.

Next, consider the detector noise. The theoretical lower limit is the noise in the beam current due to the shot effect. It has been determined experimentally that the beam noise actually measured at the

Faraday cup is about a factor of two higher than this:

N ss 2 x /2e I Af , (25) where I is the average beam current measured and Af is the bandwidth of the current measuring electronics. The higher measured value of the noise is probably the result of several random processes including the statistics governing the production process, technical noise arising from the ion beam optics, tha statistics governing the collection of ions, and Johnson noise in the current preamplifier.

The signal-to-noise ratio estimate may be calculated from Eqs.

(23) and (25). With the reasonable values for the various quantities that are given in Table 1, the signal-to-noise ratio estimate is about

3 for an eight-second integration time. This is in rough agreement with observed signal-to-noise ratios. This agreement is of limited signifi­ cance because of the guessed values of the cross sections. 54

TABLE 1. Numerical values for quantities appearing in Chapter 4. -- The dipole moment and population numbers used are for the (2,9)-*-»-(l,10) transition.

Item Value

HeH+ beam current I 3xl0~9 A 0 Beam voltage V 5000 V 7 Ion speed u 4.4x10 cm/sec

Interaction length L 30 cm

Transit time T 6.8x10 ^ sec

Laser flux 5.5xl03 W/m2

Laser electric field E 4.6xl02 V/m 0 Rotational factor (see Appendix C) 0.42 av -19 Permanent dipole moment derivative 1.43x10 C-m/m (see Appendix C) (4.29 debye/A°) \3x Jx=0

Classical oscillation frequency V 8.7x10^ sec * (see Appendix C) osc

Scale factor (see Appendix C) /r 8.3xl010 m"1

Vibrational factor (see Appendix C) 11 Fv 1.2x10" m Transition dipole moment matrix element y 7.2xl0"31 C-m (see Appendix C) (0.216 debye)

Interaction strength (with above value a 1.6x10^ sec * of E ) 0 Spread in resonance frequencies Ato 6.0xl06 sec *

Saturation flux 2.0xl03 W/m2 si Population difference (from Table 3) 0.020

Gas target parameters (l/lo=0.74) n£o" 0.3

Cross section difference 0.01a _ -14 Signal 6.7x10 A 55

TABLE 1.--Continued

Item Value

Bandwidth (8 sec integration time) Af 0.124 see"*

Noise N 2.2x10 ^ A

Signal-to-noise ratio (8 sec integration S/N & 3 time) CHAPTER 5

APPARATUS

The Doppler tuned ion beam laser spectroscopic method, the prin­ ciples of which are described in Chapter 4, requires a considerable amount of experimental apparatus. The technologies involved include those of high vacuum systems, ion source design, ion beam optics, stable infrared gas lasers, laser beam handling, and computer control and inter­ facing, as well as the mechanical and electronic techniques needed to

"glue" everything together into a functioning experiment. The topic of this chapter is the description of the equipment used in the experiment, which is shown schematically in Fig. 6, and the methods of operation.

Experiment Framework

In the Doppler tuned ion beam laser spectroscopic method, the interaction angle between the ion beam and the laser beam is critical in the determination of the Doppler shifted resonance frequency. Previous experience has shown that mounting the ion beam apparatus and the laser on separate stands is not adequate. Flection of the floor due to daily temperature changes caused the laser beam and ion beam to become slight­ ly misaligned. Also, freedom from mechanical vibrations is desired in order to make the laser frequency as stable as possible. For example, vibrations generated by mechanical vacuum pumps in the room and trans­ mitted through the floor and stand to the laser cause noticeable

56 57 frequency instability in the laser. These two requirements make a stur­ dy mechanical structure with isolation from floor vibrations desirable.

The previously-existing aluminum channel frame (see Ref. 21 ), upon which the ion beam vacuum chamber rests, was strengthened by adding diagonal members. The resulting frame resembled a truss-type structure.

The entire weight is supported at two points where air mounts, as de­ scribed below, are located. At the center the frame is securely bolted to the end of a 1.52 m by 3.66 m by 30-cm thick (six foot by 12 foot by

12-inch thick) stable optical table.* The table is constructed of a very rigid aluminum honeycomb core with top and bottom aluminum surfaces.

The laser and laser beam optics are mounted on this table. The frame­ work of optical table plus ion beam chamber frame is shown schematically in Fig. 7.

The optical table-frame combination is supported by six air mounts. 2 ™The air mounts are pneumatic. pistons that serve to isolate the experiment from floor vibrations. They are designed so as to absorb both vertical and horizontal floor motions. The six mounts are inter­ connected in a system of tubes and automatic valves which provides automatic leveling of the apparatus when weight is redistributed.

Flexing of the floor can now cause a tilt of the apparatus, but because of the rigidity of the structure, the various parts can no longer move relative to each other.

1. The optical table and air mounts were manufactured by Newport Research Corporation, Fountain Valley, California. 2. Ibid. 58

ION BEAM LINE

FABRY-PEROT FILTER

CO LASER

LASER BEAM OPTICS

I METER

ULI

-VACUUM CHAMBER FRAME

STABLE TABLE

"*NUR SUSPENSION

FIG. 7. Optical table-ion beam framework. 59

Ion Beam Vacuum System

The ion beam high vacuum system is shown schematically in Fig. 8.

It is primarily constructed of commerically available high vacuum system components. The main vacuum chamber consists of several six-inch stain­ less steel nipples''' and six-inch stainless steel multi-crosses.'' Also

2 in the beam line are two aluminum gate valves. The parts have been modified as necessary to facilitate attachment of feedthroughs, ion gauges, and other accessories. The ion source flange is electrically isolated from the rest of the chamber by a glass section. The larger parts are mated and sealed with Dependex flanges using Viton o-rings.

Smaller seals are copper gasket Conflat seals.

The chamber may be conceptually divided into three regions: the ion source region, the interaction region, and the detection region.

The interaction region is separated from the other two regions by dif­ ferential pumping apertures because the source chamber and the detection chamber are subject to high gas loads during operations. The interac­ tion chamber is pumped by two pump stacks; the source chamber and the detection chamber are each pumped by one pump stack. Each pump consists 3 4 of an aluminum gate valve, a four-inch cryogenic cold trap, and a four-inch, 800 liter/sec, oil diffusion pump5 with Mexican-hat cold cap.

1. Manufactured by High Voltage Engineering Corporation, Burlington, Massachusetts. 2. Model 5010 manufactured by Airco Temescal, Berkeley, California. 3. Model 1283-4 manufactured by Norton Vacuum Equipment Div., Newton, Massachusetts. 4. Model 278-01 Cryoclean trap manufactured by Granville- Phillips Company, Boulder, Colorado. 5. Model VHS-4 manufactured by Varian Lexington Vacuum Division, Lexington, Massachusetts. IS L,-L3 L4 D,D2 L5 B, B2 L6 D3D4 GT FC

_i.J i HI I II II I I — j

Roughing line CT CT CT CT Backing line DP DP DP DP

To To roughing backing pump pump

I meter

FIG. 8. High vacuum system schematic. -- The letters identify components as follows: A, analyzing magnet coils; CT, methanol-cooled cold traps; DP, diffusion pumps; G, gate valves; GI, glass insulator section; V, bellows valves. The letters with the arrows above the drawing iden­ tify the approximate locations of the ion beam line components: IS, ion source; Lj-Lg, electrostatic lenses; D1-D4, deflection plate pairs; Bj and B2, beam sensors; GT, gas target cell; FC, Faraday cup. The dashed lines indicate the position of coils used to cancel transverse components of the laboratory magnetic field. 61

The four diffusion pumps are cooled by an approximately 20°C mixture of water and ethylene-glycol based antifreeze. This mixture is circulated through a heat exchanger which is cooled by the university-campus chilied water system. The four pump stacks are backed by a 660 liter/min mechan­ ical vacuum pump.^ The vacuum chambers can also be rough-pumped via a 7 separate roughing line with a 140 liter/min mechanical pump. To prevent oil backstreaming from the forepumps, both forelines have both water- g cooled baffles and molecular sieve oil traps.

The usual practice in high vacuum systems is to run cryogenic cold traps at 77 K (-196°C) by cooling them with liquid nitrogen. Be­ cause of the rapidly increasing cost of liquid nitrogen, a cooling system with a mechanical refrigerator was used instead. Methanol is circulated through the four traps in series via insulated polyethylene tubing from a reservoir which is cooled by a two-stage, cascade, mechanical refrig- 9 eration system. The volume of methanol in the system is about 60 I.

The refrigerator can keep the methanol chilled to between -50 and -70°C when the vacuum system is operating. Because the trap temperature is considerably higher than in the usual liquid nitrogen cooling case, we — 9 o use a very low vapor pressure (2x10 Torr at 25 C) polyphenyl ether

6. Model ED-660 manufactured by Edwards High Vacuum, Inc., Grand Island, New York. 7. Model 1402 manufactured by Sargent-Welch, Inc., Skokie, Illinois. 8. Model 0345 manufactured by Varian Lexington Vacuum Division, Lexington, Massachusetts. 9. Model FC-100-84-P40-SV manufactured by FTS Systems, Inc., Stone Ridge, New York. It is identical to the unit used to cool the CO laser. based diffusion pump fluid rather than the more usual hydrocarbon-

based oils. Also, a high quality oil'** is used in the forepumps.

Because it takes several hours to bring the traps down to opera­

ting temperature with the mechanical refrigeration system, a different

procedure for pump-down of the vacuum system is followed than the proce­

dure used with liquid-nitrogen-cooled systems. First, the entire system is roughed out with the roughing pump and the backing pump to about 15

pm Hg pressure. The methanol refrigerator is operated for about two hours, while methanol is circulated through the traps. During this time the methanol temperature will reach about +5°C. At this stage, lower tem­ peratures are not desired because residual water vapor would prematurely condense on the traps. When this temperature is attained, circulation to the traps is stopped. In about four hours the temperature of the methanol reservoir will be about -77°C. Then the diffusion pumps are turned on. After about one-half hour of operation, methanol circulation is again started. Because of mixing of warm methanol in the traps with the cold methanol in the reservoir, the average temperature will be about

-30°C. After about eight hours, the methanol will be cooled to the oper­ ating temperature of about -72°C. At that time the gate valves to the vacuum chamber can be opened. In one to two hours, the system will be near its ultimate vacuum. Even though this procedure is lengthy, in practice it is not objectionable because the system is not cycled often.

10. Convalex-10 manufactured by CVC Products, Inc., Rochester, New York. 11. Convoil-20 manufactured by CVC Products, Inc., Rochester, New York. 63

The normal operating conditions of the vacuum system are given

in Table 2.

Ion Source

Requirements for an ion source to be usable in this type of ex- -9 -7 periment include (i) moderate beam currents (10 -10 A), (ii) low beam current noise, (iii) low ion energy spread, (iv) no (or at least small and constant with respect to source operating conditions) ion energy offsets, and (v) non-equilibrium excitation of the internal mo­ tions of the ions. Commercially available sources and sources described in the literature generally fail in one or more of these areas. Require­ ments (iii) and (iv) preclude the use of radio-frequency and arc- discharge sources. Requirement (v) usually eliminates direct current discharge sources as well. An electron bombardment source, described by Spezeski et al. (90,91) and used successfully in the Doppler tuned ion beam spectroscopy of the hydrogen molecular ion HD+ (19-22) has been modified to efficiently produce beams of molecular ions which are created through gas-phase ion-molecule reactions. The design and per­ formance of the resulting source has been described by Kyrala et al.

(92). The production mechanisms of HeH+, and the ion source design, performance and typical operating parameters are described in this section.

Production Mechanisms of HeH+

Possible mechanisms for the production of HeH+ in mixtures of

He and H^ gases bombarded with electrons are the reactions: 64

TABLE 2. Vacuum system operating parameters. — Pressures are uncorrec­ ted Bayard-Alpert ionization gauge readings, except for the gas target cell pressure which is an uncorrected Schultz-Phelps ionization gauge reading.

Item Typical value

Ultimate vacuum (no gas load) < lxlO-7 Torr

Ion source chamber operating pressure 6x10"6 Torr

Interaction region operating pressure w 5xl0-6 Torr

Gas target cell operating pressure l.lxlO"1 Torr

Target chamber operating pressure 1.6xl0-4 Torr

Backing line operating pressure 35 mTorr

Cold trap operating temperature -60°C 65

+ + He + H2 -+• HeH + H , (Ref. 93 ) (1)

He + (H*)* -»• HeH+ + H , (Ref. 94 ) (2)

+ He** + H2 -> HeH + H + e" , (Ref. 95 ) (3)

+ He + H* -*• HeH + H2 , (Ref. 96 ) (4)

4. * + * * where (H^) represents vibrationally-excited H2 and He represents electronically-excited He. Reaction (1) is exoergic by the large amount

8.3 eV (93). Because this is enough to ensure that the product molecu­ lar ion has excitation in excess of the HeH+ dissociation energy (97 ),

Reaction (1) is not a likely mechanism for production of HeH+. Reaction

(3) is likely to produce HeH+ ions in a highly excited state near the dissociation limit (66). Reaction (4) is known (96) to be much less likely than Reaction (2).

Reaction (2) is endoergic for ground vibrational state H* by approximately 0.803 eV (98 ). The implications of this fact can be understood by referring to Fig. 9. Shown there are several vibrational-

+ rotational energy levels of H2 and of HeH , drawn relative to a common dissociation energy where the three particles He, H, and H+ are separa­ ted at infinity. Notice that the v=0 level of HeH+ lies about 0.8 eV higher in energy than the v=0 level of H2. Thus in order to produce

HeH+ from H*, this much energy must be supplied either as translational kinetic energy of the reactants or as vibrational energy of the H2. It has been demonstrated by Chupka and Russell (98 ) that vibrational energy in H2 is more effective in overcoming this energy barrier than is He + H + H+ He + H + H'1

V = 6 V=5 V = 4 V=I0 V = 9 V = 3 V = 8 V=7 V = 2 10000 E V=6 o V = 5 V= I V = 4 X v=o CD V = 3 o: 14875 Ui V = 2 2 J-0 J-l J-2 J=3 1044Q LxJ V= I 20000 V=0 Do= 21353 De= J=0 J=l J=2 J=3 23544

30000 0.18 0.10 0.00

"—F / Z F + v V V H5 HeH FIG. 9. Ion-molecule reaction kinetics diagram. -- The center and right sections show several vibrational-rotational energy levels for the ground electronic states of Hi and HeH+ relative to a common dissociation limit. The left section plots the Franck-Condon factors for single-electron impact excitation of H2. After Ref. (99). translational kinetic energy . The kinetic energies of the reactants during operation of the ion source at low-to-medium pressures is less than about 1 eV, as can be judged from the 0.5 to 1.0 eV beam kinetic

energy spread, displayed as 0.5 to 1.0 V resonance linewidths (see Chap­ ter 6). The spread is probably due to the space-charge potential varia­ tion in the radial direction in the ionization/reaction channel described in the next section. Reaction (2) does not appear to have been studied experimentally in this regime. Therefore, we must rely on theory to pre­ dict the product HeH+ population distribution. Schneider et al. (99) have calculated the product HeH+ vibrational and rotational population distributions for 1 to 4 eV kinetic energies using a quasi-classical three-dimensional trajectory calculation. Their results for 1 eV ki­ netic energy are reproduced in Fig. 10. It is seen that the HeH+ reac­ tion products will be mostly in the v=0, 1, and 2 vibrational states, with rotational angular momentum J as high as 17. This is consistent with the facts that at low kinetic energies, only H* in vibrational states with v>3 can contribute to the reaction and that for v>3, the populations of the states decrease rapidly as a function of v due to the

Franck-Condon factors for production of H^. This statement can be veri­ fied by again referring to Fig. 9 where the Franck-Condon factors for

as a function of v are also shown. This result is supported experi­ mentally by the distribution inferred by Kanter et al. (75), as mentioned in Chapter 3. Table 3 gives fractional populations for (v,J) states, derived from Fig. 10.

At high ion source pressure, the mean kinetic energy of the pri­ mary reactants will probably increase because of augmented space-charge FIG. 10. Theoretical population distribution for HeH+. -- Theoretical vibrational and rotational distributions are shown. After Ref. . The lines drawn connecting some states represent transitions that can be Doppler tuned into resonance with CO laser lines. 10 12 13 14 15 16 17 O I 2

V 3

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20

J

FIG. 10. Theoretical population distribution for Hell+. TABLE 3. HeH+ theoretical population distribution. These data are taken from the distributions of Fig. 11. The numbers are the fractional populations in each (v,J) state.

J V 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 .013 .013 .013 .022 .022 .022 .035 .035 .035 .035 .035 .035 .026 .026 .026 .018 .018 .018

1 .011 .011 .011 .019 .019 .019 .030 .030 .030 .030 .030 .030 .022 .022 .022 .015 .015 .015

2 .004 .004 .004 .007 .007 .007 .010 .010 .010 .010 .010 .010 .008 .008 .008 .005 .005 .005

3 .002 .002 .002 .003 .003 .003 .005 .005 .005 .005 .005 .005 .004 .004 .004 .002 .002 .002

&to potentials, perhaps leading to formation of HeH+ ions having v>2. How­

ever, increased collisional deactivation and loss from subsequent reac­

tions will also occur. For example, the destruction of HeH+ via the

reaction

+ HeH + H2+ He + H* (Ref. 100 ) (5)

and the competing process of H* production (see section on ion source

performance) will limit the maximum obtainable HeH+ currents.

Ion Source Design

The details of the ion source design are shown schematically in

Fig. 11. The source is cylindrically symmetric with a convergent elec­

tron gun featuring an ionization/reaction channel within the anode.

Electrons are emitted by thermal emission from the 7.2-mm radius spheri­

cal surface of a dispenser cathode* with a 10.0-mm diameter tungsten

body. The surface is coated with the so-called "Type A" impregnant con­ sisting of barium oxide, calcium oxide, and aluminum oxide in the ratio 2 5:0:2. The cathode is heated by a tungsten filament wound in a double helix to minimize magnetic fields. The heater is powered by a direct current of several amps. A molybdenum focusing electrode is operated at a potential near the cathode potential. Its shape very roughly approxi­ mates an extension of the spherical cathode surface in accordance with the design principles of Pierce (101). The electrons are focused through

1. The cathodes were supplied by Spectra-Mat, Inc., Watsonville, California and by Kometco, Inc., Clermont, Florida. 2. The heaters were supplied by CM Manufacturing, Bloomfield, New Jersey. GAS INLET LINE

SUPPORT ROD (TYPICAL)

OXIDE - COATED CATHODE

FOCUS ELECTRODE ANODE BODY VENT WZAZl HOLE (TYPICAL) SPACERS

ANODE I CAP 0.5 CM IONIZATION/ MOLYBDENUM REACTION ANODE CHANNEL INSERTS

FIG. 11. Ion source schematic. -- The left half-section is rotated 22.5° from the right half to show vent hole detail. From Ref. (92). a 0.51-mm diameter hole in the anode and enter the ionization/reaction channel. The shape of the anode in this region is spherical, with cen­ ter at the center of the cathode surface sphere. Ideally the hole in the anode should be conical, converging to a small aperture. Due to dif­ ficulty of fabrication, a flat bottomed hole as shown in Fig. 11 is used.

