VELOCITY DISTRIBUTIONS IN BEAMS OF METASTABLE ATOMS EXCITED BY ELECTRON IMPACT by John Christopher Pearl A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1970 Doctoral Committee: Associate Professor Jens C. Zorn, Chairman Lecturer George R. Carignan Associate Professor Gordon L. Kane Professor Gabriel Weinreich Associate Professor William L. Williams

AC KN OWLEDGMENTS I am deeply grateful to Professor Jens C. Zorn for proposing the present problem and for his generous advice and assistance. I would also like to express my sincere appreciation to Mr. George R. Carignan, Director of the Space Physics Research Laboratory, for his unflagging support and encouragement. I am indebted to Professor William L. Williams for useful discussions and for his support. I also wish to thank Professor R. T. Robiscoe for discussions of the recoil problem and its bearing on measurements of the Lamb shift. I am grateful to Richard A. Heppner for the design and construction of the vacuum system. I thank also Dr. Denis P. Donnelly and David R. Crosby for development of the electronics used in the time-offlight experiment. I thank Martin Misakian for our many conversations about the molecular dissociation experiment and for providing me with helium velocity distributions from his machine. I am also grateful to Mr. Leslie R. Thurston for his skillful execution of the illustrations contained in this thesis. ii

To my wife, Susan, an especially warm "thank you" for her cheerful forbearance, and for typing the manuscript. I am grateful to the National Aeronautics and Space Administration for its financial support of this research. iii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................. ii LIST OF TABLES................ vii LIST OF FIGURES............... viii ABSTRACT.................... xi CHAPTER 1. INTRODUCTION.......... 1 2. THE KINEMATICS OF THE PROBLEM..... 3 2.l The Elementary Contributing Event: The Collision of an Atom and an Electron 4 2.2 Summation over Elementary Collisions: The General Velocity Distribution Function is Determined.. ~.. ~. ~ 13 3. METASTABLE VELOCITY DISTRIBUTIONS ARISING FROM MAXWELLIAN GROUND STATE GAS SOURCES.... 19 3.1 Ground State Beam Well Collimated. 20 3.2 Divergent Ground State Beam...... 37 3.3 Isotropic Ground State Gas Source... 40 4. THE RELATIONSHIP BETWEEN VELOCITY DISTRIBUTIONS AND TIME-OF-FLIGHT SPECTRA.. * * * 46 5. EXPERIMENTAL INVESTIGATION OF METASTABLE VELOCITY DISTRIBUTIONS.......... 62 5.1 Apparatus.*........ 62 5.2 Results............... 67 iv

Page 6. FURTHER FACTORS AFFECTING METASTABLE VELOCITY DISTRIBUTIONS.......... 71 6.1 Geometrical Effects......... 71 6.2 Complications Due to Several Excited States ~ ~ ~ ~ *......~ * 73 6.3 Spread in Electron Energy...... 76 6.4 Anisotropic Scattering of the Electrons 76 7. APPLICATIONS AND CONCLUSIONS...... 81 7.1 Application to Measurements of the Lamb Shift........4.. * 81 7.2 Application to H2 Dissociation Experiment.... *.. *.. 84 7.3 Measurement of Differential Excitation Cross Sections.......... 85 7.4 Other Possible Applications..... 86 7.5 Summary and Conclusions....... 87 A PPEND ICES I. ELEMiNTARY KINEMATICS OF AN ELECTRON-ATOM COLLISION. *............. 89 II. A GEOMYlETRIC VIEW OF THE kiETASTABLE VELOCITY DISTRIBUTION FUNCTION, 9 (V)....... 92 III. SOME EXPLICIT CALCULATIONS....... 100 III.1 The Introduction of Polar Variables into 9 (V).. *......... 100 III*2 A General Relation for the Cutoffs 102 III.3 The Integrated Metastable Flux at Threshold.... *...... 106 IV. PROGRAM BY DR. E. S. FRY FOR NUMIERICAL INTEGRATION OF 9(v) OVER FINITE DETECTOR WIDTH 110

Page V. "DETECTION OF M1ETASTABLE ATOMS AND MOLECULES WITH CONTINUOUS CHANNEL ELECTRON MULTIPLIEIRS"......... 121 VI. "EFFECT OF RECOIL ON THE VELOCITY DISTRIBUTION OF METASTABLE ATOMS PRODUCED BY ELECTRON IMPACT".* a *.......... 122 VII. SIMPLE SCALING RELATIONS FOR EVALUATING EFFECTS ON DISTRIBUTIONS OF DIFFERENT MASSES AND EXCITATION ENERGIES.... 125 A DDENDUM TOTAL INTENSITIES IN DIRECTIONS NOT CONTAINED IN THE PLANE OF THE INCIDENT BEAMS..... 129 LIST OF REFERENCES...... ~ ~............ 138 vi

LIST OF TABLES Table Page 7.I Comparison of experimentally determined and fitted parameters for metastable hydrogen velocity distribution 83 VII.I Deflection angles and energy excesses 128 vii

LIST OF FIGURES Figure Page 2.31 Normalized momenta used for discussion of electron-atom collision 8 2.2 Vector diagram showing variables used in determining velocity distribution when all metastables are detected 10 2.3 Vector diagram showing variables used in determining velocity distribution when only metastables within a small solid angle are detected 12 2.4 Diagram showing relation of vector variables and imposed coordinate system 17 3.1 Metastable velocity distributions for 23S helium, with detector angle 0 as parameter 22 3.2 Metastable velocity distributions for 23S helium, with electron energy E as parameter 24 3.3 Metastable velocity distribution for 23S helium, showing structure due to velocity distribution of ground state atoms 25 3.4 Variation of cutoff velocities with electron energy for 23S helium 26 3.5 Metastable intensity for 235 helium as a function of detector angle O, with electron energy E as parameter 27 3.6 Metastable velocity distributions for 23S helium, showing effect of equal simultaneous angular displacements of ground state beam and detector 31 3.7 Metastable velocity distribution for 235 helium, showing effect of equal simultaneous angular displacements of ground state beam and detector 32 3.8 Behavior of cutoffs for 23S helium with azimuthal angle p as parameter 34 viii

Figure Page 3.9 Behavior of cutoffs for 23S helium with azimuthal angle q as parameter 35 3.10 Metastable velocity distributions for 23S helium in plane defined by p- 5~ 36 3.11 Metastable velocity distributions for 23S helium at fixed detector angle 0 38 3.12 Metastable velocity distributions for 23S helium at fixed detector angle 0 39 3.13 Normalized velocity distributions for 23S helium with detector angle 0 as parameter 43 3.14 Peak shift of 235 helium velocity distributions relative to peak of Maxwellian distribution 44 3.15 Variation of peak intensity of 23S helium velocity distributions with detector angle 45 4.1 Simplified beam geometry for time-of-flight calculations 48 4.2 Time-of-flight spectra for velocity distributions of Figure 3.1 51 4.3 Range of integration for time-of-flight integral 54 4.4 Time-of-flight spectra for model metastable helium velocity distribution for finite beam width, A 56 4.5 Time-of-flight spectra for model metastable helium velocity distribution for finite pulse duration, T 56 4.6 Time-of-flight spectra for a Maxwellian metastable beam, with finite beam width,A 60 4.7 Time-of-flight spectra for a Maxwellian metastable beam, with finite pulse duration, T 61 5.1 Pictorial representation of experiment for measurement of metastable velocity distributions 64 5.2 Functional diagram of the apparatus used for time-of-flight measurements 66 ix

Figure Page 5.3 Time-of-flight spectra for metastable 23s helium showing experimental data 69 6.1 Metastable velocity distributions for 23S helium, showing effect of exciting different metastable states 74 6.2 Energy and angle dependences of 23S helium excitation function 77 6.3 Velocity distributions at the 23S helium resonances 78 6.4 Velocity distributions for 2 2S* hydrogen 80 7.1 Measured velocity distribution for H(2S) beam showing theoretical fit 82 II.l Cylindrical envelope of metastable velocities which arise from a collimated ground state beam 94 II*2 Geometrical interpretation of metastable velocity distribution function 9(V) 96 III.1 Velocity space cartesian coordinate systems used in describing the cylindrical envelope of metastable velocities 104

ABSTRACT This thesis is concerned with the velocity distribution in a beam of metastable atoms produced from ground state atoms by electron impact excitation. A relation between the velocity distribution functions in the ground state and metastable beams is derived, assuming that the electron beam is well collimated and monoenergetic, and that the scattering of the electrons by the atoms is isotropic. In the case where the ground state beam is well collimated and has a Maxwellian velocity distribution, it is shown that a measured distribution in the resulting metastable beam should be distinctly non-Maxwellian. In general, it is characterized by two peaks and by the existence of sharp cutoff velocities which are themselves very sensitive functions of the alignment of the ground state and electron beams, the position of the metastable detector, ard the energy of the incident electrons. Collimation of the ground state and of the metastable beams also strongly affects the distribution. A simple experiment is described which verifies these effects in a helium 23S beam. For the case where the ground state gas enters the electron beam isotropically, the double peaked structure disappears. However, the metastable velocity distribution xi

may still deviate significantly from that of the ground state gas. Extension of the calculations shows that non-isotropio electron scattering may affect the metastable velocity distribution strongly. A simple parametrization is derived to compare the distributions for different gases and/or different excited states. Several experiments in which such distorted velocity distributions can introduce systematic errors are briefly discussed. xli

CHAPTER 1 INTRODUCTION There are many atomic physics experiments utilizing beams of metastable atoms or molecules in which the metastables are produced by electron bombardment of ground state species, e.g., fine and hyperfine structure measurements on 22Si hydrogen (Lamb, 1950; Heberle, 1956; Robiscoe, 1965, 1968; Shyn, 1969), measurements of electron impact excitation of n= 2 states in helium (Holt, 1966; Vriens, 1968), excitation of metastable H(2S) atoms by electron impact (Stebbings, 1960), time-of-flight studies of repulsive molecular states (Leventhal, 1967; Freund, 1967; Clampitt, 1969), analysis of the kinetic properties of single and multicomponent gases (French, 1967; Crosby, 1969), and measurements of lifetimes of excited molecular states (Freund, 1969). Frequently, the velocity distribution of the metastable particles enters explicitly into the analysis of such experiments. It is surprising, therefore, that very little has been reported on the actual form of the velocity distribution to be expected from the common experimental arrangement in which ground state particles in a collimated beam are excited by a transverse beam of electrons. It is the purpose

of this thesis to investigate theoretically and experimentally the relationship between the velocity distributions of the ground state and excited atoms in such an experiment, identifying the various parameters which must be controlled in order to produce a metastable beam with a predictable velocity distribution. The thesis is organized as follows. First, the kinematics of the electron-atom collision process is discussed and the relation between the velocity distribution functions of the ground state and metastable atoms is determined; the results obtained are applied to several configurations which approximate common experimental arrangements. The use of the time-of-flight technique for measurement of velocity distributions is then considered and two systematic effects in such a measurement are discussed. Next, a simple crossed beam experiment is described which is designed to check various aspects of the calculations. The work is concluded with some remarks on the effects of various factors previously neglected in the analysis, and with a discussion of several actual experiments to which the present work is applicable.

CHAPTER 2 THE KINEMATICS OF THE PROBLEM For purposes of this thesis, an "apparatus" shall be defined by the following: i) a source of ground state atoms. The velocity distribution of these atoms may be isotropic, as within a Maxwellian gas in a large container, or may be highly directional, as in a collimated beam; ii) a perfectly collimated monoenergetic beam of electrons; iii) a detector for the metastable atoms. This will generally be assumed to subtend some given solid angle at the interaction region of the atoms and electrons. The solid angle may result from the physical size of the detector itself, or from collimation of the beam of metastable atoms after it leaves the electron beam. For purposes of analysis, the three beams may be described in terms of the velocity distribution function {tio) of the ground state atoms, the collimation of the ground state beam, the velocity to of the incident electrons, the

velocity distribution function g(v) of the metastable atoms, and the solid angle.de. of the detector. The object of the analysis is to relate the distribution functions f(VO) and g(Vi), taking into account the effects of beam oollimation. 2.1 The Elementary Contributing Event: The Collision of an Atom and an Electron Fundamental to the entire analysis is a description of the inelastic collision of pairs of particles, as viewed in a laboratory frame of reference. Problems of this nature have been considered in detail by physical chemists for the purpose of analyzing reactive scattering experiments (Toenniea, 1967; Warnock, 1968). The most extensive discussion to date of electron-atom collision kinematics is that of Rubin et al. (1969). The principal features of the relationship between the velocities of a ground state atom and of the metastable atom produced from it can be obtained by applying the laws of conservation of momentum and energy for the two particle (electron-atom) collision. Consider the collision, as seen in a laboratory-based coordinate system, of a ground state atom (velocity vO) and an electron (velocity Uo). Let the laboratory velocities of the two particles after the collision be r and i, respectively, and denote their respective masses by M and m. For convenience, all masses will be normalized with respect to the mass of the atom. In these units

the atomic mass is unity; the normalized electron mass will be indicated by the quantity J m (2.1) r M This normalization makes the atomic velocity and momentum identical; for the atom, therefore, velocity and momentum diagrams are "equivalentt. In the above units, the total normalized momentum of the system is given by VO + p X- (iU 41)W (2.2) where W represents the velocity of the center of mass of the two particle system in the laboratory frame of reference. Denoting quantities in the center of mass (CM) system by upper case letters, the two particles have incoming velocities given by Va C3 -W (VO UO) (2.3) and = (.-lv) (2.4) Taking vo and uo as the particles' velocities at large separations, the energy available to excite the atom to a

metastable state is the kinetic energy in the CM frame MVo * 2 u F.~c 0 (o.): z M + (i+) _ *!>. (ti'->.)2 (2.5) i.e., one half of the reduced mass times the square of the relative velocity. Assume that the atom is excited directly to its metastable state by the electron, so that energy E* equal to the excitation energy of the metastable level is "absorbed" by the atom. Then, after excitation, when the electron and the metastable are at large separations, the total kinetic energy in the CM frame is AE (auovo)= Ec (o, o)- E (2.6) To relate the velocities of the particles in the OM frame after the collision (quantities without subscripts), the laws of conservation of momentum V p= -U (2.7) and of energy I V2 A F (,ao, Uo_-_____i(2.8)

are again utilized. From this last relation, the speed of the outgoing metastable in the CM frame is given by 4=,B A\E( (O Vo) (2.9) An alternative derivation of this result is given in Appendix I. It is noted explicitly that the direction in which the outgoing metastable travels is not determined by the conservation laws; only the magnitude is fixed, so that, in velocity space, the vector V can terminate anywhere on a complete spherical surface of radius V (the center of the sphere is located at the tip of the center of mass velocity vector, W). Statistically, the probability that the vector V will terminate within a given element of solid angle,do, on the sphere is determined by the inelastic differential scattering cross section, -, for the excitation process. Once the direction of V is specified, however, the velocity, V, of the metastable in the laboratory system is found by the transformation v2VOW (2*10) The relationship between the normalized variables for the excitation process is indicated in Figure 2.1. The above results show that, in an experiment utilizing very narrow, perfectly collimated, monoenergetic electron and ground state atom beams, the velocity distribution of the metastables is not only spread over a range of speeds,

LL L 8IIU 0Figure 2.1 Normalzed momenta used for Fdisc gure 2 Normalized momenta used for discussion of electron-atom collision. In the laboratory system, the incoming atom and electron momenta are Vo and Pio, respectively. The same symbols without subscripts denote the momenta of the outgoing particles; the same quantities capitalized represent the momenta within the center of mass system. The motion of the center of mass itself is denoted by W. The total momentum of the system is 4.

but is also spread over a finite solid angle. Consequently, if any collimation is imposed on the beam after excitation, the distribution of metastable speeds which is observed at the detector will be altered, unlike, for example, the case of a beam formed by ideal effusion from a reservoir containing a Maxwellian gas. To assess the severity of such effects, it is useful to consider the observed distribution of particle speeds in two limiting cases. Case I: no post-collimation In this case, all of the metastables are assumed to be detected; the expected distribution of particle speeds can be easily ascertained with the help of Figure 2.2. If the metastables are uniformly distributed over the surface of the sphere (pure s-wave scattering), the fraction of metastables having speeds within dv of v is given by P(v)cdk 2ns5 n J. (2.11).4 nV2a However dS= Vd (2.12) cv sin cL L (2.13) and from the law of sines W v.i r 5;s (2014) Combining these results leads to PLJdv c -- clv (2.L5)

,Adz dv Figure 2.2 Vector diagram showing variables used in determining velocity distribution when all metastables are detected.

