THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Physics Technical Report No. 3 AN EXPERIMENTAL DETERMINATION OF THE SPIN EXCHANGE CROSS SECTION OF K39 AND Cs Thomas'E Stark ORA Project 04941 under contract with U. S. ATOMIC ENERGY COMMISSION CHICAGO OPERATIONS OFFICE CONTRACT NO. A.T (11-1)-1112 ARGONNE, ILLINOIS administered through: OFFICE OF RESEA.RCH ADMINISTRATION ANN ARBOR April 1966

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1966.

ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor Richard H. Sands, who suggested this problem, for his patience, guidance and understanding during the course of this work. The interest and help of the other members of my committee are gratefully acknowledged. My thanks, also, to Mr. G. Kessler and Mr. H. Roemer of the Physics Department Shop for their skill and cooperation in the construction of numerous pieces of equipment. I wish to thank the members of the Magnetic Resonance Group for the sustenance that their lively conversations gave me. The partial support of this research by the United States Atomic Energy Commission under Contract No. AT(ll-1)-1112 is gratefully acknowledged. This thesis is dedicated to my wife, Betty, whose patience and devotion assured the success of this investigation. ii

TABLE OF CONTENTS Page LIST OF TABLES iv LIST OF FIGURES v ABSTRACT vi CHAPTER I INTRODUCTION 1 II. THEORY 5 A. Dynamical Basis for Spin Exchange 5 B. Equation of Motion for the Density Matrix 12 C, Solution to the Equation of Motion 19 III. EXPERIMENTAL PROCEDURE AND RESULTS 30 Ao Description of Apparatus 30 B. Experimental Procedure 35 C. Results of the Measurements 41 D. Errors 49 APPENDIX. ADDITIONAL COMMENTS ON THE DENSITY MATRIX TREATMENT 54 A. Off-Diagonal Matrix Elements 54 B. Intermediate Fields 58 REFERENCES 66 iii

LIST OF TABLES Table Page I TABLE OF MATRIX ELEMENTS 38 II. EXPERIMENTAL RESULTS 49 IIIo SOURCES OF ERROR IN TfHE DETERMINATION OF THE EXCHANGE CROSS SECTIONS 50 IVo CORRECTIONS TO THE CESIUM EXCHANGE CROSS SECTION 62 Vo LINE WIDTHS FOR VARIOUS TRANSITIONS IN CESIUM 64 iv

LIST OF FIGURES Figure Page 1. Molecular potentials in the Heit:ler-London approximation, 8 2. Ratio of the effective spin-lattice relaxation time, T., to the effective spin-spin relaxation time, T2, versus the ratio of the "true" spin-lattice relaxation time, Ti, to the relaxation time for spin-exchange collisions, Txo 27 35 An equivalent circuit for Eq. (II534). 28 4. Block diagram of the EPR spectrometer. 31 5. Typical EPR spectrum of an alkali metal vapor (K39 shown). 33 60 Microwave cavity and sample tubes,. 34 7' Relaxation measurements on potassium: T1/T2 versus l/r2 44 8. Relaxation measurements on potassium: Tl/Tx versus 1/T2. 46 9. Experimental results: Peak to peak line width versus alkali density. 48 v

ABSTRACT An important parameter in many optical pumping experiments is the cross section for spin-exchange collisions. These cross sections measure the probability for interchange of spin coordinates by two electrons in the course of a collision. The accurate determination of these cross sections by optical pumping technique is hampered by inexact knowledge of the atom density in an optical pumping cell. This thesis reports a value for the exchange cross section for collisions between potassium atoms (K39). The measurements employ the standard techniques of electron paramagnetic resonance (EPR). A microwave absorption spectrum is obtained by observing magnetic dipole transitions between various magnetic sublevels of the ground state of a potassium atom, split by the hyperfine and Zeeman interactions. The spin-exchange collision rate is found by measuring the line widths of isolated transitions and the alkali density determined from an absolute calibration of the microwave spectrometer. For the latter measurement, small weighed-out crystals of copper sulphate, CuS04'5H20, are used as standards of intensity. The role of spin-exchange collisions as a broadening mechanism for magnetic dipole transitions is examined by means of the density matrix formalism. A suitable equation of motion for the density matrix is established and solutions appropriate to the Paschen-Back region of the Zeeman interaction are obtained. Expressions are obtained for the imaginary part of the complex susceptibility, which is related to the microwave power absorbed by the potassium atoms, and for the contributions of spin-lattice and spin-spin relaxation to the observed line width. 59 The spin-exchange cross sections for collisions between K39 atoms is 1.45 x 1014 cm2, with an assigned probable error of +20%. Also reported in this thesis is a value for the exchange cross section for cesium (Cs133). In the case of cesium the hyperfine coupling is sufficiently strong that the Paschen-Back region is not reached for magnetic fields corresponding to the use of X-band microwaves. This is the so-called intermediate field case and results in appreciable mixing of the electron spin states by the hyperfine interaction. Certain corrections in the density matrix treatment follow thereby and are discussed in an appendix. The measured value of the exchange cross section for Cs is 2.2 x 10-14 cm2, with an assigned probable error of +20%o vi

CHAPTER I INTRODUCTION Certain features of the scattering of one-electron atoms were studied in the precise experiments of Wittke and Dicke on the hyperfine structure splitting of atomic hydrogen. Following a suggestion by Purcell, these authors showed that the broadening effects of electron spin exchange collisions placed an effective lower limit on the line width of hyperfine transitions observed in their experiment. At about 2,28 the same time Purcell and Field examined the question of spin exchange broadening for possible astrophysical importance. A classical but surprisingly accurate model for-the exchange cross section was also provided. Later, the effectiveness of spin exchange collisions in transferring spin polarization was exploited by Dehmelt,/ who succeeded in polarizing free electrons via exchange collisions with optically pumped sodium vapor. Subsequent to this work, other investigators were able to extend the op4 tical pumping technique to a variety of atomic species for which sources of resonance radiation were not conveniently available. A partial listing of these efforts is given in the references. Additional interest in atomic collision processes accrued with the development of gaseous frequency standards and maser devices. Here again, exchange collisions were shown to have small but easily measurable 1

2 effects, not only broadening the transitions but also introducing secular perturbations which shift the center of the spectral line. The relationship between exchange collisions and frequency shifts 6 seems to have been first pointed out by Bender. More recently, exchange collisions harve been found to play a significant role in the operation of the hydrogen maser.7 Frequency shifts attributable to exchange collisions have also been measured in the optically pumped 8 rubidium-electron system. Attempts to characterize spin-exchange collisions by means of an effective cross section harve been hampered by inexact knowledge of the atom density.* In the case of alkali-alkali cross sections determined by optical pumping methods, early experimenters relied on vapor pres9 sure data, determining the density of the alkali metal vapor by measuring the temperature of their optical pumping cells. The resultant 10 cross sections were in error by as much as a factor of three. The first accurate measurements of spin exchange cross sections 11 12 were reported by Jarrett and Moos in their thesis work at The University of Michigan. Jarrett combined standard optical pumping techniques with a scanning Fabry-Perot interferometer to obtain spectral profiles of the light transmitted by an alkali vapor cell. From these measurements the integrated absorption coefficient was determined. *In an optical pumping experiment, as in the experiments reported here, the exchange cross section Qx cannot be measured directly. What is measured is the effective rate of exchange, related to Qx by the formula I/TX = NQxS where N is the atom density and nv a relativ'e velocity.

3 Knowledge of the lifetime of an alkali atom in the excited ( P) state allowed the density to be calculated. The relation between optical pumping signal strengths and the effective exchange collision rate was found by solving the usual rate equations. 9 The measurements were 85 87 carried out on a mixture of Rb5 and Rb, present in their natural abundance ratio. 13 The approach developed by Moos and Sands forms the basis for the experiments reported in the present work, and we shall on numerous 12 occasions refer to Moos' thesis for details. Briefly, the experimental method is that of electron paramagnetic resonance. An absorption spectrum is obtained by observing magnetic dipole transitions between various magnetic sublevels of the ground state of an alkali atom, split by the hyperfine and Zeeman interactions. The spin exchange collision rate is found by measuring the line widths of isolated transitions and the alkali density determined from an absolute calibration of the microwave spectrometer. This method differs in several important respects from the usual optical pumping schemes: 1. The experiments are conducted at much higher temperatures, and correspondingly higher densities. However, the alkali-alkali exchange cross sections have been shown to have a rather weak temperature dependence so that a meaningful comparison between experiments can be made. 2. A source of "pumping" radiation is not required (nor would it serve any purpose at the relatively high densities used). Consistent with the higher operating temperatures the microwave resonance technique is sufficiently sensitive to detect transitions between states whose populations differ only by the

4 Boltzmann factor. Under these circumstances the relationship between signal strength and exchange rate is relatively simpler than in the case of optical pumping. 3. The measurements are made without the use of a buffer gas and in relatively strong (o3000 gauss) magnetic fields. Individual transitions are well resolved and no simplifying assumptions need be made as to the degree of excited state mixing caused by gaseous diluents (6 ). 4. Finally, convenient intensity standards can be found which serve to calibrate the spectrometer absolutely in terms of atoms per unit volume. In the present experiment small weighed-out crystals of copper sulphate, CuSO,.5H20, were used for this purpose. The remainder of this thesis is ordered as follows. Chapter II is devoted to a theoretical understanding of the spin exchange process. By means of a density matrix formalism the role of exchange collisions in determining spectral line-widths and effective relaxation rates is examined. The question of intermediate field coupling is treated in the Appendix. In Chapter III the experimental apparatus is discussed briefly and the results of the measurements are described.