Neutral gas is fed into the ionization/reaction channel radially through two milled slots in the anode spacers and cap. The production of ions takes place in the ionization/reaction channel. The ions are then ex­ tracted through another 0.51-mm diameter aperture and are accelerated and focused into a beam with electrostatic elements not shown in Fig. 11.

The 0.51-mm diameter apertures are machined in molybdenum inserts which are press-fitted into the copper anode body. The molybdenum inserts were designed to be easily replaceable and to withstand continuous electron and ion bombardment over long periods of time.

The principal modification over the design given by Spezeski et al. (90,91) is the addition of the two anode spacers. This has the effect of increasing the length of the ionization/reaction channel by a factor of about 13 from 0.51 mm to 6.86 mm. The idea of the shorter de­ sign was to extract the ions rapidly as they were formed, before they could undergo ion-molecule reactions which would destroy the primary spe­ cies. In the present case, however, we rely on ion-molecule reactions to produce secondary ions such as HeH+. Since the rate of these reac­ tions depends on the product of the gas pressure and the volume of the ionization/reaction channel, one can increase the amount of secondary ions by increasing the length of the channel and by increasing the gas pressure. The advantage of increasing the channel length is that the 73 the system gas load may be minimized. A limit to the length occurs when the area of the exit aperture becomes so small relative to the total wall area of the channel that the rate of loss of ions by diffusion to the walls becomes comparable to the rate of production of the ions by the source.

Also shown in Fig. 11 is a typical vent hole milled in the anode body to permit venting of gas in the cathode region into the source vac­ uum chamber where it is pumped away.

The cathode is mounted on a stainless-steel or molybdenum plate via small spot-welded brackets. The cathode and focusing electrodes are held in place inside the anode body with alumina support rods and spacers and molybdenum wire retaining nuts. The insulating spacers are shadowed from emissions from the cathode, which can build up conducting films that ultimately short the electrodes, by small "insulator protector" 3 cups.

The electrical wiring of the source is shown in Fig. 12. The ion source anode is connected to the ion beam voltage supplies; the source, the flange it sits in, and the source power supplies are at the beam voltage which is typically 1000 to 10,000 V. For operator safety, the ion source is enclosed in a lucite cage and the power supplies are contained in, but insulated from, a grounded chassis. They derive their power through a 120 V ac, 30 kV isolation transformer. The tungsten heater is powered with 6 to 8 A, at about 10 V dc, a power input of

3. The plates, rods, spacers, and cups are "eV kit" components produced by Kimball Physics, Inc., Wilton, New Hampshire. CATHODE FOCUSING ANODE ELECTRODE

FILAMENT ©- SUPPLY 0-20 V 0- 0-10 A

M FOCUSING ©- ELECTRODE TO BEAM VOLTAGE SUPPLY -0- SUPPLY 0-100 MA

CATHODE SUPPLY 0-400 V 0- 0- 50 mA -0- 0-100 mA

FIG. 12. Ion source wiring diagram. -- The filament power supply is a Hewlett-Packard model 6286A; the cathode power supply is a Kepco model ABC 425M. The focusing electrode supply is shown in Fig. 13. The cathode heater filament is actually a double helical coil. The power supplies float at the beam voltage and are enclosed in an insulated, grounded rack. about 70 W, obtained from a regulated 0-20 V, 0-10 A dc power supply. 4

One of the heater leads is electrically connected to the cathode body.

The cathode-to-anode potential difference is maintained by a 0-425 V,

0-50 mA dc power supply.^ This potential is the electron bombardment voltage (EBV) and is usually 400 V. This supply provides the electron emission current which is typically 20 mA. The focusing electrode is maintained at a potential with respect to the cathode which is up to

±40% of the cathode-to-anode voltage by the homemade circuit shown in

Fig. 13. In practice the focusing electrode potential is adjusted to give the maximum ion beam current. The circuit works as follows: Op-amp

Aj and the associated resistor network makes available at the sweeper of the potentiometer a voltage in the range ±0.8% of the anode voltage (with respect to the cathode). Op-amp A2> the opto-coupler, the Darlington- paired transistors, and the high voltage transistor together make up a high voltage operational amplifier with a gain of about 50. Therefore the output to the focusing electrode is a fraction of the anode voltage, selectable within ±40%.

The source is designed to fit into a water-cooled, high-vacuum flange as shown in Fig. 14. The flange cooling water, kept at about room temperature, is fed through polyethylene tubing and is continuously recirculated through a deionizer. Also shown in Fig. 14 is the position of the ion extractor electrode relative to the ion source. Not shown is an orthogonal pair of magnetic field coils outside the flange which are

4. Model 6286A manufactured by Hewlett-Packard, Inc., Palo Alto, California. 5. Model ABC 425M manufactured by Kepco, Inc., Flushing, New York. Delco DTS-804

200V

IN4002 200V

50kft 2N3639

82pF 330kfl 2W

OANODE

FOCUSING ELECTRODE 10mA 500kft

FIG. 13. Focusing electrode power supply schematic. -- The output (FOCUSING ELECTRODE) tracks the input (ANODE) as a constant fraction of the input, selectable within ±40%. ^ 77

GAS INLET LINE

4 CM

WATER COOLED FLANGE

WEDGE ION SOURCE ASSEMBLY HIGH VOLTAGE FEEDTHROUGH

'////////$$

INSULATOR ION EXTRACTOR POST

FIG. 14. Ion source flange. -- The ion source is shown installed in the water cooled flange. Also shown are the ion source electrical feedthroughs and gas inlet line. The position of the ion extractor ele­ ment (Lj) is also shown. The anode-to-extractor distance A=26.6 mm; the extractor aperture diameter B=3.96 mm. From Ref. f91. Used by permission. 78 used to align the electron beam into the anode hole, thus compensating for inaccuracies in construction.

Ion source feed gas mixtures are mixed from pure gases in a mani­ fold, which is at ground potential and is connected to the source by a long coil of glass tubing so that there is no problem with discharges.

Gas flow into the source is restricted by a needle valve.

Ion Source Performance

The performance of this source with respect to maximizing the intensity of the molecular ion beams produced under various source con­ ditions has been reported in detail by Kyrala et al. (92). Several spe­ cies of ions including H+, H*, H*, He+, HeH+, and He* are produced simultaneously when the source is fed with a mixture of ^ and He. The more important results with respect to HeH+ are discussed here.

Ion production is sensitive to ionization/reaction channel pre- sure, electron bombardment energy, electron current, and electron focus­ ing. The extraction electric field does not affect the composition of the beam. The current ratios of the various species are dramatically affected by the transverse magnetic field used to align the electron beam into the anode hole: in practice, the field was adjusted to maxi­ mize the current of the species desired. Figure 15 shows the production efficiency for various species of ions as a function of the ionization/ reaction channel pressure when the source is fed with a mixture of 10%

+ H2 and 90% He. This is near the optimum mixture for production of HeH .

Here production efficiency is defined as the ratio of the beam current of a particular species to the cathode electron emission current. The

FIG. 15. Ion source performance. -- Shown is the dependence of ion production efficiencies Ij/Ie = ion current/electron emission cur­ rent for the source fed with a 10:90 h^.'He gas mixture and electron bombardment voltage = 200 V on the anode ionization/reaction channel pressure PA (corrected for sensitivity of pressure gauge to gas mix­ ture), for electron currents on the order of 1 mA. The mass-to-charge ratio of each ionic species is enclosed in a circle. Identifications for the m/q values shown are: 2, and He^+; 3, H*; 4, He+; 5, HeH+; 8, He?. From Ref. (92). 79

: H2 •He 10 90 VERSION HE EBV = 200

,-6

a> »—i i—i

-8

\"9

-10

-3 -2

PA (Torr)

FIG. 15. Ion Source Performance. source was usually operated near the pressure giving maximum HeH+ cur­ rent. At normal operating pressure the HeH+ current peaked at about

400 eV electron beam energy, therefore the electron gun was usually run at this voltage. The cathode electron emission is strongly affected by the focusing electron potential. In practice the focusing electrode was adjusted to give maximum HeH+ current: at this setting the emission current was typically 20 mA. -9 + Under these conditions, beams of up to about 8x10 A of HeH could be obtained, depending on the beam voltage. Less beam current could be obtained at lower beam voltage, the lower limit of useful beam _9 being about 1.5x10 A, which is about the maximum obtainable at a beam voltage of about 1500 V. As the source ages, or if the cathode becomes contaminated with water vapor or oxygen, the beam currents produced de­ cline because of reduced cathode emission current. Sometimes overnight operation with no gas being fed into the source will rejuvenate the cathode and restore previous performance. Typical life of the source before rebuilding, which includes cleaning the parts and replacing the cathode and heater, is about 4 to 6 weeks of more-or-less continuous operation.

Typical ion source operating parameters are summarized in Table 4.

Ion Beam Optics

As ions formed in the ionization/reaction region of the ion source drift to the anode exit aperture, they are extracted, accelerated, and focused into a beam. This section describes the ion beam optics and beam handling apparatus. 81

TABLE 4. Ion source operating parameters. -- Pressures are uncorrected Bayard-Alpert ionization gauge readings.

Item Value

Ion source chamber pressure:

typical value 6x10 ^ Torr

useful range 2x10 ^ to 9x10 ^ Torr

Filament current (range) 6.5-7.75 A

Electron bombardment voltage:

typical value 400 V

useful range 100-400 V

Focusing electrode voltage:

typical value -160 V

useful range Oto -160 V

Source gas mixture, He:^ 90:10

Beam currents obtained:

H+ 2 nA

H! 22 nA I H! 30 nA

He 220 nA

HeH 5 nA

He* 0.7 nA 82

It is desirable that the collection of ions which are interact­

ing with the laser beam have parallel trajectories in order to minimize

the range of interaction angles. This implies a collimated ion beam or

a beam which is going through a shallow focus at the center of the in­

teraction region. To ease the realization of this goal, the ion optical

system has been laid out so that it is symmetric about the center of the

interaction region as shown schematically in Fig. 16. The idea is to adjust the various lens powers so that the beam comes to foci at the cen­

ter of the interaction region, at the gas target aperture, and by sym- •

metry, between lens and L,..

Electrostatic elements with applied voltages that scale with the beam voltage were used because they have two very convenient proper­ ties (102). First, the charged particle trajectories are independent of particle mass, that is, the beam line affects all ions in the same way.

Second, the trajectories are independent of the beam voltage, that is, the beam remains focused and aligned as the beaffl voltage is changed.

This is necessary because alignment and focusing must be maintained as the beam voltage is swept through a resonance. The magnetic analyzer is obviously not electrostatic, but the magnetic field strength is pro­ grammed so that the action of the analyzer also scales with the beam voltage. The lenses are modified three-tube einzel lenses (103) with the center cylindrical tube having larger radius than the two outer tubes and overlapping them. The outer tubes are held at ground poten­ tial and the center tube is kept at a voltage which is derived from the beam power supply by a voltage divider network as shown in Fig. 17. ION SOURCE

I 1 1 1 O 10 20 30 (cm)

B2 I -!=^§MB beam

B,

GAS Dyg Dy-i) Dy4 TARGET i FARADAY CUP

ANALYSING MAGNETIC COILS

FIG. 16. Ion beam optics schematic. -- The ion beam is shown in three sections as viewed from the top. The various components are identified in the text. Detail at the beam sensors and B^ is given in Fig. 19 FIG. 17. Ion beam electrical wiring schematic. -- The range of output of the Kepco power supply is switch selectable. One of six electrostatic lens voltage dividers are shown. The voltage dividers are actually Kelvin-Varley dividers rather than the schematic circuit shown here. One of eight deflec­ tion plate voltage dividers is shown. The beam supply voltage is measured with a Fluke 80E precision voltage divider (100 ppm precision) and a Fluke 887AB differential voltmeter (25 ppm precision). -ay o-ioo FLUKE 50: UA 80E PRECISION ION SOURCE DIVIDER ANODE E.LECTROSTATIC IK"-1 ° LENS

FLUKE 4I0B FLUKE 887AB POWER SUPPLY DIFFERENTIAL 0-10000 V VOLTAMETER

HP 6209B - + HP 6209B STEPPING POWER SUPPLY POWER SUPPLY MOTOR 0-320V 0-320V MASTER SLAVE -A/V*

-"W® KEPCO 2 6.2 V 0PS-2000

l0-io 0-2000 V mA H-o 6- DEFLECTION PLATE FAIR

FIG. 17. Ion beam clectricnl wiring schcmatic. 00 85

Element is the ion extractor and has the form of a plate with an aperture, with a tube attached on the back. Elements and accel­ erate and focus the beam so that a focus is formed just beyond (at a point symmetric to the gas target aperture). Lens is a condenser lens which images the extraction aperture onto the beam defining aper­ ture at Bj. Deflection plate sets Dxl, D,^, and Dy2 steer the beam through the beam defining apertures at Bj and B^. They are driven such that the average voltage on each plate pair is always at ground while the voltage difference across the pair scales with the beam voltage.

The circuit is shown schematically, also in Fig. 17. Not shown in

Fig. 17 is another circuit which applies 60 Hz sine wave voltage to any of deflection plates Dxl, D,^, Dy3, or D^. This is used in alignment of the beam through the machine. A 60-Hz voltage on plate is also used to scan the ion mass filter described below. Lens L,. collimates the beam (or focuses it at the center of the interaction region). Lens

L^ refocuses it and deflection plates D^, D^, and Dy^ steer it into the gas target aperture. To preserve symmetry, lenses L<- and L^ have the same power.

The gas target is a small cell with small entrance and exit apertures through which the beam passes. To prevent backstreaming of the target gas up the ion beam path, the target cell is preceded by a shadow chamber, the aperture of which is also the differential pumping aperture for the target vacuum chamber. Details of the construction are shown in Fig. 18. The target cell pressure is typically 1.1x10 * Torr of argon. H2, and He have also been used as target gases. MOUNTING TARGET FLANGE GAS INLET

19mm 3 e—7.9mm 19mm

2.4mn 1.5mm ION Omm FARADAY > SOURCE CUP SHADOW CELL

Psl.lxlO Torr

INTERACTION GAS EXIT VENT REGION TARGET Ps5xlO~'Torr CHAMBER Ps:l.6x 10 Torr

FIG. 18. Gas target detail. -- The cell is constructed of stainless steel. The ion beam exit aperture is approximately 20 cm from the Faraday cup. Target gas is fed into the cell through a needle valve. oo c* The magnetic field coil and deflection plates produce a re­ gion of crossed electric and magnetic fields which, with the gas target aperture, acts as an ion mass filter. The deflection plate voltages and the coil current are adjusted so that the desired ionic species passes through the aperture. The deflection plate voltages scale with the beam voltage as described above. The magnetic field is made to track with the beam voltage as follows: Analysis of ion trajectories in the crossed field region shows that the magnetic field strength must track as the square root of the beam voltage if the trajectory is to remain unchanged as the beam voltage is changed. This is done by driving a square root circuit module^ with a fraction of the beam voltage. The output of the square root module voltage programs the magnetic field 2 coil current supply.

The Faraday cup at the end of the machine is a cylinder that has been filled with a spiral of crinkled foil to capture secondary elec­ trons emitted when ions strike the inside surfaces of the cup.

The details of the beam defining apertures at Bj and are shown in Fig. 19. Consider the detail at B^. The beam lateral position can be determined by the sensors in planes P^ and P^. These consist of a pair of small rigid wires which extend into the beam either vertically or horizontally. The difference in current collected by the two wires is amplified and displayed on a meter. The diameter of the beam can be

1. Model AD533 manufactured by Analog Devices, Norwood, Massachusetts. 2. Model 6286A manufactured by Hewlett-Packard Company, Palo Alto, California. CROSS HAIR OFFSET = 6.35 mm CROSS HAIR SEPARATION = 117.32cm

ION LASER-ION BEAM GAS ION BEAM SOURCE ±n OVER LAP c? 30 cm TARGET J5T

ION BEAM SENSOR SEPARATION = 121.0 cm

Pri P"2 yP P 4 P. P•> Pre *9P 'toP

B, B:

FIG. 19. Beam defining aperture detail. -- The components in planes Pj-Pio are described in the text. The interaction angle between the lnser beam and the ion beam is 10.825 milliradians.

00 00 judged by the current collected on the plate at P^. Note that the plate

at Pj has a larger aperture.

Also shown in Fig. 19 is a small prism, one face of which is

gold coated and used as A mirror to reflect the laser beam so that it

crosses the ion beam. Near the prism, in plane P^, is a cross-hair

which is used to align the laser beam. The cross-hairs at Bj and at B2

define the laser beam axis. The parts shown in Fig. 19 are all rigidly

mounted on one structure. Therefore, the laser beam defining cross­

hairs are a fixed, known distance from the center of the ion beam de­

fining aperture at P^. Thus when both the ion beam and the laser beam

are properly aligned, the interaction angle is fixed and known

accurately.

The ion beam voltage is maintained by two regulated high voltage

power supplies, operated in series as shown in Fig. 17. A 0-10 kV sup- 3 ply gives the desired high voltage. It floats above an operational 4 power supply which is configured as a high voltage op-amp with constant input voltage and resistance and variable feedback resistance. The var­

iable feedback resistor is a precision, 40-turn potentiometer^ which is

turned by a stepping motor^ on command of the microcomputer system described in a later section. Thus, the ion beam accelerating voltage is programmable.

3. Model 410B manufactured by John Fluke Manufacturing Company, Inc., Seattle, Washington. 4. Model ClPS-2000 manufactured by Kepco, Inc., Flushing, New York. 5. Manufactured by Helipot Division, Beckman Instruments, Inc., Fullerton, California. 6. Model HS-50D manufactured by the Superior Electric Company, Bristol, Connecticut. Typical operating parameters of the ion beam line are given in

Table 5.

Carbon Monoxide Laser

The source of the intense, monochromatic infrared radiation re­

quired in this experiment is a carbon monoxide laser. 12 16 We use a stable, continuous-wave C 0 laser built according to

the design of Freed (104). It is capable of operating on about 150 dif­

ferent molecular lines in the range 5.2-6.2 ym. Power in individual

lines ranges up to more than 1 W. Because of its construction, and with

the aid of stabilizing electronics, the operating frequency can be held

constant to within about 2 MHz for indefinite periods of time. The de­ sign of the laser and its associated equipment are described in this

section. Various operating parameters of the laser are summarized in

Table 7 at the end of the section.