11 Thus the distribution of particle speeds P(v) is given by the linear relation 2VW JW-1W P(v) (2.16) 0 all other values of v This result is true regardless of the relative magnitudes of W and V; it can easily be shown that W. V provided that Uo.o ~ —;2,|E - (2.17) A value of w < V merely states that the velocity vectors of the metastable particles are spread over 4n solid angle and that the speed is a single valued function of angle; W > V, on the other hand, implies that the vectors are spread over a solid angle less than 27 and that the speed is double valued. Case II: very narrow post-collimation In this case it is assumed that only metastables in a small element of solid angle d1det are observed. Referring to Figure 2.3, the only metastables which pass through the collimation system are those with velocities v, and v2, corresponding to the two surface elements dct(,) and dr(91). Of the total number of metastables produced, the fraction which falls in such an element is given by 4 (2.18) However, the surface element dff(~) is related to the element

12 areBdetected. Figure 2.3 Vector diagram showing variables used in determining velocity distribution when only metastables within a small solid angle are detected.

13 of solid angle, cldet. by fn ev cdtr) IcosAl (2.19) det. I 2 so that () cl d e (2.20) Since it is difficult to relate d-ldet. to L&dv in a simple manner, the above expression will not be reduced to these variables. The important point to notice is that, for W > V, the distribution in such an arrangement consists of metastables with Just two velocities, the relative intensities being proportional to the square of the ratio of their speeds (if W< V, there will only be metastables of a single velocitW contributing to the signal). It is concluded, therefore, that the velocity distribution of metastable particles which arises from initially collimated and monoenergetic ground state and electron beams will vary from linear (no post collimation) to one or two delta functions (strong collimation). 2.2 Summation over Elementary Collisions: The General Velocity Distribution Function is Determined It is convenient at this time to introduce the approximation that all terms of order of the electron-atom mass ratio, y( 10-3), and of order of the atom-electron speed ratio, vo/uo (normally < 5 x 10'3) are to be neglected relative to unity. The normalized momentum A and the center of

14 mass velocity W then become equal: IM+: _ ~ azW (2.21) In addition, the radius, V, of the sphere becomes independent of so and may be expressed in the simple form X Z/lU~ E"(2 22) where E is the electron energy in the laboratory frame: E=- --— u (2.23) Henceforth these approximations will always be assumed. The results of the previous section show that metastables arising from ground state atoms with a single velocity must have velocities which terminate on a sphere of radius V in velocity space. Since this sphere is centered at the tip of the center-of-mass velocity vector, W, the simplest representation of the (normalized) resulting velocity distribution occurs in the center of mass system, where it may be expressed g()- 4t t- (-V) (2.24) Here w is the magnitude of the velocity of the metastable in the CM system, and s-wave scattering is assumed. The relation between CM variables (6) and those in the laboratory system (v) is = v - V rWe(.5

15 Suppose now that the ground state gas has an arbitrary velocity distribution function, (vO), defined so that f L3 o i (2.26) Upon excitation by electrons, atoms having velocities within c v, of vo will become metastables with the velocity distribution represented by the sphere of the previous section. In addition, the number of metastables produced will be proportional to the electron flux, e a to the density, n of ground state atoms in the interaction regionl, and to the total metastable excitation cross section, G*. Consequently, the velocity distribution function of the metastable atoms, S(v), is given to within geometrical factors by.i) n f3. 4PT* 3 i(w-Y) (2.27) where the integration is taken over the portion of ground state velocity space which contributes to metastables with velocity -, and s-wave scattering has been assumed, as before. To reduce this expression to usable form, it is necessary to convert the delta function from CM variables (w) to laboratory variables (V,). This is accomplished using the 1 Note that the density is the relevant quantity. Therefore, when discussing excitation of a ground state beam formed by effusion from a reservoir containing a Maxwellian gas, the quantity n0oa(o)d3co will be proportional to no Wo Qep(-v~/o2) (where ~ is the most probable atomic velocity in the reservoir), rather than to nov3 exp(-wvl/X), which actually represents the flux of atoms in the beam.

16 relation S(w-V7.) ( ) (2.28) i I dvi where the quantities v0o are the solutions of w(v0, ) = V. To determine w(vo), it is convenient first to rearrange the vectors in Equation 2.25 to define a vector B which is constant for a given value of o (see Figure 2.4) B v -rl0o- W + VO (2.29) From this it immediately follows that w BZ + Z C 2 B v Cos 49 (2.30) The roots, Vo,, of the expression w(vo)= V are then vO- Cos9 i / b VL sin S (2.31) If B <V, only v,+ is physically meaningful. In addition, it is found by differentiation of Equation 2.30 that dwA dwo J {I~'VI (2.32) where the carats indicate unit vectors. After substituting these results in Equation 2.27, it is found that the velocity distribution function, g(V), for metastables arising from a ground state gas with velocity distribution function {(vo) is given by g(>')= e n.rV_, 9 (,,o -U,,. 1) +1 ~ ~,) (2.33)

17 Figure 2.4 Diagram showing relationship between vector variables and an imposed coordinate system. The normalized momenta of the incoming atom and electron are Vo and,uG, respectively. The vectors'I and -w represent the total and center of mass contribution to the momentum of the outgoing metastable atom.

18 or..... A J d V-B (2.34) where the range of integration, flo, is determined by the degree of oollimation of the ground state gas beam. If B<V, only the first term in the integral appears. A geometric interpretation of the various factors occurring in the integrand of Equation 2.34 is given in Appendix II.

CHAPTER 3 METASTABLE VELOCITY DISTRIBUTIONS ARISING FROM MAXWELLIAN GROUND STATE GAS SOURCES The relationship between the velocity distribution function 9(i) of metastable atoms and the velocity distribution function o(Vo) of the ground state atoms from which the metastables are produced is given by Equation 2.34. In the present chapter, the predictions of this result will be investigated for various degrees of collimation of the ground state gas beam. In evaluating Equation 2.34 it will be assumed that the ground state beam is formed from a gas with a Maxwellian distribution of velocities. Taking account of the normalization condition of Equation 2.26, the appropriate form for {(Vo) is 0 -'/ CCwo) t.0Ze e (3.1) where o-L- (k is Boltzmann's constant and T is the absolute temperature of the gas). In this case, the velocity distribution function for the metastables may be written C" -VAo/ z 1a 2 9(a):= g,:'V" a.hod w,,o.~+ e' V, (3.2) 19

20 The explicit form for this integral is given by Equations III.9 and III.10 of Appendix III. 3.1 Ground State Beam Well Collimated Suppose that the ground state beam is confined to a small solid angle duco. The integral in Equation 3.2 then degenerates to the integrand itself. From the geometric arguments of Appendix II, the distribution function of the metastables, 9(9v), is expected to have the following properties: i) it is zero outside of two well defined cutoff speeds, v; ii) within the cutoffs, g(V) is generally nonzero; it becomes large near the limiting values of v; iii) the magnitudes of the cutoffs are very sensitive functions of the observation angle 0. 3.1.1 General Characteristics of the Metastable Velocity Distribution in the Plane of the Incident Beams Consider now the excitation of the metastable 23S state of helium (E*- 19.8 eV). Suppose that the detector is located in the plane of the incoming beams (cp= 00). Then, for reasonable conditions of beam orientation and detector position (see Section III.2 of Appendix III), the distribution function g(v) takes the form

21 e.4AV [~. -Vo I L -(1- xi' V9(V')Lw i r X i[ os-vsin(b-o) ~ (3.3) O otherwise where vo~ = v c os ( -) -~ ~o L i t iot - VUoSin -C sin (- sin L7/ v 6 iry"~tn I-YCin (3.4) and vi = | Sinm i\] rst (I-\ C (3-5) (see Appendix III). It is emphasized here that Equation 3.3 assumes isotropic scattering of the electrons in the CM frame,. Figure 3.1 shows several distributions, taking the detector angle ) as the parameter (the numerical scale on the ordinate represents the magnitude of using 9.. using the arbitrary normalizatione i. This same scaling is used consistently throughout the thesis). All of the prethe cutoff positions is due to the cosecant factor in Equation 3.5. It is interesting to notice how the distributions "ride over" the Maxwellian distribution in the integrand (actually, there is an additional factor of v2 since the ob

22 X103 >1~ 6 e He* CJHeH >~ 5IV) 0=72~ 0:800 Helium 23 S QC72~ 0:80~goo 3 9: 900 >= E: 2[0 >VV 0 1 2 3 4 NORMALIZED METASTABLE VELOCITY 3 19 MI Figure 3.1 Metastable velocity distributions for 23S helium, with detector angle as parameter. The ground state beam is assumed to be perfectly collimated and to have a Maxwellian velocity distribution.

23 servable quantity is proportional to v2g(v); this is why the underlying "envelope" appears to peak closer to v/c]. -f than to wc - i). Figure 3.2 indicates how the velocity distribution at a fixed angle 0 varies with electron energy. The widening of the distribution with increasing energy results from the corresponding increase in the radius V of the cylindrical envelope on which the metastable velocities terminate (see Appendix II). Figure 3.3 shows that if the energy of the electron is sufficiently large, the cutoffs may become separated by enough so that the Maxwellian-like envelope mentioned in conjunction with Figure 3.1 can appear as a bump within the distribution found for a single set of parameters. The way in which the width of the distribution (cutoff separation) depends on energy is shown in Figure 3.4 for various detector angles 0. 3.1.2 The Integrated Metastable Intensity1 Figure 3.5 indicates the variation of the total metastable intensity (integral of the velocity distribution over all velocities) with detector position 0; the electron energy excess hE (energy above threshold) is the parameter (see Section III.3 of Appendix III). These curves indicate a substantial nonuniform behavior for the metastable intensity pattern in the plane of the incoming beams, even though the scattering is isotropic. Qualitatively, this can be understood as follows. 1Intensities for 0o are shown in the addendum, page 129.

24 y9 -xlo3 E =20eV 8 e He -'' 7 e >18:z W vlelium 23S, Q5= 00 (1) 0 =72~ E =21eV LiJ E 25 eV 1 0 0.5 1.0 1.5 NORMALIZED METASTABLE VELOCITY V Figure 3.2 Metastable velocity distributions for 238 helium, with electron energy E as parameter. The ground state beam is assumed to be perfectly collimated and to have a Maxwellian velocity distribution.

25 2.0 x102 1Helium 23S,:= 900 cp: p0 q:> — 0~ 0-14 ~t~ He ee=72";>!~3 He. I THe E =40eV 1.5 e-:) ZD c~ -" 0.5 0 1.0 2.0 NORMALIZED METASTABLE VELOCITY 17 oo Figure 35.3 metastable velocity distribution for 23S helium, showing structure due to Maxwellian velocity distribution in the (collimated) ground state beam. Note that ordinate scale differs from that of Figures 3.1 and 3.2.

26 6 He* He C/ ~Helium 23S, / ~ = 90':~'{8 gc = 0" -,, 3 L) F. 2 1=72" _0 50 Q.' I,,I 0 20 21 22 23 24 25 INCIDENT ELECTRON ENERGY E(eV) Figure 3.4 Variation of cutoff velocities with electron energy for 23S helium; detector angles is the parameter. The ground state beam is assumed to velocity distribution.

We* Welium 2351 Wle AE5 L)U Lu~~~~~~~~~~~~~~~~~~~~~1 0-1 50 60 70 80 9 POLAR ANGLE AT DETECTOR DEGREES Figure 3.5 Total mnetastable intensity for 238 helium as a function o detector anglee.9 The electron energy increment above thresbold,tL.E, Is th Parameter. The ground state beam is assumed to be perfectly collimated andt have a Maxwellian velocity distribution. It is emphasized that isotropio s c at t ering of the electrons is also assumed.

28 The cylindrical portion of metastable velocity space whioh contains the allowed metastable velocities (see Figure II.1 of Appendix II) is composed of an infinity of spheres. Each sphere is weighted according to the ground state velocity distribution (vo); the most populated portion of ~(Vo) is mapped into a densely populated sphere in the metastable space. For a given increment of metastable speeds, the greatest contributions to the metastable velocity distribution are made near the circumference of the cylinder. To lowest order, therefore, it is reasonable to expect that the total metastable intensity will exhibit maxima near angles 0 determined by the points at which the most densely populated sphere is tangent to the cylindrical envelope. In fact, this argument predicts the positions of the bumps in the curves of Figure 3.5 within 20%, provided that the extra factor v2 is also included, giving the most weight to atoms with speeds roughly 42 times the most probable speed in the ground state beam. Although the curves of Figure 3.5 are normalized to the same peak height, an estimate of the relative "average" amplitudes can be made using an approximation suggested by Equation III.28 of Appendix III. There, since hVVVI(};oN) is finite, it appears reasonable to suppose that to lowest order in V the quantity VI(o,0;o,V)/C* is a function of 0 and VI only (the cross section is taken over from the right side since it is generally a strong function of the excess energy, and hence of V). This should be a good assumption

29 as long as V is resonably small, and as long as 0 (and I ) is chosen in the range where the metastable intensity in the limit V-*O has a representative value, e.g., in the vicinity of the peak of the AE-O curve. Since MVcI, the above condition reduces to the approximation wher (E, F(at a E) (3.6) where (I) indicates a representative intensity. This result may be given the following physical interpretation. The total metastable production is proportional to *1. These metastables are spread over a cylinder in velocity space which is characterized by a dihedral angle roughly proportional to the radius V of the cylinder (see Figure II.1 of Appendix II). If (i) is a representative intensity of the metastables over this angular range, then the total number of metastables is proportional to V(). Dividing by (t4 for normalization gives Equation 3.6. 3.1.3 Sensitivity of the Velocity Distribution to Beam Misalignments One important objective of the present work is to assess the effects of various misalignments on the velocity distribution of the metastable atoms. From Figure 3.1 it is seen that the velocity distribution is a very sensitive function of detector angle 0; this has already been explained in terms of the cosecant factor in the equation for the cutoffs (Equation 3.5). Because this factor also contains the angle

30 2, the distribution should show about the same sensitivity to misalignment of the ground state beam. Thus, misalignment of either the ground state beam or of the adet3ct r will introduce strong shifts in the metastable velocity distributions Since the argument of the cosecant is I-E, it is to be expected that equal simultaneous errors in 0 and i should not produce a very large effect on the distribution (assuming that sin l 1). Comparison of the velocity distributions in Figure 3.6a with each other and with the E= 25 eV curve of Figure 3.2 shows that this expectation is reasonwell substantiated for angular errors up to * 100 centered at A=90o. For very large total angular deviations, however, the shape of the velocity distribution is more distorted (Figure 3,6b). Figure 3.7 shows that this distortion can become extreme if the energy of the incident electrons is sufficiently large. It is worthy of note that equal simultaneous errors in 0 and 6 are equivalent to a misalignment of the electron beam in the plane of the ground state beam and the detector. It is therefore concluded that a small misalignment of the electron beam in that plane will not affect the metastable velocity distribution nearly as seriously as an equal misalignment of either the ground state beam or the detector alone. The discussion so far has been concerned with velocity distributions in the plane of the incident beams (q=0);

31 4 - x o3 x0 Helium 23S, C 3 He* u ~-e = 180 E = 25eV 2 - >18 ai~~~~~~~~~~~~r~8' CD~~~~~~, —~~~~ ~=1000 "-" = 82~ 0O - 0 0.5 1.0 1.5 a C 4 - x103 > lHelium 235S 3 E = 25eV Lii _.J'H2 He e-'J 2500 fit 0 =320" O. — 1- =1300 0 0 0.5 1.0 1.5 b NORMALIZED METASTABLE VELOCITY U, O4 Figure 3.6 Metastable velocity distributions for 23S helium showing effect of simultaneous angular displacements of ground state beam and detector. The ground state beam is assumed to be perfectly collimated and to have a Maxwellian velocity distribution.

32 7 - x102 >18 He* >18 a He- 510 5/e 8 360 e:D I.. 4 Helium 23S ~ -e: 150 3 E= 30OeV 1290 =114~ F- 2C1 +:10 E O- - 0 0.5 1.0 1.5 2.0 2.5 V NORMALIZED METASTABLE VELOCITY Figure 3.7 Metastable velocity distributions for 23S helium showing effeet of simultaneous angular displacements of ground state beam and detector. The ground state beam is assumed to be perfectly collimated and to have a Maxwellian velocity distribution.