CHAPTER II THEORY A. DYNAMICAL BASIS FOR SPIN EXCHANGE The dynamical basis for spin exchange between one electron atoms may be conveniently discussed from the point of view of the Heitler-London theory of homonuclear diatomic molecules. As a preface to the density matrix calculations which appear in section B of this chapter, and in order to introduce the cross section for spin exchange, the nature of the scattering of alkali metal atoms is discussed. At large separation the alkali atoms are described by appropriate atomic orbitals UA(1) and UB(2), respectively. Here 1 and 2 are labels for the electrons and UA for example, is a hydrogen like wave function centered on nucleus A. As the two atoms are slowly brought together an interaction Hi2 is turned on. So long as the interaction is symmetric in the electron coordinates (i.e., H2-=H21) any linear combination of products UA(1) UB(2) and UA(2) UB(1) can be used to describe the system, the only requirement being that the total wave function at most change sign upon interchange of electrons. Thus, th order wave functions for this problem are d f7r) tW ) + (W) ) and (II.1) 5

6 S A Here, N and N are normalization factors which take into account deviations from orthogonality. They are given by (II. 2) where L = JUA(l)UB(l)UA(2)UB(2)dTldT2 measures the overlap in the electron clouds. The energy of interaction as a function of internuclear St ordeA th separation is computed in 1s order using S and 4 as 0 order wave functions and treating H12 as a perturbation. The result is where K = UA.(l)UB(2)Hl2UA(l)UB(2)dTldT2 and J = fUA(l)UB(2)Hl2UA(2)UB(l) dTldT2 are the Coulomb and exchange integrals, respectively. As yet no mention has been made of the electron spin, and indeed since the interaction is wholly electrostatic the spin plays no direct role. However, the Pauli Exclusion Principle requires that the total wave function, including spin, be antisymmetric in the electron coordinates. Accordingly, the molecular state associated with VS(R) is (anti-parallel spins) and with VA(R), Z (parallel spins). Calculations of the sort described above have been carried out by 14 Rosen and Ikehara for hydrogen and the alkali metals. Hydrogenic atomic orbitals with effective n,Z were used and the integrals J,K,L given in graphical form. The interaction Hamiltonian was taken to be

7 e e X -Z t- e _ e _ e_ (TI.4) The qualitative form of the interaction potentials is shown in Fig. 1, showing the typical repulsive character of the 3 state and the tendency of the Z state to form stable molecules. From their numerical work, Rosen and Ikehara computed the molecular dissociation energy and equilibrium internuclear separation. Comparison with experimental values was judged to be excellent, tending to support the claim that the main features of the potentials were given by theory, at least for small separation of the atoms. The significance of the potentials VS and VA for the scattering of one electron atoms lies in the very different boundary conditions they impose on the scattering process. Depending on the orientation of the electron spins, the atoms will scatter from one or the other, or both., of these potentials. From the point of view of the Faxen-Holtsmark theory, different scattering cross sections result from the different phase shifts introduced into the scattered wave by these potentials. To see how the possibility of spin exchange collisions arises, consider two well-separated alkali atoms and assume that the (outer) electrons be found with equal probability in any one of the spin.states () i (Z()) tcld it

VA:. 3 0 w z z INTERNUCLEAR -- /-SEPERATION 0 Fig. 1. Molecular potentials in the Heitler-London approximation. Fig. 1. Molecular potentials in the Heitler-London approximation.

9 where ac and 3 are the usual spin-up, spin-down functions. If U(1) and U(2) are spatial wave functions for these electrons, a properly (anti) symmetrized wave function describing the pair of atoms at infinite separation (or time t -oo) is,t~ C i I lA (1t)Z)C (J)- 2 l(2) a (1)( (21 (In.6) S A In terms of the functions 4~ and f of (II.1), and the spin functions S (i/ e ma rewrte 6 the orm we may rewrite (IL6) in the form -i _ l 0i- +r^ C! ~ (IIo7) S A Due to the different potential functions V and V, the spatial S A parts, 4 and 4, evolve differently. in time as the two atoms approach and recede from one another. Depending on the values of the angular momentum 2 and relative kinetic energytk2= 2gE, where p. is the reduced mass, the phase shifts introduced into the scattered wave function are given by +0o -_o0 and'o (11.8)'r(\A-i I \V(RPdt respectively. Thus, after the collision and when the atoms are again at infinite separation (or time t = +g), the wave function for the pair can be written

10 g\2tc1- r^ D( 2< i X (119)A Considering only relati-re phase changes, (IIo9) may be writte n i the form of Eqo (lIIo6) with'the result _ e (il - 1 2j) + ~ (Z (ITI.C) where C.-|(<5/ r) - L Mt These results may be succinctly sunmarized by introducing the Dirac Pauli matrices. Then^ t P1j i() and( o Q ~ e I) ~ e (e -)& &oA> texchangtercha=e of electron spin coordinates. The peroatir P2 tuh'i)n Pauli mat~rices~ Th~er?'~ = Pi2 ( and,F~l~iieP12

11 The results obtained so far pertain only to the scattering of the gth partial wave for two atoms of relative energy E = ~ k2/2j1. The quantities and (II.14) may be interpreted as the "direct" and "exchange" partial scattering amplitudes, respectively. In terms of them, Eq. (II.10) and (II.13) may be written 4-Y ( ) + L2 A4 (II.lOa) jI^ i = e+ X q~ i4 (IIo13.a) In the Faren-Holtsmark theory the scattering amplitudes of the 17 various partial waves are linearly superposed yielding 4(g) e _( 2 -4( t, O) F,7(o )= T =(2Q+I C?(Lt,@ ) (1.15) Thus, an immediate generalization of (Io.13) is where M(k) = Fd(~) + Fx(Q) vlo-2. The total cross section for spin exchange is then Q% = ^^ \I iY F.6jL d(om6) (1.1I6) -e= o

12 and comparison with the usual expression for this cross section17 shows that o where b7 and b5 are the quantum mechanical phase shifts for scattering from the spherically symmetric triplet (V ) and singlet (VS) potentials, respectivelyo For the purposes of this paper, the operator MI(k) is more convenient and will be used in subsequent calculations. Bo EQUATION OF MOTION FOR THE DENSITY MATRIX A detailed examination of the role of spin exchange collisions in determining the lifetime of an electron in a given spin state followso To facilitate the interpretation of the experimental results a model is required on which to base the observations. The treatment of spin exchange given here follows the discussion of Witt'e and Dicke on this same problem. The objective of the calculation is to establish a relationship between observed spectral line width and the cross section for spin exchange. Since the details of the motion of the atomic spins are of primary interest, rather than the gross deflections of the atoms themselves, it will be sufficient to regard the atoms as being in definite but unspecified states of relative momentum before and after the collision. Attention may be confined to the atomic spins, taking into account their 18 statistical character by means of the density matrix formalism. Furthermore, in so far as it is improbable that successive collisions

15 would involve the same pair of atoms, the atoms may be regarded as statistically independent and hence uncorrelated. Consider, then, two alkali atoms, chosen at random from an assembly of N such atoms, and presumed about to collide. The density matrices used to describe the incoming atoms are obtained as followso Let L1 and *2 be the (total) spin functions for atoms 1 and 2, respectively. These can be expanded in terms of spin functions li>, which are used to label the rows and columns of the density matrix. Thus'- \,) {L OQ' 1L> and the corresponding density matrices are given by With the assumption that the atoms are not correlated by previous collisions, the wave function for the combined system prior to a collision is a,,=,~i, i a~c a _, lLJ so that the density matrix for the pair is (11.II7) In matrix notation p12 = Pl(p2, where the symbol) denotes the direct, or outer product of the matrices pl and p2~

L4 According to section A of this chapter, the wave function for this pair of atoms after a collision is given by Therefore, after the collision the pair of atoms is described by the density matrix P(9 _'^ O p \ Mtl) (IIo18) c Finally, the density matrix for one of the atoms iS recovered by taking the trace over p12 with respect to all quantities bearing the indices of the other atom. Thus L Tf = 17- -l(Vl^~\Qi p -=w I R 1 = rt tA pIx R With some labor the operator M (Eq. 115) may be substituted in these equations with the result that ).j - (#-3Z ) (f e Q 24,, /J> 4(. ( (,- 33) ~|.<>T L \~7 (I. 19) using the fact that TR(l)pl = TR(2)p2 = and allowingiin the last term, lsy to form a cyclic permutation of X)YZ~ For simplicity, the labels on cPe(k) have been dropped.