Optical Cavity and Mechanical Design

The optical cavity is half-symmetric, one mirror having 3.0-m radius and one mirror being a diffraction grating in the Littrow config­ uration (order number m=-l) which serves as a plane mirror. The cavity

length is 1.513 m. The output mirror is a 3.0-m radius, 95% reflectiv­ ity at 5.6 pm, germanium mirror with an antireflection coating on the flat back surface. The grating* is a 150 lines-per-mm, gold-coated original grating ruled on a copper blank. It is blazed at 6 ym with blaze angle 26° 45'. The efficiency in Littrow configuration for light

1. The grating was manufactured by Bausch and Lomb, Inc., Rochester, New York, and is their part number 35-53.00-880. 91

TABLE 5. Ion beam line parameters

Item Value

Total beam current « 300 nA

Beam voltage 1500-10,000 V

Beam defining apertures:

diameter 4.76 mm

separation 1.219 m

Interaction length « 30 cm (laser-ion beam overlap)

Ion mass analyzer resolution Can separate masses differ­ ing by 1 up to about mass 16

Ion beam current noise <2x shot-noise limit polarized perpendicular to the grooves is between 90 and 95 percent in the range 5.0-6.4 ^jn. The grating is mounted in a micrometer-driven

mechanism, by means of which a particular laser line is chosen. The out­ put mirror is mounted on a piezoelectric ceramic cylinder that is used to control the cavity length and thus the laser frequency.

The output mirror and the grating are mounted in bushings which are housed in granite end plates. The granite plates are spaced by four,

1-inch-diameter, Invar LR-35 stainless steel rods which are covered with several layers of thermal, acoustic, and magnetic shielding materials.

Many of the mirror- and grating-mount pieces are also constructed of the ultra-low thermal expansion coefficient Invar material.

The mirror and grating are internal components, that is, they are inside the gas volume of the laser. The discharge tube is construc­ ted entirely of quartz. It features a cylindrical nickel cathode in a side-arm at the center of the tube. There are two cylindrical nickel anodes at the ends of the tube. The tube inside diameter is 12 mm.

Around the 12-mm tube is a co-axial cooling jacket through which coolant is flowed. Around the cooling jacket is a vacuum jacket which serves as an insulator for the coolant. All the vacuum seals on the laser (except for permanent glass-to-metal seals on the discharge tube) are copper- gasket Conflat seals to minimize air permeation into the laser which can contaminate the laser gas.

Laser Power Supply

The frequency of an infrared gas laser typically varies with discharge current at the rate on the order of 1 MHz per mA. Because most commercially available high voltage power supplies are available with voltage regulation rather than with current regulation, and because we wanted to operate two lasers with one set of power supplies, it was decided that each laser should be provided with a current regulator.

The requirements for the laser controller were (i) that it provide current-regulated power at about 5000 V with current selectable from

0 to 50 mA, (ii) that the current noise be small, (iii) that during op­ eration of the laser, the anodes be near ground potential, (iv) that a voltage greater than about 8000 V be available for initially starting the discharge, and (v) that any abnormal condition which might result in operator injury or equipment damage be detected and the laser be immedi­ ately shut down automatically.

A laser controller design to accomplish these objectives is shown in Figs. 20-24. Figure 20 gives the schematic of the laser cur­ rent regulator circuit. It is an improved version of the circuit given by Posakony (105). The laser discharge current is controlled by chang­ ing the cathode-to-grid voltage of vacuum tube V^. This is done by keeping the grid voltage fixed while changing the cathode voltage. The cathode voltage is determined by the voltage drop across the cathode resistance, which is the sum of and Rj fixed resistances and the variable resistance of transistor T^. Current regulation is obtained as follows: Op-amp compares a fixed voltage obtained from the 10 kfi potentiometer with the voltage across the current sampling resistor I^.

A difference in these voltages represents a departure from the desired discharge current. The error signal is amplified and passed to transis­ tor T2> changing its resistance, which changes the cathode voltage, +15

43011 J —vw— Modulation IN4002; k TIL3I! I • [TIL8I Input I.SkXl >-15 To Loser Cathode

2 F 220kft !nu30V "T,— Tantalum—I— Eimac -15 +15 IN4002 3-400Z +15 (8163) lOOfl Low Noise

Temp. Stabilized LM IN4002 +15 Zener 10.000 v 399H |470pF Oo 240J_ ,o 120 Vac o o 40J 2N3439 lOkfi 10 Turn Triad ,2N3439 F-I0U I0KV Insulation -15 + 15 +15

'Precision Wirewound o +15 Volt XIN4002 XI/X2 o o TIL3I V- TIL8I I Watt, Reguloted Input o o 200fl 1% o o Power o Supply IN4002 2N3439 o o

Triad F-7X IN4002 -15

Local Common To -High = -High Voltage Voltage

FIG. 20. Laser current regulator circuit schematic. -- Local common on this diagram is connected to the negative high voltage power supply. The ±15 V power supply is referred to this voltage. The 1/8 IV resistor on the grid of tube Vj acts as a fuse. 95

which brings the current to the desired value. Also shown are circuits

for doubling the current range of the regulator by halving the current

sampling resistance and for coupling a modulation signal into the regu­

lator circuit.

Figure 21 shows the current regulator circuit in the complete

laser discharge circuit, including power supplies and switching relays.

For reasons of safety of equipment and operators, it is desirable to

keep the laser anodes near ground potential. This is because there is

a relatively short path through the laser gas between the anodes and the

bushings that house the laser optics and which must occasionally be

touched by the operator for adjustment. A discharge from an anode to a

bushing could damage the laser optics or could elevate the bushing to a

dangerous potential. The anode resistor values are chosen to bring the

anodes to ground potential when the normal current is flowing through the circuit. The additional power supply and anode resistor are needed to allow a high initial voltage across the laser tube when first start­ ing the laser. Meter measures the average anode voltage with respect to ground. Meter M^ measures the difference in potential between the two anodes. This is brought to zero with rheostat . The LED current sensors between the anode resistors and the anodes, and the high voltage relay are connected to the logic circuit as inputs and output, respec­ tively, as discussed below. The X1/X2 switch and relay are used to double the current range of the circuit. The XI setting is used during normal operation of the laser. The positive high voltage power supply FIG. 21. Laser current controller circuit schematic. -- The anode power supply voltage is normally about 5250 V so that the laser anodes are near ground potential. The cathode power supply voltage is normally about 7000 V. The voltage drop across the series regulator tube is about 1700 V, so that the laser tube voltage is about 5300 V. Meter measures the anode voltage imbalance; measures the average anode voltage; measures the discharge current. lOOkA lOOW Anode Current lOOkA lOOW

N.O.

IMA Anode \ IN4002 XI/X2 HC MA

NO. Loser Tube iMA i—jh: i—w\A- Anode Current lOOkA lOOW Cathode Voltage-Regulated _High Power Supply Voltage Reloy 0-6.5KV

o XI X2

1ST **24 Unregulated on 5/8 O.D. Power Supply Ferrlte ring O-IOKV 002 Anode 2

N.O.* 0-30 mA V Current +5 Regulator Circuit N.C.

IN4002

FIG. 21. Laser current controller circuit schematic. o is a 0-6500 V, 0-60 mA voltage-regulated supply.^ The negative high voltage supply is an unregulated power supply. 2

The logic circuit which controls the electrical connections to the laser is shown in Fig. 22. The logic is implemented in transistor- transistor logic (TTL) on a circuit board inside the laser controller chassis. The laser discharge is started by pressing the start button.

This turns on the high voltage to the discharge tube by closing the high voltage relay for several seconds as determined by the 74122 monostable multivibrator circuit. If during this time the laser discharge starts normally, the sensor conditions will all be correct and the high voltage stays on. If not, the high voltage goes off and the start button must be pressed again. The condition that the high voltage stay on and the laser keep running is that the eight inputs, including six sensor inputs, to the two NAND gates all be high.

There are three types of condition sensors as shown in Figs. 23 and 24. Current flowing the PZT circuit, or current flowing between a bushing and the point where it is grounded indicates a discharge between a laser anode and a bushing, a potentially dangerous condition. Such currents are detected by sensitive relays which will trip if as little as 3 mA is detected. The PZT current and bushing current sensors are located in a separate enclosure which sits on top of the laser. Another undesirable condition results if one of the two discharge paths goes out.

If the current through either anode drops below a pre-set value, the

1. Model RE-6506 manufactured by Northeast Scientific Corpora­ tion, Acton, Massachusetts. 2. Model 820-125 manufactured by Hipotronics Incorporated, Brewster, New York. To 47/iF 500Kfi_ High Voltage -£<2- Relay Computer 33k ft +5 HIGH Storl 74122 Red VOLTAGE Indicator 2N4I24 ON

470ft

IN4002 TIP29C High Voltage On

To PZT Relay PZT Connected

Bushings Yellow PZT Anode 1 Grounded Indicator CONNECTED Sensor

Anode 2 470ft > Sensor Computer; Connect Stop PZT Manual 2UN4002 Disconnect Q TIP29C PZT 1 To Bushing Relay Sensor +5 3 t. PZT 2 Green BUSHINGS Sensor w Indicator GROUNDED Bushing 1 Sensor

Bushing Ground Sensor KIN4002 Bushing Manual Flooto TIP29C

FIG. 22. Laser current controller logic circuit schematic. -- The logic is implemented in transistor-transistor logic using series 74LS00 components. To To PZT Relay Bushing Relay A n A n ECG 5143 o N.C. I O N.C. o +5 O o o ->+5 o o o PZT Current o Bushing Current o Sensor (High o Sensor (High for No Current for No Current *IN75I 1N75I t LM5 LM5 Sensitive Sensitive Relay Relay To PZT Driver/Amplifier

FIG. 23. PZT/bushing current sensor circuit schematic. -- These condition sensors are located a chassis box located on top of the laser. The relays will be tripped by as little as 3 mA current. 100

IN4002

To Anode To Laser Resistors Anode TIL3I

TIL8I

Anode Current Sensor 2N4I24 74L5I4 (High=OK)

FIG. 24. Laser anode current sensor circuit schematic. -- These condition sensors are located in the main chassis box between the anode resistors and the connections to the laser anodes. 101 sensor shown in Fig. 24 is tripped. The anode current sensors are lo­ cated within the main controller chassis.

The logic circuit controls the laser through high voltage 3 vacuum-tube relays, switched by the TIP29C power transistors. One relay is shown in Fig. 20, where it switches the high voltage to the laser. Two other relays are not shown. They are located in the chassis box on top of the laser which also houses some of the condition sensors.

One of these relays connects the laser PZT to the PZT driver/amplifier.

The other connects the bushings to ground. The PZT can be disconnected, and the bushings can be disconnected from ground by manual switches on the front panel of the controller.

Because intense electrical noise is generated inside the con­ troller chassis when the high voltage to the laser is switched on or off, special precautions must be taken. First, to prevent spurious signals from entering the logic circuit and causing erratic behavior, all signal lines leading onto the logic circuit board must be heavily filtered and clamped, as shown in Fig. 22. Also, to prevent disruption of other ex­ perimental equipment, the chassis is well shielded. All ventilating holes, etc. are covered with screening to minimize radiation leaking out of the controller chassis.

The performance of the laser current regulator was checked by observing the frequency spectrum of the current fluctuations when the regulator was attached to a 100 kfi load resistor (the laser discharge

3. The relays are manufactured by Kilovac Corporation, Santa Barbara, California. We use model number H-19, a DPDT, 20 kV glass envelope relay. tube could not be used conveniently because then the current pick-off

resistor would have to float at high voltage). Fluctuations at the

fundamental and first few («5) harmonics of the 60 Hz line frequency

were present at the level of about 1 pA. (This is probably due to ca-

pacitive coupling in the filament transformer of the series regulator

tube.) Higher harmonics were present at much reduced amplitudes. Using

1 pA as a current-fluctuation rms amplitude, the resultant laser fre­

quency fluctuations due to discharge current fluctuations is about 1 kHz,

which is much smaller than other sources of laser frequency jitter (see

a later section of this chapter).

Laser Cooling and Gas Filling Systems

Because of the partial population inversion of the medium (see, for example, Ref. 106 ), the CO laser operates much more efficiently at

low temperatures. In fact, many lines will lase only if the discharge

tube is cooled considerably below room temperature. This is accom­

plished by circulating chilled methanol through the cooling jacket of

the discharge tube. The methanol is circulated through insulated poly­

ethylene tubing from a reservoir which is cooled by the evaporator coil

of a two-stage, cascade, mechanical refrigeration system.* The refrig­ erator has sufficient heat-extraction capacity to keep the circulating methanol cooled to about -55°C. This is in the useful range of operat­

ing temperatures. The cooling of the reservoir may be regulated by a

1. This is a model FC-100-84-P40-SV system manufactured by FTS Systems, Inc., Stone Ridge, New York. It is identical to the unit used for cooling the high vacuum system cold traps. 103

temperature sensor and a feedlock loop which cycles the refrigerator on

and off. Lower temperatures than the refrigerator is capable of can be

obtained by flowing liquid nitrogen through auxiliary coils in the meth­

anol reservoir. The limit of about -90°C is reached when the methanol

becomes so viscous that it is difficult to pump through the system.

The laser gas mixture is given in Table 6. The mixture is pre­

pared using the gas filling system shown in Fig. 25. Pressures in the 2 gas fill system are measured by an electronic pressure transducer that

is sensitive to differential pressure. The fill station is pumped by an

oil diffusion pump with a cold trap chilled to about -40°C by a mechani­

cal refrigerator. All the vacuum seals in the system are copper-gasket

Conflat-type seals. No oil or grease is allowed in the system since

minute quantities of these adversely affect the laser action. Likewise

the gas bottles are attached by glass-blowing them onto the system:

greased glass vacuum joints are not acceptable. Metal bellows high vac­

uum valves are used exclusively in the system.

The pure gases from which the laser gas is prepared are commer­ cially available with 99.99% or better purity. The procedure for fil­

ling the laser prevents cross-contamination of the pure gases by allowing filling of the fill station manifold from only one gas bottle at a time.

Unfortunately, this procedure makes necessary a wait while the pure gas in the manifold mixes with the gas in the laser. A wait of about 20 minutes minimum is needed each time an additional gas is added to the mixture.

2. This is model 5010 pressure transmitter manufactured by Computer Instruments Corporation, Hempstead, New York. 104

TABLE 6. Laser gas mixture. -- Gas purity is 99.99% for CO, 99.995% for Xe, 99.999% for N,, and 99.9999% for He.

Gas Partial Pressure

CO 0.6 Torr

Xe 1.5 Torr

2.5 Torr N2 He 14.0 Torr

Total 18.6 Torr CO LASER DISCHARGE TUBE

FLEXIBLE BELLOWS

DIFFERENTIAL PRESSURE GAUGE GAS FILL MANIFOLD O-IOO TORR

•TO MECHANICAL TRAP REFRIGERATOR COLD TRAP = -30° C

OIL DIFFUSION PUMP CO Xe He TO MECHANICAL 1 FORE PUMP

FIG. 25. Laser gas fill system schcmatic. -- The gas fill manifold and all gas bottles are Pyrex with blown-glass joints. The diffusion pump stack is used to evacuate the laser and gas fill system before and during fills. The base pressure is

Laser Frequency Control

In this experiment it is essential that the laser operate at a

stable, known frequency. This goal is achieved in several steps. As

mentioned above, the CO laser operates on a large number of lines between

5.2 and 6.2 pm. Tuning the diffraction grating selects which line or

lines will lase. The grating drive has been calibrated so that it can

readily be adjusted to a desired molecular transition. This was done by

initially identifying several CO lines by means of a one-meter monochro-

mator, and correlating the identified lines with the grating drive

micrometer reading. A least-squares fit of this data to a model cali­

bration function resulted in a list giving the micrometer setting for

each CO transition.

Passive Stabilization. Once the laser is operating on the cor­ rect molecular transition, its frequency is still uncertain by the width

of the gain curve. In CO at -50°C the gain line is broadened inhomoge-

neously, primarily by Doppler broadening, and is about 120 MHz wide.

By changing the length of the laser via the PZT mirror mount, the lasing

frequency may be set anywhere within this frequency band. As the length

of the laser changes due to thermal expansion, the frequency drifts.

The sturdy mechanical construction and temperature-compensated design

reduce the drift rate to about 1 to 4 MHz per hour. This is the limit

of the passive stability of the laser.

Active Stabilization. To obtain further improvement in frequency

stability, active stabilization of the laser to some fixed frequency is

required. A convenient and sufficiently stable reference frequency can 107 be obtained from the CO molecules themselves, namely the center frequency of the transitions. Then the frequency stabilization can be achieved by locking the laser frequency to the center of the gain curve. This can be accomplished by using a feedback circuit as follows: The PZT on which the laser output mirror is mounted is "dithered" at a frequency of about 3 kHz with a small amplitude. If the average point of operation is on the slope of the gain curve, as shown in Fig. 26(a), the laser amplitude will be modulated at the dither frequency. A lock-in ampli­ fier detects this ac modulation and converts it to a dc signal which is amplified and returned to the PZT. This shifts the average point of op­ eration. When the operation point approaches the flat top of the gain curve, for example, the ac amplitude modulation component at the dither frequency vanishes and the laser is locked to that point. Properly choosing the phase of the lock-in allows locking to either a maximum or to a minimum of the gain curve.

In practice the situation is complicated by the fact that there are usually two, or occasionally three, CO laser lines lasing at the same time. Then the laser output power as a function of cavity length is a superposition of curves like the ones shown in Fig. 26(b), each having the maxima and minima in different places. If the maximum (line center) of the transition of interest occurs where the other transition has a large slope, then the maximum of the composite curve will be displaced from the true line center as shown in Fig. 26(b). The effect can be large enough to introduce frequency shifts in the laser frequency of up to a few tens of megahertz. A solution to this problem is to use a 108

a: LU I Q_

OC LU CO < (D 00

PZT VOLTAGE

(a)

PZT VOLTAGE (b)

FIG. 26. Laser stabilization scheme. -- (a) Single CO line lasing case: (i) Unlocked system generating an error voltage, (ii) System locked in single-mode operation, (iii) System locked in two-mode opera­ tion. Cb) Two CO lines lasing simultaneously: (i-ii) Individual line power curves, (iii) Total power curve, (iv) Locking error due to presence of the second laser line. 109

Fabry-Perot interferometer as a filter to eliminate the interfering

transitions.

Figure 27 shows how this is accomplished using two locking loops with a common radiation detector. The two loops operate at well separated dither frequencies, so their actions are quite independent.

One loop locks the Fabry-Perot interferometer to give maximum transmis­ sion of a particular CO transition; other frequencies are not passed.

As the frequency changes, the Fabry-Perot filter tracks the frequency, always giving maximum transmission. With the interfering frequencies eliminated, the second locking loop locks up the laser as described above.

For most of the CO transitions the gain curve is broader than the inter-mode spacing. Therefore it is possible for the laser to run in two longitudinal cavity modes simultaneously. This happens when the laser power is near a minimum, as shown in Fig. 26(a). Since a minimum is a much sharper feature of the gain curve than a maximum, a tighter

lock, requiring less dither, can be obtained by locking there. In this case, the output consists of two frequencies, separated by the inter- mode spacing of — = 99.1 MHz and positioned symmetrically about the

line center. Since the frequency range over which two modes will lase simultaneously is very small, typically 2-3 MHz, very good frequency stability can be achieve by this method.

