COLLISION THEORY AS APPLIED TO THE CALCULATION OF A RELAXATION TIME by Sc. KATHERINE STEPHANIE NIELSEN (Hons.), U n i v e r s i t y o f B r i t i s h Columbia, 1 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f CHEMISTRY We accept t h i s t h e s i s as to t h e r e q u i r e d conforming standard THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1969 In p r e s e n t i n g an this thesis advanced degree a t the the Library I further for shall in p a r t i a l University of British make i t f r e e l y agree that f u l f i l m e n t o f the permission available for for extensive s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e by his of this written representatives. thesis I t is understood for financial permission. Department of CHEMISTRY The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date gain MARCH, 1969 Columbia shall Columbia, I agree for that r e f e r e n c e and Study. copying o f this thesis Head o f my D e p a r t m e n t o r that not requirements copying or p u b l i c a t i o n b e a l l o w e d w i t h o u t my ABSTRACT An e x p r e s s i o n T^, f o r the s p i n - l a t t i c e o f a d i l u t e monatomic gas relaxation time, can be d e r i v e d s t a r t i n g from the quantum-mechanical Boltzmann e q u a t i o n . The r e a l difficulty i n c a l c u l a t i n g the r e l a x a t i o n time f o r a p a r t i c u l a r system lies i n the e v a l u a t i o n o f the t r a n s i t i o n o p e r a t o r which appears i n the e x p r e s s i o n f o r T^ In t h i s p a r t o f the t r a n s i t i o n o p e r a t o r , distorted-wave In t h i s model the is one, V ^ , and the o t h e r , the the e s t i m a t e d by a specific d e s c r i b e d by %^ are isotropic rigid the a n i s o t r o p i c nuclear spin interaction p o t e n t i a l , describes relevant approximated by a collisions governed by two p o t e n t i a l s : potential, is the Born a p p r o x i m a t i o n (DWBA). The monatomic gas model. t^, thesis, V^. sphere dipole-dipole The l a t t e r interaction c o u p l i n g between the degenerate n u c l e a r s p i n s t a t e s o f the atoms and the t r a n s l a t i o n a l degrees o f i n the gas. explicit The former ( i s o t r o p i c ) potential form o f the r i g i d sphere d i s t o r t e d governs T.. ^ breaks up i n t o two terms, the wave. A f t e r the DWBA t r a n s i t i o n o p e r a t o r i s i n t o the e q u a t i o n f o r the r e l a x a t i o n t i m e , freedom substituted the e x p r e s s i o n the " d i a g o n a l " and for "non-diagonal" contributions. expression for At this stage the explicit is sufficiently complicated that, in order to finish the calculation, analytical approximations to the diagonal and non-diagonal terms are made. These approximations may be succinctly described by stating that they result in two separate evaluations, a linear and a quadratic one, for the overall relaxation time. The magnitude of a small parameter 2 -1 c , which appears in the exponential term of T^ , is used as the basis for neglecting certain contributions to the' integrals which arise in estimating T^. The linear and quadratic . approximations yield numerical factors of 3;50V] and 2.56 !n +he e x c e s s /oh respectively, for the relaxation time. These values are to A be compared with the factor of 2 obtained elsewhere. iv TABLE OF CONTENTS Page Abstract ii Acknowledgment v CHAPTER I INTRODUCTION 1 CHAPTER II THEORY OF THE TRANSITION OPERATOR.. 5 CHAPTER I I I CALCULATION OF (t| " 1 + 30 ) Jg CHAPTER IV RELAXATION TIME - T " CHAPTER V A LINEAR APPROXIMATION CHAPTER VI A QUADRATIC APPROXIMATION CHAPTER V I I SUMMARY... BIBLIOGRAPHY 55 1 TO T J 70 1 TO T " . . . 