15 As an example of the use of Eq. (II.19) consider a single scattering event between two "nonequivalent" spin-1/2 particles. For the states li> take the usual spin up, spin down functions a and p. Suppose that particle 1 has been prepared, in some way, so that it is entirely in the spin up state c. The density matrix for this particle is then I 0D 94' o) On the other hand, suppose that particle 2 is thermal; that is, there is a Boltzmann distribution among the states Q, Bo Then 0 o 0I O with Dll + P22 = 1. The polarization of these particles is given by Z = TR(aip) so that Z1 = 1 and Z = Z To find the density matrices after a single collision apply Eq. (II.19)o The result is, for the polarizations, tantl conlusion to be drawnro this eaple is tat altou An important conclusion to be drawn from, this example is that although exchange collisions rearrange the populations of the spin states and C c hence alter the polarization, the total polarization Z1 + Z2 = ZL + Z2o That is, exchange collisions act as a sort of relaxation mechanism for the state populations, but not in the sense in which relaxation is generally understood: spin exchange collisions do not dissipate the total (spin) angular momentum of the system~

16 So far only the effects of a single exchange collision have been considered as a possible mechanism of change for the density matrix. To find the average effect of many such collisions, acting in competition with other types of collisions and with externally applied driving forces. it is necessary to find, and solve, an appropriate equation of motion for the density matrix, Such an equation of motion will be ob19 tained by following the procedure of Karplus and Schwinger. Assuming that the atomic gas is sufficiently rare, and the collision cross sections sufficiently small, collisions occur with a mean frequency 1/T7 where rcI the mean time between collisions, is long compared to the time tc of a collision. Let p be the density matrix for a representation atom in the gas. Except for a small time interval te the density matrix is free to evolvre in time according to the usual equation of motion. ^^t ^l^^i (11.-20) where' is the spin Hamiltonian including externally applied fields if any. Furthermore, if tc is short compared with the characteristic periods of the Hamiltonian, changes in p due to H during the collision itself may be ignored. Thus a single collision has the effect of providing an initial condition to the solution of Eq. (120). If to is the time of the last collision the density matrix will at all later times depend at least parametrically on too Formally, the solution can be written as p(t,to), where p(toto)Epc is the initial value of the

17 density matrix, The form of Pc defines the type of collision that has occurred. To find the effect of many collisions an average over the distribution of collision times to is taken: t ()' = l (tit tto to (II.21) 1-a* where P(to)dto is the probability that the last collision occurred in the interval to to to + dto. For collisions which occur randomly at a mean rate 1/Tc) -^to ~~dto- = e / 2 (11.22) To find the equation of motion satisfied by p(t) we differentiate Eq. (II.21), using Leibnitz rule, and Eq. (IIo20), to find X =- 9 l^ )]- at(P-rL) (II.23) In the case of spin exchange collisions pc is given by Eq. (IIl19) and 1/Tc = 2NQcvo N is the number of atoms per unit volume, v the average k2 speed. Qc will be taken to be j/k (2e+1), the maximum partial cross c section associated with the turning on of the exchange operator f. O~1p Seemingly little justification for this choice can be made, except that it be noted that the "direct" amplitude f. does not con27 tribute to scattering in which the spin state changes. In this work only those states are observed in which such a spin change occurs. However, we cannot set f =0, for it will be seen later that the inter

18 ference of the "direct" and "exchange" amplitudes leads to a frequency shift. 8 Since the operator Ml admits of the possibility that in a given collision no exchange occurs, Q, cannot be identified with the exchange cross section. Spin exchange does not provide a mechanism for restoring thermal equilibrium to a spin system, Energy added to the spin system by external fields cannot reappear at the walls of the container but rather is distributed by collisions in such a way as to equalize the populations of all the levels. In order to allow for relaxation we must specifically include relaxation collisions in the equation of motion. This may be done by repeating the averaging procedure, Relaxation collisions are presumed to occur at a mean rate 1/T1, with the initial condition on the density matrix being p(toto) = po p is the density matrix describing thermal equilibrium among the spin states at a (kinetic) temperature To The result is _f ^ D -l- (pi- (II.24) With the Hamiltonian described below, (II.24) is the equation of motion which will be solved. It should be noted that this equation of motion is only correct where no correlation exists between the process of relaxation and that of spin exchange. No correlation is expected here.

19 C, SOLUTION TO THE EQUATION OF MOTION In the presence of a static magnetic field Ho along the Z-axis and a microwave frequency magnetic field 2Hlcosnt along the X-axis, the Hamiltonian for the alkali atoms is 7 ZUbdo ^ ^t76Z, 2 x 7' j,)) (II.25) where,o is the Bohr magneton and A measures the strength of the Fermi contact interaction (hyperfine interaction). For our problem the terms proportional to the nuclear g-value are of little interest, since gI " 1/2000 gso Also, 2H1 << Ho so that Hi(t) may be treated as a perturbation The eigenvalues ofo, for S = 1/2 and arbitrary field strengths 20 H0, are given by the Breit-Rabi formula - l |- z-x-x 7 -(- ( II26) -A1It-+ — z++~ ~ X ]. -H - 4-1 4.+ x A \ M is the projection of the total angular momentum on the Z-axis, X = 2gSJoHo/A.(2I+l), and F = I+So The corresponding eigenfunctions (normalized) are easily found to be

20 r = it-I-4s-~> = II,- > = IT-(L ~'-r (IIo27a) for the matrix b g of dime n 2(2I+ i representation p is a diagonal matrix~ the diagonal matrix elements 4 —! beg -ul te c puai of1 the,orresponding energ where = X 21+1 I (II 27a) Xr\ - d~ Z TL~ 4(Tcr~t\+ +Lt)7 The energy eigenfunctionls are used to construct the density matrix p for an alkali atom, the matrix being of dimension 2(2I+1). In this represestation p is a diagonal matrix, the diagonal matrix elements being equal to the fractional population of the corresponding energy states. A case of particular interest is one for which the microwave frequency is close to the resonant frequency for transitions between a pair of states'The density matrix will then havre off-diagonal matrlix elements corresponrding to the excitation of these levels by the microwave fieldo Although the spin exchange term in Eq. (II.24) couples this pair of leTvels to the remaining levels in the atom, other off-diagonal matrix elements will be nege neg ble so long as the various transitions are well-resolved (see Appendix)

21 The solution of the equation of motion will be examined when the static field is large, so that the Zeeman interaction dominates the hyperfine splitting. For large values of the parameter X the eigenfunctions become simple products of the nuclear and electronic spin functions and the energy levels become linear in the applied field HoO The selection rules for magnetic dipole transitions are AMS = +1, AM = 0. The microwave frequency is assumed near the resonant frequency for transitions between the states II,+ 1/2> and II,- 1/2>o The corresponding off-diagonal. matrix elements in p will be denoted by P12 and P21 P12o MS + The diagonal elements will, be denoted by p PM (-I < M < I). From Eqo (IIo19) the matrix elements of pc are X - t 4)-4z p( (p where Z - S Pm Z p; the matrix elements of LHp] are then M M m m,4A )_ a 2i4. iYLs ((VA -yZ Q <I,-r\It\P1II\-'^ =-1ZiaS 0^'l'ttX88(0~p1 <I>+ 2iPR4T >- 2tt1h +pE) ZIILyp]\)4^^ 2 ^ "^ - ZÆE #MwC P)- p(1/P -I)0