The Fabry-Perot filter represented in Fig. 27 consists of two

25-mm diameter germanium flats, 1/100 wave at 10.6 um surfaces, with

99% reflective coatings and antireflective coatings on the backs. The flats are spaced about 5 cm apart by Invar LR-35 spacers. The spacing FIG. 27. Laser frequency locking loop schematic. -- Two independent locking loops with a common radiation detector are used as described in the text. The PZT elements and the circuits driving them in the laser and Fabry-Perot filter are actually bi-polar, rather than the simplified diagram shown. CO LASER

TRACKING FABRY-PEROT FILTER

DETECTOR x 50 LOCK-IN

xlOO x 50 LOCK-IN

FIG. 27. Laser frequency locking loop schematic. Ill is adjustable and is chosen so as to allow piezoelectric scanning of a complete free spectral range. The resolution is sufficient to give par­ tial resolution of two longitudinal laser modes separated by 99 MHz.

One mirror is mounted on a three-element piezoelectric aligner/ translator.* Thus the filter can be aligned, as well as tuned, electrically.

The radiation detector shown in Fig. 27 is a thermoelectrically 2 cooled lead-selenide photo-resistive detector. Its integral thermo- electrical cooler will cool it to -20°C. It has a peak detectivity in l. _ i D*=1.0xl0 cm-Hz2 W . The detector is biased by a 90 V battery. The detector signal is amplified by a X10 or a X100 ac-coupled preamplifier.

The lock-in amplifiers shown in Fig. 27 are commercially avail- 3 able single-card phase sensitive detectors with a homemade reference frequency oscillator and phase control unit.

The signal integrators and X50 high voltage amplifiers shown in

Fig. 27 are integrated into one homemade unit called a PZT driver/ amplifier. Since the laser PZT resides inside the low pressure laser gas and discharges can occur quite easily between one of the PZT sur­ faces and the grounded bushing, it is necessary to minimize the voltage excursion of both of the two PZT electrodes with respect to ground.

The voltage excursions can be minimized, while maintaining a constant

1. Model PZ-80 manufactured by Burleigh Instruments, Inc., Rochester, New York. 2. Model OTC-11-52 manufactured by Optoelectronics, Inc., Petaluma, California. 3. Model 4110-X manufactured by Evans Associates, Inc., Berkeley, California. 112

PZT electric field by driving the two electrodes with opposite sign,

symmetrically about ground potential. Since the PZT element is bipolar

in its response (that is, extending for one sign of applied electric

field, and contracting for the other), both signs of electric field

should be available.

A schematic of the circuit is given in Fig. 28. The error input

signal is integrated by op-amp Aj, then summed with a modulation input

and a dc offset input by op-amp The signal and its negative are

then each fed into a high voltage amplifier which drives the PZT. There

is also an asymmetric voltage limiting circuit which prevents damage to

the PZT by applying too large a voltage. It is asymmetric because the

damage thresholds are different for voltages applied in the forward and reverse directions, the damage threshold in the reverse direction being

the depoling voltage.

Frequency Calibration. It is not enough that the laser frequency

be stable: it must also be a known frequency. Many absolute frequency

measurements of a large number of transitions in several isotopes of CO

have been made (for the latest, most complete compilation, see Ref. 107).

A least-squares fit, using a Dunham series expansion can quite accu­

rately extend these calibrationas to unmeasured lines, providing fre­

quency calibration for all CO lines of interest. Unfortunately, the

center of the gain curve in the CO laser is not an absolute frequency standard. This is because the collection of CO molecules used as a reference are subject to considerable environmental influences. It is known, for example, that the line centers shift with laser gas pressure

and discharge current (108). It is easily conceivable that shifts FIG. 28. PZT driver/amplifier circuit schematic. -- Inputs are an error signal input, which gets integrated; two modulation inputs; and a potentiometer-settable dc offset. Outputs are two PZT voltages with opposite signs, and two outputs, scaled down from the PZT outputs, for monitoring by oscilloscope. The limiting circuit limits the total output voltage (PZT POS OUT to PZT NEG OUT) to +600 V and to -300 V. There are two modes of operation of the amplifier circuits, a high voltage mode (output ^20 V) where the outputs are driven by the high voltage transistors, and a low voltage mode (output ^20 V) where op-amps A^ and A^ drive the outputs through the 0.5 F capacitors. In the low voltage mode the frequency response drops to half-power output at approximately 10 kHz. In high voltage mode the output waveform is severely distorted above about 20 Hz. IN4002 20ku LOCK IOOW ADJ 3 Mil > w/ IN4007 100mA PZT 2MA )POS / E-JSMS 820kA VF OUT +450 XiTOkn 30MA 5H0pF 40J IW IOOMA 270a 2N4I24 IN40OT lOOkA HN4002 N ERROR TIME 47kA CONSTANT <4TWl FD300 MCD-2 soon UN4002 ECG-165 IN4007 2W W

3.3kA 5/iF V- iooa 330kA luF?IOOA 2000V 2W IN4007 10-20 pF -450 ON 270kA lOK << IMA IMA IMA IMA IMA 560kA 500kA +450 MODULATION 4 • INPUTS | T 20kA NEG 1 SELECT IOOW LIMITING

I.2MA IN4007 10k AI* 2.7kA F W ON 270kA 820kA V IOKAI/5 5-K)pF 33KA IN4007

2N2I24 lOOkA IN4I48' r i JN4002 \ 330kA MCD-2 FD300 ;i N4002 ECG-165 N4007HJ OFFSET lOkA S< 33kAl ~ V- _X 5*iF 2000V IN4002 450 IMA IMA IMA IMA IMA -450 100mA AW

FIG. 28. PZT driver/amplifier circuit schematic. 114 dependent on gas temperature and gas mixture also exist. Preliminary measurements (108) have established shifts of about 1 MHz per mA with discharge current changes and about 2 MHz per Torr with gas pressure changes. No other measurements of such effects have been reported. The most comprehensive of the absolute CO frequency compilations (107), which incorporates data taken by several authors, presumably under different conditions, does not even mention the possibility of these effects.

Therefore, while the center of the gain curve does provide a convenient frequency locking reference, it does not provide an absolute frequency calibration because it may vary by an unkown amount with the laser operating conditions.

The laser operating parameters are summarized in Table 7.

Laser Beam Optics

The laser beam leaves the output mirror of the laser as a diverg­ ing gaussian beam that appears to be emanating from a virtual beam waist within the laser cavity. To obtain maximum overlap between this beam and the ion beam, the laser beam mode must be transformed so that it focuses to a waist of the proper size at the center of the interac­ tion region of the ion beam line. That is, a beam with a given waist at one position must be transformed to have a specified waist at another position. The gaussian mode matching formulae given in the literature

(109) do not solve this particular problem, but rather the less restric­ tive problem of transforming from one waist size to another, but not with a particular distance specified between them. A method for solving the present problem (with a specified distance between the waists) has 115

TABLE 7. Summary of laser parameters.

Item Value

Cavity length 1.513 m

Output mirror Reflectance = 95% Rj=3 m, R2= «°

Grating 150 lines/mm, 6 pm blaze, efficiency > 90%

Mode limiting apertures:

grating end 5.59 mm diameter

mirror end 6.99 mm diameter

Discharge tube i.d. 12 mm

Discharge length 1.18 m

Cathode-anode voltage 5.3 kV

Discharge current 24 mA

Electrical power input « 130 W

Coolant temperature -20°C to -70°C

Spectral range 5.2-6.2 ym (1923-1613 cm"1) (> 150 lines)

Output power (single line, single mode) up to 1.0 IV

Frequency stability (free running) ^ 4 MHz/h drift

Frequency stability (single-mode operation) < 2 MHz jitter

Frequency stability (two-mode operation) < 500 kHz jitter

Lifetime of gas fill pa 300 h 116

been suggested by Siegman (110; p. 326) but he gave no formulae, so they are

developed here. The original beam is traced forward from the waist, and

the desired beam is traced backward from the position of the desired

waist using the formulae (110; pp. 308-509)

(0(z) = U) (6)

r(z) = z (7)

where z = ttlo /A is the "Rayleigh range," z is the distance from the nK O waist, is the "spot size," and A is the wavelength. At some point

between the two waists, both beams may have the same spot size. (If

such a point does not exist, mode matching is not possible with a single

lens.) Since a lens cannot change the beam spot size, but only the wave-

front radii of curvature, this is the point at which a matching lens

must be placed- If the original waist spot size is cooj at position z=0

and the desired waist spot size is ojq2 at position z=d, then the posi­

tion at which the two beams are of equal size is given by

2 . 2 ,,2 o z = - 1 ± m + (m -1) l-j-. (8) I 2 m -1i

where m = is the "magnification," fQ = — "0j<»>02 Kogelnik's

(109) "characteristic matching length," and the sign is chosen to give

a z such that 0

determine the focal length necessary to affect the transformation from

one beam mode to the other from

L + _L ?7 + 77=fT • O) "l r2

The solution is

r r f = i 2 . (10) rr 2

If a single thin lens (or mirror) is not used, the standard formulae for thin lenses in series may then be used to calculate the positions of the individual elements.

Since it is desirable to make the focusing system as achromatic as possible over the wavelength range of the CO laser (and even over a range determined by other lasers such as CO2 lasers operating at 10.6 ym), it is expedient to use reflective optics rather than refractive optics. In particular, spherical mirrors can be used in the place of lenses. However, spherical mirrors must be used off-axis and this introduces aberrations into the system - primarily astigmatism. The reason that aberrations are of concern is that deviations of the wave- fronts at focus from the ideal plane waves will lead to instrumental line broadening due to increased range of interaction angles. Astig­ matism can be minimized by using a concave and a convex spherical mirror in series. The contribution to the astigmatism from one off-axis mirror can then be nearly completely compensated by an opposite contribution from the other, while the combination retains substantial focusing power. 118

Five-centimeter diameter, gold-coated mirrors, concave and con­ vex, with 1-meter radius of curvature were used for focusing. They were separated by 26.5 cm, giving an effective focal length for the pair of

94 cm. This would very nicely focus the laser beam (which has a vir­ tual waist with spot size Woj=0.58 mm a distance of 70.6 cm behind the output mirror) to a waist with spot size u) 2=1-70 mm at the center of the ion beam interaction region, which is 4.039 meters from the laser output mirror. The off-axis angle for the spherical mirrors was ?»9°.

More detailed aberration analysis (111) of this layout reveals that the residual astigmatism is ^1/15 wave and that spherical aberration and coma are negligible.

In addition to focusing the laser beam properly, it must be aligned to pass through the center of the interaction region at the proper angle. The alignment of the CO laser beam through the ion beam is done as follows. First, the optical axis of an alignment telescope* is brought into coincidence with the axis defined by the two cross-hairs in the ion beam interaction region. Then, a visible helium neon laser beam is made coincident to the alignment telescope axis. Finally, the

CO laser beam is aligned to the helium neon laser beam. The various beams are merged by using mirrors mounted on hinged mounts so that they can swing into and out of the infrared beam. Since accurate reposition­ ing during the alignment procedure is necessary, a hinge with very

1. Model D-275 manufactured by Davidson Optronics, Inc., West Covina, California. 119 little play must be used. A suitable mount was designed using flexural

2 pivots as the hinges.

Each axis (the infrared laser beam axis, the visible laser beam axis, and the alignment telescope axis) must be brought into coincidence with the axis defined by the two cross-hairs in the ion beam interaction region. Since each such alignment has four degrees of freedom, two mirrors, each with two angular adjustments, give a sufficient number of adjustments to accomplish the alignment. We use a combination of high 3 precision mirror mounts manufactured by Newport Research Corporation 4 and by Oriel Inc. Tests in our laboratory have shown that these two mounts have superior stability against thermally-induced drifts and ex­ perience very little unwanted motion during adjustment.

For steering the CO laser beam, infrared-enhanced high reflec­ tivity flat mirrors^ are used.

Signal Processing, Data Acquisition, and Computer Control

As described in Chapter 4, obtaining useful signal-to-noise ratios in this experiment requires rather long integration times. When searching for resonances with relatively poorly known frequencies, searches of a few hundred hours may be required. Such time scales ne­ cessitate some degree of experiment automation. This section describes

2. Manufactured by Bendix Corporation, Electric and Fluid Power Division, Utica, New York. 3. Model 600A-2, Newport Research Corporation, Fountain Valley, California. 4. Model 1770, Oriel Corporation, Stamford, Connecticut. 5. Model 20D04 with ER.2 coating manufactured by Newport Research Corporation, Fountain Valley, California. 120 the signal processing, computerized data acquisition, and computer ex­ periment control used in the experiment.

Signal Processing

The raw signal is the fluctuating ion beam current collected in the Faraday cup at the end of the beam line. A resonance is indicated by a change in the current due to a change in the beam survival in the gas target, as discussed in Chapter 4. Since the change is rather small, typically of few parts in 10^ of the total beam current, the laser beam is mechanically chopped at a frequency near 1 kHz and the Faraday-cup signal is synchronously detected. The resonance signal is then the change in beam current that is coherent with the chopping of the laser beam. The signal is masked by statistical noise (shot-noise) in the beam current, but can be extracted from the noise as it travels the fol­ lowing signal processing path.

The ion beam current collected in the Faraday cup is fed to a high input impedance preamplifier. This is a simple, inverting, current- 9 to-voltage op-amp circuit with a feedback resistance of 1x10 £2, and an

FET-input op-amp (112). The large resistance, combined with unavoid­ able input and cable capacitance gives the amplifier a rise time of about 1 msec, which is barely acceptable since the signal is modulated at just less than 1 kHz. The output, which follows the input as 1 V/nA is fed into a lock-in amplifier* with an input filter tuned to 1 kHz.

The reference signal for the lock-in is obtained from a photodetector

1. Model 128A manufactured by Princeton Applied Research Corporation, Princeton, New Jersey. 121 that is illuminated through the laser beam chopper wheel. The sensitiv­ ity and output-filter time constant of the lock-in are adjusted to keep the fluctuations from occasionally overloading the lock-in circuits.

The output of the lock-in amplifier is fed to a homemade inte­ grating digital voltmeter (IDVM). A simplified schematic is shown in

Fig. 29. The analog-to-digital conversion and integration is done by counting the output pulses of a voltage-to-frequency encoder for a fixed period of time. The number of counts at the end of a time period is latched and displayed by seven-segment LED displays. The count is also presented to a 12-bit digital-to-analog converter which drives the sig­ nal input of a chart recorder for making resonance traces during a run.

Microcomputer System Hardware

The experiment is controlled by a microcomputer system based on an S-100 bus "personal" microcomputer. In order to take advantage of an excellent software package, we chose a Z-80 based"1 system. Themain- 2 frame of the system is an IMSAI 8080 chassis, motherboard, front panel 3 and power supply. The CPU board is an Ithaca Audio 4 MHz 2-80A board. 4 With a bus termination board, the system will reliably operate with a clock speed of 4 MHz. The computer has 32 kilobytes (1 kilobyte = 1024

8-bit bytes) of programmable random access memory residing on two 16- kbyte, 250-nsec, static RAM memory boards.^ There is also approximately

2. Manufactured by IMSAI Manufacturing Corporation, San Leandro, California. 3. Now known as Ithaca Intersystems, Ithaca, New York. 4. Manufactured by Bill Godbout Electronics, Oakland, California. 5. Model 1625 manufactured by Dynabyte, Inc., Palo Alto, California. FIG. 29. Integrating digital voltmeter schematic. -- The logic is implemented in transistor-transistor logic (TTL) using 7400 series com­ ponents. The boxes labeled DV600 and DV610 are a 10 kHz clock signal generator module and voltage-to-frequency encoder modules, respectively They are manufactured by Hybrid Systems Corporation, Burlington, Massachusetts. The 12-bit digital-to-analog converter module is a model DAC-12QZ manufactured by Burr-Brown Research Corporation, Tucson, Arizona. 122

FIG. 29. Integrating digital voltmeter schematic. 123

16 kbytes of erasable, programmable read only memory (EPROM) which con­ tains the software operating system and the BASIC programming language and resides on a 32-kbyte capacity ROM board.^ The command console is a model 35 Teletype, which is interfaced to the computer via a 20-milliamp current loop and a serial I/O port on a serial/parallel input/output 7 board. A perforated paper tape punch and high speed reader are inter-

g faced through parallel I/O ports on another I/O board.

The experiment is interfaced to the computer through two inter- 9 face boards. The first is another parallel I/O board through which the integrating digital voltmeter is interfaced. The IDVM status and data can be read through parallel input prots, and commands to set the count period, start counting, etc. can be sent to the IDVM through parallel output ports. (Alternatively, an EPROM programmer,which is used to write programs and data into the EPROM chips, can be attached as an ac­ cessory to the same I/O ports.)

The computer controls the ion beam voltage by turning a poten­ tiometer connected to the beam power supplies (as described in another section of this chapter) with a stepping motor. The stepping motor is interfaced to the computer by a homemade stepping motor interface board.

A detailed description of the board has been given by Farley etal. (113).

6. Model 2708/2716 EPROM manufactured by Ithaca Audio, now known as Ithaca Intersystems, Ithaca, New York. 7. Model 10-4 manufactured by SSM, Santa Clara, California. 8. Model PI0-4 manufactured by IMSAI Manufacturing Corporation, San Leandro, California. 9. Altair 88-4PI0 board manufactured by MITS, Inc., Albuquerque, New Mexico. 10. Model EP-2A-79 manufactured by Optimal Technology, Inc., Earlysville, Virginia. 124

The circuit schematic is reproduced in Fig. 30. The board contains the

logic and circuitry necessary to energize the four motor windings of a

standard 4-phase, 6-lead stepping motor in the proper sequence for step­

ping the motor in either direction, in full steps or half steps. The

board can be disabled and reset by external button and limit switches,

or by software.

Also available, but not used in the present experiment are an

audio cassette recorder interface*"* and a homemade analog input/output

board based on a Burr-Brown model SDM853 data acquisition module 12

(analog-to-digital converter) and Burr-Brown model DAC80-CBI-V digital-

to-analog converters.

Microcomputer Software

The microcomputer operating system is the ZAPPLE monitor.*

ZAPPLE provides several commands which are useful for system software

and hardware development as well as handling all system input and output

functions. ZAPPLE resides in read only memory. ZAPPLE was modified to

be compatible with the hardware with the aid of an assembler program."