1 80 109 115 V ACKNOWLEDGMENT I wish t o thank Dr. R. F. S n i d e r f o r t h e s t i m u l a t i n g and i n f o r m a t i v e y e a r s t h a t I spent as h i s r e s e a r c h s t u d e n t . I a l s o wish t o acknowledge the f i n a n c i a l support which I r e c e i v e d both from t h e N a t i o n a l Research C o u n c i l o f Canada and t h e Chemistry Department Columbia. o f the U n i v e r s i t y o f B r i t i s h CHAPTER I INTRODUCTION In the Boltzmann equation approach to the theory of nuclear magnetic relaxation in dilute monatomic gases, Chen and Snider''' have derived a general expression for the spin-lattice relaxation time, T . Their expression for T^ ^ involves the transition operator, t, which arises naturally in the collision term of the modified quantum2 mechanical Boltzmann equation of Waldmann 3 and Snider . If a rigorous, analytical evaluation of t for the relevant interaction potential could be obtained, then i t would be possible to evaluate T^ ^ completely and explicitly for a fluid relaxing because of this particular interaction potential. The collisions in this thesis are governed by two potentials which affect the scattering to a different extent. 4 Thus a "distorted-wave Born approximation" (DWBA) can be used to estimate the anisotropic part, t^, of the transition operator. Such an approximation leaves t^ with rigorous distorted waves governed by the isotropic rigid sphere potential, V_, and with a s t r i c t l y linear dependence on the anisotropic 2 dipole-dipole nuclear spin interaction potential, V . It is the purpose of this thesis to use the DWBA transition operator in order to obtain an approximate analytical expression for the relaxation time, T 1 due to V^. A concise review of scattering theory for molecules with degenerate internal states is given in Chapter II. This is completed with a derivation of the DWBA to the anisotropic part of the transition operator. In the next chapter this transition operator is explicitly calculated for the rigid sphere potential and the dipole-dipole nuclear spin interaction potential. It is at this point that the work in this thesis differs from that of Chen and Snider. They performed essentially the same calculation; that is, they used the same two potentials as used here, but in their treatment they approximated the distorted waves by plane waves. The rigid sphere potential entered into the problem only as a lower limit on the radial integration in t^, thus neglecting a l l scattering effects due to the isotropic potential. Such a plane wave approximation simplified the integrations in t^ considerably. The purpose of treating the wave functions more realistically is to see i f there is a significant change in the value obtained for the relaxation time. Because the plane wave approach of Chen and Snider is so much simpler 3 m a t h e m a t i c a l l y than, f o r example, t h i s more e x t e n s i v e treatment, the improvement o b t a i n e d by a more exact method would have t o be c o n s i d e r a b l e t o outweigh involved i n evaluating t more p r e c i s e l y . the time and In f a c t , effort i t was found t h a t the s i m p l e r treatment g i v e s 78% o f the v a l u e f o r the r e l a x a t i o n time t h a t i s c a l c u l a t e d by the more e x t e n s i v e treatment. Because o f the p a r t i c u l a r chosen in this thesis, the d i s t o r t e d waves c o u l d be expressed e x a c t l y i n terms o f a p a r t i a l wave expansion. s c a t t e r i n g problem rigid A n e a t e r s o l u t i o n to t h i s particular would be a C a r t e s i a n e v a l u a t i o n o