22 Let EI EI == ^o? - 2<il|lJx>/Ti gSio/ i and yHl = l; then a) - ( b) - = LtC>, (pl-pl2.) - ^ (l+38 7 pr - o-d) pap. - ^o^~i +L7.t(fl r) - 4 T S — D,?tO-( P ~-'! 2 " e) (^ CD' (4 -Al ZO^ ^ ^ where, in Eqso e) and f) mrI.o A.lso, Z is the total polarization of the gas and Zm = p1 - pm is the partial polarization associated with m *m the pair of states |M, + 1/2> and |M, 1/2>o From Eqso e) and f),A4d A-. I_ - p, — ^^|^- p,;, 1: - p _L+ = go 0 0 1 el"

23 According to (II.16) the spin exchange partial cross section is the mean rate for exchange collisions is then _ 7Q t1 b= -? (1I.30) Substituting into (IIo29) and summiing over all M#I: Mnt iI Tx $-^ vi {,?^, r c(II.31) \ML I <-^* ^\ > Finally, we substitute for Z = ZI + ) Zmo In the high temperature m#I approximation (kT~>>TiH) the equilibrium distribution of population is nearly uniform over the spin states; thus, Nm _ 1/2I+1 and Z _ (2I+1) ZI. The steady state solutions to (II.31) are iM\ *PJL Using these results the remaining Eqs. a)-d) can be combined to give.~=. + ~p ~-(Z A) -' (II.33) V1 - t(8z- fz.) = + ^ +^ ^1LkI - 02L? I

24 where _ - T 2 i+ Tj (11.54) = TI + z,__L T Equations (I.533) may be compared with the well-known Bloch equations of nuclear magnetic resonance theory. There, as here, T2 is the spin-spin relaxation time responsible for the spectral line width; T1 is the spin-lattice relaxation time governing the return of a disturbed spin system to thermal equilibrium, The quantity A in these equations represents a frequency shift in the line center; A is given by ~^*~r ((II3.5) X-Ip Y and ZI are, except for multiplicative factors, the components of magnetization of the sample. The steady state solution to Eqs. (Iio33) may be obtained by transforming to a coordinate system which rotates with the frequency of the applied rf magnetic field and in the sense of the Larmor precession of the spinso In doing this certain high frequency nonresonantterms which oscillate like cos2o)t appear in the equations. These terms are due to the counter rotating component of the rf magdnetic. field, which has been taken to be linearly polarized, and give rise to the Bloch-Siegert

25 21 shift of the line center. This effect will be ignored. The solutions obtained in this way are part of the standard 22 literature on magnetic resonance. In the rotating coordinate system the components of magnetization in phase and out of phase with respect to the microwave field are denoted by u and v, respectively. It is convenient to express the solution in terms of the components of the complex rf susceptibility Xrf = X-ix", where,r~X" + c 22 One finds ( II.6) where Xo is the static susceptibility and T1 and T2 are given by (II3.54). The average rate P at which power is absorbed per unit volume by the gas from the microwave field is ^ - XIzTr t Z"Xil 2 4(IIo37) Although the equations have been derived for a specific pair of spin states, all the results do in fact apply to any other pair of states of the form IM,+1/2>. In particular the frequency shift A is always in the same direction, determined only by Z- 0(1/2I+l)Z and sin~(&-5e) Since Z < 0, measuring the sense of the frequency shift

26 will also measure the relative values of 5 and 5. Unfortunately the very small (thermal) polarization precludes such a measurement in the -12 present case (A/coo 10 )1 Of considerable importance here are the conclusions embodied in Eqso (IIo34), (II.36), and (II3.7). As a function of microwave frequency Eq. (Ii.37) predicts a Lorentzian line shape with half-width at half power points equal to 1/72, so long as y2H T <<1. (The latter quantity is called the saturation factor and can generally be made small by suitable reduction of the microwave power). Through Eq. (II.34) it is clear that both exchange and relaxation contribute to the observed line width. The effective spin lattice relaxation time can be found 22 from a measurement of the saturation factor Y H1T1T2 and the observed line width then corrected, The effectiveness of spin exchange in coupling a given pair of levels to the remaining levels of the system is illustrated in Fig. 2, where T1/T29 the measured variables, are plotted versus Tl/Tx, the variables of physical interest. Eq. (II.34) may also be interpreted through the electrical network analogue shown in Fig. 53 It should be noted that broadening effects due to unavoidable field inhomogeneities are not included in Eq. (II,34). The effect of an inhomogeneous dc field (Ho) may be included by defining I ^ At? (I.358)

27 50 40 I 7/2 5/2 30 -- 3/2 O / 1/2 20 I0 9 x7 4 2I 2 3 49 1 ~ 2 3 4 5 6 7 8 9 10 T. /TX Fig. 2. Ratio of the effective spin-lattice relaxation time, T1, to the effective spin-spin relaxation time, T2, versus the ratio of the "true" spin-lattice relaxation time, T1, to the relaxation time for spin-exchange collisions, Tx.

21 REMAINING PAIRS OF LEVELS LEVELS OF INTEREST (IPAIR) 22.I + MICROWAVE ok Spin System" POWER SOURCE T1 "Spin System" 4. ~T,? ~T(li g t"Latt ic Fig.. An equivalentt ice ( Fig. 3. An equivalent circuit for Eq. (II.34)..

29 Here l/T2 is the effective spin-spin relaxation rate due to the inhomogeneous field, l/T2 is given by Eqo (II.34) and 1/T2 is the measured relaxation rate. In a similar way, several distinct mechanisms may contribute to the spin-lattice relaxation rate, l/Tlo For example, both atom-atom collisions and collisions with the walls of a sample container will be presentS The total relaxation rate is then given by, = 1f S+ T (IIo39) where l/Tcd is the collision rate with the container walls and l/T! an atom-atom collision rate, Equation (II537) forms the basis for the measurements to be discussed in the next chapter. There a simple modification of that formula which arises because of the experimental technique will be discussedo The formulae which support (II.37) apply only when the magnetic field Ho is largeo The problem is somewhat more complicated in case the Zeeman and hyperfine interactions are comparable. The solution of this intermediate field problem is relegated to the Appendix.

CHAPTER III EXPERIMENTAL PROCEDURE AND RESULT$ A.o DESCRIPTION OF APPARATUS A functional diagram of the microwave spectrometer and associated electronics is shown in Fig. 4. The relationship between the displayed signal voltage and the microwave power absorbed by the sample has been 23,12 examined in detail by many authors. The following is a list the salient features of the apparatus. 1. The A.FoC. system stabilizes the spectrometer against small frequency changes and in doing so suppresses any dispersive components in the signal (i-oeX'). A resonance spectrum is obtained by slowly varying the magnetic field. 2o Crystal detector bias is obtained directly from the klystron by shunting a small amount of power around the microwave bridge (sometimes called homodyning). This is in contrast to the more usual practice of introducing a slight unbalance in the bridge by means of the adjustable cavity iris, The homodyne system has the advantage of extending the system linearity to very low power levels, on the order of 5 microwatts in the present design. 3. The microwave cavity (Fig. 4) used in this work resonates in the TE10o4 mode at a frequency of about 9.2 kmc/sec. The loaded Q of the cavity is nominally 1500. Quartz dewars are located in'the cavity to position a paramagnetic sample in the maximum microwave magnetic field and to allow temperature control of the alkali sample and standard sample. 4o The spectrometer employs high frequency amplitude modulation of the magnetic field and phase sensitive detection. In this work the modulation frequency is 100 kc/sec. The displayed signal is in the form of the derivative of the absorption prof ile 30

KLYSTRON REFLECTOR REFLECTOR AUTOMATIC MEHA AL ^ VOLTAGE FREQUENCY F" ^ -- 1 -4 — L- -LI CRYSTAL A = SIGNAL _100 IKC SIGNAL NARROW BAND COU' ------ ~~1- + AFC AMPLIFIER ATT. SEPARATION 1 1 1'-I. ISI POWER I i LOCK-IN p SUPPLY I \ i DETECTOR C and MECHANICAL VN 0 FIELD b'SWEEP_ k_ 0 __ FIELD MODULATION CAVITY and COILS- I_| V,G___________ 1 20 ----— 100 KC F~Ig.Boc igrmo teOSC. I Fig. 4. Block diagram of the EPR spectrometerN

32 12 24 Moos and others have given formulas relating the signal voltage at the recorder to the parameters of the spectrometer and paramagnetic sampleo A typical recorder tracing, in this case one of the resonance lines observed in potastsium, is shown in Fig. 5. If AV is the peak to peak voltage difference on the recorder, one obtains2 JV- \/ /3 z(i s+O) T ]^;^ (III l) where y - 2<i | Ix | j>/h, N is the number of unpaired spins per unit volume and T the absolute temperature. The filling factor L is given by (III.2) -cAv where 2H1(x) coscot is the microwave magnetic field in the cavity. Similarly, Hm cosut is the modulation applied to the dc magnetic field and ^ _A-p (III.5) AR is the spectral line width, in gauss, measured between the points of maximum slope on the absorbtion curve (Fig. 6). AH is related to 1/.Tp,2 the line width in frequency units, by Fh 3_ J ) (ino4) where aco/MH is the slope of the transition frequencyo* +*(ao/6H)=7 in strong magnetic fields, where the hyperfine levels of the alkali atoms diverge linearly with the field strength. If the Zeeman interaction is smaller than, or of the same order of magnitude as, the hyperfine interaction, magnetic dipole transitions involve both reorientation and changes in size of the dipole moment. acuo//H can be computed from the Breit-Rabi formulas for the energy. Further consideration of this intermediate field case is given in the Appendix.