Also residing in read only memory is a version of the BASIC programming

language. In addition to the usual features, this version of BASIC has

11. Manufactured by Tarbell Electronics, Carson, California. 12. Manufactured by Burr-Brown Research Corporation, Tucson, Arizona. 1. ZAPPLE monitor version 1.11, Technical Design Labs/Xitan, Inc., Princeton, New Jersey. 2. T.D.L. Z-80 Relocating Macro-Assembler, version 1.2, Technical Design Labs/Xitan, Princeton, New Jersey. 3. T.D.L. ZAPPLE BASIC, version 3 (12 K version), Technical Design Labs/Xitan, Princeton, New Jersey. FIG. 30. Stepping motor driver/interface circuit schematic. -- Shown is one of two interfaces that are incorporated on an S-100 bus circuit board. The board contains the logic to energize the four windings of each motor in the proper sequence for either half-stepping or full-stepping, in either the clockwise or the counter-clockwise direction. The board can be disabled and reset either by software or by hardware via the panic and clear buttons shown or motor limit switches mounted externally. The I/O port address of the board is determined by the jumpered address network JAN. The prefix 74LS has been omitted on all applicable integrated circuit designations. The values of components shown are: resistors-ohms; capacitors-microfarads. P.S. is an 8 V, 20 A dc power supply. From Ref. (113). Used by permission. • sv 7407

IN4002

OCX) STEP __ RESET I

WmOTroaN 04-0"DtC.O'L DATA OttTA LINES LINES S-100 BUS

FIG. 30. Stepping motor driver/interface circuit schematic. facilities for reading and writing input/output ports; examining and altering memory locations; manipulating characters strings; dumping pro­ grams to, and retrieving them from an external storage device; and inter­ facing to machine language routines via CALL statements. Floating point arithmetic is done with 14 digits of precision.

The experiment control/data acquisition program is written in

BASIC. It is structured as an endless loop in which the program waits for a command from the computer console, then decodes the command and executes it, then returns to waiting for another command. The various commands are useful for setting up and testing various components of the experiment and for actually making data collection runs. A listing of the control program is given in Appendix C. Data is collected by execu­ ting the following sequence the necessary number of times: First, the ion beam voltage is incremented by outputting the proper commands to the stepping motor. Next, the IDVM is signaled to begin counting. The program then enters a waiting state until the IDVM signals that the in­ tegration period is over and data is available. (Optionally, the com­ puter will spend this time punching to paper tape, in a specified format, data acquired during the previous integration period.) When the IDVM signals that data is available, the program will read the data, do some preliminary processing, and then store it in memory. Then the ion beam voltage is incremented and the cycle repeats. At the same time, a trace is being drawn on the chart recorder automatically (not under computer control). 127

Computer Programs

Subtle, but essential components in designing and carrying out this type of experiment are computer programs. One program, the micro­ computer program that operates the experiment is described in the pre­ vious section. Other useful computer programs involved in the experiment are outlined below.

One very important program is the "matching" program that deter­ mines where to look (in terms of ion beam voltage and CO laser line) to find resonances. It is written in FORTRAN and runs on the University of

Arizona DEC-10 time-sharing computer system using the F40 FORTRAN com­ piler. Very briefly, it generates a list of theoretical molecular ion transition frequencies and a list of CO laser line frequencies and then searches for a "match." A match is a molecular ion transition which can be Doppler-shifted into resonance with a CO laser line. The program output lists matches sorted by molecular ion transition or by laser transition, and gives the ion beam voltage and laser grating micrometer setting corresponding to each match.

Another useful FORTRAN program is used to extrapolate raw experi­ mental data to "zero" operating conditions. For example, there could be a shift in the observed resonance voltage as the ion source conditions are changed. This program can aid in determining the true resonance voltage when the ion source conditions are hypothetically reduced to nonperturbing values. This program makes extensive use of the CURFIT non-linear least-squares fitting program given by Bevington (114).

Other miscellaneous programs were used during the construction of the experiment - for such things as calculating laser beam propagation and transformation, calculating strengths and deflections of truss structures, etc. CHAPTER 6

EXPERIMENTAL RESULTS AND DISCUSSION

Portions of the infrared spectrum of the X^E+ ground electronic 4 + state of HeH have been obtained (65) using the Doppler tuned ion beam

laser resonance method described in Chapter 4. The frequencies of five

transitions between 1700 and 1900 cm * have been measured to ±0.002 cm

In this chapter, these experimental results are presented and discussed.

Experimental Procedure

Searches for transitions were made beginning with the laser-

molecule match found by the matching program. As input, the program

used the best available theoretical calculations'' for the HeH+ transi­ tions chosen. The laser was tuned to the appropriate CO line and locked there. Because of the difficulty in keeping the laser locked in two- mode operations, as discussed in Chapter 5, for the long period of time necessary to find a resonance, the laser was locked to the peak of the

(multiline) gain curve where a lock could be maintained indefinitely.

The ion source gas reservoir was filled with the proper mixture of gases and the source gas was turned on. The ion beam was set at the correct voltage and the beam line was adjusted to give maximum HeH+ current. A search was then made under microcomputer control, with the beam voltage

1. At first this was the Born-Oppenheimer calculations of Dabrowski and Herzberg (31), but before the first resonance was found the more accurate adiabatic results of Bishop and Cheung (59) became available.

129 130

being incremented in 0.1 V steps, and with 8- or 16-sec integration time

at each voltage setting. The length of a scan was limited to about

950 V because of errors of the mass analyzing magnet circuit in tracking

the beam voltage.

The search for the first resonance took about 300 h of scanning

and covered a range of 5425 V, corresponding to 0.44 cm Each succes­

sive resonance was easier to find, since with each measurement a more

accurate prediction of the next transition frequency could be made.

This prediction was based on a Dunham-type expansion (115) of the

experimental-theoretical differences, using the transitions already

measured. In this way, rapid convergence was achieved so that the fifth

resonance found required only about 1.5 h of searching. Signal-to-noise

ratios obtained under these search conditions (8-sec integration time,

0.1 V steps) were greater than about 3:1.

After each resonance was found, a slower, more closely sampled

scan was made to acquire data for measurement. For these scans, 0.05-V

steps were used and the integration times were'32 or 64 sec per step.

Signal-to-noise ratios were now greater than about 10:1. A typical resonance signal is shown in Fig. 31. Scans under these conditions

took 2 to 3 h.

As will be discussed below, the uncertainty in the measurement,

that is the calibration error, is greater than the observed linewidths.

In view of this fact, it may be argued that the signal was sampled at

more points than is necessary. For the sampling density shown in Fig.

31, for example, the line center could probably be determined to better than 1/10 of the linewidth, a greater accuracy than can be justified —I—I—I—I—I—I—I—I—I—I—I—I—I— 6055 6050 6045 Beam Voltage (V)

FIG. 31. Chart recorder trace of the (1, ll)«->(0,12) resonance. -- Voltage step size is 0.05 V. Integration time is 32 sec per step. Focused laser flux was approximately 5x10^ W/m-. HeH+ beam current was 7x10"® A. Peak signal is 1.5x10" of the beam current. FWHM is 5.6 MHz. From Ref. (65). 132 since the calibration error is much greater than this. This objection has some merit if the only information being sought is the line center.

However, these resonances are composites of hyperfine transitions which are split by small, but unknown amounts. The purpose in taking such densely sampled data was to look for partially resolved hyperfine struc­ ture in the resonances.

In fact, the resonance in Fig. 31 does display a slight asym­ metry, being skewed out on the high voltage side. All the resonances observed exhibited this behavior to some degree. Apparently this is not due to hyperfine splittings, however, since the observed asymmetries are always to the higher voltage side of the resonance, regardless of whether the laser is directed parallel to (Doppler down-shifted) or antiparallel to (Doppler up-shifted) the ion beam. Such a situation could arise from an asymmetrical distribution of initial ion velocities, but not from a hyperfine structure induced asymmetrical distribution of the resonance frequencies. Therefore, the correct explanation of the observed asymmetries probably involves the initial ion velocity distribution.

Method of Data Analysis

The output of the experimental procedure given in the preceding section is a chart recorder trace. The beam voltages at the beginning and at the end of the run were accurately measured and recorded, pro­ viding an accurate voltage calibration for the chart recorder trace.

The center of the resonance line was estimated "by eye." It was not difficult to do this to within 0.15 V, which corresponds to about 1 MHz 133 in frequency. The line center estimates thus obtained are the "measured resonance voltages" listed in Table 8.

As discussed in more detail in the following section, there are various physical effects which can introduce errors into the measurement.

In particular, there are effects which can cause the ions to have an energy which is different from the beam voltage. Examples might be a potential difference at a plasma boundary in the ion source, and space- charge-induced potentials both in the ion source and in the ion beam itself. These effects can be eliminated by taking data under various conditions and extrapolating the results to some suitably defined set of "zero" conditions. This was done as follows: The (2,£))•«->-(1, 10) res­ onance was measured under various ion source conditions, for a total of over 20 runs. The results were fitted to the following simple formula with a least-squares procedure. The fitting function is:

v A1 + A2X4 + A3X][X4 + A4X2 + A5X3 + A6X5 , (1)

where X^ is the ion source chamber pressure in 10 ^ Torr, the electron bombardment voltage, X^ is the focusing electrode voltage, X^ is the electron emission current in mA, X^ is the HeH+ ion beam current in nA, and the A.-'s are fitting coefficients to be determined. This Is multilinear model for the resonance voltage shifts due to operating con­ ditions was chosen for simplicity and for lack of a better functional form. In Eq. (1), the A^ term is a constant. The A^ term models a shift due to electron space-charge-induced potential in the ion source ionization/reaction channel. The A^ term models the effects of TABLE 8. Extrapolation of data to zero conditions. -- The conditions under which the resonances were made are given, and the raw and corrected resonance voltages are given.

Transition Ion source Electron beam Focusing Electron HeH beam Measured Corrected chamber voltage electrode emission current resonance resonance pressure voltage current voltage voltage (v,J)~(c',J') (Torr) (V) (V) (mA) (nA) (V) (V)

(1.11)-<->-(0,12) 6.7x10" 400 -160 21.9 4.95 6050.40 6050.56

-6 (1.12)-^(0,13) 6.6x10 400 -140 23.5 5.65 8265.02 8265.13

-6 (2.8)*-*(l,9) 7.3x10 400 -148 22.5 2.55 7869.70 7870.33

-6 (2.9)^(1,10) 5.4x10 400 -160 24.7 2.90 6174.17 6174.22

-6 (2.10)^(1,11) 7.8x10 400 -150 22.5 3.95 4412.66 4413.24 135 space-charge due to positive ions in the ionization/reaction channel.

The term models penetration of the electron bombardment electric field into the ionization/reaction region. The term models the volt­ age shift due to penetration of the focusing electrode electric field into the ionization/reaction channel or shifts due to ions being formed at different points within the channel. The A^ term models the effect of space-charge depression in the ion beam. The signs chosen for the terms and parameters are not important (as long as they are internally consistent) since the fitting procedure will assign signs to the A^ coefficients as necessary.

The extrapolation was done by calculating 6V where

v . » = v , + 6V , (2) corrected measured '

t ie where ^correctecj ' resonance voltage that would be measured in the limit as all the parameters go to zero. The fitting program returned the following form for 6V:

6V = -A2X4 - A3XiX4 - A4X2 - A3X3 - A6X5

= -(0.0741)X4 + (0.0111)X1X4 + (0.0030)X2

+ (0.0029)X3 - (0.1188)XS . (3)

This formula was then used to correct all the measured resonance volt­ ages as shown in Table 8. 136

Analysis of Experimental Linewidths and Uncertainties

Full widths at half maximum of the observed resonances range

from 5 to 8 MHz (0-00017 to 0.00027 cm . Major contributions to the

linewidths are ion kinetic energy spread, laser spectral width (laser

frequency jitter), and interaction angle spread due to both the ion beam

and laser beam divergence. Lesser contributions are laser power broad­

ening and transit time broadening. The individual contributions are

estimated in Table 9. Also listed in Table 9 are estimated contribu­

tions to the transition frequency measurement uncertainty. The most

important of these are contributions arising from the laser frequency

uncertainties.

Linewidth Contributions

The ion kinetic energy spread contributes to the linewidth by

generating a range of beam voltages over which there are ions in reso­ nance with the laser. This energy spread is a property of the ion source

and not the ion beam line. It has not been measured in the present

experiments. In Table 9 a reasonable value of 0.5 eV has been assumed.

This is an approximate upper limit since a larger value would broaden

the resonances beyond the widths observed.

With the laser locked to the peak of the gain curve, the PZT

voltage dither necessary to maintain a lock was about 5.1 V, correspond­

ing to 1.7 MHz. The laser would not drift outside of this modulation band while locked, so this number is the laser spectral width contribution. TABLE 9. Estimated contributions to experimental linewidths and uncertainties. -- Assumed beam energy is 5000 eV and assumed transition frequency is 1900 cm 1 (5.7xl07 MHz).

Item Assumed Contribution Contribution Fraction of value in MHz in cm~l transition frequency

A. Contributions to linewidth (FWHM)

Ion kinetic energy spread 0.5 eV 4.2 1.4xl0~4 7.3xl0~8

Laser spectral width 1.7 MHz 1.7 5.7xl0~5 3.OxlO"8 -3 -4 Ion trajectory angular spread 4x10 rad 3.6 1.2x10 6.3xl0~8

Laser beam angular spread 1.2x10 ^ rad 1.1 3.6xl0-5 1.9xl0"8

Laser power broadening 0.8 MHz 0.8 2.7xl0"5 1.4xl0~8

Transit time broadening 1.5 sec 0.7 2.3xl0"5 1.2xl0~8

Hyperfine structure 500 kHz 0.5 1.7x10"' 8.7xl0~9

Quadratic sum « 6.0 « 2.0xl0"4 1.05x10" TABLE 9--Continued

Item Assumed Contribution Contribution Fraction of value in MHz in cm"* transition frequency

B. Contributions to experiment uncertainty (calibration errors) -3 Laser frequency uncertainty 50 MHz 50 1.7x10 8.8x10 (stabilization accuracy)

-4 -8 Laser frequency uncertainty 5 MHz 1.7x10 8.8x10 (spectroscopic data)

-4 -8 Laser frequency shifts with operating 5 MHz 1.7x10 8.8x10 conditions -4 -7 Potential offsets 1.5 V 12.5 4.2x10 2.2x10

-4 -8 Beam voltage calibration (100 ppm) 4.0 1.4x10 7.2x10

Extrapolation to zero conditions 0.3 V 2.5 8.3x10 4.4x10 -8 (uncertainty)

-5 -8 Line center estimate from chart 0.1 V 0.8 2.7x10 1.4x10

Quadratic sum *60 SO.002 9.2x10 139

The angular spreads in the ion trajectories and the laser beam divergence both contribute to the linewidth by providing different amounts of Doppler shift for the same beam voltage. The laser beam angular spread is given by the diffraction angle for the laser focal spot size assumed. The ion trajectory spread is half the angle defined by the beam defining apertures. This is a compromise between the best case of parallel ion trajectories and the worst case allowed by the aper­ tures; it should represent a good estimate of the true situation which has a space-charge limited ion beam waist at the center of the inter­ action region.

Laser power broadening has been determined experimentally to be very small. The resonances reported here were taken with 100 to 155 mW of laser power. For one resonance several runs were taken as the laser power was systematically reduced to as low as 8 mW; thi? resulted in only a slight (r*10?O reduction in the observed linewidths.

These estimates lead to a composite linewidth estimate of 6 MHz which is in excellent agreement with the observed linewidths of between

5 and 8 MHz.

Frequency Measurement Uncertainties

Although the transition frequency measurements are very repro­ ducible (reproducibility of the measured resonance voltage for a given transition was better than 0.25 V, corresponding to 2 MHz, when attempt­ ing to match the operating conditions of a previous run), there are still errors in the calibration of the frequency measurements. Table 9 also lists estimated contributions to the transition frequency measurement 140

uncertainty. It can be seen that the calibration errors are consider­

ably larger than the experimental linewidths.

The largest contribution to the calibration error is the laser

frequency uncertainty. The principal cause for this is the fact that

the laser was locked to the peak of the multi-line gain curve as ex­

plained in Chapter 5. The result is that the laser frequency could dif­

fer from the line center frequency by several tens of megahertz. Lesser

contributions arise from the uncertainties in the laser spectroscopic

data and in shifts of the laser frequency with operating conditions.

Assuming the values of 1 MHz/mA and 2 MHz/Torr obtained by Laguna-Ayala

(108), an uncertainty of 5 MHz due to operating conditions seems reasonable.

Another important systematic error is that due to potential off­ sets. That is, the voltage measured by the voltmeter is not the ion kinetic energy in electron volts because of unmeasurable voltage drops across dissimilar conductor junctions (and perhaps plasma boundaries in the source). Magnitudes of potential offsets are not well documented, but a value of as much as 1.5V does not seem to be too high.

Although the beam voltage can be measured with a relative pre­ cision of only 0.01% (100 ppm), this affects the Doppler shift which is only 0.1% of the transition frequency, so this is not a major contribu­ tion to the calibration error.

As described in the previous section, the raw data was corrected by extrapolating to "zero" conditions. The uncertainty in the extra­ polations is also a small contribution to the calibration error. 141

The resulting calibration error is about 1 part in 10^ of the transition frequency or 60 MHz or 0.002 cm

Presentation of the Results and Comparison with Present Theory

The experimentally observed transition frequencies are presented in Table 10, along with the CO laser frequencies and corrected resonance voltages which are used to calculate them. A comparison with the most accurate theory is also made. The theoretical values are obtained from the adiabatic energy levels of Bishop and Cheung (59). Bishop and

Cheung conservatively estimate that their energy levels are in error by less than 1 cm ^ Because of cancellation, the errors in derived transi­ tion frequencies should be somewhat smaller. At this level is appears that the experiment and theory are roughly consistent. It may be- inter­ esting to note that the theory seems to be in better agreement with ex­ periment for larger rotational quantum numbers. This trend seems to hold for both the v=l to 0 and the v=2 to 1 vibrational bands. This is the opposite of what one might expect from this type of variational cal­ culation (118). However, since the magnitude of the trend is less than the uncertainty of the theoretical values, the significance of the trend is questionable.

The theoretical results quoted in Table 10 are in the adiabatic approximation. Thus, the effects of nonadiabaticity, relativity, and quantum electrodynamics have been neglected. These effects have not yet been calculated for HeH+, but order-of-magnitude estimates of them can be made, starting with values which have been calculated for other simple molecules. The nonadiabatic transition frequency corrections TABLE 10. Summary of the measurements of HeH vibrational-rotational transition frequencies. -- The 12cl&o laser frequencies and the corrected resonance voltages on which the measurements are based are also given. The value used for the HeH+ mass is 5.009879737(49) amu (Mc2=4666.710 MeV) as calculated from data given by Cohen and Taylor (116). The interaction angle 6' is given by TAN0'=0.010825.