33 the effects of out-of-plane misalignment will now be considered. The cutoffs provide the most sensitive indicator of the distortions of the velocity distributions which result when q0O. The general expression for the cutoff velocities is given by Equation III.15. Figures 3.8a and 3.9a show how the cutoffs vary as a function of detector position, taking the azimuthal angle, p, as parameter. The most striking feature of these results is that the cutoffs no longer move out to infinity with increasing 0, but have a definite upper limit for a given T. This merely reflects the fact that the plane described by $p0 cuts the cylindrical envelope obliquely, so that as the angular position Gof the ray to the detector is increased, the ray eventually passes outside of the envelope. Thus, whereas distributions corresponding to uQp always widen as e is increased (see Figure 3.1), the distributions will generally widen and then narrow again if p#0. This behavior is indicated by the plots of cutoff separations shown in Figures 3*8b and 3.9b, e.g., for 5= 2 If the plane in which the detector moves is far enough out of plane, the resulting velocity distributions may show very little variation in width over a relatively large range of ). Figure 3.10 shows such a case (compare Figure 3.1). If the detector is kept at a fixed angle 0, and q is slowly increased, a value?max will eventually be reached where the ray to the detector becomes tangent to the cylinder. This value of ~ is given by Equation 111.16. The ef

O0c: oo0c+I- 1: r Cz P' H. V+ - v_ CD -, NORMALIZED SEPERATION BETWEEN CUTOFFS V+ o~ w Po (: O, NORMALIZED CUTOFF VELOCITIES IP+ a ~4 0) rU C) U' C) U- C - CD~ — "-': nPo, C+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~T (D IO 0 CD 0'sv Bq~~~~~~~~~~~~~~~~~~~~~~~~~2 **~~~~~~~~~~~~~~~~~~~'d z 0 so 9) J 1 -1 iEg 4 I \ c cD ~~~~~~~'0~~ P~~~~~ H ~3.~-e 0 c C'O Orz ~~~-r ~~ ~E C ) O I-J (D 1c )m -'10 0. CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I C I G-) (JI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'T~)(~I — ScD'-b'i 2 "Is al -4 a.4 -1 — CC wo m Co (n0 ~ ~rO~1[) cc+ CDC C. a <:( PO 0 ml U rri~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i Oc+ CA'.-, 0-'~ 0 CDO~~~~~~~~~~~~~iI\)~~~~~~~~~~ ~'. D Q 0C J I CD' C~ D W- N ) C:)1 Irb t"~ ~' c+ t.., ~~~~~~~-~~~~~~~~~~~~CD~~~~~~~~~c1DN 14aItI C) o — o o o c* I he o * c ~..J 0 I-' (.0 ~~~~ ~ ~~~~~~~~~~~~~~~~C) aC CD CD N ca'D

opeSmrIOO ~t.oe.Jae d pu uTTll8emxW pemnsae mseq e9;vs punos. *jjJoqno ueetteq uoT~rws~dee pezTIslImou:q GeAanmo'seeTooi -etA jjono pezvllmuou:e saeAJn'*Je9emujd se 6 eolues leqnm -Tzv qR m nTlerq G aJoj sjjo~no 30 JOTAsqea 6'* 0enL1 q (s33930) ae d01l)313l IV 319NV VlOd?z 06 09 OL 09 09 OS J)o= /)d — I -- bi o 6'6 = 06 09 OL 09 09 I F/T TS Fn!VH OG O 0 OL 09 0; 7-.

36 3.0 -X 103 >138 0=780 C> 2.5 [ I Helium 23S1 >18 ~ =90~ O 1.5 = 750 >- e NHe* CD 1.0 V H e.0 I —.0 LL 0 1.0 2.0 3.0 NORMALIZED METASTABLE VELOCITY 20 oc Figur 3e.10 Metastable velocity distributions for 2 5 helium in plane defined by P= 5; detector angle G is parameter. Ground state beam assumed Maxwellian and perfectly collimated.

37 fect on the velocity distribution is a gradual narrowing, as indicated by the curves in Figure 3.11; for this choice of 3,)~ it is found that x= 4.588~. This effect is qualitatively similar to decreasing the value of AL (compare Figure 3.2). The behavior of the metastable velocity distribution arising from a well collimated ground state beam can be summarized as follows. The cutoffs (which determine the "position" of the distribution in the velocity coordinate v) are sensitive functions of 0,i andcp. The width of the distribution is a sensitive function of qp and AE; it is also usually sensitive to 0 and $, except when in proper combination with. The distribution is relatively insensitive to misalignment of the electron beam in the plane of the ground state beam and the detector. 3.2 Divergent Ground State Beam If the ground state beam is not perfectly collimated, then the integration indicated in Equation 3.2 must be performed. The resulting metastable velocity distribution will therefore tend to be smoother than the distributions discussed in Section 3.1. Figure 3.12 shows the effect of finite detector width on the metastable velocity distributionl (since the effects 1I I would like to thank Dr. E. S. Fry for the use of his computer program (which is included as Appendix IV) to evaluate these integrals; numerical results from this program were also used in Figures 3.5, 5.3 and 7.1.

38 9 x103 8 >18 ct) )6 - -< 4 CD) I- - C/):= 1eV 0.. I W. I Im 23I, 720 2leV 0 0.8 0.9 1.0 1.1 1.2 1.3 NORMALIZED METASTABLE VELOCITY V 4,6 C Figure ~.11 Metastable velocity distributione for 2 S helium at fixed detector angle 0; azimuthal anglel is parameter. Ground state beam assumed Maxwelli an and perfectly collimated.

METASTABLE VELOCITY DISTRIBUTION rt(X) g(2 ) -- -, ru C) ru C ru ~(l ( m m CD CD CD CD cD- CD N F 0 CDJ 0 C C L >C>C ~~ ~ ~ ~ ~~ m~'O~- c::) — i ----— T —-,,,,,,,, ,,,, )C~ Or I" _____ m C' q:> I:> I::>::: _!:: (D D t p..w c+ 1 1'44 0~~~~~~~~~~~~~~~~ so C:::'10-.Pm ~-_.. Pi C+ ( D m YO(D "-"C)'1000~ ~ o o ( e C s CDC tip rc..j' C:> co cc o iP' oiF CD pi0+ P -' I I-.0D 0 CD o Qn Q) 0I D', I -1 c - c+ -b * (o O

40 of an error in I are roughly the same as those of an error in @ - see Equation 3.5 and Figure 3.6a - this is approximately equivalent to the distribution arising from a ground state beam of angular divergence L=A4O, assuming the detector width to be negligible). The curves are normalized to represent intensity per degree of detector width. Qualitatively, the effects are as expected. The peaks are gradually smoothed out, and the width of the distribution increases. For sufficiently wide collimation, the distribution takes on a relatively smooth, single-humped character. Other similarly integrated distributions are shown in Figures 5.3 and 7.1. 3.3 Isotropic Ground State Gas Source The ultimate in divergent ground state beams is a completely isotropic gas source; such a situation exists for an electron gun in a gas filled container. Interestingly enough, in this case the integral in Equation 3.2 can be evaluated analytically, as will now be shown. The integration is most easily performed in a coordinate system with origin at the center of the contributing sphere (see Appendix II), i.e., centered at the tip of the vector B. Defining a polar coordinate t which is related to the variable 9(Figure 2.4) by the law of sines, the solid angle element dwo can be transformed to a corresponding element in the new coordinate system. Making this transforma

41 tion, Equation 2.34 takes the simple form where d~ow is the solid angle element in the system with origin at the tip of the vector B, and the integral is taken over the entire sphere (J47rn). From Equation 2.29 with w =V it follows that v0 =- BL +VL -2 BV co0S (3.8) With the Maxwellian form of 4(vo) (Equation 3.1), Equation 3.7 becomes 2t r B..BV-2B Vcoss J nodp - ndte %inh rsZ (3.9) e 7e'l.3 2 B The dependence of the result on v is of course hidden in the quantity 13 which is obtained from B -- - poU (3.10) One remarkable prediction of Equation 3.9 concerns the behavior of 9(v) at threshold (V= 0). In this case the quotient involving the hyperbolic sine becomes equal to unity. If the vectors v and yu are perpendicular (~ 2) then it follows that

42 L VI V - o cs' a: — e e (3. 11) i.e., the distribution function 9(i) is Maxwellian once again. The velocity distribution which results from Equation 3.9 at threshold is plotted for several values of 0 in Figure 3.13; the curves have been normalized to identical peak values. As the angle of observation 0 shifts from zero to 1800, it is seen that the distribution shifts from "wide and fast", through Maxwellian to "narrow and slow". Figures 3.14 and 3.15 show, respectively, the behavior of the peak position and of the peak intensity (assuming T* is constant) of the velocity distributions obtained from Equation 3.9. Over the full range of 0, the shift in peak position and the variation in peak intensity are seen to be substantial.

43 8: 1800. 90 0~ 3>18 He* CO NLU ~~~~~~~~~e CD 0 1.0 2.0 3.0 V NORMALIZED METASTABLE VELOCITY X Figure 3.13 Normalized velocity distributions for 238 helium; detector angle 0 is parameter. Arrows indicate peak positions. The Maxwellian ground state gas is assumed to enter the electron beam diffusely; the electrons are assumed to have threshold energy.

44 < 1.3 E-40eV 1.2:E20 H e* 1.1 1.0. 10 30 60 90 120 150 180 rY(, e (DEGREES) __~ I E:4OeV w 0.8: Figure 3.14 Peak shift of 235 helium velocity distributions relative to peak of Maxwellian distribution; electron energy E is parameter. The Maxwellian ground state gas is assumed to enter the electron beam diffusely.

45 m E=40eV C, Co E-20eV Z LI 0 30 60 90 120 150 180 0 (DEGREES) Figure 3.15 Variation of peak intensity of 233 helium velocity distributions with detector angle 0; electron energy E is parameter. The Maxwellian ground state gas is assumed to enter the electron beam diffusely.

CHAPTER 4 THE RELATIONSHIP BETWEEN VELOCITY DISTRIBUTIONS AND TIME-OF-FLIGHT SPECTRA Frequently in atomic physics, and in the measurements made for the present work in particular, velocity distributions are measured by means of time-of-flight techniques. However, it is known that systematic effects in time-offlight systems can complicate the analysis of such data (Zorn, 1967; Alcalay, 1969). Consequently, a simplified analysis of the relation between velocity distributions and their corresponding time-of-flight spectra is presented below. Given a beam of neutral atoms with arbitrary velocity distribution function and a burst of electrons for excitation, the present problem is to determine the flux of metastable atoms at a downstream detector as a function of time. The assessment of effects due to the duration of the electron pulse and to the finite extent of the excitation region is of particular interest here. The problem stated above can be broken into two parts: the determination of the metastable velocity distribution function given the distribution function of the ground state 46

47 beam; and the determination of the metastable flux at the detector given the metastable velocity distribution function. The first of these problems has been treated in Chapter 3; the second will be discussed below. The analysis is based on the following assumptions: i) The metastable lifetime is infinite (no metastables decay while travelling from the electron gun to the detector); ii) The fraction of the neutral beam which is metastabilized is negligible (the neutral particle density is constant over the region of excitation); iii) The electron density is uniform over the crosssection of the electron beam; iv) The electron pulse is rectangular in time (zero rise and fall times; constant intensity over an interval T). These assumptions, along with certain reasonable physical requirements, lead to a simple relation between the detected metastable flux and the velocity distribution of the metastable atoms. The geometry to be considered for the analysis is shown in Figure 4.1. If the metastable atoms were all produced instantaneously in a volume of negligible dimensions, then the resulting time-of-flight distribution could be determined from the dispersion of the "packet" of metastable atoms as follows. Let the detector be located a distance L from the ex

48 DETECTOR A, PATH OF METASTABLES INTERACTION VOLUME Figure 4.1 Simplified beam geometry for time-of-flight calculations.

49 citation region; the transit time t to the detector for a metastable with speed v is then t=-L (4.1) V Due to spatial dispersion, metastables having speeds within dv of v will strike the detector over a time increment of magnitude dt- L dv (4 2) so that dC V Ld L t (4.3) Let the velocity distribution of the metastables be vgS(v). Then v A(v)dv metastables will strike the detector within the indicated time interval; using Equation 4.3 this can be written f(hV)dVo tim i(cmdt o(4 4) The coefficient of the time increment ct on the right side of this expression represents the rate at which metastables strike the detector at time t R*( L,t) L (4*5) This states that, in the limit of small interaction volume and short pulse duration, the metastable flux at the detector differs from g only by a factor proportional to t4.

50 Transforming the metastable velocity distributions of Figure 3.1 as indicated in Equation 4.5 yields the time-of-flight distributions shown in Figure 4.2. In an actual experiment, the interaction volume is of finite size (characterized by the electron beam widthA), and the electron gun pulse is of finite duration (T). While these effects do not alter the form of 9(v), (see Seotion 6.1), they do change the time-of-flight distribution because the metastables are no longer all created at the same instant and because different metastables must travel different distances to the detector. To account for these effects, Equation 4.5 for the metastable flux must be integrated over both the width, A, of the electron beam and over the duration, T, of the electron pulse. Before carrying out this general program it is of interest to see what effect each of these factors has on time-of-flight distributions individually. first consider an electron beam of width A which is pulsed instantaneously. Let the coordinate describing a lateral position in the electron beam be I (see Figure 4.1), and suppose that this is measured from the edge of the beam at which the metastables leave; let 1 be taken in a positive sense if it is measured opposite to the motion of the metastables. Then the distance a metastable created at I must travel to reach the detector is L+L, where 04 J A. Introducing this path length into Equation 4.5, normalizing with respect to the beam width, and integrating over the width of

51 5 x 102 He X i 3 F 0:800 | Helium 23S1 c1j I 1 = 9QO p0 <: 2 i- I\ I i\ E = 20.8eV E: | l= l 372~ | ~ L = 28.6cm = 72~ 0 84e jI e:600 0 100 200 300 400 500 TIME OF FLIGHT (psec) Figure 4.2 Time-of-flight spectra corresponding to the velocity distributions of Figure 341. It is assumed that the electron beam is of zero thickness and that the pulse is instantaneous. The ground state beam is assumed to be perfectly collimated and to have a Maxwellian velocity distribution.

52 the beam gives R"*(Lt;htI (L1) 9(L (4.6) where the zero in the argument of R indicates that T=O. Introducing a new variable of integration defined by V -L (4.7) t allows Equation 4.6 to be reduced to R (Lt;,O) A i v 9g(v) (4.8) t This expression can be used to determine the effect that finite A will have on time-of-flight spectra corresponding to velocity distributions such as those in Figure 3.1. The salient features of the actual distributions are the sharp cutoffs and the singularities. The quantity responsible for the singularities is the radical in the denominator of g(v) (see Equation 3.3 or III.9 ); to within a factor which is inessential for the present discussion, this radical can be written in the simple form /Vi2-(w-~)t' where vc and v. are respectively the average speed and the halfwidth of the distribution V V* V.. (4.9) as- _

53 The function in the numerator of Equation 3.3 (or III. 9) is relatively smoothly varying, and does not affect the character of the distribution drastically if the energy of the incident electrons is near threshold. For this reason, and because it greatly simplifies the integration of Equation 4.8 without losing any of the essential features of g(V), it is convenient to replace the numerator of the actual distribution by the function 7t; this leads to the following qualitatively accurate but simple model of the metastable distribution function ~(v; i-_-v -v C'./ iV2~ V VV-aV Av(y V 4 (4.10) O o6.herw0se Integrating Equation Equation 4.8 using this distribution leads to an expression of the form R*(L,t;AO): Lv si(l>n(VC)j (4.11) v'(Lt;A) Here the limits V* are the boundaries of the interval common L L+A to the intervals - \ — < and vc-v <VA v~.v& (the six t t possible cases which result are illustrated in Figure 4.3). To study the behavior of Equation 4.11 for conditions similar to those of the experiment to be discussed in Chapter 5, the following parameters are chosen

54 CASE Vc-VA Vc+V, L L+ t t I I L L+LA t t L, t L +A LL L+A t t' I t L Figure 4.3 Range of integration for time-of-flight integral, accounting for finite beam width,A. The appropriate range, indicated by the heavily shaded portion of the axis, is the interval common to the intervals t < v L+ and u,-va.<v<vc+V. t ~t

55 vc =.3 5 9 x o0 cm Isec V.62.0 x Os cmwlsec (4.12) L 2 8.6 Ad The quantities'V. and va correspond to the predicted parameters for a 23S helium beam described by -= 90~, p= 00, 072~, E = 25 eV and a gas temperature of 3000 K. The distortion of the model helium time-of-flight distribution due to finite electron beam width A(assuming the electron pulse duration to be negligible: T O) is illustrated in Figure 4.4. Consider next the other limiting case of vanishing beam width (A:O) and finite pulse duration T. Let the time tat which a metastable is created be measured from the instant the electron beam is turned on; then the time it takes a metastable created at t to reach the detector is t-t, where 0Ot LT. Inclusion of this effect in Equation 4.5 leads to the following expression for the metastable flux at the detector R*(L,t; o,T) = - (t —)4 (tT L f+-L-~T (4. 13)

56 A: 0 cm Model Helium Distriburion 1.0 Vc = 1.359 x 105 cm/sec =~ 1 VA=.620 x 105cm/sec jm_ I 11 T = Opsec ~ x-.6 L.L = 28.6cm x 3~~~~~~3 6 1A:Ocm 3 1w=E ~~~~~~~~~~610 0 __t 0 100 200 300 400 500 TIME OF FLIGHT (pSec) Figure 4.4 Time-of-flight spectra for model metastable helium velocity distribution, showing effects of finite electron beam width, A. T=Opsec Model Helium Disrriburion 1.0 Ve =1.359 x 105cm/sec VA =.620 x 105 cm/sec, —=- LA = Ocm.6 t353 L = 28.6 cm X4 1__r 405 TlT:afo40-m58 o d a O. sec 82 1.16 [j13.53 wit Eq o 15135.27 I-. —6~~~~~~~~~~ ~~~~~~40.58:5 81:16 135.27 0 0 0 100 200 300 400 500 TIME OF FLIGHT (,sec) Figure 4.5 Time-of-flight spectra for model metastable helium velocity distribution, showing effects of finite duration, T, of electron pulse. Values of T are related to the values of A in Figure 4.4 in accordance with Equation 4.15.