I' WF F I I I I Irl I ILI I 4 1 B 1 1~- r 1 1 1l 1 1 1 1 1 I I1 HFtL-T -~ e~-s _-> 4 1 l m< 11 111 m +oi -~LeXX S- t- H -m - --- ---.1 ~~~~ 0 4 - t -- |; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~'-= ~~~~~~~~~~~~~~~~~ _- r — X F + =l =4-0 * - 0=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —----- L~ — ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~CCO.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. Co Coit_',>~~~~~~~~ * -- I\ - __.,__ _ __1__.-I —;~ — -; —- ~ ~ C~-~ ~ — c-+-I- i~- - 1II II_1-I IIII =-,T= —._-__-=i —--'-t-_t =._:= —_-.-_=r t t - t.=t__:|__r —._ — - = —= I L. -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ —— 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I T CD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' = = t=t._,~ -~=;~r $= t - >:~';. _ - -s _ t - t -_ — xI — =- I1- ==- - --- ~- in ---;i- — ~_I~ —- ~- - |= -| t- - - -i-t > | - -= —- - = - I =- =i *. _, r - — =1I!- t.t-=t| t _ a — tf - _ r =:.: g - _ _; L.= __ _- 4 __ _ -1 __ig Tyical i spctu of an alkal| |i me talt- vao (K1 sh —— own- ) — -.-~ I. -:-_1 ——.-!- =~~-1 - = r5 IlII' X. z: C t -_ t::t i —-- 1- 0 -X; —-— l - + —tL I X X X == = X -- -.. - --—:- __ —- — 1- - -- X — t -: 1! 1 I, It I~-t,-t- -, -T- -— 2 2 I _r I 1 -I = - _-_ —._ 1> — r - 0 -i — iI I I X - -- - - ~ -- I-T1-~-: —r-1LI. L. 1 —;0 H =1=1~ 1 —-~-I -T=IJ- I-'I t lI - ~ — -t —t —-- I t — tll - I -- tt-WI- -4 I I~~~~~~~~~~~~~~~~~~ - ~~~ —Fig —' Typica EPR- spcrI of-t- arl alai ea aao (3 ho

Fig. 6. Microwave cavity and sample tubes. 34

35 2 2 Equation (III.1) assumes that the saturation factor y HIT1T2<l. The factor A is a (known) constant incorporating the cavity Q, amplifier gain settings, incident power, etc. More explicitly, where P incorporates the conversion efficiency of the crystal detector and power losses in the microwave bridge. p is a constant characteristic of the spectrometer. The experimental determination of the spin exchange cross';section is based on the two-fold application of Eq. (III.l): first, to the magnetic resonance spectrum of the alkali metal vapors and to the particular transition being studied there; and second, to the spectrum of Cu ions transition being studied there; and second, to the spectrum of Cu ions in crystalline copper sulphate, CuSOi.5f20O. In this experiment small (1-2mg) single crystal specimens of copper sulphate were used to establish, by direct comparison of signal voltages and line widths, the density N of the alkali metal vapor at a given temperature. B. EXPERIMENTAL PROCEDURE Before proceeding to a discussion of the measurement technique some understanding of the copper sulphate spectrum is needed, since this material is to be used as a primary standard of intensity. The ++ 2 lowest state of a free Cu ion is a D state, corresponding to an occupied 3d shell minus one electron. The effect of crystalline electric fields and spin orbit coupling is to split the degeneracy of this state, leaving behind as the lowest state a pair of levels which diverge

36 25 linearly in an applied magnetic field.5 The next highest levels are about 12000 cm-1 away and are not occupied at room temperatures. Although there are two non-equivalent sites for the Cu++ ions in the last lattice, a strong exchange interaction between the sites tends to average out the spectral differences of the crystalline environments, As a result the effective number of levels [previously (2I+1)(2S+1)] is 2 and a single spectral line is observed. However, the exchange interaction is sufficiently weak that the position and width of this line depends on the orientation of the crystal axes with respect to the mag++ netic field, Associated with the two Cu sites in the lattice are two crystalline (tetragonal) axes which are coplanar and nearly 90~ aparto The g-value for the dc magnetic field perpendicular to the plane of these axes has been measured to be 2.09 ~ 0.01, while in the plane of 25 the axes the g-value varies from 2.23 to 2.28. The particular usefulness of copper sulphate as an intensity standard stems from the fact that the exchange narrowed spectrum has a 26 Lorentzian line shape, and that the number of spins contributing to.2 the signal can be assayed by weighing the sample. The product AV(AH), which is a measure of the area under the absorbtion curve, may be taken directly from recorder tracings for comparison with a similar product for the alkali absorption curve. If the alkali and Cu++ signals were not Lorentzian such a direct comparison would not be possible,

37 In practice small crystals of copper sulphate were selected and weighed on a microbalance. Typically these crystals weighed on the order of 1.5 mg, corresponding to 3.6x10 spins. The crystals were affixed, with a small drop of Duco cement, to a thin Pyrex filament and a thin coating of paraffin applied to prevent the loss of waters of hydration. A crystal so mounted is shown in Fig. 4. A small piece of cay bisdiphenylene P-phenylallyl, which gives an intense and narrow signal at g=2, was attached to assist in locating the crystal in the maximum rf magnetic field in the cavity. The crystals were oriented in the dc magnetic field so that the g-value in that direction was 2.09 + 0.01. Thus co/)H, in Eq. (III.1) is 2.09 p. /-. The rf field, which is perpendicular to the dc field, then lies in the plane of the tetragonal axes. The g-value in that direction was taken to be 2.25 ~ 0.03, so that y = 2.25 9 o/f. This orientation of the crystal corresponds to a minimum line width of about 30 gauss. It is also the orientation which gives the maximum signal to noise ratio. The quantities aw/aH and 7, as they pertain to the alkali spectra, are discussed in the Appendix. Table I presents a compilation of all the relevant matrix elements for the alkali transitions studied in this work, together with the data on the transitions studied by Moos.2

38 TABLE I TABLE OF MATRIX ELEMENTS Alkali Transition Field (Gauss) y*-,** _ cjo/H, 39 K9 (2,-33)-(2 2) 5560 gJ/l gl/h ( 4) -3 ). 4- 4) 5790 0. 96gko/t 0o.92g$i0/i Cs1 ( 4, 2 )(35 -3) 4160 0.82glo/ 0.635g0/i (4,- )(53,-2) 2480 o0 7 1g: Zo/0 o0 37 go/A Rb85 (35 -1)(2,0) 4100 go$ o/A guo/A Rb 87 (2,oe)o( o-1) 2900 0.84gooo e/ 6 o eoo o Cu 3 -- 5120 2.25 1o/a 20a9 jo/ *For the alkali metals, g = 2.00. **g A/h = 2.80 M cps/gauss +Reference 12 A typical alkali sample container is shown in Fig. 4. These samples were prepared by vacuum distillation of the metal into the containers. The sample tubes consist of precision bore (3/16" ~ 0.002" I.D,) quartz tubing approximatly 2-1/2" long. Taking into account the variation in rf field strength over the length of the sample, the effective volume 3 of the container is 0.51 cm, The tubing is sealed to a capillary quartz stem which leads to a small reservoir volume at the top of the sample (Figo 4)o It had been intended that this thesis also include measurements on 23 the exchange cross sections for Nao However, because of the extreme chemical affinity of the hot sodium vapor for the walls of the sample tubes these measurements were not possible. Sodium transitions