HeH transition CO laser transition Observed Observed HeH Theory Exp.-Theory resonance transit ion assumed voltage frequency^ frequency (corrected) (v,J)-w(v',J') (v,J)-Kv',J') (cm"1) (V) (cm 1) (cmr -1-,) (cmr -1-*)

(1,11)^(0,12) (10,12)^(9,13) 1858.,8984 6050.,56 1855,.905 1856.,152 -0..247

(1,12)^(0,13) (13,19)^(12,20) 1755.,2748 8265.,13 1751,.971 1752. 198 -0.,227

(2,8)^(1,9) (8,16)^(7,17) 1893..5149 7870..33 1896 .992 1897..139 -0..147

(2,9)^(1,10) (11,19)->(10,20) 1805..2860 6174..22 1802.349 1802.,492 -0..143

(2,10)^(1,11) (16,12)->(15,13) 1707..8920 4413..24 1705.543 1705..684 -0..141

Calculated from the data of Ref. (1 7)-

^Assigned uncertainty is ±0.002 cm * (see text).

CAdiabatic calculation of Ref. (59). Uncertainty is probably about +0.2 cm 143

(119) for are about 1 cm The effects in HeH+, which is isoelec- tronic with 1^, should be comparable. The relativistic corrections to vibrational frequencies in are typically +0.024 cm * (120). The radiative corrections in are typically-0.022 cm * (121). In HeH these effects should be several times larger because of the larger charge

2 of one nucleus. A rough scaling using a Z nuclear charge dependence indicates that both these effects could be about 0.1 cm * in magnitude.

Thus, the neglected effects are about the same size as the present theo­ retical uncertainty. A definitive determination of the accuracy of the present adiabatic theory will therefore have to await calculation of the neglected effects for HeH+. The comparatively high experimental preci­ sion will allow an accurate check on such theoretical results.

Comment on the Implications of the Present Work

It is hoped that the results presented here will stimulate the theoretical work indicated in the preceding section. In view of the considerable complexity of the theoretical problem, as compared to the one-electron ion HD+, it seems quite likely that the present experimen­ tal results will present a substantial challenge to theoretical quantum chemists in matching the experimental precision.

It is also hoped that these results will help stimulate and fa­ cilitate searches for HeH+ in extraterrestrial regions. In any event, the present results should aid greatly in clarifying the ambiguities in the identification of certain infrared spectral features as discussed in Chapter 1. Note that the present measurements are accurate enough to allow the determination of velocities of moving HeH+ sources to within 500 m/sec, if vibrational-rotational transitions are observed, or 10 km/sec, if purely rotational transitions are observed, provided of course that the astronomical observations have sufficient precision.

It may be that detection of HeH+ will be possible only via purely rota­ tional transitions as suggested by Flower and Roueff (38). The present results have some bearing on this possibility also, since they may be used to generate more accurate estimates of the frequencies of these transitions than is presently available.

Comment on Possible Future Work

This chapter has presented experimental results for five transi­ tions in the v=l to 0 and the v=2 to 1 vibrational bands. Figure 10 indicates that a few additional transitions might be observable with the present laser and ion source. Some of the accessible transitions are in the v=3 to 2 vibrational band. Here, only the relatively poor

Born-Oppenheimer frequencies of Dabrowski and Herzt;erg (31) have been available as a starting point for searching for these resonances. This would entail long search times to find these resonances (probably longer than that which was required to find the first resonance we observed).

Possibilities for future work include measurement of these as yet unseen transitions and also more accurate measurement of the pres­ ently reported transitions, these being made with the laser locked up in two-mode operation. The latter measurements should be an order of magnitude more precise than the present results. APPENDIX A

COMPREHENSIVE LISTINGS OF THEORETICAL WORK ON HeH+

The purpose of this appendix is to gather together, in one place, a comprehensive listing of theoretical work on the structure of HeH+.

Since this dissertation involves the structure of HeH+ (through spec­ troscopic interaction) rather than reaction properties and collisional interactions, theoretical studies in those sorts of areas are not included.

Much of the theoretical work on the structure of HeH+ has been ab initio calculations carried out using the variation principle. It is these calculations that are of interest here for comparison with the present experimental results. Other approaches include approximate treatments and model theoretical treatments. The later approaches were often the result of applying to HeH+ an approximation that was developed as a tractable method for use with more complicated molecules. The ap­ plication to HeH+ was to test the method by comparison with more rigor­ ous results. Since the division into ab initio and approximate approaches is somewhat arbitrary, references to both types of work are given in the following tables. Table 11 lists, in chronological order, treatments judged to be ab initio, while Table 12 lists approximate and model theoretical treatments. The listings are thought to be compre­ hensive; they are the results of several manual and computerized litera­ ture searches conducted between August, 1979 and February, 1980.

145 146

TABLE 11. Comprehensive listing of ab initio theoretical treatments of HeH+ with method of treatment (in chronological order).

Reference Method of Treatment

G. Glockler and D. L. Fuller, Applied Heitler-London method J. Chem. Phys. _1, 886 (1933). to HeH+.

J. Y. Beach, J. Chem. Phys. 4_, First variational calculation 353 (1936). of HeH+: essentially LCAO.

C. A. Coulson and W. E. Duncanson, Seven different variational+ Proc. Roy. Soc. (London) A165, 90 calculations applied to HeH . (1938).

S. Toh, Proc. Phys. Math. Soc. Extended James and Coolidge- Japan 22, 119 (1940). type wave functions.

A. A. Evett, J. Chem. Phys. 24, Extension of Toh's wave func­ 150 (1956). tions.

A. C. Hurley, Proc. Phys. Soc. Modified MO calculation with (London) A69, 868 (1956). empirical corrections.

R. Bhattacharya, Proc. Natl. Inst. Simplest LCAO calculation. Sci. India A27, 185 (1961).

B. G. Anex, J. Chem. Phys. 38, Configuration-interaction 1651 (1963). SCF calculations.

H. H. Michels and F. E. Harris, Open-shell single-configura­ J. Chem. Phys. 39, 1464 (1963). tion calculation.

H. Conroy, J. Chem. Phys. 41_, Configuration-interaction cal­ 1341 (1964). culation including correlation terms.

S. Peyerimhoff, J. Chem. Phys. £3, Hartree-Fock LCAO calculation. 998 (1965).

L. Wolniewicz, J. Chem. Phys. 43, Generalized James-Coolidge 1087 (1965). basis set.

J. Goodisman, J. Chem. Phys. 43, Modified James-Coolidge basis 3037 (1965). set.

F. E. Harris, J. Chem. Phys. 44_, Ellipsoidal wave functions. 3636 (1966). 147

TABLE 11-Continued

Reference Method of Treatment

H. H. Michels, J. Chem. Phys. 44, Linear combination of configu­ 3854 (1966). rations.

J. R. Hoyland, J. Chem. Phys. 45^ SCF with configuration- 466 (1966). interaction

J. R. Hoyland, J. Chem. Phys. 47, SCF for IT states. 49 (1967).

L. Piela, Int. J. Quantum Chem. Rayleigh-Schrodinger pertur­ 5, 945 (1969). bation theory.

P. R. Certain, J. Chem. Phys. 55, Exchange perturbation theory. 3045 (1971).

K. E. Banyard, M. Dixon, and A. D. Electron correlation effects. Tait, J. Phys. B 5, L160 (1972).

T. Yamada, H. Sato, E. Ishiguro, Electron configuration in HeH and T. Takezawa, J. Phys. Soc. ground state. Japan 32, 1595 (1972).

G. Pouzard and L. Pajol, Theor. MO-SCF for HeH . Chim. Acta (Berl.) 26, 187 (1972).

A. Macias, J. Chem. Phys. 57_, ab initio harmonic force con­ 1364 (1972). stants .

J. M. Peek, Physica 64, 93 (1973). Quasi-bound states.

V. K. Kelkar, K. C. Bhalla, P. G. Hartree-Fock SCF calculation Khulochandani, Mol. Phys. 26^ 221 (1973).

T. A. Green, H. H. Michels, J. C. Configuration-interaction in Browne, and M. M. Madsen, J. Chem. *1 states. Phys. 61, 5186 (1974).

T. A. Green, J. C. Browne, H. H. Matrix elements in states. Michels, and M. M. Madsen, J. Chem. Phys. 61, 5198 (1974).

W. Kolos, Int. J. Quantum. Chem. JW> Extends Wolniewicz's calcula­ 217 (1976). tions to larger R. 148

TABLE 11.--Continued

Reference Method of Treatment

A. K. Chandra and K. L. Sebastian, Bond-formation using MO Mol. Phys. 31_, 1489 (1976).

W. Kolos and J. M. Peek, Chem. Phys. Extends Wolniewicz's calcula­ 12, 381 (1976). tions to smaller R.

T. A. Green, H. H. Michels, and J. C. Configuration-interation Browne, Jr., J. Chem. Phys. 64, 3951 studies in excited states. (1976).

R. I. Price, Mol. Phys. 33, 559 Quasi-bound states. (1977).

I. Dabrowski and G. Herzberg, Calculate vibrational- N.Y. Acad. Sci. Ser. II 38, 14 rotational energy levels in (1977). the Born-Oppenheimer approxi­ mation for the ground elec­ tronic state.

R. I. Price, Chem. Phys. 3]^, Quasi-bound states. 309 (1978).

T. A. Green, H. H. Michels, and Configuration-interaction J. C. Browne, J. Chem. Phys. 69^ studies in excited states. 101 (1978).

D. M. Bishop and L. M. Cheung, Most accurate ground state J. Mol. Spectrosc. 75, 462 (1979). potential curve; vibrational- rotational energy levels in the adiabatic approximation. 149

TABLE 12. Comprehensive listing of approximate and model theoretical treatments of HeH+with method of treatment (in chronological order).

Reference Method of Treatment

H. Preuss, Mol. Phys. 8, 233 (1964). Covalent + ionic bond varia­ tional calculation.

J. D. Stuart and F. A. Matsen, One-center LCAO with configu­ J. Chem. Phys. 41_, 1646 (1964). ration interaction.

G. A. Gallup and M. S. McKnight, SCF using 1-electron, 2- J. Chem. Phys. 45, 364 (1966). nucleus wave functions.

P. Politzer, J. Chem. Phys. 45, Exchange and polarization 1856 (1966). modeling.

M. E. Schwartz and L. J. Schaad, Gaussian-type orbitals J. Chem. Phys. 46, 4112 (1967).

A. A. Frost, J. Chem. Phys. 47, Floating spherical Gaussian 3714 (1967). orbitals.

A. A. Wu and F. 0. Ellison, Scaled-atomic orbitals J. Chem. Phys. 48_, 1103 (1968).

L. L. Combs and. L. K. Runnels, One-center MO. J. Chem. Phys. 49, 4237 (1968).

L. J. Bartolotti and J. Goodisman, Bare-nucleus perturbation J. Chem. Phys. 49, 4237 (1968). theory.

K. E. Banyard and M. R. Hayns > One-center calculation with Mol. Phys. 15, 615 (1968). angular dependence.

J. B. Moffat, Theor. Chim. Linear combination Gaussian- Acta (Berl.) 10, 447 (1968). type orbitals.

F. Grein and T. -J. Tseng, Configuration-interaction Theor. Chim. Acta (Berl.) with exponential orbitals. 12, 57 (1968).

J. A. Keefer, J. K. Su Fu, and One-center MO. R. L. Belford, J. Chem. Phys. 50^, 160 (1969). 150

TABLE 12.--Continued

Reference Method of Treatment

T. F. O'Malley, J. Chem. Phys. 51, "Quasistationary" electronic 322 (1969). states; projected atomic orbi­ tals.

M. R. Hayns and K. E. Banyard, Variational calculation with J. Chem. Phys. 52, 1609 (1970). angularly dependent nuclear charge.

K. E. Banyard and C. C. Baker, One-center configuration- Int. J. Quantum'. Chem. 4_, 431 (1970). interaction .

IV. C. Mackrodt, J. Chem. Phys. 54, "Molecular puff" perturbation 2952 (1971). theory.

A. B. Anderson and R. G. Parr, Poisson equation for vibra­ Theor. Chim. Acta (Berl.) 26, tion potential. 301 (1972).

B. Klahn and W. A. Bingel, Theor. "Electron density quotient" Chim. Acta (Berl.) 28, 53 (1972). calculation.

C. Bottcher, J. Phys. B 6, 2368 Model potential calculation. (1973).

R. J. Bartlett and E. J. BrSndas, Multidimensional partitioning J. Chem. Phys. 59, 2032 (1973). in configuration-interaction.

L. M. Haines, J. N. Murrell, B. J. Gaussian cell MO. Ralston, and D. J. Woodnutt, J. Chem. Soc. 70, 1794 (1974).

C. D. H. Chisholmand K. B. Lodge, Perturbation theory. Mol. Phys. 28, 249 (1974).

M. Aubert, N. Bessis, and G. Bessis, Prolate-spheroidal orbitals. Phys. Rev. A 10, 61 (1974).

M. B. Milleur, R. L. Matcha, and LCAO-SCF-MO (artifically ex­ E. F. Hayes, J. Chem. Phys. 6£, cludes some atomic orbitals). 674 (1974).

R. F. Stewart, D. K. Watson, and Time-dependent Hartree-Fock A. Dalgarno, J. Chem. Phys. 65^, 2104 approximation. (1976). 151

TABLE 12.--Continued

Reference Method of Treatment

L. L. Combs and C. P. Miller, Int. J. One-center exponential orbi- Quantum Chem. _1£> 455 (1976). tals.

J. W. Johnson and R. D. Poshusta, Gaussian orbitals. Int. J. Quantum Chem. 1_1, 885 (1977).

W. Butscher and H. -H. Schmidtke, Density matrix bond calcula­ Chem. Phys. 30, 41 (1978). tion .

J. Aguilar, C. Murez, and H. Variable screening model. Nakamura, Chem. Phys. Lett. 53, 174 (1978).

J. Aguilar and H. Nakamura, Chem. Variable screening model. Phys. 32_, 115 (1978).

F. S. Levin, Int. J. Quantum Chem. Scattering theory channel Quantum Chemistry Symposium _12, coupling method applied to 109 (1978). molecular structure calcula­ tion . APPENDIX B

VIBRATIONAL-ROTATIONAL SELECTION RULES AND APPROXIMATE TRANSITION

DIPOLE MOMENT CALCULATION

In the dipole approximation (83; pp. 15-16) for the interaction

Hamiltonian for a molecule and an electromagnetic field is

V , = -p ,•£ cos fit , (1) ab ab o as used in Chapter 4. In this appendix, the derivation of the vibrational-rotational transition selection rules arising from this in­ teraction is outlined, and an approximate calculation of the vibrational- rotational transition dipole matrix element is done.

For a molecular system, the dipole moment operator is given by

M = V Z e it - R. , (2) Z—i a a • ^ ^ ' a ^ where Z is the atomic number of the ith nucleus, R is the position of a ct r the ath nucleus, and is the position of the ith electron. The sums are over all the nuclei and all the electrons in the system. The transition dipole moment is defined as

"ab 5 • (3) where the 1"s are the complete molecular wave functions, including both nuclear and electronic portions. Quantity (3) would be used in the

152 155 calculations of Chapter 4 is an evaluation of it was available. Since one is not, I will proceed to derive an approximation which can be used in its place.

In the Born-Oppenheimer approximation, ¥ = $, where ip is the electronic wave function and $ is the nuclear wave function. Consider­ ing only vibrational-rotational transitions, the transition matrix ele­ ment reduces to

= (4) Pab <*al"K> ' where

V = 52 Zae£a - <^11 eRj^) . (5) a i

The integration implied in Eq. (4) is over nuclear coordinates only, and the integration implied in the second term in Eq. (5) is over electronic coordinates only. The second term in Eq. (5) is not independent of the nuclear motion because the electronic wave function ip depends parametri- cally on the internuclear distance. The dependence of y on the internu- clear separation R can be represented by a Taylor series expansion about

the equilibrium separation RQ:

vg = V°g + _ X + 9s1'2'3' (6) ' x=o where g designates one of the three orthogonal directions in the rota­ ting, molecule-fixed reference frame,and x=R-Ro. In the particular case of a diatomic molecule, there can only be one component of p, the one along the molecular axis, and hereafter designated by Equation (6) gives the dipole moment in the rotating, molecule-fixed reference frame. 154

What is needed is the dipole moment in the laboratory-fixed frame where

the light is polarized. The component parallel to the laser field po­

larization vector in the laboratory-fixed frame (call it the z-direction)

is then

V = Z y (7) z zg vg

where £ is the cosine of the angle between the two directions. z9 The task now is to perform the indicated averaging over nuclear

wave functions: J „ /aP„\ p = $ ? v + X (8) z < al|_ zg °g hg (~df)0 . I*b> •

To a good approximation for these purposes, |$) = |v)|j,M) where v, J,

and M are quantum numbers with their usual meanings. Here |v) refers

to the molecule-fixed, rotating frame, and |J,M) refers to the lab-

fixed frame. Then

V = p°

x|v'>

= y° FRCJ,M,J',M')6VV, + FR(J,M,J',M') Fyfv.v'), (9)

where the rotational factor FR(J,M,J'and the vibra- K zg tional factor FY(v,v')=(vjx|v'). The first term on the right-hand side of Eq. (9) leads to purely rotational transitions, and the second term leads to vibrational-rotational transitions. Evaluating the rotational factor Fr leads to the selection rules (55; pp. 21-22) 155

AJ = +1 ,

AM = 0 (10)

Taking a particular case (J-KJ+1) for definiteness, it can be shown (55;

p. 22) that ? 2 (J + l)*"-M FR(J,M,J+1,M) = (11) (2J+1)(2J+3)

Note that Fn depends on the quantum number M. However, the molecules in K the beam have not been prepared in pure states with definite values of

M. Rather, they are in a mixed state, which is an incoherent superposi­

tion of all M states, with each M state being equally likely. It is

desirable to calculate a single, average transition dipole moment that

can be used in the laser-molecule interaction calculations of Chapter 4, so that sums over the M states will not have to be carried along there.

The proper average over the incoherent superposition of M states is a root-mean-square average. That this is the proper averaging procedure

can be seen from the following considerations: Since it is ultimately

the square of the transition dipole moment that enters the transition probability, one first squares the contribution from an M state, multi­ plies by the probability of having that M state, and then sums over all

M states. The square root of the result is the desired average, that is

(12)

where the probability of having any M state is 1/(2J+1). Evaluating

Eq. (12), one obtains 156

(13)

(Note that the above averaging procedure is correct only if all the de­ generate transitions are unsaturated. If some are saturated, there is no recourse but to do the interaction calculation in Chapter 4 for each of the M levels, and then add up the results at the end.)