57 which differs in form from Equation 4.8 only in the exponent of v in the integrand. Inserting the model distribution 4.10 and carrying out the integration leads to the expression V+ (L,t;T) R (L t;o,T) sin( t -T (4.14) V-(L)t;T) The indicated limits, Vb, are the boundaries of the interval which is common to the intervals L. v t- and \i, \ v t t-T vc+v1 (the resulting six cases are qualitatively the same as those of Figure 4.3; however, (L*A)/l is replaced by L(t -T). For purposes of illustrating the behavior of Equation 4,14, the parameters given in Equations 4.12 are again used. Time-of-flight distributions for several values of the pulse duration T are shown in Figure 4.5. Each T of Figure 4.5 is related to one of the A's of Figure 4.4 by the expression T- a n (4* 15) In this sense, a beam width A.scm in Figure 4.4 is equivalent to a pulse duration T 13.53 Asec in Figure 4.5. Distributions in the two figures which are related in this way have identical widths. Comparing such related distributions, it is seen that the peak corresponding to the fast side of the velocity distribution is displaced more in the curve

58 parametrized by T than it is in the curve parametrized by A. This is to be expected, since a temporal displacement (the pulse duration, T) is uniform on a time-of-flight scale for all particles, while a spatial displacement (the beam width, A) translates into a time-of-flight interval which varies inversely with the velocity of the particles, and is therefore least for the fastest atoms. The generalization of Equation 4.5 to include both finite A and finite T is given by T T ) 0 Introducing the variable.L+)t (4.17) in place of T, and performing the integration over I leads to the expression AT R*(Li;ATit AFk) LF (L+A) (4.18) L L A -:P ~I'. F T F~ t-TI' where so (~x)= jdv vr(w-~9() (4(v)9 (4.19) Here it has been assumed that 9(v) has no cutoffs, so that the limits are not dependent on such quantities as Vc* VA Calculated time-of-flight spectra assuming 9(v) is Maxwellian (as might occur in the case of a diffuse ground state gas

59 at 0: 900; see Equation 3.11) are shown in Figures 4.6 and 4.7 (note differences in scales on the abscissas of the two figures). Perhaps the most remarkable aspect of these curves is the absence of flattening at the peaks even when the beam width, A, or the pulse duration, T, become comparable respectively to the flight path, L, or to the most probable flight time.

- Helium >< T = 1.0psec ~_I L = 28.6cm COi ir t=- 0.1cm:Z 0 100 200 300 400 TIME OF FLIGHT (,sec) Figure 4.6 Time-of-flight spectra for a Maxwellian metastable beam, showing effect of finite electron beam width,A.

Helium A 0.1cm L 28.6 cm LU iLE ~~~~~~~~~~~~~~100 LLI 30 4 T~~~~~~l.O~~ I Osec LU cm -~J 0 100 200 300 400 500 TIME OF FLIGHT (psec) Figure 4.7 Time-of-flight spectra ffor a Maxwellian metastable beam, showing effect of finite duration, T, of electron pulse,

CHAPTER 5 EXPERIMENTAL INVESTIGATION OF METASTABLE VELOCITY DISTRIBUTIONS Because the predicted forms for the metastable velocity distributions produced from collimated ground state beams are rather striking, experimental verification of their principal features should not be difficult. A simple time-of-flight experiment suffices for this purpose. Basically, the experiment utilizes a resonably monoenergetic pulsed electron beam to excite ground state atoms to metastable states by electron impact; the rate at which metastables arrive at a detector is recorded, providing time-offlight spectra which can be compared with theoretical predictions. 5.1 Apparatus The experiment is conducted within two identical 70 liter chambers. The chambers share a common flange, enabling the system to be differentially pumped. Each chamber contains a liquid nitrogen cold trap and is connected to a 4 inch oil diffusion pump; these pumps exhaust into a common foreline which is connected to a rotary forepump. Base 62

63 pressure in the system with liquid nitrogen in the traps is typically 1 x 10'7 torr. The first chamber is used in forming the ground state beam, The gas effuses into the chamber from a copper pipe of 1.6 mm inside diameter, and is collimated by an adjustable slit system in the flange between the two chambers, The path length and slits are typically adjusted to give a divergence of about 30 in the ground state beam (see Figures 5.1 and 5.2); typically, the density in the resulting beam is 10ll atoms/om3. The gas flow is controlled by a Nupro very fine metering valve which is fed from a glass storage bulb. The second vacuum chamber contains the electron fgun and the detector (see Figure 5.1). The electron gun is a simple three element device, utilizing a directly heated thoriated tungsten ribbon cathode, an accelerating grid of tungsten mesh, and a stainless steel anode, The cathode is heated by a direct current of 4 to 6 amperes at about as many volts. With 20 volts applied to the grid-plate combination, the electron current through the interaction volume defined by the electron and ground state atom beams is about 100 microamperes; it is estimated that the electron energy spread over this region is between 0.5 and 1.0 eV. The detector is a Bendix "Channeltron" continuous dynode electron multiplier (Donnelly, 1969; see Appendix V). Ultraviolet photons or metastable atoms striking the cathode eject electrons by photoelectric and Auger processes, re

64 (I_ _ AXIS OF ROTATION DETECTOR "CI'ANNELTRON" GRID. ANODE e SOURCEA M ESN GEARS0." - e COLLIMATING SLIT /~GAS ~ WORM' AND SOURCE SPUR GEARS Figure 5.1 Pictorial representation of experiment for measurement of metastable velocity distributions.

65 spectively, with an efficiency of a few per cent. To reduce background counts arising from free electrons or ions produced in the electron gun, the detector is enclosed in an aluminun box composed of three electrically isolated sections. The first section is positively biased to repel ions; the second, located between the first section and the oathode of the electron multiplier, is biased negatively to repel electrons; the remainder of the box is grounded. The biased portions of the box have aligned 6.3 mm slits, which in combination with the beam-detector separation of 28*6 cm, give a detector width of approximately 8 minutes of arc. The entire detector assembly is mounted on a radial arm, enabling it to be rotated about the interaction region in the plane defined by the incident electron and atom beams; the rotation is controlled by a worm and spur gear combination with a 100-1 turns ratio, the worm being turned by a shaft which passes through the chamber wall via an O-ring seal. The pulsing of the electron gun and the conditioning of data for storage are performed by the electronics system indicated schematically in Figure 5.2. Data conditioning involves the sorting of pulses from the electron multiplier according to their times of occurrence after initiation of the electron gun pulse; this is accomplished by time-topulse-height conversion, as follows. The entire system is controlled by a recurrent pulse generator. The raw data

66 RAMP E R IN LINEAR A OUT GENERATOR GATE I: RECURRENT GATE L ANALYZER PULSE L GENERATOR DISCRIMINIATOR DETECTOR 1.6mrm 2.0mm1t/ E.UC %.i 9cm GAS SOURCE LECTRON t* 5.9cm - 14.6cmQ ELECTRON GUN Figure 5.2 Functional diagram of the apparatus used for time-of-f light measurements.

67 pulses pass through a preamplifier, an amplifier and a discriminator to the input of a linear gate (I). The linear gate is operated in the anticoincidence mode, the gate input being a pulse which is synchronized with the electron gun current pulse. This gate eliminates pulses from the spectrum which correspond to the arrival of ultraviolet photons at the detector (these photons arise from the rapid decay of non-metastable states excited by the electron bombardment). The remaining pulses are used to gate the second linear gate (II). This is operated in a coincidence mode, the input being a linear ramp voltage which is initiated by the start of the electron gun pulse. The output of the second linear gate is then a series of pulses, each having an amplitude proportional to the time interval between the firing of the electron gun and the arrival of the corresponding metastable at the detector. After further shaping, these pulses are stored in a Nuclear Data series 1100 multichannel analyzer. 5.2 Results Helium was chosen as the test gas because its small mass and high excitation energy lead to a relatively large angular spreading of the metastable "beam" (see Sections 3.1 and 3.2). The experiment outlined in Section 5.1 was performed using the following parameters: Electron pulse duration T = 3Msec Width of electron beam A\= 0.5 cm

68 Path length from interaction region to detector L =28.6 cm Pulse repetition rate 1600/sec Number of analyzer channels 128 Nominal angle between ground state and electron beams — 90~ Divergence of ground state beam;,t3~ Angular width of detector A~O0~8' The background pressure during the experiment was generally less than 2 x 10-7 torr,. Sample time-of-flight data for the 23S metastable state of helium are shown as points in Figure 5.3 (these results have been published: see Pearl (1969), included as Appendix VI). The solid curves represent theoretical predictions based on the results of Chapters 3 and 4. Curves a assume perfect collimation of the ground state beam; curves b allow for a 30 divergence in the beam. Curve c shows how the distribution would appear if the metastable velocity distribution were the same as the assumed Maxwellian velocity distribution in the ground state beam. Although the fits are by no means perfect, the salient features of the predictions are experimentally established. These include i) the narrowness of the distributions ii) the double peak structure iii) the sensitivity of the "position" of the distributions to the detector angle 0.

69 Helium 23S, 3 90~ E = 20.55eV He* 3b e _ 11 \; 6-77.6~ $,- X2.,1 0b: 70.4~ O0 100 200 300 400 500 600 TIME OF FLIGHT (,usec) Figure 5.3 Time-of-flight spectra for metastable 23S helium. Points represent experimental data. Curves a are theoretical for perfect collimation; curves b are theoretical, accounting for 30 angular divergence in the incoming, ground state beam. Curves a and b are calculated assuming that the electron energy is 0.75 eV above threshold. Curve o is theoretical if recoil effects are neglected, i.e., it represents the spectrum of the full Maxwellian velocity distribution in the ground state beam.

70 The discrepancies between the data and theoretical curves b in Figure 5.3 are probably due in large part to the following factors. First, the actual energy distribution of the electrons in the electron beam is unknown. A reasonable estimate of the energy spread is *0.5 eV; this would tend to widen the distributions. Second, it was not possible to determine, except at 0=900, that (P=0, i.e., that the detector tracked perfectly in the plane of the incident beams. As indicated in Figures 3.8 and 3.11, nonzero values of p tend to narrow the distributions. Finally, the resonance structure in the excitation function of 23S helium (see Section 6.4) can greatly affect the shape of the velocity distribution.

CHAPTER 6 FURTHER FACTORS AFFECTING METASTABLE VELOC ITY DISTRIBUTIONS The analysis of metastable velocity distributions presented in Chapters 2 and 3 is based on the simplest possible model of an experiment employing a metastable beam. The purpose of this chapter is to discuss the influence on the velocity distribution of several other effects which would play a part in a real experiment. 6.1 Geometrical Effects Principal among factors to be discussed under this heading are the effects associated with a finite interaction volume. Consider the case of perfectly collimated beams. It is emphasized that if the interaction volume becomes finite, but the unidirectional character of the beams is preserved, there is no alteration in g(v), a fact which remains true even if the spatial distribution of particles (atoms and electrons) in the beams is nonuniform over the beam cross sections. This is because the metastables produced in each differential volume element are all created under identical kinematic conditions. A measured time-of71

72 flight spectrum of such metastables, however, may be strongly affected (see, for example, Figure 4.4). This is due to the fact that different metastables travel along different paths (both in length and in direction) to reach the detector. The spatial integration implied to account for this effect will lead to variations in the observed velocity distribution which are qualitatively similar to those indicated in Figures 3.12 and 4.4. An error in positioning the center of rotation of a movable detector relative to the point of intersection of the electron and ground state beams will result in apparent shifts of the velocity distributions due to transit time effects, and in shifts and distortions due to accompanying errors in angular measurements. Divergence of the electron beam (or shearing or cyclotron motion, if it is magnetically confined) requires another angular integration. The principal corrections due to such effects will occur for the electrons with the greatest outof-plane motions, since these have the largest azimuthal angles, P (compare Figures 3.6 and 3.11). In the case of diffuse ground state "beams", the interaction region is determined by the shape of the electron beam, e.g., a cylinder. A detector with a fixed viewing angle will therefore see an intensity pattern as a function of O which is is increased by a factor of csc over that shown in Figure 3.16.

73 6.2 Complications Due to Several Excited States It is obvious that if two or more (long lived) metastable states of a given atom are excited by electron impact, then the observed total velocity distribution will be a superposition of the distributions due to each state individually. However, it should also be noted that any optical levels which are excited and which can cascade into a given metastable state must also be considered. The treatment of each such state is identical to that for the directly excited metastable levels, since the same kinematic arguments apply; the only change is that the excitation energy of the level in question must be used in the calculations. The influence of recoil due to the photon emitted in the cascade process is negligible. Figure 6.1 shows two velocity distributions. The solid curve is the distribution which would result from excitation of the helium 2s 3S, state alone by electrons of energy E =25 eV (5.2 eV above threshold); the dashed curve represents the distribution which would result from excitation of the 3p 3p' state alone (assuming subsequent cascadinp into the 2s 3S, level) by electrons of energy E=28.2 eV (5.2 eV above threshold). The widths of the distributions are identical; the principal difference between them is a slight shift, which is proportional to the square root of the energy of the incident electron (these results follow from the sum and difference of Equations 3.5). Excluding this shift, and assuming the s-wave scattering approximation,

74 -—'- 4 X103 >18 Helium cn., - 90~ se* Co 00 >18 3 720 E-E*= 5.2eV -o2 - e- -—: 2 e E* 23 eV r 0. I I I I! 0 ~0.5 1.0 1.5::E E = 28.2e ~ V 4,8 NORMALIZED METASTABLE VELOCITY _, Figure 6.1 Metastable velocity distributions for 23S helium. The solid curve is the distribution for direct excitation of the 2s 3S level by 25 eV electrons; the dashed curve is the distribution arising from direct excitation of the 3p 3p; level followed by cascade to the 2s 3Si metastable state. The electron energy increment above the respective thresholds is constant. A perfectly collimated Maxwellian ground state beam is assumed.

75 velocity distributions for two states are qualitatively the same if excited by electrons of a fixed energy above the thresholds of the states (see also Appendix VII). A converse relation is also useful: if two states are excited by electrons of a single energy, the resulting velocity distributions will be qualitatively the same as distributions arising from the excitation of a single state by electrons of two energies, provided that the energy increments above threshold are made to correspond. The three distributions in Figure 3.2 for the excitation of the helium 23S state by electrons of energies 0.2, 1.2, and 5.2 eV above threshold, can therefore be reinterpreted: they are qualitatively the same as the total distribution which would result from the excitation of three states, A, B and C, by electrons of a single energy, provided that the energies of the states are related by EB- EA= 1.0 eV, EC-EA= 5.0 eV, and that the incident electrons have energy E:+ 0.2 eV (energies taken as positive from the ground state). Consequently, Figure 3.2 qualitatively represents a metastable distribution, considering the lowest level to be metastable, and allowing for cascades from the two higher levels. The effect of cascades, therefore, is to fill up the central region of the distribution arising from the lowest state with the narrower distributions corresponding to the higher levels.