39 were observed using small tubes made from special alkali-resistant glass (Schott No. 8243). These tubes lasted about 10 min, far too short a time for any measurements to be made. In addition, the dielectric loss of this glass is quite high and rather strongly temperature dependent. At the required operating temperatures (-400~C) the Q of the cavity was reduced below operating limits. The capillary and reservoir serve as a method of gettering impurities introduced when the sample tube is sealed off from the vacuum -4 system. Although the pressure was never larger than 10 torr at the time of tip-off, without the capillary and reservoir of cold alkali metal it was found that the pressure in the sample tube subsequently -2 rose as high as 10 torr. The exact nature of these impurity gases is unknown but their presense is felt since they can, and do, provide relaxation collisions which broaden the alkali spectrum. The impurities are apparently gettered by the cold alkali but are promptly released when the sample is heated to its operating temperature (-~350~C in the case of potossium). To produce a more permanent gettering action some of the alkali is moved through the capillary into the reservoir, which is located to one side of the axis of the bulb so as to be out of the stream of hot air blowing past. This metal remains relatively cool during an experimental run and so long as the capillary remains open, effectively getters the impurities released by the hot alkali belowo Using this technique the spin lattice relaxation time T1 was increased, in

40 potassium, by as much as a full order of magnitude. Little, if any, change was noted in the case of cesium, presumably because at the lower temperatures employed for it the impurities were not released. Experimentally, a copper sulphate crystal and alkali sample were properly positioned in the two halves of the microwave cavity. An additional piece of precision quartz tubing was slipped around the sulphate crystal to equalize the amounts of quartz in the two halves. This was done to minimize any unwanted differences of the microwave field distribution. Hot air flowing past the alkali sample brought the alkali vapor pressure to a point where detectable microwave absorption occurred. The temperature was determined by a calibrated copper constantan thermocouple fastened to the outside of the sample tube. The temperature of the copper sulphate crystal was determined by a thermometer before and after an experimental run. Despite the presence of high temperatures not more than 1 in. way, the temperature of the crystal was found to remain nearly constant at about 30~C. When the system had stabilized, the alkali resonance was located and slowly scanned by varying the dc magnetic field. The field sweep was provided by a motor driven Helipot potentiometer which added an error voltage, varying linearly in time, to the reference voltage of the magnet power supply. The sweep rate was determined, in units of gauss per ino of recorder chart paper, by measuring the magnetic field with a proton magnetic resonance probe. A similar procedure was followed

41 to obtain permanent records of the Cu spectrum. Also recorded on the chart paper was the amplifier gain settings, modulation amplitude, power incident on the cavity and the temperature of the alkali cell. Additional information on the details of the experimental procedure can be found in Reference 12. C. RESULTS OF THE MEASUREMENTS ++ Equation (III.1) applies equally to the Cu and alkali resonances_/^< - n ^?1} A W\ Al, \5U. - 4 00 0 L The modulation coils shown on the cavity in Fig. 4 were especially constructed to provide as uniform modulation as possible over the active length of the alkali sample. Experimentally (4, 2e For the ratio of the filling factors we have where vA = 0.51 cm is the active volume of the alkali sample. Thus t\1_ -4 i_ r. j-, (..-.t (mI.5) AV/X TA 2.1w bu f C (88 ) (A1 where nu = N cuvcu is the total number of copper ions. Using Eqo (IIo 5) the variation of the spectral line width of the alkali resonances with density was studied. In agreement with the theory of Chapter II, the

42 line width in all cases varied linearly with the density. The results are shown in graphical form in Figo 9. It will be noticed that these plots do not extrapolate to zero line width at zero density. This residual broadening may be attributed to two effects: inhomogeneous magnetic fields in the gap of the electromagnet and relaxation collisions caused by collisions with the wall of the sample tube and with impurity atomso The magnet inhomogeneity was approximately 50 milligauss over the sample. The spin lattice relaxation time T1 was determined for cesium and potassium by progressive saturation of the resonance. Making allowance for the fact that the microwave field is distributed over the sample, 12 one finds jm Q - 4(\ 7_ (Ciii6) where A. = 7 H'1 1i2 AV-1 and AV2 are the peak to peak signal voltages 2 2 at two very different power levels PlV2L and P2aV2, respectively. The 2 magnitude of H could be determined either from the measured Q of the cavity* or by the saturation of a substance with known Tlo** Both methods *For a rectangular cavity operating in a TE'M mode, the relationship between the loaded Q of the cavity, QL, and the magnetic field strength HI (gauss), is where Pg is the power incident on the cavity, Vo the resonant frequency and Vc = AxBxC is the cavity volume, C is the length of the cavity in the direction of propagation (Z-direction) and A is the heighth of the cavityo **Small samples of the free radical ay-bisdiphenylene-(-phenylallyl are very useful for this purpose. This substance is characterized by Ti=2 and a line width AH f 1/2 gauss, It is easily saturatedo

43 were used. Numerical values of T1 and Tx were obtained by use of Eq. (II.34). The relaxation measurements were subject to rather large experimental errors, primarily due to fluctuations in the cavity Q during an experimental run and uncertainties on the order 2 of 20% in Hi. Nevertheless, these measurements did sho-s that for line widths AH > o2 gauss, the spin lattice relaxation made a contribution to the total line width no larger than 20%. For potassium, this was true only if the self-gettering bulbs shown in Fig. 4 were used, Measurements on a typical sample bulb of potassium (the bulb was of the type shown in Fig. 4) are summarized in Fig. 6. Plotted there is the ratio of the measured relaxation times, T1/T29 versus 1/T2. Throughout this experiment a lower limit to the usable line width was set by the inhomogeneity in the magnetic field produced by the large electromagnet (Fig. 3), The data plotted in Figo 7 has been corrected for a field inhomogeneity of 0.05 gauss, distributed over the volume of the sample. This correction was characteristic of the Varian 6" magnet (ring-shimmed) used in this work. Using Eq (IIo38) with AH = 0.05 gauss is the line width due to inhomogeneous fields and y/2t = 2.8 Mc/gauss is the gyromagnetic ratio for the potassium spins.

44 10 9 87 6 5- / 4-g 3 2 I_.2..I i I I i. 0.1.2.3.4.5.6.7.8.9 1.0 I (107sec' ) Fig. 7. Relaxation measurements on potassium: T1/T2 versus 1/T2.

45 By means of Fig. 2, or, equivalently, Eqs. (II.34), one obtains the ratio TL/Tx, which has been plotted versus 1/T2 in Fig. 8. One sees from this graph that for narrow lines (1/T2 small) the spinlattice relaxation is beginning to contribute significantly to the observed line width. This means that as the vapor density is reduced the rate of atom-atom collisions (and hence the exchange rate) gradually loses out to the various T1-processeso In particular, the life time in a given spin state is determined, in this limit, by collisions with the walls of the sample tube. For a line width of 0.1 gauss (l/T2 = 1.5xlO sec-1) one estimates from Fig. 8 that Tl/Tx - 2. From Eq. (II34), one finds 7 - ~ T1 or On the other hand, if wall collisions are the dominant spin-lattice relaxation mechanism (T1 TL), the relaxation rate can be estimated from elementary kinetic theory. Because of their exposure to a resonant microwave field the atoms in the sample tube are not distributed according to a Boltzmann distribution over the spin stateso Instead. the atoms have absorbed a small amount of energy from the microwave field. In the steady state the rate at which energy is absorbed may be equated to the rate at which this energy is removed by means of wall collisions. In this experiment, the sample tubes are cylindrical, about 2-3 ino long and about 3/16 in. diameter. Neglecting end effects, the average time

46 40 36 32 28 24 = 20 16 12 4 - - 0.1.2.3.4.5.6.7.8.9 1.0 (107 sec-') Fig. 8. Relaxation measurements on potassium: TL/TX versus 1/X2.

47 between wall collisions is where D is the diameter of the sample tube and V the average speed of 4 the atoms.* For potassium at 300~C, F r. 5.7x10 cm/sec. This gives -4 Although this estimate of the wall-time is perhaps the shortest that can be expected, the agreement with the measured relaxation rate is considered good. It is conceivable that additional relaxation mechanisms are at work here but their appearance is masked by the large experimental error. The spin-exchange cross section Qx can be related to the slope ATH/SN (Fig. 9) through Eqs. (11.30), (I1134) and (IIo.4), XQ =^ 2H ) I S) (Iiy7) v, the average speed of the alkali atom at temperature T, was calculated from standard kinetic theory. The quantities entering into Eq. (III7), together with the calculated cross sections, are given in Table II.** Also included are the experimental results obtained 12 87 by Moos on Rb85 and Rb I KTa **&C/8u is given in Table I, page 38.