f the sphere w a v e f u n c t i o n . More g e n e r a l l y , the most u s e f u l s o l u t i o n to any c o l l i s i o n problem would be a good analytical C a r t e s i a n approximation t o the d i s t o r t e d wave f o r a g e n e r a l isotropic potential. In Chapter explicitly IV the r e l a x a t i o n time i s w r i t t e n down f o r the t d i s c u s s e d above. U n f o r t u n a t e l y , the e x p r e s s i o n which i s thus o b t a i n e d f o r TV ^ i s s u f f i c i e n t l y c o m p l i c a t e d t h a t an exact e v a l u a t i o n cannot be c a r r i e d Consequently, further. i n Chapters V and VI a " l i n e a r " and a " q u a d r a t i c " approximation r e s p e c t i v e l y , are c a r r i e d out a n a l y t i c a l l y i n o r d e r t o e s t i m a t e the remaining sums and i n t e g r a t i o n i n A c e r t a i n s m a l l parameter, 2 c , which appears T^. i n the e x p r e s s i o n 4 f o r TV* was used as a guide i n p e r f o r m i n g the integrations i n the " l i n e a r " and " q u a d r a t i c " a p p r o x i m a t i o n s . example, 300 © for 12? Xe, c K . , expansions 2 is o f the o r d e r o f 1.25 Because, x 10 -4 for at are made i n terms o f t h i s parameter and 2 cut o f f at terms l i n e a r i n c . o f the approximate e v a l u a t i o n s In the l a s t to T ^ i s 1 c h a p t e r a summary given. CHAPTER II THEORY OF THE TRANSITION OPERATOR A b r i e f resume o f s c a t t e r i n g t h e o r y o f molecules w i t h degenerate i n t e r n a l s t a t e s w i l l be g i v e n i n t h i s chapter. and The g e n e r a l purpose o f t h i s summary i s t o discuss the t r a n s i t i o n o p e r a t o r , but the u l t i m a t e introduce aim i s to e x h i b i t the e x p l i c i t o p e r a t o r e q u a t i o n f o r t h e a n i s o t r o p i c t r a n s i t i o n o p e r a t o r f o r a system i n t e r a c t i n g through an i s o t r o p i c and an a n i s o t r o p i c p o t e n t i a l . not the whole a n i s o t r o p i c i n t e r e s t here, but o n l y or, e q u i v a l e n t l y In f a c t , i t i s t r a n s i t i o n o p e r a t o r which i s o f a " l i n e a r i n anisotropy" approximation, a " d i s t o r t e d - w a v e Born a p p r o x i m a t i o n " (DWBA) to the whole a n i s o t r o p i c t r a n s i t i o n operator. A two body c o l l i s i o n problem i n v o l v i n g a c e n t r a l p o t e n t i a l which i s a l s o an o p e r a t o r i n i n t e r n a l s t a t e can space, always be reduced t o a pseudo one-body s c a t t e r i n g problem. S i n c e the c e n t e r o f mass i s not a f f e c t e d by the c o l l i s i o n , t h e problem can be expressed e x c l u s i v e l y i n terms o f the s t a t i o n a r y state Schroedinger equation w r i t t e n namely in relative coordinates, 6 (2-1) where jJL i s the reduced mass o f the two m o l e c u l e s , r e l a t i v e coordinate r ^ - r ^ , and ^ is r. i s the t o t a l r e l a t i v e o f the system p l u s the energy o f the i n t e r n a l s t a t e s . Hamiltonians internal V^ n t (r) state the energy The and 3\, ' ao?e o p e r a t o r s s t r i c t l y i n the int ''int s t a t e space o f molecules 1 and 2 r e s p e c t i v e l y , w h i l e is . an o p e r a t o r i n both p o s i t i o n space and i n t e r n a l space. The c o r r e s p o n d i n g f r e e p a r t i c l e S c h r o e d i n g e r e q u a t i o n f o r molecules w i t h i n t e r n a l s t a t e s A. j t is 2. (2-2) where 'I 5 A (2-3) 7 and - /2 i ^ - A 3 -A - /* 3 .A The = <AIJP>Z> JL label -jlk-A JL . U> <jp y JL = lA/\ (2-4) (2-5) actually stands f o r the p a i r of quantum numbers