76 6.3 Spread in Electron Energy Since the width of the velocity distributions is a strong function of electron energy (see Figure 3.2), a spread of energies in the electron beam should have a marked effect on the observed metastable velocity distribution. Qualitatively, the principal effect should be of washing out the peak structure (see Rubin, 1969). 6.4 Anisotropic Scattering of the Electrons Throughout this thesis, it has been assumed that the scattering of the electrons in the center of mass system is isotropic. It is known, however, that even for helium s-wave scattering does not occur for more than a few tenths of an electron volt above threshold; in fact, the excitation function for the 23S state exhibits s-, p-, and d-wave resonances at 19.95 +.05, 20.45:f.05, and 21.00 *.05 eV respectively, as shown in Figure 6.2 (Ehrhardt, 1967; 1968; Andrick, 1968). The inclusion of such variations in the determination of the metastable velocity distribution function requires that the factor dw be included in the scattering distribution, Equation 2.24. The same factor evaluated at the appropriate value of Vo therefore appears before each of the terms in the integrand of Equation 2.34. The effects that such terms can have on helium metastable velocity distributions are shown in Figure 6.3. Figure 6.3a shows how the velocity distributions would appear at the energies of the resonances if the scattering were

77 Helium 235 - 100 X,' I 9300 / 900 8 s 20 21 22 23 24 Stolenerg/e [eV] --- 1,0 0 I f 11 1 - I 0 - I IT 2I2IPI I I 0 o 2I t IOo I,,11 I _ II I I - II 45 I I ~o 30 60 90 ~ 0~ 30 60 50 o 20 Streuf/enA/ From H. Ehrhardt and K. Willmann, Z. Physik 203, 1 (1967) 2Figure 6.2 Energy and angle dependences of 2-S helium excitation function.

-, 25 x 103 1-lelium 23S, E 19.95eV(S-WAVE) C14 E 20.45eV(P-WAVE) >18 11 )I cp -0* E 21.0OeV(D-WAVE) b13 e 720 E 19-95 eV 15E 20.45eV >;1O''I I r ((.E:21.00ev (I W 0 Li i 0 0.8 1.0 1.2 1.4 0 0.8 1.0 1.2 1.4 NORMALIZED METASTABLE VELOCITY (a) (b) Figure 6.3 Velocity distributions at the 235 helium resonances, a: distributions assuming isotropic scattering at all electron energies; b: distributioii assuming pure s-, p- and d-wave scattering at the 19.95, 20.45 and 21.00 eV resonances respectively. Perfectly collimated Maxwellian ground state beam assumed.

79 isotropic in the center-of-mass system; Figure 6.3b shows a more realistic evaluation, assuming that the resonances involve completely pure s-, p- and d-wave scattering of the electrons. Comparison of the resulting p- and d-wave distributions with the corresponding s-wave results shows that the former are characterized by relatively few metastables in the middle of the distributions, and by an excess of them at the extremes. At very high electron energies, strong forward scattering of the electrons occurs in the center-of-mass system; the first Born approximation provides a description of this behavior. Applying the Born approximation to the forbidden lsa-2s transition in hydrogen leads to a very simple expression for the differential scattering cross section. Figure 6.4 shows several calculated velocity distributions for atomic hydrogen, assuming direct excitation to the 2s 2=S state. The curves of Figure 6.4a assume isotropic electron scattering; those of Figure 6.4b assume the scattering is described by the first Born approximation. The strong forward scattering of the electrons, with the concomitant small deflections of the metastables, results in a pronounced shift of the metastable velocity distributions to low velocities*

80 8 x 102 6 E = 2Oe 4 4 S - WAVE b 3 2 50eV,-a- (a)' 8 xlOz,, 8 x0 H4ydrogen 2 2S, c H I=90~' C. cp 0~ -J~,~~~~~~~ e =810 6 6 >e T = 2800 K Ll> \E =20eV - BORN 2 - 30eV 0 1 2 3 4 NORMALIZED METASTABLE VELOCITY v (b) Figure 6.4 Velocity distributions for 2 2n~ hydrogen. Ground state beam assumed perfeotly collimated an Maxwellian a: a-wave scattering of electrons assumed; b: scattering assumed to be described by the Born approximation.

CHAPTER 7 APPLICATIONS AND CONCLUSIONS 7.1 Application to Measurements of the Lamb Shift It has been emphasized by Robiscoe and Shyn (1970) that departures from a Maxwellian velocity distribution in metastable hydrogen beams provide a source of systematic error in several determinations of the Lamb shift. The above authors present results of a measurement of the velocity distribution of metastable hydrogen atoms in the apparatus used for a recent determination of AE-S (Shyn, 1969). The deviation from Maxwellian is substantial. Using Ulexp (-U ) as a convenient analytical approximation to the data, Robiscoe and Shyn determined that an upward shift of about 0.04 MiHz was indicated in the value of the Lamb shift (1057.86t0.10 MHz (2 r)) previously determined in a level crossing experiment by Robiscoe (1968). The current status of the theory and experiments on the Lamb shift has been summarized by Appelquist and Brodsky (1970). The theory of Chapters 2 and 3 can be applied to the velocity distribution data of Robiscoe and Shyn. A fit using the program of Appendix IV is shown in Figure 7.1. A comparison of the parameters used for this fit with those quoted by Robiscoe and Shyn is presented in Table 7.I. The 81

82 T = 28000Kuexp(u) o = 6.80 x 10' 5cm/sec U4exp(-u2) U 4 4, 100 emu ) FITTED DISTRIBUTION: 75 95cc I / I \ \\ ~~6c= 3.80 I-=e = 90 / / / \ \ E* = 10.2ev vP i I I I ~~~~E = 12.0ev 50 -./...... EXPERIMENTAL DATA LU LIJ 0 5 10 15 20 H* VELOCITY v(105cm/sec) ADAPTED FROM ROBISCOE AND SIYN (1970) Figure 7.1 Measured velocity distribution and t eoretical fit for H(2S) beam. The U2exp(-U2) and U4exp(-U2) distributions are included for comparative purposes.

TABLE 7.I Comparison of experimentally determined and fitted parameters for the metastable hydrogen velocity distribution from Robiscoe and Shyn (1970) E _E 0 AO __ _ Experimenta 1l parameters not 11.4 eV 2.9t 0.5 eV 90~ 0.764~ measuredl 96.3O0.30 (Robiscoe, 1970) Fitted parameters 10.2 eV 1.8 eV 900 AOL+A = 3.80 95.10 (this work) 1T. W. Shyn (private communication) estimates this to be several degrees.

84 fit is very sensitive to the parameters AE,-O, and Al4A0 (see Figures 3.2, 3.1 and 3.12 respectively); the values stated for the fit could only be varied within narrow limits without producing noticeable deviations from the measured velocity distribution. The parameter &i is not included in the data of Robiscoe and Shyn; however, Shyn (private communication) estimates it to be several degrees, which is in qualitative agreement with the measurements made by Rebane (1970) on the same apparatus. Although the assumptions used for calculating metastable velocity distributions in this thesis differ from those suggested by Robiscoe and Shyn, both works are in agreement that atomic recoil will have a substantial effect on velocity distributions in beams of light metastable atoms. 7.2 Application to H2 Dissociation Experiment Leventhal et al. (1967) have investigated the electron impact induced dissociation of molecular hydrogen: H + e - H (zS)+ -+eI (7.1) It is oossible to learn about the relevant dissociating state of the molecule by studying the intensity and energy distribution of the metastable atoms. It is found that there are two groups of metastables which result from this reaction: one centered at an energy of 0.32~ 0.05 eV and the other with an energy of 4.7~ 0.7 eV. In an extension of this experiment, M. Misakian (private communication) has

85 found that the angular distribution of the "slow" metastables is probably influenced strongly by the kinematics of the electron-molecule collision itself. Using a diffuse source of helium for calibrating the experiment, Misakian has found a strong angular dependence in the scattered intensity of helium metastables. Comparison with the predictions shown in Figure 3.16 shows good agreement only if the factor sin (see Section 6.1) is not included when reducing the data. However, it is possible that the ground state gas distribution is not perfectly isotropic in the electron bombardment region and this may be the cause of the discrepancy. 7.3 Measurement of Differential Excitation Cross Sections As is well known (Stebbings, 1960; Vriens, 1968; Bederson, 1968), it is possible in principle to use electronatom recoil kinematics to study the differential cross section for electron impact excitation of metastable states. However, due to various complications, such as the inability to distinguish cascade contributions from those due to direct excitation, the energy analysis of the incoming and outgoing electrons is generally more suitable for this work (Ehrhardt, 1968). Nonetheless, the work of Stebbings et al., in which total intensities of metastable hydrogen atoms were measured as a 2uncteon of detector angle and electron energy, shows nicely the effects mentioned in Section 3.1.2, as modified by the gradual onset of strong forward scattering of the electrons. Considering their data in the light of Figure

86 3.5 shows an initial broadening of the maximum (along with a curious shift to lower values of 0), followed by a strong shift to large 0 accompanied by a decrease in intensity at small angles, in agreement with qualitative expectations of the effect of strong forward scattering. 7.4 Other Possible Applications It would be both of intrinsic interest and a good check on the calculations to be able to resolve the helium 23S resonances (Figure 6.2) by observation of the changes in metastable helium velocity distributions (Figure 6.3). To observe the d-wave resonance, it would be necessary to quench the singlet metastables (Fry, 1969), allowing the triplet moetastable to be observed for electron energies above 21 eV (the excitation function for the 23P state is a very shallow function of energy near threshold; see Ehrhardt, 1968). A very useful application of the kinematic effects discussed in this thesis might be as a diagnostic tool for studying Eround state gas conditions (French, 1967; Zorn, 1967; Pichlik, 1970). The simple velocity distribution given by Equation 3.9 (or more generally by Equation 3.7) indicates that excitation of gases by electrons of controlled energies might be useful as a temperature probe. Analysis of velocity distributions in streaming gases might also be accomplished. In order to minimize uncertainties in the metastable distributions due to imperfectly known differ

87 ential cross section behavior, it might be best to operate at high electron energies where the Born approximation applies; this would have the advantage of increasing counting rates due to cascade contributions, while minimizing the effect of different excitation thresholds. The decrease in total cross section with increasing energy would be detrimental, but simple high current electron guns could be employed to compensate for this. 7.5 Summary and Conclusions This thesis has been concerned with the determination of the velocity distribution of metastable atoms produced from ground state atoms by electron impact excitation. The analysis, which is based on the transformation of the collision kinematics from a center of mass coordinate system to a laboratory coordinate system, yields the following interesting results. The velocity distribution of metastable atoms resulting from crossed electron and Maxwellian ground state atom beams is either void or is distinctly non-Maxwellian with the following features: i) it has two peaks; ii) it has upper and lower cutoff speeds; iii) the magnitudes of the cutoff speeds are very sensitive to such factors as beam alignment, detector position, and electron energy;

88 iv) it is very sensitive to the form of the differential cross section for impact excitation. Consideration of a diffuse Maxwellian ground state gas which contains an electron beam shows that the velocity distribution of the resulting metastables is still generally non-Maxwellian, although the double humped property is absent. The form of the distribution which is observed is still a fairly sensitive function of detector orientation, but the sensitivity to electron energy is relatively less than in the case of collimated beams. In many experiments utilizing beams of metastable atoms, the present results are relevant as a possible source of systematic error. In addition, the sensitivity of the velocity distributions to various parameters of the beams of ground state atoms and electrons suggests that metastablization by electron impact can be used to monitor kinetic properties of the ground state gas.

APPENDIX I ELEMENTARY KINEMATICS OF AN ELECTRON-ATOM COLLISI ON The object of this appendix is to show by straightforward algebra that the velocities,'i, of metastables formed from ground state atoms with a single velocity, %o, must terminate on a sphere in velocity space and conversely, i.e., that metastables with a given velocity are produced from ground state atoms with velocities terminating on a sphere. Considering the former case, the normalized laws of conservation of momentum and energy may be written respectively as: v +yu =, Rio~ h +<>o - e (I.1) +)A' v V U _E' ~ (1.2) By assumption vo and uo are known, so that the quantities on the right are known also, Solving Equation I.1 for p)A and squaring yields r.us 2'0-") (I.3) Multiplying Equation 1.2 by ) and substituting Equation 1.3 gives (btf) v-2, _ ttal +Bo Ao= C (I.4) 89

Taking Bo as the polar axis of a spherical coordinate system, so that',- = ov allows Equation 1.4 to be reduced to - ( - o I (1.5) This equation, involving the components of the velocity, r, of the metastable, is that of a sphere. Furthermore, since = (tjL)W this sphere is centered at the tip of the vector W, representing the velocity of the center of mass of the electron-atom system. Manipulation of the quantities in the right side of Equation I.5 shows that the radius of the sphere is given by y Z Uz M+r (Iw6) in agreement with Equation 2.9. Turning to the converse problem, it is convenient to rearrange the conservation laws as follows 4 Ha, V -'PLk. = (I.7) ~V I- A L:+ZE A (I.8) where it is now assumed that r and uO are known. Proceeding as above leads to the expression showlng that metastables with velocity r arise from ground state atoms with velocities vo terminating on a sphere which -. is centered at the tip of the vector l — B. The radius of

91 this sphere is found to be given by -:,' ~ 2, uo -)I Et (I1o0) Expanding Equation 1.6 in powers of A and Vo/u, and expanding Equation I.10 in powers of and v/u., it is found that to lowest order V~R e e. EC (1.11) where E. Imnu is the kinetic energy of the electron in the laboratory frame.

APPENDIX II A GEOMETRIC VIEW OF THE METASTABLE VELOCITY DISTRIBUTION FUNCTION, (iQ) The representations of Figures 2.1, 2.2 and 2.3 provide a convenient means for visualizing the properties of the distribution function of metastables which arise from a monoenergetic beam of ground state particles. It is informative to extend this geometrical point of view to the case where the speeds of particles in the ground state beam are distributed over a range of values. Returning to Figure 2.1, it is recalled that the position of the center of the sphere is determined by the vector sum of vo and.u. If, now, the magnitude of Vo is changed, but its direction remains unaltered, then the new sphere which results will be centered on a line passing through the center of the original sphere and parallel to vo. Consequently, if the gas in the ground state beam has a range of speeds but a fixed direction of motion, then the corresponding family of spheres will be strung along this line. Because V is constant, the envelope of these spheres is a right circular cylinder with axis parallel to vO. Thus, where the velocity distribution of metastables corresponding 92

93 to a single sphere and V4W is spread over a cone in velocity space, the distribution corresponding to an entire cylinder (if \V4)Ausin[) is spread over roughly half of a lune with axis vo (see Figure ll). Consequently, a properly placed detector of infinitesimal solid angle will observe metastables with Just two speeds v if the ground state beam contains atoms with a single speed, but will observe a continuous range of speeds (between well defined cutoffs) if the distribution of speeds in the ground state beam is continuous. The above discussion has emphasized the determination of the range of velocities, v, available to metastables produced from ground state atoms with known velocities, Wo. However, a better insight into the form of S(V) (Equation 2.34) can be gained if this point of view is reversed, i.e., if the range of velocities, Vo, of ground state particles which contributes to a single known metastable velocity, v, is determined. As has already been mentioned, a monoenergetic, collimated ground state beam gives rise to metastables with velocity vectors which may terminate anywhere on a well defined spherical surface in velocity (or normalized momentum) space; for purposes of identification, this sphere will be referred to as the "outgoing" sphere. It is shown in Appendix I that a similar relation holds if the roles of the velocities Vo and v are interchanged, i.e., that metastablee having a given velocity v can only be produced from ground

94 2V - ENVELOPE OF OUTGOING ~ - -. -..-'- | I \ METASTABLES LUNE CONTAINING - OUTGOING METASTABLES posinl'0 Figure 11.1 Cylindrical envelope of metastable velocities which arise from a collimated ground state beam

95 state particles having velocities vo which terminate on an altogether different sphere; this sphere will be referred to as the "contributing" sphere. To the present order of approximation, the outgoing sphere is centered at the tip of the center of mass velocity vector W, while the contributing sphere is centered at the tip of the vector B; in addition, the radii of the two spheres are equal (see Figure II.2). Thinking now in terms of the contributing sphere, it is possible to glean a good deal of qualitative information about g(v) from the geometric representation. For concreteness, consider the ground state beam to be collimated within a solid angle cLw,; let the atoms in the beam have a distribution function f(ao), and suppose that this beam passes at right angles through an electron beam. Consider only metastables which leave the interaction region in some given direction defined by the unit vector v; suppose, for si.mplicity, that v lies in the plane of the incoming beams and that it makes an angle G with the electron beam. Then salient features of 9g() can be obtained by considering three different cases for the magnitude of o (see Figure II.2). Case I: very small (very large) values of v. If v is sufficiently small (large), then the vector B m -uo, will be such that the contributing sphere does not intersect the ground state beam at all. Consequently, there are no ground state particles available to contribute metastables with speed v; the distribution function g(v) is

-~B C(a se 4T - -:d wCs V+ -) 88 8 = B - V X ) 6v - W+V d v-= d- V given a value of vo, given a value of v, wnai values of v may occur? what values of vo con ribuve? Now co.nsider a beam of ground state atoms having an arbitrary distribution of speeds f(vo) Then, for a given value of v, there are three possibilities for conrriburing values of vo. b Case I Case tI Case [I Fi.... - rUo itreaio =of e vlct dutor uo Bibrt\ 9round state ground state vo ground state beam beam beam Sphere does nor intersecl- beam; Sphere intersects beam twice;, hus two Sphere and beam tangent; thus thus no me.gsrables with speed v. values of vo(vo_ and v0+) contribute to one vAre of vo contribures, and merasrables wilh speed v. Con!ribution value of v is maximum possible, proportional ro area of sphere inrer- call it v+. From geometrysecred by d(o,.e., ro v'/ V^ ] s e Fir V+ = [o.v] sec v o Figure II.2 Geometrical inter.oretatoon of metastable velocity distribution function g(~).