A: K39- 8(H) =0.059 per 1015atoms SNA 1.0 0: Cs133-8:(H) =0.055 per 1015atoms.NA K39 f.8 A~ 9/=P'Cs 133.7 n.6.5.4 m.3 2 3 4 5 6 7 8 9 10 I1 12 13 14 NA(10-15) Fig. 9. Experimental results: Peak to peak line width versus alkali density.

49 TABLE II EXPERIMENTAL RESULTS Species ToK v(cm/sec) 5AH/5N(gauss/10 atoms) Qx(cm ) K 39(=3/2) 650o 5-92x104 0.60 l.4xo-14 (2, -2)^:2, -2) Rb85(I-5/2) 550 3568x14 0.68* 2,58x10o-14.(5, -1)( 2, 0) Rb 87 (I=3/2) 550 5-63x1o4 0.98* 2. 63xl0o:14 (2, O)1,-1) 4 -14 Cs (I=7/2) 550 2.96x10 0.58 2..16xl0 14 (4, -2)3,.-3 ) *Reference 12 **See Appendix (Part B) for a discussion of corrections which apply to the intermediate field case. D. ERRORS An extensive discussion of experimental errors has been given by 12 Moos. In the present work only minor improvements were made in the apparatus, none of which significantly affected the resultant errorin-measurement. Most certainly the most important error arises from improper positioning and orientation of the small copper sulphate intensity standards. Within a given run the scatter of data points is relatively small and random, but a large variation in slope can easily arise from the systematic mis-orientation of the sulphate crystals. Great care was taken to minimize this possibilityo

50 After taking account of other sources of error, such as variation in temperature of the sulphate crystals, inaccuracies in field sweep calibration, distortions in the microwave field due to the quartz 12 dewars and thin metal films within the sample tubes, etc., the cross sections quoted in Table III are assigned a 70% probable error of ~20%. TABLE III SOURCES OF ERROR IN THE DETERMINATION OF THE EXCHANGE CROSS SECTION Source Size Random ~ 10O Effect of films and quartz tubing ~ 5% Calibration of field sweeps 2% Measurement of temperatures - 2% Copper sulphate line shape -10% Determination of sample volume ~ 5% 7 and -/'H for CuS045Hi20 5% Determination of HM(ef) ~ 5 Miscellaneous ~ 5% Table IV (see Appendix) presents a compilation of the sources of error, together with the probable error assigned to each. Only a brief discussion of these errors will be given here, for a detailed disk cussion the reader is referred to the thesis by Moos.l

51 The copper sulphate crystals used in this work introduce two important sources of error. Because of a slight mis-orientation of the 2 crystals in the microwave field the quantities 7cu2 ac</H)cu and (AH)cu, all appearing in Eqo (II.5), acquire a corresponding variation. Assuming a ~10~ tilt in the crystal axes, the quantity 7cu/2,w/~H)cu is assigned probable error of ~5%. (AH)cu, the peak-to-peak line width, was measured with an accuracy of ~2o% Calibration of the mechanical field sweep was carried out with a proton magnetometer and frequency counter. Care was taken to always use the central position of the motor-driven potentiometer in the field sweep unit. This minimized the nonlinearities in the circuit. Spectral line shapes are an important parameter in this experiment. Both the alkali metals and copper sulphate are expected to show Lorentzian line shapes. In the case of the alkali metal vapors the Lorentzian line shape follows from the assumption of collision broadening, whereas for copper sulphate this line shape is a consequence of a rather pronounced exchange narrowing. The importance of the spectral line shape lies in the fact that the number of atoms contributing to a signal is proportional to the total area under the absorption curve. For a Lorentzian line shape approximately 80% of this area lies within three line widths (peak-to-peak) of the center of the absorption. This means, for example, that if the absorption dropped more rapidly than a Lorentzian, at points removed by more than three line widths from the

52 center, one would have under estimated the spectral intensity by some 20%, which has the effect of increasing the computed cross sections by the same amount. Carefu.l measurements of the spectral line shape of copper sulphate and the alkali gases, show that the line shapes are quite accurately Lorentzian out to a least three line widths. A reasonable estimate of the uncertainty in the density measurements, due to deviations from the assumed line shapes, is-l0%. Temperature measurements on the alkali samples were made by means of a copper-constantan thermocouple and precision potentiometero The thermocouple was located outside the cavity but adjacent to the small.L puddle of alkali metal which was feeding vapor to the sample bulb below (Fig. 5). The temperature of the copper sulphate crystals was measured with a mercury thermometer. Good thermal shielding of the sulphate crystals was provided by the quartz dewar around ito Water cooling of the cavity also reduced the heat transfer. The extreme error associated with these temperature measurements is taken to be ~5%0 Having the source of vapor located downstream in the flow of hot air past t;he sample tube minimized the formation of thin metal films on the tube walls. However, occasionally the molten metal would break away and fall into the microwave cav'ity. This had the effect of imfmediately lowering the cavity Q and terminating that portion of t4he rurio The sample tube was removed'and the alkali metal restored to the top of the tube by heating in a Bunsen flame. After placing the tube

53 back in the cavity and awaiting the return of thermal equilibrium, the run could be resumed. The measurement procedure entails the comparison of signals obtained from different absorbing samples located in each half of the same microwave cavityo An accurate comparison of signals requires the field distributions in the two halves be the same. This in turn requires that the symmetry of the cavity be maintained when the quartz sample tubes are inserted. For this reason a similar piece of quartz tubing was slipped around the copper sulphate crystal. It was found that the rf field intensity did not change by more than ~5% (probable error) when different sample tubes were used. The field distribution within the cavity was probed by means of a small DPPH sample attached to a drawn out Pyrex fibero When corrected for the (known) distribution of the modulation field, these measurements showed that the microwave field distribution was the theoretical distribution, except in the vicinity of the "chimneys" (Fig. 5)o There, the fields were perturbed somewhat and penetrated the opening at the base of the chimney. These fringing fields were taken into account in the determination of the alkali sample volume. The probable error associated with the determination of the sample volume, and Hm(eff), is ~15+.

APPENDIX, ADDITIONAL COMMENTS ON THE DENSITY MATRIX TREATMENT Certain features of the density matrix treatment given in Chapter II need further discussion. In Section A of this appendix the off-diagonal matrix elements corresponding to spin-states which are not resonantly coupled to the microwave field are shown to be unimportant, so long as the spin states are well resolved. In Section B approximate solutions to the equation of motion are obtained for the case of intermediate field strength (hyperfine coupling on the order of or less than the Zeeman interaction). A. OFF-DIAGONAL MATRIX ELEMENTS In Section C of Chapter II the equation of motion for the density matrix was solved with the assumption that only two off-diagonal matrix elements were nonvanishing. These matrix elements corresponded to the excitation of a given pair of energy levels by a microwave field. It is not necessary to retain this assumption, although under the conditions of the present experiment it is justifiableo The justification will now be given. For simplicity only the case I = 1/2 (hydrogen) will be considered in detail, The states are labelled in order of decreasing energy; that is, aean 1, aepn = 2, Pepn = 3 and pean = 4. With this labeling, and for magnetic dipole transitions AMS = +1, AMI = 0, the density matrix has the form 54

55, 0 ( 0 0 0 PK Using this density matrix and Eq. (II19) one finds c ao The diagonal matrix elements of p are unaffected by this new form, (i.e., the pii are still given by the expressions in Chapter II)o bo The off-diagonal elements are altered and, in the present notation, are given by +, 2 - (p.,.-Y) zl +^ Pt-(i) tt - (p +3j)( - weeZ=Pl + a2Q\ -V Pa 3 1 where Z pll + p22 - p3 - p44 and N = pl + p22 + P3 + p44 = i. Neglecting the very small frequency shift A (the expression for A is essentially unchanged, Eq. (IIo35)) transforming to a coordinate system rotating with the sense of the Larmor precession at a frequency o, the equations of motion assume the form

56 I +AX 2S-' tudE~tF gh-A AS~ =. tA-.. where AA = o A = - 1/TX /2TX, 1/T2 1/2TX + 1/Aoi.A. wo w&, Aci, = wo- wJ, / / /2 Also, U and V are the out of phase and in phase components of the magnetization in the rotating frame, respectively. Z is the component of magnetization along the applied field (Chapter II). These equations represent the motion of two spins systems (A and B) in "magnetically different environments" exchanging magnetization at the rate l/rxo Similar equations have been proposed by McConnell* in order to deal with a similar problem in nuclear magnetic resonance. The approximations made in Chapter II are represented by the equations VB = UB = 0. The problem now is to determine under what conditions these approximations are valid. A. general solution will not be attempted. Instead, only the case of relatively weak microwave excitation will betreated (i e., no saturation). For sufficiently o o weak rf ZA L 1/2 Z0 where Zo = ZA + Z7 is the total equilibrium H.oMo, McConnell, J. Chem. Phys. 28, 430 (1958).