Now consider the vibrational factor Fy(v,v'). This can easily be evaluated in the approximation that harmonic oscillator wave func­ tions are used for the vibrational motion. For vibrational states that lie well down in the potential well, this should be an adequate approxi­ mation. The following selection rule is immediately obtained from the properties of the harmonic oscillator wave functions (45; p. 80):

Av = ±1 (14)

(It should be noted that if either more terms are included in the expan­ sion (6), or anharmonic wave functions are used, additional transitions with Av=±2,3,... become allowed.) Again taking a particular case

(v->-v-l) in which the selection rule holds for definiteness, one obtains

(45; pp. 76-78):

where X=/ex, Hy(X) is a Hermite polynomial of the vth degree, and the normalization factor

(16) 157

Here, E=2TTMV /H, where M is the nuclear reduced mass and v is the ' osc osc classical vibration frequency of the harmonic oscillator. Expression

(15) is easily evaluated if one compares it to the following expression, which is the integral of the square of the normalized harmonic oscilla­

tor wave function and is by definition of normalization equal to unity:

N N — I e"X H (X) H (X) dX e 1 (17;) V V ifI v v

Therefore, N 11 1i v-l1 V v(v r > V— 1) = — -jp- 2 /c v

= /V/2E . (18)

Therefore, the transition dipole moment matrix element for vibrational-rotational transitions to be used in Chapter 4 is

Pab = av

s where (F^)av i given by Eq. (13), and Fy is given by Eq. (18). Numer­ ical values for these quantities for a particular transition and the value of (3p^/9x)o derived by Dabrowski and Herzberg (31) are given in

Table 1 and used in the calculations in Chapter 4. The resulting value of y given in Table 1 is 0.216 debye and is a reasonable value since it is the same order of magnitude as the transition moments calculated by more rigorous methods for other simple diatomic molecules such as HD+. APPENDIX C

LISTING OF THE MICROCOMPUTER EXPERIMENT CONTROL PROGRAM

On the following page begins a computer-generated listing of the microcomputer experiment control/data acquisition program. The program is written in a version of the BASIC programming language,* and is in­ terpreted and executed on the Z-80 based microcomputer (see description in Chapter 5).

1. T.D.L. ZAPPLE BASIC, version 3 (12K version), written by Technical Design Labs/Xitan, Princeton, New Jersey.

158 30 CLEAR 400 40 REV=3.5 50 DIM MC(6),MU(6),ML(6),MP(6) 51 DIM CM$(16) 52 DIM N(100) 53 DIM SIG(800) 60 NUL$=CHR$(0):BEL$=CHR$(7):LF$=CHR$(10):CR$=CHR$(13) 61 SI$=CHR$(15):ESC$=CHR$(27):SP$=CHR$(32):ASK$=CHR$(42) 62 DOT$=CHR$(46) 100 GOSUB 40000:REM INITIALIZE SYSTEM 110 CM$(1)="HELP" 120 CM$(2)="DUMP" 130 CM$(3)="CAL IDVM" 140 CM$(4)="INIT VOLT STEP" 150 CM$(5)="CAL VOLT STEP" 160 CM$(6)="MOVE VOLT STEP" 170 CM$(7)="SET BEAM VOLTS" 180 CM$(8)="SEARCH" 190 CM$(9)="SCAN" 200 CM$(10)="INIT LASER STEP" 210 CM$(11)="MOVE LASER STEP" 220 CM$(12)="SET LASER" 230 CM$(13)="NOISE" 260 CM$(16)="STOP" 300 OUT 255,15:INPUT">";CI$ 310 OUT 255,0 320 FOR J=1T016 330 IF CI$=CM$(J) THEN GOTO 400 340 NEXT J 350 PRINT"WHAT?":GOTO 300 400 ON J GOTO 1000,2000,3000,4000,5000,6000,7000,8000,9000,10000 401 ON J-10 GOTO 11000,12000,13000,14000,15000,16000 1000 REM HELP COMMAND 1005 PRINT"CURRENTLY RECOGNIZED COMMANDS:" 1010 FOR I=1T016:IF CM$(I)<>"" THEN PRINT I;CM$(I) 1020 NEXT I 1999 GOTO 300 2000 REM DUMP COMMAND 2010 GOSUB 54000:REM OUTPUT COLLECTED DATA 2999 GOTO 300 3000 REM CAL IDVM COMMAND 3010 PRINT"NOT IMPLEMENTED." 3999 GOTO 300 4000 REM INIT VOLT STEP COMMAND 4001 MN=1:OUT 255,15 4010 INPUT"WHAT IS VOLTAGE STEPPER POSITION";VS:OUT255,0 4020 Y=FNSC(1,VS):MC(1)=Y 4030 IF MC(1)>=ML(1) AND MC(1)<=MU(1) THEN GOTO 4060 4050 PRINT"VOLTAGE STEPPER OUT OF RANGE." 4060 GOTO 4999 4999 GOTO 300 160

5000 REM CAL VOLT STEP COMMAND 5001 MN=1 5010 IF VS>=0 THEN GOTO 5030 5020 PRINT"VOLTAGE STEPPER NOT INITIALIZED.":GOTO 300 5030 ND=FNDR(1,ML(1)):GOSUB 55000 5040 PRINT"VOLTAGE STEPPER = ";ND; 5045 OUT 255,15SPRINT BEL$ 5050 INPUT"WHAT IS BEAM VOLTAGE";VL:OUT255,0 5060 ND=FNDR(1,MU(1)):GOSUB 55000 5070 PRINT"VOLTAGE STEPPER = ";ND; 5075 OUT 255,15:PRINT BEL$ 5080 INPUT"WHAT IS BEAM VOLTAGE";VU:OUT 255,0 5090 SL=(VU-VL)/(MU(1)-ML(1)):REM VOLTS/COUNTER 5100 XL=(VU-VL)/(FNDR(1,MU(1))-FNDR(1,ML(1))):REM VOLTS/DIAL 5110 PRINT"ONE DIVISION ON VOLTAGE STEPPER DIAL = ";XL;" VOLTS." 5120 VC=1 5999 GOTO 300 6000 REM MOVE VOLT STEP COMMAND 6001 MN=1 6010 IF VS>=0 GOTO 6030 6020 PRINT"VOLTAGE STEPPER NOT INITALIZED.": GOTO 300 6030 OUT 255,15:INPUT"MOVE TO WHERE";ND:OUT 255,0:GOSUB 55000 6999 GOTO 300 7000 REM SET BEAM VOLTS COMMAND 7001 MN=1 7003 IF VC>=0 GOTO 7010 7005 PRINT"STEPPER NOT CALIBRATED.":GOTO 300 7010 OUT 255,15:INPUT"NEW BEAM VOLTAGE = ";BV:OUT 255,0 7020 ND=FNVD(BV):GOSUB 55000 7029 XX=FNDR(1,MC(1)) 7030 BV=FNVB(XX) 7031 PRINT"NEAREST VOLTAGE VALUE IS ";BV 7040 PRINT BEL$ 7999 GOTO 300 8000 REM SEARCH COMMAND 8002 MN=1 8010 IF VS>=0 GOTO 8030 8020 PRINT"VOLTAGE STEPPER NOT INIT.":GOTO 300 8030 IF VC>=0 GOTO 8090 8040 PRINT"VOLTAGE STEPPER NOT CALIBRATEDGOTO 300 8090 OUT 255,15 8110 INPUT"WHAT IS STARTING VOLTAGE";VI 8120 INPUT"WHAT IS ENDING VOLTAGE";V2 8130 INPUT"WHAT IS VOLTAGE INCREMENT";V3 8135 OUT 255,0 8160 M3=INT(2.0*ABS(V3/SL)+0.5)/2.0:IF M3=0 THEN M3=0.5 8180 IF V2-V1 < 0 THEN M3=-M3 8190 V3=-M3*SL 8200 PRINT"NEAREST VOLTAGE INCREMENT IS ";V3 8210 NS=INT(ABS((V2-Vl)/V3)+0.5)+l 8213 OUT 255,15:INPUT"WHAT IS INTEGRATION TIME IN SECONDS";IT:OUT 255,0 161

8215 IC$="A":GOSUB 49000:REM SETUP IDVM 8218 VR=ABS(FNVD(V1)-FNVD(V2)) 8221 MT=VR*0.065 8224 PR=4.4::ID=0.6 8227 TS=PR+MT+NS*(IT+ID) 8230 TM=INT(TS/60):TH=INT(TM/60):TM=TM-TH*60 8240 PRINT"THIS SEARCH WILL TAKE APPROX. ";TH;" HOURS ";TM;" MINUTES." 8245 V=V1-V3:ND=FNVD(V1):G0SUB 55000 8300 FOR IK=1 TO NS 8302 IF INP(255) <> 240 THEN GOTO 8380 8310 V=V+V3:ND=FNVD(V):GOSUB 55000 8330 GOSUB 43000:REM READ IDVM 8350 NEXT IK 8360 PRINT BEL$ 8370 GOTO 300 8380 PRINT "OPERATOR ABORT OF SEARCH.":GOTO 300 8999 GOTO 300 9000 REM SCAN COMMAND 9010 MN=1 9020 IF VS>=0 GOTO 9040 9030 PRINT"VOLTAGE STEPPER NOT INITIALIZED.":GOTO 300 9040 IF VC>=0 THEN GOTO 9060 9050 PRINT"VOLTAGE STEPPER NOT CALIBRATED.":GOTO 300 9060 OUT 255,15:INPUT"WHAT IS RUN LABEL";R$ 9065 INPUT"WHAT CURRENT (NANOAMPS) DOES LOCK-IN F.S. REPRESENT";SF 9070 INPUT"WHAT IS STARTING VOLTAGE";VI 9080 INPUT"WHAT IS ENDING VOLTAGE";V2 9090 INPUT"WHAT IS VOLTAGE INCREMENT";V3 9100 OUT 255,0 9110 M3=INT(2.0*ABS(V3/SL)+0.5)/2.0:IF M3=0 THEN M3=0.5 9120 IF V2-V1 < 0 THEN M3=-M3 9130 V3=-M3*SL 9140 PRINT"NEAREST VOLTAGE INCREMENT IS ";V3 9150 NS=INT(ABS((V2-V1)/V3)+0.5)+l 9155 IF NS > 800 THEN GOTO 9600 9160 OUT 255,15:INPUT"WHAT IS INTEGRATION TIME IN SECONDS";AT 9161 IT=AT:IC$="A":GOSUB 49000:REM SETUP IDVM 9165 OUT 255,0 9170 VR=ABS(FNVD(V1)-FNVD(V2)) 9180 MT=VR*0.065 9190 PR=4.4:ID=0.6 9200 TS=PR+MT+NS*(AT+ID) 9210 TM=INT(TS/60):TH=INT(TM/60):TM=TM-TH*60 9220 PRINT"THIS SCAN WILL TAKE APPROXIMATELY ";TH;"HOURS 11 ;TM;"MINUTES." 9230 V=V1:ND=FNVD(V1):GOSUB 55000 9240 OUT 255,15 9250 INPUT"WHAT IS TRUE STARTING VOLTAGE";V4 9260 INPUT"WHAT IS ION BEAM CURRENT IN NANOAMPS";B4:OUT 255,0 9270 GOSUB 47000 9271 S$="LASER-ION BEAM SPECTROMETER DATA OUTPUT FILE." 9272 S$=S$+CR$+LF$+LF$+LF$ 9273 GOSUB 48000 9274 S$="RUN LABEL: "+R$+CR$+LF$+LF$ 9275 GOSUB 48000 9276 Z$=STR$(V4):WL=10:GOSUB 41000 9277 S$="TRUE STARTING VOLTAGE IS "+Z$+CR$+LF$ 9278 Z$=STR$(B4):WL=10:GOSUB 41000 9279 S$=S$+"STARTING BEAM CURRENT IS "+Z$ 9280 S$=S$+CR$+LF$+LF$ 9281 GOSUB 48000 9282 S$=" STEP VOLTAGE 11 9283 S$=S$+"SIGNAL NORM SIGNAL"+CR$+LF$+LF$ 9284 GOSUB 48000 9290 FOR IK=1 TO 800 9295 SIG(IK) = 0:NEXT IK 9310 GOSUB 43000:REM READ SIGNAL 9320 SIG(1)=SG 9400 FOR IK=2 TO NS 9401 KM=IK 9410 V=V+V3:ND=FNVD(V):GOSUB 55000 9460 GOSUB 43500:REM READ SIGNAL & PUNCH 9470 SIG(IK)=SG 9480 IF INP(255) <> 240 THEN GOTO 9560:REM OPERATOR ABORT 9500 NEXT IK 9505 OUT 255,15 9510 INPUT"WHAT IS TRUE ENDING VOLTAGE";V5 9520 INPUT"WHAT IS ION BEAM CURRENT IN NANOAMPS";B5:OUT 255,0 9530 V6=(V5-V4)/(NS-1) 9535 B6=(B5-B4)/(NS-1) 9538 Z$=STR$(NS):WL=10:GOSUB 41000 9539 S$=Z$:Z$=STR$(SIG(NS)):WL=40:GOSUB 41000 9540 S$=S$+Z$+CR$+LF$:GOSUB 48000 9541 Z$=STR$(V5):WL=10:GOSUB 41000 9542 S$=LF$+LF$+"TRUE ENDING VOLTAGE IS "+Z$+CR$+LF$ 9543 Z$=STR$(B5):WL=10:GOSUB 41000 9544 S$=S$+"ENDING BEAM CURRENT IS "+Z$+CR$+LF$ 9545 GOSUB 48000 9546 GOSUB 47000 9550 GOTO 300 9560 PRINT "OPERATOR ABORT OF SCAN—-":OUT 255,15 9570 INPUT"WHAT IS TRUE VOLTAGE AT ABORT TIME";V5 9575 INPUT"WHAT IS ION BEAM CURRENT AT ABORT TIME";B5:OUT 255,0 9580 V6=(V5-V4)/(KM-1) 9585 B6=(B5-B4)/(KM-1) 9588 Z$=STR$(KM):WL=10:GOSUB 41000 9589 S$=Z$:Z$=STR$(SIG(KM)):WL=40:GOSUB 41000 9590 S$=S$+Z$+CR$+LF$:GOSUB 48000 9591 Z$=STR$(V5):WL=10:GOSUB 41000 9592 S$=LF$+LF$+"TRUE ABORT VOLTAGE IS "+Z$+CR$+LF$ 9593 Z$=STR$(B5):WL=10:G0SUB 41000 9594 S$=S$+"ABORT ENDING BEAM CURRENT IS "+Z$+CR$+LF$ 9595 GOSUB 48000 9596 GOSUB 47000 9597 GOTO 300 9600 PRINT"MORE THAN 800 POINTS REQUIRED. MUST RE-DIMENSION SIG(N)." 9610 GOTO 300 9999 GOTO 300 10000 REM INIT LASER STEP COMMAND 10010 PRINT"N0T IMPLEMENTED." 10999 GOTO 300 11000 REM MOVE LASER STEP COMMAND 11010 PRINT"NOT IMPLEMENTED." 11999 GOTO 300 12000 REM SET LASER COMMAND 12010 PRINT"NOT IMPLEMENTED." 12999 GOTO 300 13000 REM NOISE COMMAND 13010 OUT 255,15:INPUT "WHAT IS INTEGRATION TIME IN SECONDS";IT 13012 OUT 255,0 13015 IF IT=0 THEN GOTO 300 13020 IC$ = "A" : GOSUB 49000 13025 OUT 255,15:INPUT "HOW MANY SAMPLES";NN:OUT 255,0 13040 FOR I = 1 TO NN 13050 GOSUB 43000 13060 N(I) = SG 13070 NEXT I 13080 NB = 0 13090 FOR I = 1 TO NN 13100 NB = NB + N(I) 13110 NEXT I 13120 NB = NB/NN 13130 DN = 0 13140 FOR I = 1 TO NN 13150 DN = DN + (N(I) - NB)+2 13160 NEXT I 13170 DN = SQR(DN/(NN - 1)) 13180 PRINT "MEAN IS ";NB;" STANDARD DEVIATION IS 11 ;DN 13190 PRINT BEL$:GOTO 13010 14000 GOTO 350 15000 GOTO 350 16000 REM STOP COMMAND 16010 FOR MN=1 TO 1 16020 GOSUB 53000:NEXT MN 16030 PRINT"SESSION ENDED. RETURN TO BASIC.":OUT 255,0 16040 END 40000 OUT 255,0:REM SYSTEM INITIALIZATION SUBROUTINE 40001 PRINT"LASER-ION BEAM SPECTROMETER CONTROL PROGRAM REV ";REV 40002 MU(1)=7800:ML(1)=200:MP(1)=67:REM MOTOR #1 VALUES 40003 IS=32:10=37:IL=33:IM=35:REM IDVM VALUES 40010 OUT 32,0:OUT 33,0sOUT32,60:REM INITIALIZE IDVM LSD PORT 40020 OUT 34,0:OUT 35,0:OUT34,60:REM INITIALIZE IDVM MSD PORT 40030 OUT 36,0:OUT 37,255:OUT 36,4:REM INITIALIZE IDVM CONTROL PORT 40040 QQ=INP(33):REM READ IDVM PORT TO INIALIZE 40050 FOR MN=1 TO 1 40060 GOSUB 52000:NEXT MN 40310 VS=-1:REM FLAG FOR VOLT STEP NOT INIT 40320 VC=-1:REM FLAG FOR VOLT STEP NOT CAL 40330 TI=0:REM FLAG FOR TITLE NOT GENERATED 40500 DEF FNSC(MN,DR):REM GIVES MOTOR COUNTER FOR DIAL READING 40510 ON MN GOTO 40520 40520 X=8000-2.0*DR 40521 X=INT(2.0*X+0.5)/2.0:FNRETURN X 40590 FNEND 0.0/0.0 40600 DEF FNDR(MN,MX):REM GIVES DIAL READING FOR MOTOR COUNTER 40610 ON MN GOTO 40620 40620 XX=(8000-MX)/2.0:FNRETURN XX 40690 FNEND 0.0/0.0 40700 DEF FNVD(BV)=INT(0.5+FNDR(1,ML(1))+(BV-VL)/XL) 40701 REM GIVES VOLTAGE STEPPER DIAL READING FOR BEAM VOLTAGE 40710 DEF FNVB(VS)=VL+(VS-FNDR(1,ML(1)))*XL 40711 REM GIVES BEAM VOLTAGE FOR VOLTAGE STEPPER DIAL READING 40999 RETURN 41000 REM RIGHT JUSTIFY SUBROUTINE 41010 ZS$="":IF LEN(Z$)=WL THEN RETURN 41020 FOR Z=1 TO WL-LEN(Z$) 41030 ZS$=ZS$+" " 41040 NEXT Z 41050 Z$=ZS$+Z$ 41060 RETURN 43000 REM READ IDVM SUBROUTINE 43060 OUT IS,52:OUT IS,60:OUT 255,240 43070 WAIT IS,128 43075 OUT 255,0 43080 DM=INP(IM):DM=(DM AND 15)+((DM/16) AND 15)*10 43090 DL=INP(IL):DL=(DL AND 15)+((DL/16) AND 15)*10 43100 SG=(DM*100 + DL - 4999.5)/500 43110 RETURN 43500 REM READ IDVM WHILE PUNCHING 43510 OUT IS,52:0UT IS,60:0UT 255,240 43520 Z$=STR$(IK-1):WL=10:GOSUB 41000 43530 S$=Z$:Z$=STR$(SIG(IK-1)):WL=40:GOSUB 41000 43540 S$=S$+Z$+CR$+LF$ 43560 GOSUB 48000 43570 WAIT IS,128 43575 OUT 255,0 43580 DM=INP(IM):DM=(DM AND 15)+((DM/16) AND 15)*10 43590 DL=INP(IL):DL=(DL AND 15)+((DL/16) AND 15)*10 43600 SG=(DM*100 + DL - 4999.5)/500 43610 RETURN 47000 REM PUNCH LEADER/TRAILER SUBROUTINE 47005 OUT 255,2 47010 FOR W=1 TO 400 47020 IF INP(6) AND 128 <> 0 THEN GOTO 47020 47030 OUT 7,0:NEXT W 47035 OUT 255,0 47040 RETURN 48000 REM PUNCH CHARACTER STRING SUBROUTINE 48010 IF LEN(S$)=0 THEN RETURN 48015 OUT 255,2 48020 PO$=LEFT$(S$,1) 48030 IF INP(6) AND 128 <> 0 THEN GOTO 48030 48033 OUT 7,ASC(PO$) 48040 S$=MID$(S$,2) 48045 GOTO 48010 48048 OUT 255,0 48050 RETURN 49000 REM SET IDVM CONFIGURATION SUBROUTINE 49010 IF IC$="A" THEN C=16 ELSE C=0 49020 OUT 10,(C+0.5+LOG(IT)/LOG(2.0)) 49030 RETURN 51000 REM MOTOR STATUS ERROR SUBROUTINE 51005 OUT 255,255 51010 IF ST AND 2 THEN PRINT"'MOTOR";MN;"PANIC STOPPED." 51020 IF ST AND 128 THEN PRINT"MOTOR";MN;"CW LIMIT HIT." 51030 IF ST AND 64 THEN PRINT"MOTOR";MN;"CCW LIMIT HIT." 51040 IF ST AND 32 THEN PRINT"MOTOR";MN;"CW LIMIT HIT." 51050 IF ST AND 16 THEN PRINT"MOTOR";MN;"CCW LIMIT HIT." 51060 PRINT"MOTOR STATUS ERROR ABORT.":GOTO 300 52000 REM RESET MOTOR SUBROUTINE 52010 OUT MP(MN),128:RETURN 53000 REM DISABLE MOTOR SUBROUTINE 53010 OUT MP(MN),8:RETURN 54000 REM OUTPUT COLLECTED DATA SUBROUTINE 54010 REM DATA IS OUTPUT SO THAT IT CAN BE READ WITH THE FOLLOWING 54002 REM FORTRAN FORMAT: 54003 REM I10,3X,D17.10,3X,D17.10,3X,D17.10 54005 GOSUB 47000:REM PUNCH LEADER 54010 S$="LASER-ION BEAM SPECTROMETER DATA OUTPUT FILE." 54020 S$=S$+CR$+LF$+LF$+LF$ 54040 GOSUB 48000:REM PUNCH SOME 54050 S$="RUN LABEL: "+R$+CR$+LF$+LF$ 54060 GOSUB 48000:REM PUNCH SOME MORE 54070 S$=" STEP VOLTAGE " 54071 S$=S$+"SIGNAL NORM SIGNAL"+CR$+LF$+LF$ 54100 GOSUB 48000:REM PUNCH SOME MORE 54105 V=V4-V6 54106 B=B4-B6 54110 FOR IK=1 TO KM 54120 V=V+V6 54121 B=B+B6 54122 FS = SIG(IK) * SF / B 54130 Z$=STR$(IK):WL=10:GOSUB 41000 54140 S$=Z$ 54150 Z$=STR$(V):WL=17:GOSUB 41000 54160 S$=S$+" "+Z$ 54170 Z$=STR$(SIG(IK)):WL=17:GOSUB 41000 54180 S$=S$+" "+Z$ 54190 Z$=STR$(FS):WL=17:G0SUB 41000 54200 S$=S$+" "+Z$+CR$+LF$ 54210 GOSUB 48000 54220 NEXT IK 54230 GOSUB 47000 54240 RETURN 55000 REM STEPPER MOTOR DRIVER SUBROUTINE 55001 PN=MP(MN) 55002 BM=1:IF MN-2*INT(MN/2)=0 THEN BM=2 55010 DC=FNSC(MN,ND):IF DC = MC(MN) THEN RETURN 55011 IF ML(MN)<=DC AND DC<=MU(MN) THEN GOTO 55020 55012 PRINT"CAN'T MOVE MOTOR ";MN;" OUT OF RANGE.":GOTO 300 55020 SD=1:IF DC 1 THEN GOTO 55070 55073 IF ST<>1 THEN GOSUB 51000 55075 OUT PN,P 55077 MC(MN)=MC(MN)+SD 55080 NEXT K 55085 OUT 255,0 55090 IF 12=0 THEN RETURN 55095 P=1.5+SD*0.5:IF BM=2 THEN P=16*P:P=INT(P+0.5) 55100 ST=INP(PN):IF ST AND 1 <> 1 THEN GOTO 55100 55110 IF ST<>1 THEN GOSUB 51000 55120 OUT 255,3:OUT PN,P:0UT 255,0 55130 MC(MN)=MC(MN)+SD*0.5 55140 RETURN 65000 STOP LIST OF REFERENCES