97 therefore zero for all such values of v. Case II: intermediate values of v. For a range of values of v between those mentioned above, the vector B will be such that the contributing sphere intersects the ground state beam. Normally there will be two such intersections, corresponding to ground state velocities Vo, and vo_. In addition, the contribution to the metastable signal will be proportional to 4(vo), and to the portion of the sphere, d&h, intersected by the solid angle element dwo. Expressing cidt in terms of dco leads at once to a. (1.I1) c~a-t v. ct o Case III: cutoffs The limiting values of v between Cases I and II occur when the vector B is such that the contributing sphere Just touches the ground state beam. Under these conditions Vo is perpendicular to V and the corresponding values of v (cutoffs) are found from simple trigonometry (Figure II.2) to be vt j. o t\is se- lO (II.2) In addition, since vo is orthogonal to V at the cutoffs, the denominator of Equation II.1 vanishes there. The results of this geometrical investigation can be summarized as follows. Given a well collimated ground state

98 beam (velocity distribution function 4 (o) ) which intersects a monoenergetic electron beam at right angles, the velocity distribution function g(i) of the metastables will have the following properties in the plane of the incident beams: i) it will vanish identically for all but a well defined range of values of v; ii) within this range, metastables of any given value of v will arise from two values of vo; each value of vo will contribute to S(V) in proportion to v.'(Vo), and will be weighted by the factor to. since this quantity blows up at the limits of the allowed range of v, the distribution is singular there; iii) the cutoff values of v are given by Equation II2. The secant factor indicates that the "position" of the nonvanishing portion of S(V) will be a very sensitive function of the observation angle G, moving off to infinite values of v as the detector moves from a direction looking into the electron beam around to a direction looking into the ground state beam. Qualitatively, the same results are valid for beams which intersect at any angle. If the beams are considered to intersect at an angle 3 instead of at right angles, then Equation II.2 becomes Vt. _r A i b m,, \, __ _ v~

99 If the detector is no longer considered to lie in the plane of the incident beams, the arguments of Cases I and II are still valid, and Case III is only complicated by the necessity for a more cumbersome generalization of Equation II.2 (see Appendix III).

APPENDIX III SOME EXPLICIT CALCULATIONS The purpose of this appendix is to introduce explicit polar variables into the expression for g(v) (Equation 2.34). Cutoffs and integrated intensities are then discussed in terms of these variables. III.1 The Introduction of Polar Variables into g(~) The velocity distribution function 3(v) for the metastable atoms produced by electron impact excitation is given by (Equation 2.34) -* (VA, o. (V. 0) q=J o 74 4(a0) A V 4. I 0o Here V, = Eo iV t-' sii (III.2) where the variables are defined in terms of the vector diagram of Figure 2.4. For purposes of calculations, the vector notation is inconvenient; the natural variables for the problem are the polar variables which are also indicated in Figure 2.4. It is therefore useful to introduce these into 100

101 Equations III.1 and III.2. The most straightforward way to do this is to introduce the orthogonal coordinate system shown in Figure 2.4, in terms of which the important vectors are 5V ~ v (ss~nVIn Cp St"cos cost) (III*3) y,,,Uo-.o(A o, o. d ) (III.4) \v V0( O, i coa) (III.5) The absence of an azimuthal variable,5, in the definition of %o represents no loss of generality; the important parameter is the deviation of the vector V from the plane containing Vo and u, which is adequately described by (p. If, however, for purposes of integration it is convenient to use the variable S, it can be included by merely replacing by q- in all of the expressions derived below. From the above expressions, it is possible to write = (inrsin p, v-tin 0) costs v wco,%3 -ioA (III*6) Using this relation it follows that = —z (sints; cas qf + s )U Lo% (II%.7) and

102 = 5YGif 3inc 4 zF[jy;uniin t v( inO43osicoC. o<.0% v31)1 (1118) Consequently, 9( a) takes the explioit form f (., ) (II.09) se 4 xVI[V-' V',,ien, siq, Sn where Vo a |,i'i *v (slin c01 cos.-icos* 1,o = -v (,,e 4 ico,, tc C +(3 o ia ) - puo <ta ~ivil~._~ | ilt, -- - - -I ( II I.10), -[ [,~osir,} +,, ~ (;,e),.,,, { <, f - c45 e S~, n)]" III.2 A General Relation for the Cutoffs If the ground state gas enters the electron gun as a well collimated beam, then the envelope of the velocity vectors of the resulting metastables is a semi-infinite cylinder with a hemispherical cap on the end (see Appendix II). This cylinder defines the limiting metastable speeds which can be observed in a given direction (the cutoffs). Since these are important for the discussion of the metastable velocity distribution, the cylinder will now be described in terms of the variables relevant to the problem, i.e,* polar coordinates. Because the present results are to be applied to nearly orthogonal ground state and electron beams, it is legitimate to consider only the portion of the envelopes determined by the following restrictions: it is assumed that

103 ii) sin v Together, these assumptions are sufficient (though not necessary) to limit consideration to cases where the cylindrical envelope does not enclose the origin of the velocity space, and to those portions of a given envelope which are not "near the end", i.e., which are exterior to the sphere which defines the end cap (see Figure II.1). The cylinder is most simply described in the cartesian velocity coordinate system (x,y,z) shown in Figure III.1. Tere the axis of the cylinder is parallel to the y-axis, so the equation for the cylinder is X + ttX5) -N (III.ll) and y is arbitrary. The natural variables are polar coordinates ( v, ), 9 ) defined with z' as the polar axis. The transformation between the two systems is x v s;n C A XJ_ %in %in iOUP + -v co 0 c-% ( III12) In addition iv = Lao s;n D (III 13) Consequently, the equation for the cylindrical envelope becomes

104 CYLINDRICAL ENVELOPE OF METASTABLE ELOCITIES X,X'/ I' ~y Figure III.1 Velocity space cartesian coordinate systems used in describing the cylindrical envelope of metastable velocities.

105 V( -ovsinE)Iosa~h rpV*Sivt; ~ -FLLAsls 42) (III*14) If a detector is located along a ray from the origin which is defined by the angles e,cp, then Equation III.14 can be solved to determine the minimum and maximum speeds Vi (cutoffs) which can be observeds rialt a brout t <p (-Th lmi v aluep t Ccur hie) valid for a range of azimuthal angles 9 which lies symmetrically about p=O,0 The limiting value, occurs when,, the line to the detector Just grazes the cylinder, i.e., when the two cutoffs coalesce. Thus,,,Ma is the value of 4p which makes the radical in Equation III.15 vanish. This condition leads to = cs -D cs.ia nsin.o) 0-% eo C os - f (III.16) W/ t[(NML~o)-lljtnQE~s;bjl -';-i cGoD;.s' I

106 For values of >?, the distribution 9(V) vanishes for all v. III.3 The Integrated Metastable Flux at Threshold A useful quantity in an experiment is the total metastable intensity (integrated flux) in a given direction. In principle, this can be obtained by integrating the metastable velocity distribution over all velocities: l (Gv v 950) (III.17) It would be particularly useful to evaluate this quantity for the distribution arising from a collimated ground state beam since this is the situation of present experimental interest. Unfortunately, the general integral III.17 cannot be expressed in closed form. However, it can be easily evaluated at threshold, as will be shown below. To conform to the case of most immediate concern, the following assumptions will be made i) the ground state gas enters the electron gun as a perfectly collimated beam ii) the velocity distribution function in the ground state beam is Maxwellian ill) the detector is in the plane of the incoming beams. The intensity per unit solid angle can then be written

l07 1 e 4 V,Ae ( I I18) where V4. I.. (vol" e +VL_ e v-(111.19) Specializing the results of Equations III 9 and III.10 to the case 9-0O allows this to be rewritten',..-vlt' V -C, I, -Lsz'' (- e ) _(Q) _ oV) -_ _ _ _ (III.20) where VO*!, 4(.......Z1..2. Considering values of V44/u. it is observed that the radical in Equation III.21 is generally much smaller than the leading terms; in addition, the limits of integration are nearly equal. It is therefore a valid approximation to replace vo0 in Equation III.20 by the single value given by the first two terms in Equation III*21. Furthermore, the terms in the numerator of Equation III.20 can be evaluated at the mean value of v, call it v>), <()> /=,uo;,i c.(1-G ) (II1.22) and then taken outside of the integral. Equation III20 then becomes

108 T(e, *,0;v) _ 2r uoV IZOt ZG siA, 4 ( 4-) %.ex)1 p 4 }, se \Q4ij. K (e i;v) (.23) where u + ce) \( (e,:o;V) j _(11.III.24) f2 EiL~II" CSi 2-.) Introducing the new variable of integration W - V | *tuo~inii V SiY,.i-Oj] tIII.25) leads quickly to the iLzsult K0 (O O )V-tt csc(.4-e) (111.26) The angular dependence of the intensity very close to threshold (Equation III.18) is therefore summarized by combining Equations III.23 and III*26 3 24,,;V), &T rL e 4 Ui 4 %%, E n1 S i O(f-) ~ expi- )JLt Sni-cscli (111.27) Since this is independent of V (the next term in the expansion for J( (,,O;V) is of order v) it is concluded from Equation III.18 thlat ~i~ UY(o,, o;V) no - Ia ( 31 J I) o; O) (III.28) i.e., that the product W(E,i%,V)) is non-zero but bounded. Considering orthogonal ground state and electron beams

109 (ia.- 1 csc(iI-o) = sec e), the angular behavior at threshold of the metastable intensity in the plane 0 is found from Equation III*27.;(,e' o0.o)-4.ta,,sc, exePt,[-ct.n'e] (III*29) In normalized form, this is the equation for the curve parametrized by E-=O in Figure 3.5.

APPENDIX IV PROGRAM BY DR. E. S. FRY FOR NUMERICAL INTEGRATION OF g(v) OVER FINITE DETECTOR WIDTH The velocity distribution function for metastable atoms arising from a collimated ground state beam is discussed in Section 3.1. All calculations presented there assume that both the ground state beam and the detected portion of the metastable beam are well collimated. The program presented below integrates g(v) over a detector of finite angular width A )-A, assuming the detector to be in the plane of the orthogonal incident beams (i=2, 9O ). Using a Gauss-Chebychev quadrature of order n =6, the program evaluates A cUrZ, cci dn U cosir _ where rj= ~X~_~jc~ _ UzsinV " X t~u (IVe2) T. U C. T,- UF ~o~~ ~ rJ ~: —— h= t[ ti n) ii ^- |/lluO 9 1 1a(IV.3 )>

111 - rc~ e IJtT (Iv*4) (Iv.5) By x ( iv. 6) and d=i/ is the most probable atom velocity in the Maxwellian reservoir from which the ground state beam is formed. INPUT: The data cards must have columns 1 and 73-80 blank. A data set begins with &M and ends with &END. A run consists of as many sets as desired. The input variables and their default values are: - - variable - - - - default - - E The metastable excitation energy E*(eV) no default DELE The incident electron energy in excess of E*(eV) no default PSI The observation angle, 4I (degrees) no default DELPSI The spread in observation angle, alf (degrees) no default

112 - - variable - - - default T The atom source temperature (degrees K) 300.0 DU The spacing between the values of Uf at which the integral is evaluated 0.005 MASS The atom mass (at. mass units based on C12) 4.0026 FRAC When the value of the integral falls to FRAC times its peak value, the integral is not evaluated at any additional points UL. 0.001 UICH For plots, this is the U axis scale factor in units of U per inch. If UNCH 0.0, the axis is drawn 8.0 inches long and a "nice" scale factor is determined (U is dimensionless) 0.0 VNCH Same as UNCH but for V (km/sec) 0.0 TOFNCH Same as UNCH but for TOF (usec) 0.0 PATH The machine length, used to determine the TOF spectrum (cm) 1.0

113 The following input variables have only two possible values T or F (true or false): - - variable - - - default - - PRNTU T The U spectrum is printed out F The U spectrum is not printed out F PRNTV Same as PRNTU but for V F PRNTOF Same as PRNTU but for TOF F PLTU T The U spectrum is plotted F The U spectrum is not plotted F PLTV Same as PLTU but for V F PLTTOF Same as PLTU but for TOF F LABEL T The plot is labeled F The plot is not labeled T There must be a comma after every input variable except the last one in a data set. There must be at least one blank space after &M and before &END. Once an input variable has been given a value, that value is retained throughout the run or until it is changed. If an input variable is not specified, the default value is used. OUTPUT: When PRNTU is requested, the sum of FU(U) * DU is printed out. This sum is taken over all values of U and

114 approximates an integral over the velocity distribution. The rest of the output is self-explanatory. A listing of the program, written in FORTRAN IV, follows.

__ C T H PRSOGRAN, USFS1 k AtUS S-CHFFYC(IHFV OIIADRATURE TfO-ll NTFGV\TF TFH- F RY MOOl C, MIFTASTAR-F VELCITY DISTRItH T IO (O\VER AN ANGLE DFLPSI CENTERED FRY M002 C, A T PSI. THE FLF-CTRIN AND TIC CIFENT ATOm MOMENTA ARE ASSUMED TO FRP. Y M003 C. BE PERPENDI CULAR. FRY M004 C_ FRY MOO D I M EN SIN 0.( 1,000),FU ( 1000),\I( 1000), FV ( 1000) FRY M006 ___ lr lOl 1) T F PS I, SUM PS I, K ARK iU.IF,, E, DE L Ei Tv PS I, DE LP S I I MASS S L ABE L FRY M007 NAMFL I S T/ M/ F, DFL F I T,I0U, P S TI, DFLPSI T MASS, FRACt UNCH\/ VNCH TOFNCH P ATHIFRY MOO 1. L AfBE Li PLTU.Jg PLT \I PLTTOF, PRN\.lTIJ, P RPNTV, PRNTOF FRY MOO0 REAL K<,KAMSjMtSSjAXFtO FRY MOIG LOG, I C AL L,LA LPLTU., PLTVPLTTOF,PRN\ TU, PRN\TVPRNTOF FRY MOll INTEGEFR H FRY M012 PL TI=. FALSEF. I__ FRY M013 PLTV=*FALSE FRY M014 PL TTOF=,FALSqE. FRY M015 PRNTJ=.FALSF, FRY M016 PRN T\/= FALSE._ ______ FRY MOIY PRNTO F= FALSE. FRY MOIR LABEL=.= TRUE. FRY M019 IN NC H = 0. FRY M020 VNCH=._ FRY M021 TOFNCH=0. FRY M022 PATH=1 _ __ FRY M023' FRAC=.001 FRY M024 T= 3 00. FRY M02 5 MASS=4.00260 FRY M026 PT1=1.772454 FRY M027 [) F G=. n 17453 3 FRY M028 00 =. 0 0 5 FRY M029 X1=.9659258 FRY M030 __ __=_ __7_ __7_ _ __6 FRY M031 X3=.?588) 90 FRY M032 1 EAD (,MFND=500) _ FRY M033 SIJM=0. FRY M034 M X F t1=0. FRY M035