57 magnetization. Defining GA = UA + iVA and GB = UB + iVB, the first four equations of motion can be combined to give, in steady state, [^ - A~nGfc - ^ C 6 - \I LO, S~ These equations can in turn be combined to yield cX5G1 +- G cGt =^ GA whence R-t;Gb -4 +b e i &-L^ I+LAC When the frequency of the microwave field is very near the resonant iA frequency for system A (i.e~, 3 _ co), then Letting I/T2 + l/" -- 1/ir one finds Taking real parts this gives 2 1 (2 B A 2 T Thus UB << UA so long as T (D ) > lo That is, = X/2H1 so long as B. the resonances are well resolved: c(o >> 2o the effective exchange rate is small: T >> 1 5 both 1 and 2 prevail~

58 B A The difference o - Co is easily calculated from the Breit-Rabi formula for hyper-fine structure in a magnetic field. One finds, in high field, where A is hyperfine coupling constant. Since 1/T = 1/Tx, in the present case, the condition that only those off-diagonal matrix elements are nonvanishing that correspond to levels nearly resonant with the microwave field is The case I > 1/2 yields this same condition. Of all the alkali metals potassium (K ) has the smallest hyperfine splitting (iA _ 4.6x108 cps). The measured exchange rates are on the order of 1/Tx ~ 10. Thus the approximations made in Chapter II are valid to a very high degree. B. INTERMEDIATE FIELDS The equation of motion for the density matrix, though nonlinear, is, in the so-called high-field case (X' 1), separable and a steady state solution readily found. However, in the weak or intermediate field case (X 1) the equations are nearly intractable and the general steady state solution valid for all X has not been found. The intermediate field case is of interest for two reasons. First, the X-band microwave transitions in Cesium 133 and Rubidium 87 fall in

59 this region, and there is the possibility that in fields insufficient to decouple the electron and nucleus, the "effective" collision rate for spin exchange between specified energy states is thereby affected. Second, even if residual hyperfine coupling is not important, the solution obtained in Chapter II must be corrected for the changes which occur in the magnitude of the dipole moment (ioe., y) as the static field is loweredO Although a complete solution to the intermediate field problem has not been obtained, still it is possible to examine the equations and show that when there is no saturation (cl <<.1, Z Z~) the effective spin-spin relaxation time is given by (II534) to a very good approximation. Information concerning the effects of hyperfine coupling is contained in the off-diagonal matrix elements of the density matrix. For I 1/2, using the wave functions given by (II.27), and with the state labeling of Section I of this appendix, one calculates + - -+ 4(1+ A 4 PZ - ~ -, i{z - ~^ (r,-, -(Z-io).t +'2Le 40 i 2 + |-A t -Kl 4 ~~+~t. - -

60 fQ3 = ^p33i - ~(4(It ) 3 +? (I- i) i A(, t \ (> - 2 + Z) 2 c IoC i 0- + i - 4 In these equations For 2the + 1f lo A ilk A X For the matrix elements of [p] one finds |Dl p]f|4>'= nmOk2PLvx A(&- L

61 t1 |''0< (E, -1E) + JkL/, 4cY, t) Defining /where d Also, 04,-0, 4 E,-E4- k~o The equation of motion can be written 01~ = -L1 txto 9, pl-b i (e-^ pi, ) - (np. -ps) i - *4, X (,-8 -C>C(p4 - )- n( P4z~ =-i ~J( -.0t - - i LI6t _ L Ek LAo0 - L., O L t (5L- ( >,< - ) (-@) - p For sufficiently low fields (x << 1), AD Bo 1/ 12 and 0 0 Pi P ii ~ Then Z Z O and i. / % uO + L LQq f Ap -. P ~ +~ L01)-p- (L Td 41 Pi 1 i v -)- - P where =?p~lk _ i- Zq 41

62 So X - + I=-P-+ IX j _ = t(if -p - to K 2, m t(g - t'1~B - with T t m b c t These equations may be compared with Eqso (Iio33) and (Iio34) of the texto The effect of hyperfine coupling is exhibited by the factor 9/8 in the equation for the effective exchange rate~ Over a wide range of magnetic fields the effective exchange rate is essentially unaffected by the hyperfine coupling, as this is represented in the eig enrfunctionso Calculations of the sort described above have been carried out for the transitions of interest in cesium and the results tabulated below o TABLE IV CORRECTIONS TO THE CESIUM EXCHANGE CROSS SECTION 2I1 1Transition X T l T 1 (4, -1)0-3, -2) o075 a=0.8l (4,-, 2)e(3,-3 ) 1.26 a=o.8445 (4, -5 ) *(4', -4) 1.78 ca=o.94

63 With regard to changes that occur in the size of the magnetic dipole moment in intermediate fields, one is required to use the correct eigenfunctions. The gyro-magnetic ratio is 7 = 2/K < i xl xj >, with x = gs5to0x. Then with AM M/ + 77. The quantities XM and AM are given by Eqso (IIo27a)0 The mixing of the electron spin-states by the hyperfine interaction in intermediate fields leads, then, to three important corrections to the high-field analysis (Chapter II). 2 2 l1 Through 7y 2 = 2 | < i >lplf >1 the transition probability for magnetic dipole transitions between a given pair of energy levels (i-f) decreases with decreasing field strength. 2. In contrast to the behavior at high fields, the energy levels in intermediate fields have curvatureo Hence 7y 7 c/gH and the relationship between line width measured in units of the magnetic field (gauss) and line width measured in frequency units is 35 In intermediate fields the connection between the spin-spin relaxation time (72) and the mean time between exchange events (Tx) is not simple. Assuming that the spin-lattice relaxation (T1) makes a negligible contribution to the line broadening, the relationship is I T j t -l hi " ial^ S \

64 where a is a numerical correction factor arising from the complicated mixing of the electron spin-states. In this work the (F,MF) = (4,-2) - (F',MF) = (3,-3) was used for the determination of an exchange cross section. For this transition, and for a value of X = 1.26 appropriate to the magnetic field used, the value of a is a=0.84. If this same transition had been studied in the high-field limit (K band microwaves), a = 1. There is however, little experimental evidence that the numerical correction a, which should be applied to the line width data, is correct. In cesium there are three transitions which can be observed at X-band frequencies (9.3 KMC). With the cesium gas at constant temperature (constant density) the line widths of each of these transitions was measured. The results are summarized below. TABLE V LINE WIDTHS FOR VARIOUS TRANSITIONS IN CESIUM 1 2 I 1 Transition AH a/aH 1/72 XT2 = 2I+1 Tx ()- (4,-3)<-(4,-4) o.41 0.92 glt/h 5.9x106 6.3xo06 (2) ~(4,-2-(3,3) 0.50 0.6 go/ 4.8x106 5.7x106 (3):(4, -1)(3,-2) 0.67 0.37 glIo/1 3.7x106 4.6x106 (1) (4,-3)3 (4,-4) 0.35 5.0x106 5.3xo06 (2): (4,-2)~(3, -3 ) 0.42 4.1x106 4.9 (3): (4,-1 )(3,-2) 0.55 3.xl06 3.8 (1) (4, -3)4(4,-4) o0.19 2.6x106 2.8 (2) (4,-2)-(5,-3) 0.25 2.4x106 2.9 (35): (4,-1) -,-2) 0.40 2.2xl06 2.7

65 According to the theory the numbers in the far right hand column, being essentially the product of the density and the exchange cross section, should, at any given temperature, be the same for all three transitionso The agreement with theory is quite good for the narrower line widths but there is progressive deterioration of the agreement with increasing temperatureo The reason for this is not understoodo It is possible that the contribution of the spin lattice relaxation to the line width is considerably more complex than Eqso (II34) might indicateo Although measurements of the spin lattice relaxation time (Tl) showed that T1 was approximately the same for each of the transitions, one is not justified in merely subtracting out this (apparent) contribution to the line width0 It is believed that the difficulty could be resolved if a complete numerical solution to the equations of motion for intermediate fields was available, including the effects of spin lattice relaxation The experimental value for the exchange cross section of cesium has been corrected by dividing the measured cross section by the factor c=0o84. This correction is the one appropriate to the (4,-2)<-^(35,-3) transition,

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