1. G. Herzberg, Bull. Am. Phys. Soc. 25, 15 (1980).

2. Phillip Schewe, ed. Physics News in 1979 (American Institute of Physics, New York, 1979), p. 71.

3. E. Herlost and IV. Klemperer, Astrophys. J. 185, 505 (1973).

4. W. D. Watson, Rev. Mod. Phys. 48, 513 (1976).

5. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (John Wiley and Sons, Inc., New York, 1954).

6. G. Herzberg, Q. Rev. Chem. Soc. 25_, 201 (1971).

7. A. Carrington, P. G. Roberts, and P. J. Sorre, Mol. Phys. 35_> 1523 (1978).

8. A. Carrington, D. R. J. Milverton, and P. J. Sorre, Mol. Phys. 35, 1505 (1978).

9. A. Carrington, D. R. J. Milverton, P. G. Roberts, and P. J. Sorre, J. Chem. Phys. 68, 5659 (1978).

10. D. Buhl and L. E. Snyder, Astrophys. J. 180, 791 (1973).

11. N. Klemperer, Nature 227, 1230 (1973).

12. S. Green, J. A. Montgomery, and P. Thaddeus, Astrophys. J. 193, L89 (1974).

13. T. A. Dixon and R. C. Woods, Phys. Rev. Lett. 34_, 61 (1975).

14. R. C. Woods, T. A. Dixon, R. J. Saykally, and P. G. Santo, Phys. Rev. Lett. 35, 1269 (1975).

15. R. J. Sakally, T. A. Dixon, T. G. Anderson, P. G. Santo, and R. C. Woods, Astrophys. J. 205, L101 (1976).

16. R. J. Sakally and K. M. Evenson, Phys. Rev. Lett. 43, 515 (1979).

17. H. G. Dehmelt and K. B. Jefferts, Phys. Rev. 125, 1318 (1962).

18. K. B. Jefferts, Phys. Rev. Lett. 20_, 39 (1968); Phys. Rev. Lett. 23, 1476 (1969).

167 168

19. William H. Wing, George A. Ruff, Willis E. Lamb, Jr., and Joseph J. Spezeski, Phys. Rev. Lett. 36, 1488 (1976).

20. William H. Wing, George A. Ruff, and Joseph J. Spezeski, in Contri­ buted Papers of the Fifth International Conference on Atomic Physics, R. Marrus, M. H. Prior, and H. A. Shugart, eds. (University of California, Berkeley, 1976), p. 412.

21. Joseph J. Spezeski, Ph.D. dissertation, Yale University, 1977 (unpublished).

22. William H. Wing, in Laser Spectroscopy III, J. L. Hall and J. L. Carlsten, eds. (Springer-Verlag, Berlin, 1977), p. 69.

23. J. T. Moseley, P. C. Cosby, and J. R. Peterson, J. Chem. Phys. 65^ 2512 (1976).

24. D. C. McGilvery and J. D. Morrison, J. Chem. Phys. 67, 368 (1977).

25. T. R. Hogness and E. C. Lunn, Phys. Rev. 26_, 44 (1925).

26. Francis William Aston, Mass-Spectra and Isotopes (Longmans, Green 5 Co., New York, 1933).

27. Francis William Aston, Isotopes (E. Arnold § Co., London, 1923).

28. Arthur H. Snell, Frances Pleasanton, and H. E. Leming, J. Inorg. Nucl. Chem. 5, 112 (1957).

29. S. Wexler, J. Inorg. Nucl. Chem. 10, 8 (1959).

30. R. Raiz Von Frentz, K. Luchner, H. Micklitz, and V. Wittwer, Phys. Lett. 74A, 301 (1974).

31. I. Dabrowski and G. Herzberg, Trans. N.Y. Acad. Sci. 38, 14 (1977).

32. K. M. Merrill, B. T. Soifer, and R. W. Russell, Astrophys. J. 200, L37 (1975).

33. J. Norman Scrimger, R. P. Lowe, J. M. Moorhead, and W. H. Wehlau, Publ. Astron. Soc. Pac. 9£, 257 (1978).

34. Theodore P. Stecher and James E. Milligan, Astrophys. J. 136, 1 (1962).

35. Michael W. Werner, Astrophys. J. 139, 1014 (1964).

36. H. H. Michels, J. Chem. Phys. 44, 3834 (1966).

37. J. H. Black, Astrophys. J. 222, 125 (1978). 169

38. D. R. Flower and E. Roueff, Astron. Astrophys. 72_, 361 (1979).

39. K. M. Sando, R. Cohen, and A. Dalgarno, Seventh International Conference on the Physics of Electronic and Atomic Collisions, 973 (1971).

40. R. Johnsen and M. A. Biondi, Icarus 2_3, 139 (1974).

41. F. N. Edmonds, Astrophys. J. 125, 535 (1957).

42. R. J. Fallon, E. A. Mason, and J. T. Vanderslice, Astrophys. J. 131, 12 (1960).

43. F. A. Jenkins, Report of Subcommittee f (Notation for the Spectra of Diatomic Molecules) of the Joint Commission for Spectroscopy, J. Opt. Soc. Am. 4_3, 425 (1953).

44. E. Wigner and E. E. Witmer, Z. Physik 5j_, 859 (1928).

45. Gerhard Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd ed. (Van Nostrand Reinhold, New York, 1950).

46. K. Heitler and F. London, Z. Physik 44_, 455 (1927).

47. F. Hund, Z. Physik 42, 93 (1927).

48. R. S. Mulliken, Phys. Rev. 32_, 186 (1928).

49. George Glockler and D. L. Fuller, J. Chem. Phys. _1, 886 (1933).

50. Sigrid Peyerimhoff, J. Chem. Phys. 4_3, 998 (1965).

51. W. Kolos and L. Wolniewicz, Rev. Mod. Phys. 35, 473 (1963).

52. M. Born, Nachr. Akad. Wiss. Gottingen 1 (1951).

53. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1956), pp.166-212.

54. M. Born and R. Oppenheimer, Ann. Phys. 84^, 457 (1927).

55. Charles H. Townes and Arthur L. Shawlow, Microwave Spectroscopy (Dover, New York, 1975).

56. L. Wolniewicz, J. Chem. Phys. 43, 1087 (1965).

57. W. Kolos, Int. J. Quantum Chem. H), 217 (1976).

58. W. Kolos and J. M. Peek, Chem. Phys. 12, 381 (1976). 170

59. David M. Bishop and Lap M. Cheung, J. Mol. Spectrosc. 75, 462 (1979).

60. James M. Peek, Physica 6£, 93 (1973).

61. Richard B. Bernstein, Chem. Phys. Lett. 2S_, 1 (1974).

62. Roger I. Price, Mol. Phys. 33, 559 (1977); Chem. Phys. 2>1_, 309 (1978).

63. H. -P. Weise, H. -U. Mittman, A. Ding, and A. Henglein, Z. Naturforsch. 26a, 1122 (1971).

64. H. -P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979), pp. 300-301.

65. David E. Tolliver, George A. Kyrala, and William H. Wing, Phys. Rev. Lett. 43, 1719 (1979).

66. J. Schopman and J. Los, Physica <48, 190 (1970).

67. J. Schopman, J. Los, and J. Maas, Physica 51, 113 (1971).

68. J. Schopman and J. Los, Physica 5_1, 132 (1971).

69. J. Schopman, P. G. Fournier, and J. Los, Physica 63, 518 (1973).

70. P. G. Fournier, G. Comtet, R. W. Odom, R. Locht, J. G. Maas, N.P.F.B. Van Asselt, and J. Los, Chem. Phys. Lett. 40, 170 (1976).

71. "Search and Discovery," Phys. Today 22_ (5), 17 (May, 1979).

72. Zeev Vager and Donald S. Gemmell, Phys. Rev. Lett. 37, 1352 (1976).

73. Zeev Vager, Donald S. Gemmell, Bruce J. Zebransky, Phys. Rev. A 14_, 638 (1976).

74. D. S. Gemmell, P. J. Cooney, W. J. Pietsch, A. J. Ratkowski, Z. Vager, B. J. Zabransky, A. Faibis, G. Goldring, and I. Levine, in Proceedings of the Seventh International Conference on Atomic Collisions in Solids, Moscow University, Moscow, U.S.S.R., 1977 (to be published).

75. Elliot P. Kanter, Patrick J. Cooney, Donald S. Gemmell, Zeev Vager, Werner J. Pietsch, and Bruce J. Zabransky, in Nucl. Instrum. Methods Hamilton Conference (to be published).

76. H. Micklitz, Institut fUr Experimentalphysik, Ruhr-Universitat Bochum, personal letter, December, 1979). 171

77. A. Carrington, Proc. R. Soc. Lond. A367, 433 (1979).

78. H. J. Andra, A. Gaupp, and W. Wittman, Phys. Rev. Lett. 31_, 501 (1973).

79. H. W. Kugel, M. Leventhal, D. E. Murnick, C. K. N. Patel, and 0. R. Wood II, Phys. Rev. Lett. 35, 647 (1975).

80. J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley 5 Sons, New York, 1975), p. 522.

81. S. L. Kaufman, Opt. Commun. 1_7, 309 (1976).

82. S. M. Trujillo, R. H. Neynaber, and Erhard W. Rothe, Rev. Sci. Instrum. 37, 1655 (1966).

83. Murray Sargent III, Marian 0. Scully, and Willis E. Lamb, Jr., Laser Physics, (Addison-Wesley, Reading, Massachusetts, 1974).

84. Norman F. Ramsey, Molecular Beams (Oxford University Press, London, (1956).

85. G. W. McClure, Phys. Rev. 130, 1852 (1963).

86. E. S. Chambers, Phys. Rev. 139, A1068 (1965).

87. J. W. McGowan and L. Kerwin, Can. J. Phys. 42^ 972 (1964).

88. H. C. Hayden and R. C. Amme, Phys. Rev. 172, 104 (1968); R. C. Amme and H. C. Hayden, J. Chem. Phys. 42, 2011 (1965).

89. J. F. Williams and D. N. F. Dunbar, Phys. Rev. 149, A62 (1966).

90. J. J. Spezeski and W. H. Wing, in Contributed Papers of the Fifth International Conference on Atomic Physics, R. Marrus, M. H. Prior, and H. A. Shugart, eds. (University of California, Berkeley, 1976).

91. Joseph J. Spezeski, George A. Ruff, and William H. Wing, Rev. Sci. Instrum. 49_, 1327 (1978).

92. George A. Kyrala, David E. Tolliver, and William H. Wing, Int. J. Ion Phys. Mass Spectrom. 253, 367 (1980).

93. H. Gutbier, Z. Naturforsch. 12a, 499 (1957).

94. M. Hertzberg, D. Rapp, I. B. Ortenburger, and D. D. Briglia, J. Chem. Phys. 34, 343 (1961).

95. S. E. Kupriyanov, Zh. Eksp. Teor. Fiz. 4£, 467 (1965), Soviet Physics JETP 21, 311 (1965). 172

96. V. Aquilanti, A. Galli, A. Giardini-Guidoni, and G. G. Volpi, J. Chem. Phys. 43, 1969 (1965).

97. L. Friedman and T. F. Moran, J. Chem. Phys. 42, 2624 (1965).

98. William A. Chupka and M. E. Russell, J. Chem. Phys. 49^, 5426 (1968).

99. F. Schneider, U. Havemann, L. Zulicke, V. Pacak, K. Birkinshaw, and Z. Herman, Chem. Phys. Lett. 37, 323 (1976).

100. R. Johnsen and M. Biondi, J. Chem. Phys. 6_1, 2112 (1974).

101. J. R. Pierce, Theory and'Design of Electron Beams (Van Nostrand, New York, 1954).

102. K. -J. Hanszen and R. Lauer in Focusing of Charged Particles, A. Septier, ed. (Academic Press, New York, 1967), pp. 267-268.

103. Karl R. Spangenberg, Vacuum Tubes (McGraw-Hill, New York, 1948), pp. 348-349.

104. Charles Freed, IEEE J. Quantum Electron. QE-4, 404 (1968); Appl. Phys. Lett. 1J5, 485 (1971); and personal communications.

105. Michael J. Posakony, Rev. Sci, Instrum. 43, 270 (1972).

106. Colin S. Willett, Introduction to Gas Lasers: Population Inversion Mechanisms (Pergamon Press, New York, 1974), pp. 313-342.

107. R. M. Dale, M. Herman, J. W. C. Johns, A. R. W. McKellar, S. Naglev, and I. K. M. Strathy, Can. J. Phys. 57, 677 (1979).

108. A. Laguna-Ayala, M.S. thesis, University of Arizona, 1979 (unpublished).

109. H. Kogelnik, Bell System Technical Journal £3, 334 (1964).

110. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 326.

111. R. A. Buchroeder, Optical Sciences Center, University of Arizona (personal communication).

112. W. H. Wing and T. M. Sanders, Jr., Rev. Sci. Instrum. 38, 1341 (1967).

113. J. W. Farley, A. H. Johnson, and W. H. Wing, (to be published in J. Phys. E).

114. Philip R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969), pp. 232-242. 173

115. J. L. Dunham, Phys. Rev. 41^, 721 (1932).

116. E. Richard Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2_, 663 (1973).

117. A. W. Mantz, J. -P. Maillard, W. B. Roh, and N. Narahari Rao, J. Mol. Spectrosc. 57, 155 (1975).

118. David M. Bishop, Department of Chemistry, University of Ottawa, personal letter, August 1979.

119. David M. Bishop and Shing-kuo Shih, J. Chem. Phys. <54, 162 (1976).

120. David M. Bishop and Lap M. Cheung, Phys. Rev. A 18, 1846 (1978).

121. David M. Bishop and Lap M. Cheung, J. Chem. Phys. 69, 1881 (1978).