ALP. z. --; ).T ( T/ S..S c) FRY MO3(n 2..,...-....T(F/T/ 4/!S'- ) FR_ _Y M037 Kr-F! F-/: FRY Mn3R KA -gr T t...A)i/ ) ___FY T 1F_039 P-K P -;: —f (. ) FRY M040 P ] =( -)P f i -f; E -, g I /?. )'!) G..F: _.... FY..M.............1..041 P2 (PSI+f FLi T/. )- -F, FRY M1(42 S:'l=-l.'xi~pl _ __F_______ __ ___- _ FRY M043 CP?-e Tf!(;) FRY M44 i.I<,A.=tI-P( S(p1'(K+1. )-SORT(K ) K) _ _....FRY M04.5 RKK R-' ( S ( ",+1. ) +)SORT(K.) ) FRY iv1)046 I!'iI pKi —RKc? _ _ FRY M047 _ A= RKB F/ ( 2.;'f')t) +UM' I f ) FRY -M048 TFF(SP1.LT \) SP1=A___ - __- - FRY M049 i4 X = R K R /'' FRY M050.YR I TF (, ]f).....E,?.ELF:.,r -. LPH.,_ _ __Iv'ASS, f...........'_....... ^!tIT F 6(v,no) FflELtT.AfPH4:AASSF~L I9,PSTC ___ FRY M051 J= I N'T ( lI v/DNti ) FRY M052 i=FIl,LA T(, )', -J FRY_ M053 _ = I =INT( (l t l, A-.M), I) / l) t+ 3 FRY M054 0, T TF (J.GT. 1o0) l=100_... FRY M055 On 4 -=1,=i' FRY M056 U I )= I+FL 1AT ( -... -D............. FRY M05 UJF=t( I) FRY M058 LA S=T _FRY M059 IF (SP2.LTRKA/IJF) GFl Tn 3 FRY M060 IF (SP!.rT, KR/KB F) (;rl Tf' 3 FRY M061 P=P1 FRY M062 PR=P2 -RY M063 I F ( S P1 T A/) F) PAARS ( RKA / F ) FRY M064 IF (SP2.rT,.RKFB/tJF) PF/=ARSTfi,(R KR/IJF) FRY M065

DIFPST,5*( Pks-PA) FRY M06( SUlMPSIT. F:(PF+P __ FRY M067 CHER=F(Xl)+F(-XI)+F(X2)~F(-X2)+F(X3)+F(-X3) FRY M068 F_ I J I H F B 1) I F P S F / PFII1 / R K / R K / 6. I J F J: F FRY M069 SlJM=S kiJ+FRU.J( I ) FRY M070 IF (J I GT MAXFU) MA XFtJ=Ft(I) FRY M071 IF (FIJ(I).LLT FRAC "IMAXF1J) G0 TO 5 FRY M072 GO TO 4 FRY M073 3 FM(iI)=(. FRY M074 4 CON T I NUE FRY M07r5 C RFADOUT OF Uj SPECTRUM FRY M076 5 IF (.NDT. PRNTU) GO TO 6 FRY M077 S J) M = S U M* D t-J FRY Mv07 WRITE(6,v11l) U.MIN, UMAX,qSU(M FRY M079 WRJTE(6,O2) (Ii( I ),FM( I), I=1,LAS) FRY MOO 6 IF (.NDT. PLTM) GO TO 7 FRY M081 KEY=1 FRY M082 CALL PLOTR(U, FU.p, IINCHLASKEY) FRY M08-3 C CALCULATION AND READOUT OF VELOCITY SPECTRUM FRY M084 7 IF(.NOT.(PRNTV,OR,PLTV)) GO TO 10 FRY M085 ALP =AL-PHAl t 00001 FRY M086 00 8 1~1,LAS FRY M087 FV(I)=FUJ(I)/ALP FRY M088 8 V(I)=U(I)*ALP FRY M08__ IF( NDT.PRNTV) GO TO 9 FRY M09( VMIN= JJMIN*ALP ___ FRY M091 VMA X=IJMA X *AL P FRY M092 WRITE(6,1O3) VMIN, VMAX FRY M093 WR I TE 69 102 ) (V ( I ) t FV( I ) w1=19LAS) FRY M094 IF (.NOT. PLTV) GO TO 10 FRYM095 9 KEY=2 FRY M096, CALL PfOTR(VFV,VNCHLAS,,KEY) FRY M097 C CALCULATION AND READOUT OF TOE SPECTRUM FRY M09F 10 IF(.NOT.(PRNTOF.OR.PLTTOF)) GO TO 1 FRY M099 ALP=AL-PHHA,*000001 - FRY M10C OF) 11 I=1,LAS FRY M101

N= LAS + -I - FRY M102 \/(I)=PATH/(i(N)/AIf ___ _ FRY M10 3 I \/ I) =FUI( N)j AL P: —H( )i(N) /PATV FRY MIN4 TF(.NnT.PR!\ T ['lF) GO TAI 1? _ FR-Y MTO TIIr f'\J = PPi T H U I' A X /A L P FRY M106 TM A X = P TH/ TH I I II T N/AtL P FRY -I-lOT WRITF(6(,1O4) TMIN,,TVqN.A.XPATH FYMO f Iv;,`f1?1 T T ~ii.X,Pa r t-rFRY MiOS) IR I TF ( 6, 1O5) (V( IT FV ( T), TI=I 1 L _ _ AS _ FRY MI09 IF (.NF)T. PLTTr)F) (G TA I FR Y MilOI 12 KFY= 3 FRY M1Il C ALL PLOPTR ( \/,FVp, TOFNC HLA, KFY) FRY MI12 GO TO 1 FRY M3 II C, FRY M114 C_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ FRY M 11 5 100 FORMAT('1 T H R, FS HO L D ENERY (F\/)='7.3 5Xt X C ES FNER GY (EV)=' FRY Mi 11 1F7.3,5X,'S SoUtRCF TFfvIPFRATU.RF( FF C-C- K)='F1 FR.Y MIll 2/1HO,5X,ALPHA( CMiv/SFC l)=IPF1 1.4' I ATOMrCi Ci AMUi) OPF7.4 FRY M Il /lHO,5 X, (IABSFRVAT VA N I VEPIlN,\' F)EG 7D kRES, C ENIMTERH-A AT'F7, _3 FRY M119 41 nFGREES.') FRY M120 I1 01 F, R M A T ( 1. IH O, I 01XM I NIMJMIIF 64 5 X qM A X I M UM ti' F 6. FRY M121 15X,' THE SUM OIF FIJ(M) DIi IS,F7*4, FRY M122 2/1/5 ( 6X, U SXXt I Fi'(U)'_, 6X)) FRY M123 10n2 FORMAT( I-H /5(3XvF6.53,F.5,t5X)) FRY M124 10 3 FHtl P MAT( 11-HO/6X, ALL VA L U ES OF \ AR E IN K L M E T FiP,/ FC 2 0X FRY - iM-12-5 I IV!i\i T N I M i Nl V=' F 7.4,5 X F / M/ A X I = F 7.4 // /(6 X,' 5 X'F V (V )'6X)) FRY M126 In/4 FF)PMIAR A T(1H/H X,X A L L VA LUE S [i)F TU FPE I N S EC. I /O1X, IIMUM T=', FRY -M127 lF4. 2,5X9,'M1AXIMNI! T='IF.2t5X, I'FLIGHT PATH(CMA)=lF6.2// FRY M128?/5(5X,' TF'-",F 93X,'FTOF( I ) I,5 X) FRYUI12q I15 FOKfA T( IH /5(3X,F6.2tFi0.6,4X)) FRY M130 5nn CALL SYSTEM FRY M131

ENn FRY Ml 32 __ FUNCTION F( X) ________ FRY F001 C THIS FUNCTIOn EVALUATES THE INTFGRAND OF THE CHEBYCHEV TRANSFORMED FRY F002 C VEi.OnCI TY DI STR I RIBUTInON INTEGRAL. _ FRY F003 COMMON D)IFP, SIUMPvi PSvKA., RKTI.JF FRY F004 RFAL KA FRY F005 SP=SIN(D I F PSI X+SUIMMPS I) FRY F006 C P =S QR T ( 1.-S P S P) FRY F007 ETA=KA-I.J F SP/RK FRY F008 ETAl=1,-ETAMETA FRY F009 IF (ETAI.LE..0000001) (O TO 1 FRY P010 ETA 1=S( RT(_ETAl) _ ___ __ ___ FRY FOl Tl,=U)F*-'CP+RK" ETA1 FRY F 012 T12= T1*T1 FRY F013 T2=UFJFC P-RK- ETA1 FRY F014 T2?= T2?T2 _______FRY F015 X1=SQRT( 1.-X4 XX) FRY F016 F=(T22,EXP (-T22_) ~T12*EXP (-T12))*X1/ETAI-CP FRY F017 I-' RETURN FRY F018 I WRITE(6,r100) UJFETA FRY F019 F=(T22*EXP(-T22)+T12*EXP(-Tl2))*CP FRY F020 100 FORMAT(HOP/' ERROR AT UJ(I)='F5.3,p I ETA='E13.7) FRY F021 RE TU R N FRY P022 __ END __ _ __ __FRY F023 SUIBROUTINE PLnOTR( XYDx, L,,KEY FRY POOl C THIS SUBROIUTINE SPECIFIES THE PLOTTING FRY P002 COMMON DIFPSItSU)MPSIKARKtiFEDELtETPSIDELPSIMASSLABEL FRY P003 D)IMENSIONX(1000) Y(1000) PRY P004 LOGICAL LAfBEL FRY P005 N =L FRY P006 IF(DX.LT,.0000001) GO TO 3 FRY P007 DXI=DX FRY P008 XA=FLOaT(INT(X(l)!:10.))I103. — FRY P009 XB=FLnAT(INT(X(N)': lO.)+1)/10. FRY POlO XLNGTH=(XB-XA) l n —x FRY P011 I F ( X LNG TH. LT. I 5.5) GO TO 4 FRY P012

XL-NCITH=-1 *5 FRY P013 X 1X= 5. Af) X XXa __ __ FRY P014 DO 1 T= 1 FRY P015 IF (XU+ -T ) LT. XPIX) GC TA 2 FRY P01I CON T HAS FRY P017 2 N=,-! T FRY P0I rO TAi 4 FRY P019 3S__ _CALL RSCALF( 1.,XAOX1,X( 1) N, 1) F 1 Xl ___ __l Xr 9N. FRY P020 XLNCTH=T FRY P021 4 CALL PSCALF( A I 1. IYADY, Y( 1),N, 1) FRY P022 CALL PL Tr-"FS (XA fF1, YI,Y, 1Y.,I 1I FRY P023 (C( TOf (5,6,7)qKFY FRY P024 5 C ALL PAXIS(l,91.,91HIJ-1,XXLNTHt0,9,XAX1.5) FRY P025;o TO __ FRY P026 6 CA.LL PTXIS( 1.,1.,9HV\(KM/SEC),-9,XLNC1TH,TH.? XA.,nXl,.5) F RY P027 CO TO ( P FRY P028 7 CA 4 L.L PA XJT S ( 1I,1.,99HT C. F (I I S EC ),-9, X LNTH,0, X A,DX 1,5 FRY P029 2 CALL PAXJS(~1, 1].H 0, Y A __DY __ IFRY P030 CALL PLINF (X(1),Y(1),N,1,0,0,1) FRY P031 IF(.N-OT. LAPFL G) TO_ 9 FRY P032 CALL PFPNMPHR ( 1I,, 15,F,0.,'= 2 FRY P033 C7, A L L PFNM\IR(-0., 0., -.15,O FLF,0.,n,,,''I nELE=' WF2.3,) FRY P034 CA\L L PFNNRI(R-0 R -0.,-.15,T,0.,''' T-=',WF4.1' ) FRY P035 C(ALL P F lfAR RR(1.,t 7.,. 5,9 P S I,0.,1 PShhIWF2. 3"3 FRY P036 CALL PFN R (-fll,-0,-.1 5,FLPSII 0.,I ELPSI='I'W, F2 3' FRY P037 CA L L F PNM8R(,R -0.,. M I'IS S 0 I~1 A SS='',F 1I.' ) FRY P038 CALL PLTAN, FRY P039 R F TIRN FRY P040I END: FRY P04] 196 LINES PSIINTFD

121 APPENDXX V Reprinted from The Review of Scientifle Inetruments, Vol. 40, No. 9, 1242 (September 1969) Detection of Metastable Atoms and Molecules with Continuous Channel Electron Multipliers* DENIS P. DONNELLY, JOHN C. PEARL, RICHARD A. HE1'PNER, AND JENS C. ZORN Randall Laboratory of Physics, The University of Michigan, Ann Arbor, Michigan 48104 (Received 14 February 1969; and in final form, 9 May 1969) BEAMS of metastable molecules can be rapidly and our Channeltrons has been resuscitated in this manner efficiently detected by allowing them to strike the after it had failed from a corona discharge that occurred cathode of a windowless electron multiplier. In this note when the system vacuum was lost with 2.5 kV still applied we report on the use of a continuous channel electron to the multiplier. A noisy or inoperative multiplier should multiplier' (Bendix "Channeltron") in this application. not be discarded before a few of these "home remedies" Compared to other multipliers, the continuous channel have been tried. multiplier has the advantages of lower noise, simpler The continuous channel multipliers are made of glass circuitry, and far smaller size for a given cathbde area. tubing which can be broken if too much strain is exerted We have detected thermal beams of metastable helium, by the mount; once the multiplier is in place, however, hydrogen, neon, argon, krypton, xenon, and molecular nitrogen in our experiments. Although we have not measured the yield of Auger electrons per metastable incident FIG. 1. The Channeltron outon the multiplier cathode, we judge that the yield is +2000 put is picked up by a small 8mm 3000V DC metal button and fed to the comparable to that achieved2 on the tungsten or platinum 0input of an emitter follower surfaces used in the conventional Auger detectors. The wMeg which is mounted together with the multiplier inside the vacdetection efficiency for a given metastable species seems 0005,f the multiplier inside the vacircuit 6kvf um chamber. The circuit to remain within 10' 6 or so of an equilibrium value in 200 50 39 serves as an impedance transt0 ipO 2 former so that the signal pulses spite of the frequent exposure of the multiplier to humidc I -2-605 EU can be brought out through air and backstreaming pump vapors. f 470K IN726 long, low impedance leads. The usual background signal, or dark current, in the I Channeltron multiplier is 0.1 counts,/sec or less, so it is feasible to do experiments with beam signals as low as the assembly of multiplier plus amplifier on a single board 1 count/sec. To maintain a low background count rate it is not overly delicate. It is useful to know that pieces of is essential to keep uv radiation (as from bare filaments) a broken Channeltron can sometimes still.be used for and ions (as from vacuum gauge tubes) out of the multi- electron multiplication, albeit with lower gain, if new plier. Windows in the vacuum system need not be covered electrical connections for the dc voltage are made to the during the experiments, however, since the multiplier resistive glass tube with conducting paint. does not respond to visible light. We are grateful to John Carrico, Christopher Tosswill, The arrangement shown in Fig. 1 is a simple, effective Charles Hendee, Barry Kletke, and James Svenson of way to couple the output of the Channeltron to a remote the Bendix Research Laboratories for the loan of several pulse height analyzer. Channeltron multipliers and for suggestions concerning Sometimes the Channeltrons will develop erratic gain their use. We thank David Crosby for his contributions or a very high noise level. This can come from mistreat- to this research. ment such as overloading due to high counting rates or from causes which are less obvious. W\e have uWork suall fpporteund Wk oed in part by the National Aeronautics and Space Administration. it possible to restore a multiplier to good operating con- 1 M. Kaminsky, Atomic and Ionic Impact Phenomena on Metal dition by flushing it with acetone or ethyl alcohol and Surfaces (Academic Press Inc., New York, 1965), pp. 292 ff. 2 G. W. Goodrich and W. C. Wiley, Rev. Sci. Instrum. 33, 761 letting it rest on the shelf for a few days. In fact} one of (1962); D, S, Evans, ihid. 36, 375 (1965).

122 APPENDIX VI Reprinted from Physics Letters, Vol. 30A, No. 3, 145 (6 October 1969) EFFECT OF RECOIL ON THE VELOCITY DISTRIBUTION OF METASTABLE ATOMS PRODUCED BY ELECTRON IMPACT * J. C. PEARL, D. P. DONNELLY and J. C.'ZORN Randall Laboratory of Physics, Unziversity of Michigan, Ann Arbor. Michigan 48104, USA Received 3 September 1969 WVe have shown both experimentally and by a simple kinematic argument that the velocity distribution in a beam of metastable atoms produced by electron bombardment of ground state atoms deviates strongly from the velocity distribution of the ground state atoms. The velocity distribution of metastable atoms often enters explicitly into the analysis of resonance [1] and collision [2,3] experiments done with atomic beams. However, there have been few direct measurements and no correct theoretical description of the velocity distribution in a beam of metastable * Research supported in part by the National Aeronautics and Space A:dministration. 3- 0 776' 2- 1 a a /. 0==\* 8-s 70.4' 0 100 200 300 An00;U IME OF,IGHT (s~c) Fig. 1. Time-of-flight spectra for helium. Points represent experimental data. Curves a are theoretical accounting for the angular divergence of the incoming, ground state beam, assuming E-E * = 0.75 eV. Curve b is theoretical if recoil effects are neglected. 145

123 Volume 30A, number 3 PHYSICS LETTERS 6 October 1969 atoms formed in the common experimental arrangement where ground state atoms in a collimated beam are excited by a transverse beam of electrons. In our experiment a pulsed electron beam is directed at right angles (4) = 90~) through a collimated beam of thermal energy helium atoms. The metastable atoms are detected [41 with an electron multiplier (Bendix Channeltron) which can be rotated about the interaction region in the plane defined by the incoming electron and atom beams. Samples of data taken at two different angles 0 (angles measured from direction of electron beam) are plotted as points in fig. 1. Striking features of these results, when compared with what would be found if recoil effects were negligible (curve b in fig. 1) include the narrowness of the distributions, the presence of two peaks, and the sensitivity to deflection angle of the position of the distribution. The principal features of these distributions can be understood from kinematic considerations. If the gas beam effuses ideally from a thermal source (most probable speed a), if the ground state beam is perfectly collimated, if the detector is small, and if the electron scattering is s-wave, then conservation of momentum and energy determine the following velocity distribution for the metastable atoms in the plane of the incident beams v2 v2 exp(- v2+/a2) + v2- exp(- v2 y/a2) f(v) 2 s — - v (1) V [ r2 - {(m/M)uo sin 0 - v sin (4) - 0)}2]2 for Vm V < vmax where -max = ~Uo 0 <4)2 v =L Mu sin ~ V] cosec (- 0) V< (muo/M) sin ) (2) min and m v V OS U V2 (3) vO0 = v cos (~ - i) - Muo cos 0 i { - ["o sin D - v sin (~ - 0)]2}' (3) Here V = [2m(E- E *)]2/M; m and Mare the masses of the electron and atom respectively; E * is the energy of the metastable state; and uo and E are the speed and kinetic energy, respectively, of the incident electron. The distributionf(v), which is zero outside the limits given in eq. (2), is in disagreement with functions previously presented [7,8]. In the present experiment the detector subtends only a very small angle (A 9 008'), but the divergence of the incoming beam (A4) t 30) is significant. A numerical integration of eq. (2) over the beam divergence yields curves a of fig. 1. The present study of velocity distributions in beams of metastable atoms should facilitate the interpretation of experiments such as the fine and hyperfine structure measurements in 22S1 hydrogen [1,5], measurements of electron impact excitation of n = 2 states in helium [6], time-of-flighf studies of repulsive molecular states [7], and analysis of the kinetic properties of single and multi-component gases [8]. We are indebted to R A. Heppner, E. S. Fry and W. L. Williams for their assistance and encouragement. The importance of velocity distributions to the measurement of atomic fine structure was emphasized to us by R. T. Robiscoe References 1. W. E. Lamb Jr. and R. C. Retherford, Phys. Rev. 79 (1950) 549. 2. R. F. Stebbings, W. L. Fite, D. G. Hummer and R. T. Brackmann, Phys. Rev. 119 (1960) 1939. 3. B. Bederson, Atomic interactions, Part A, eds. B. Bederson and W. L. Fite (Academic Press, Inc., New York, 1968) pp. 89-95. 4. D. P. Donnelly, J. C. Pearl, R. A. Heppn,-r and J. C. Zorn, Rev. Sci. Inst. (to be pubished). See also pp. 85-87 of ref. 3. 5. J.W. Heberle, H. A. Reich and P. Kusch, Phys. Rev. 101 (1956) 612; R. T. Robiscoe, Phys. Rev. 138 (1965) A22 and Phys. Rev, 168 (1968) 4. 146

124 Volume 30A, number 3 PHYSICS LETTERS 6 October 1969 6. H. K. Holt and R. V. Krotkov, Phys. Rev. 144 (1966) 82; L. Vriens, T. F. M. Bonsen and J. A. Smit, Physica 40 (1968) 229. 7. M. Leventhal, R. T. Robiscoe and K. R. Lea, Phys. Rev. 158 (1967) 49; R. Freund and W. Klemperer, J. Chem. Phys. 47 (1967) 2897; R. Clampitt and A. S. Newton, J. Chem. Phys. 50 (1969) 1997. 8. J. B. French and J. W. Locke in Rarefied gas dynamics, ed. C. L. Brundin (Academic Press, Inc., New York, 1967) p. 1461; D. A. Crosby and J. C. Zorn, J. Vac. Sci. Tech. 6 (1969) 82. 147

APPENDIX VII SIMPLE SCALING RELATIONS FOR EVALUATING EFFECTS ON DISTRIBUT IONS OF DIFFERENT MASSES AND EXCITATION ENERGIES The relations for g(v) presented in Chapters 2 and 3 apply generally to atoms of arbitrary mass and to states of arbitrary excitation energy. Nearly all of the figures presented, however, have been for 23S helium. The effects of using different gases (or isotopes) have not been indicated; the effects of exciting different states have only been considered briefly (Section 6.2). An informative means for simultaneously assessing the influence these factors have on metastable velocity distributions is the following. Curve c of Figure 5.3 represents the time-of-flight spectrum for 23S helium if recoil effects are totally absent; the spectrum is characteristic only of the thermal distribution of velocities in the ground state gas. Scaling the time axis with the most probable velocity of the atoms in the reservoir therefore yields a curve which is representative of all gases. The same normalization, however, does not endow curves a with a similar universal character, since they are strongly dependent on recoil effects. These circumstances provide a simple method of comparing the degree to which 125

126 beams of metastables produced from various gases are distorted by recoil: relations between E*, E, O and M are determined which produce fixed "shifts" (represented, for example, by those of curves a) relative to the universal curve O. Curves a are characterized by their cutoffs. Let the cutoff velocities be normalized with respect to the most probable velocity in the ground state gas. For given shifts it is now possible to determine the angles and energy excesses which will produce the same relative shifts with respect to curve c for gases of arbitrary mass and excitation energy. Consider the incident beams and the detector to be coplanar, let i be fixed, and suppose that the source gases have the same temperature. Then, for a given spectrum of type a, i.e., for fixed v./at and v /d, the cutoff relations (Equation 3.5) can be inverted to give _E At %n'I:const. - (VII 1) and Si.n(~-)) =conSt.= 2 s'n Z (VII.2) where (vII.3) oA

127 and E is the energy of the incident electron. Equation VII.1 says that, for states of fixed excitation energy E a fixed energy excess AE will produce a spectrum with the same "shift" relative to curve c regardless of the mass of the atom or molecule. Equation VII.2, however, indicates that the angle at which such a spectrum is observed does depend on the mass of the excited particle. Comparing the helium cutoffs of curves a in Figure 5.3 with predictions for other gases and states yields the results shown in Table VII.I.

TABLE VII.I Deflection angles and energy excesses which lead to the same relative structure in the time-of-flight spectra as is indicated for helium by curves a and c in Figure 5.3. Atomic Excitation Deflection Angle Energy Mass Energy E) fae St~te Number E" (eV) ( (aO) slow AEF(eV) He(23S1) 4 19.82 12.40 19.60 0.75 H2(c 37u) 2 11.86 13.6O 21.50 0o45 D2(o37R) 4 11.86 9.60 15.00 0o45 N2(A3Lt) 28 6.16 2.60 4.10 0.23 N2(a 13) 28 8.54 3.10 4*80 0.32 N2( E3k) 28 11.87 3.60 5.60 0o45

ADDENDUM TOTAL INTENSITIES IN DIRECTIONS NOT CONTAINED IN THE PLANE OF THE INCIDENT BEAMS Normalized curves for total metastable intensities (integrals over the velocity distributions) which arise from a perfectly collimated Maxwellian ground state beam have been presented in Figure 3.5. These results, however, are calculated assuming that the detector is located in the plane of the incident beams; the "structure' in the curve for =AE.50 eV has been explained qualitatively in terms of the nonuniform distribution of metastables along the cylindrical portion of velocity space within which all velocity vectors of the metastables must terminate. It is of interest to Investigate the intensity distribution assuming that the detector is located in a plane which does not contain the incident beams, i.e., for values of $ O. Such results are presented in Figure AD.1 for the excitation of the 23S metastable state of helium by electrons with energy 50 eV above the excitation threshold (note that the curves are normalized to their peak values, as in Figure 3.5). Two features are worthy of note: 129

c-H me* >.= - 20~ _ ~e LI00 66 t:r 0 I I I c:,.HI-lelium 2S~00 iE- 50 eV 50 60 70 80 90 DETECTOR ANGLE e (DEGREES) OC Figure AD.1 Total metastable intensity for 23S helium as a function of detector angle O. The azimuthal angle q is the parameter; the electron energy is 69.8 eV. A perfectly collimated Maxwellian ground state beam is assumed. Electron scattering is assumed isotropic.

131 i) the curves for cqp#O terminate abruptly. These cutoffs occur when the line of sight from the interaction region to the detector is "tangent" to the cylinder which contains the velocity vectors of the metastables; ii) the "cutoff points" do not always occur at the peaks of the intensity curves. This is a result of the nonuniform population of metastables within the cylinder. A different presentation of the intensities is given for electron energies of 69.8, 25 and 21 eV in Figures AD.2, AD.3 and AD.4 respectively. Here the abscissa represents the azimuthal angle p of the detector, and the polar angle G is the parameter; each solid curve, therefore, corresponds to scanning the detector around a cone with its axis along the electron beam. The curves are symmetric about ps=0~. The ordinate scale differs from that of most figures in the body of the thesis by not including the factor i/V, which is prooortional to //-E-Ei (see the discussion of pages 28 and 29). As in the previous figure, each curve terminates abruptly when the line to the detector passes outside of the populated cylinder; the dashed curves represent the loci of these "cutoff points", and reflect the velocity distribution in the ground state beam. In fact, for a ground state beam with a Maxwellian velocity distribution, the plotted quantity (essentially VI ()O,,~ 1j)of page 28 and Section III.3 of

132 X10o2 25 (P > He LLJ Q II 8 5 P >- 15- \ ilelium 23S, V| \ E= 69.8eV =I- / 8 \ S-wave electron scattering co 900 LU Ca 0 o 70 0 10 20 30 40 AZIMUTHAL ANGLE cp (DEGREES) Figure AD.2 Total metastable intensity for 23S helium as a function of the azimuthal angle p of the detector; the polar angle 0 is the parameter. The energy of the incident electrons is 69.8 eV. A perfectly collimated Maxwellian ground state beam is assumed; a-wave scattering of the electrons is also assumed. The dashed curve represents the locus of intensities at the cutoffs (see text).

133 x102 25- ctp C He ~~~~I e >- 15 770 Helium 23S, |/ /\I- I E= 25.0eV Z | / 82~ \ S- wave elecrron c.mm,, 0,, scarrering t75~ CV) c ___ G 5 8 650 0 5 10 15 20 s AZIMUTHAL ANGLE cp (DEGREES) Figure AD.3 Total metastable intensity for 23S helium as a function of the azimuthal angle p of the detector; the polar angle 0 is the parameter. The energy of the incident electrons is 25.0 eV. A perfectly collimated Maxwellian ground state beam is assumed; s-wave scattering of the eleotrons is also assumed, The dashed curve represents the locus of intensities at the cutoffs (see text).

134 X182 25 cp _c\J~~~~~~e He > ] He:f>H 0 79~ LU t~L, | 77~7 e 15F 1 o' Helium 23S, - |J81~ \ E = 21.0eV,75~ S-wave electron scattering 10 \ 90. \ LIU I 700 a 5 I -N 83;~ _i -,65 7- C4~ -84 u/ III I 0 2 4 6 8 10 E AZIMUTHAL ANGLE cp (DEGREES) Figure AD.4 Total metastable intensity for 23S helium as a function of the azimuthal angle q of the detector; the polar angle O is the parameter. The energy of the incident electrons is 21.0 eV. A perfectly collimated Maxwellian ground state beam is assumed; s-wave scattering of the electrons is also assumed. The dashed curve represents the locus of intensities at the cutoffs (see text).

Appendix III) takes the following form NTo,2,ca'- — v-1 eI a (AD.1) where the quantities vo and v are determined from Equations III.10 and III.15 respectively, using Equation III.16. The above three figures show that, under the assumption of isotropic electron scattering, the total intensity may vary strongly with the azimuthal angle I; for large energies, the intensity may be substantially greater for values of ~q0 than it is in the plane of the incident beams (compare, for example the intensity at p=l10O (0840) with that at A= 0~ in Figure AD.2). The strong effect which nonisotrooic electron scattering can have on the metastable intensities is seen by comparing Figures AD.4 and AD.5. The results shown in these figures are calculated for electrons of energy 21.0 eV, corresponding to the energy of the d-wave resonance in the excitation function for the helium 293 state. While the intensities evaluated under the assumption of isotropic electron scattering (Figure AD.4) depend only weakly on A, those made for d-wave scattering (Figure AD.5) show large peaks and valleys. To summarize, it is seen from the results shown in the total metastable intensity which occur in the plane of the

136 X102 79,. 25P\ -, 5 0 750 9 le He "', 20 820 e *w 8 = 790 l*~u.> 15 / \ AHelium 23 S 15 =2.eV 10 - 2 4 6 D-wave electrons is 21.0 V A perfectly collimated Maxwellan of intensities at the cutoffs (see text),U 5 700 o 0 2 4 6 8 10 AZIMUTHIAL ANGLE ~ (DEGREES) Figure AD.5 Total metastable intensity for 23S helium as a function of the azimuthal angle o of the detector; the of intensities at the cutoffs (see text).

137 incident beams (of. Figure 3.5 and the intensity variation with O at P= 00 in Figure AD.5) are generally not representative of the variations which occur for 9( O0 Because of the concentration of metastables toward the "outside" of the populated cylinder in velocity space, the out-of-plane intensities can vary strongly even if the electron scattering is isotropic (Figures AD.2, AD.3 and AD.4). In addition, the inclusion of structure in the scattering cross section is seen to have a strong effect as well (Figure AD.5). 5/5/70

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139 Holt, 1966 H. K. Holt and R. V. Krotkov, Phys. Rev. 144, 82 (1966) Lamb, 1950 W. E. Lamb, Jr. and R. C. Retherford, Phys. Rev., Y., 549 (1950) Leventhal, 1967 M. Leventhal, R. T. Robiscoe and K. R. Lea, Phys. Rev., 1_8- 49 (1967) Pearl, 1969 J. C. Pearl, D. P. Donnelly and J. C. Zorn, Physics Letters,.OA, 145 (1969) Pichlik, 1970 R. Pichlik and J. C. Zorn, Bull. Am. Phys. Soc., 15, 617 (1970); J. Vac. Sci. Tech. (in press) Rebane, 1970 T. Rebane, Ph.D. Dissertation, University of Michigan (1970) Robiscoe, 1965 R. T. Robiscoe, Phys. Rev., 138, A22 (1965) Robiscoe, 1968 R. T. Robiscoe, Phys. Rev., 168, 4 (1968) Robiscoe, 1970 R. T. Robiscoe and T. W. Shyn, Phys. Rev. Letters, 24, 559 (1970) Rubin, 1969 K. Rubin, B. Bederson, M. Goldstein and R. E. Collins, Phys. Rev., 182, 201 (1969) Shyn, 1969 T. W. Shyn, W. L. Williams, R. T. Robiscoe and T. Rebane, Phys. Rev. Letters, 22, 1273 (1969) Stebbings, 1960 R. F. Stebbings, W. L. Fite, D. G. Hummer and R. T. Brackmann, Phys. Rev., 119, 1939 (1960) Toennies, 1967 J. P. Toennies, Recent Advances in Aerothermo-Chemistry, voly. 1 (AGARD Conference Proceeding, No. 12), I. Glassman, ed. (Technical Editing and Reproduction Ltd., London, 1967) Vriens, 1968 L. Vriens, Te F. NM Bonsen and J. A. Smit, Physica, 40, 229 (1968)

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UNIVERSITY OF MICHIGAN 3 9015 03022 6909Il llll II 3 9015 03022 6909