i , d. of molecule 1. The former quantum number labels l states with d i f f e r e n t energies and the l a t t e r one labels the X degeneracy. The solution to Eq. (2-2), (P. » . ( r ) , i s a wave k, I — function i n position space, but s t i l l an abstract vector i n internal state space. eigenfunction F i n a l l y , the ket 1x^,1is an s i g of the internal state Hamiltonian for a p a i r of molecules, i . e . , is & JL. JL. fy « (2-6) With the substitution of Eq. (2-6) into Eq. (2-2), the l a t t e r 8 equation ' can be w r i t t e n as Jl* sL (2-7) l l i / 3 Because the set o f s t a t e s form a complete orthonormal s e t o f v e c t o r s i n i n t e r n a l s t a t e space, ^ ? ( r ) , the s o l u t i o n to Eq. (2-1), can be expanded i n terms o f them i n the manner (2-8) When the p a i r o f o p e r a t o r s as g i v e n i n Eq. i . , ^7 mt (2-6), the r e s u l t i s Ufa act upon "jF(r) **int i f . <i»€. / x „ 0 . =C (2-9) ?Cr) It can be seen from the l a s t equation n e c e s s a r i l y an e i g e n f u n c t i o n o f that jf(r) 31 + int 3~l C int i« i s not 9 Since the total energy of the system is conserved, can be expressed as ^ ~ -fl- + ^ ~ -+ j/ (2-10) e where the left hand side of the equation refers to the total energy of the system before collision, and the right hand side, to the total energy after collision. Only elastic scattering will be discussed in this thesis; that i s , none of the translational energy of the system is transferred to the internal states (and vice versa). Consequently, the kinetic and internal state energies are separately conserved and, in fact,£^ becomes independent of the internal state labelling. Since the internal states are degenerate, they can s t i l l change within the internal.state energy shell. Now the right hand side of Eq. (2-9) is a multiple of "5" (r), and for elastic scattering, Eq. (2-1) becomes (2-11) The wavenumber k is related to the total energy and the 10 internal energy £ by a f = £ JL . (2-12) The wave vector k_ i s defined i n terms of the relative linear momentum JD by means of the equation (2-13) where g i s the relative velocity of the two molecules. For elastic scattering the wavenumber must be the same before and after the c o l l i s i o n , i . e . , /AI - IA'I -A. If Eq. (2-11) i s rewritten to look l i k e an (2-14) 11 inhomogeneous differential equation, A. 5 , (2-15) then: by standard Green's function techniques, Eq. (2-15) can be expressed as the integral equation (2-16) The Green's function G(r,r ) is a solution of the equation 7* +e (2-17) where 0^(r-/) i s the three-dimensional Dirac delta function. The s olution 3?, (r) to Eq. (2-15) is completely defined by asymptotic boundary conditions imposed on i t by the collision process. ^(j.) must essentially be the sum of an incoming p l a n e wave p l u s an outgoing s p h e r i c a l wave. The e x p l i c i t boundary c o n d i t i o n i s scattered thus A. (2-18) i ^ ^ (j_) d e s i g n a t e s t h a t jT^(_r) c o n s i s t s of a free p a r t i c l e i n c i d e n t wave p l u s an o u t g o i n g s p h e r i c a l s c a t t e r e d wave. subscript i is The a reminder t h a t the incoming p l a n e wave has a p a r t i c u l a r internal state Analogously, associated with j (j_) d e s i g n a t e s it. the sum o f a f r e e p a r t i c l e outgoing wave i n i n t e r n a l s t a t e i , p l u s an incoming s p h e r i c a l wave. The p a r t i c u l a r solution " (-** £\>4d \ . w ^ 4 - A ' / ) **** .f-w (2-19) to E q . (2-17) s p h e r i c a l wave is chosen because i t represents ( o r i g i n a t i n g at t^}. partially satisfies Since G ^ an o u t g o i n g (r_,r-^) o n l y the a s y m p t o t i c boundary c o n d i t i o n s on 13 "^(r_), E q . (2-16) the cannot represent inhomogeneous d i f f e r e n t i a l the e n t i r e equation. solution Consequently, of if another term were added to the r i g h t hand s i d e o f E q . (2-16) such a way as to f u l f i l l condition, solution is the plane wave p o r t i o n o f the boundary E q . ( 2 - 1 8 ) , then the e x p r e s s i o n to E q . (2-15) would be complete. fov*$^ } the (r) as a - + The a p p r o p r i a t e term the p a r t i c u l a r f r e e p a r t i c l e wave f u n c t i o n satisfies ^(r) which equation. 6 M = o. •—- AjJL Eq. (2-20) scattering is, in of course, process. (2-20) % j u s t E q . (2-^) f o r an e l a s t i c Now the complete formal s o l u t i o n to E q . (2-15) is -r <+> ( a* . T W -4^ (2-21) 14 The manipulations in the preceding paragraphs become more transparent when Eqs. (2-15), (2-16), (2-20), and (2-21) are cast into operator form: (2-22) -4. 5 (2-23) (2-24) and (2-25) The free particle Hamiltonian H^ which acts on wave functions in position representation is 4 " < (2-26) 15 Formally, the Green's f u n c t i o n G i s j u s t the i n v e r s e o f the appropriate differential operator. For example, with respect to Eq. (2-17) G can be w r i t t e n as G = (E-HJ -1 (2-27) where E, an e i g e n v a l u e o f H^, o f the i d e n t i t y o p e r a t o r . However, when ( p h y s i c a l ) boundary c o n d i t i o n s are i n c o r p o r a t e d they a r e i n Eq. (2-25), i s u n d e r s t o o d t o be a m u l t i p l e i n t o the Green's f u n c t i o n , as then t h e s o l u t i o n o f Eq. (2-17) i s chosen t o be i n the form o f Eq. (2-19) and G becomes The o p e r a t o r expression f o r G ^—^ i s i€ r . 1 = ( -H E n G^. + (2-28) away from the i n f l u e n c e o f t h e i n t e r a c t i o n p o t e n t i a l V ( r ' ) , can be expanded i n the f o l l o w i n g manner: 16 A. - A. (2-29) If this asymptotic expansion is used in Eq. (2-21), then -V* * 7/>4^ - A V4')£ 4<W A (2-30) where k_' — kr\ Comparison of Eq. condition on ? _K , 1(r), — Eq. (2-30) (2-18), with the boundary allows the scattering amplitude f^ (which is s t i l l an abstract vector in internal state space) to be identified as S 3 , JL<o'/A!><A?/v/'f A <+) . (2-31) 7 ^ where Eq. ( 2 - 5 ) was used without internal states. K. p_'l The bra represents the asymptotic final state of momentum p_' into which the particle has been scattered. Eq. 4^ (2-21) can be written in the form Lk^ l ^ p ^ / (2-32) 18 where SX.^ i s t h e M o e l l e r wave o p e r a t o r . _ T L p r o p e r t y t h a t i t takes t h e i n i t i a l <r| , £ i > into the f u l l can be used amplitude ( + ) has t h e incoming p l a n e wave s c a t t e r e d wave _Z" ^ . ( r ) . Eq. (2-32) i n Eq. (2-31) i n o r d e r t o express t h e s c a t t e r i n g as a m a t r i x element between t h e asymptotic incoming and asymptotic o u t g o i n g p l a n e wave s t a t e s , namely (2-33) The o p e r a t o r VJfL. i s s u f f i c i e n t l y important t o be g i v e n t h e symbol, t , d e n o t i n g t h e t r a n s i t i o n o p e r a t o r . Note t h a t t i s an o p e r a t o r i n both momentum space and i n t e r n a l state while t — g "but s t i l l an i s a m a t r i x element i n momentum space operator i n i n t e r n a l If state space space. Eq. (2-25) i s r e w r i t t e n i n terms o f t h e M o e l l e r wave o p e r a t o r , then t h e Lippmann-Schwinger i n t e g r a l equation f o r _ f i _ i s obtained, v i z . (2-34) 19 An Eq. i n t e g r a l e q u a t i o n f o r t can be w r i t t e n down by s u b s t i t u t i n g (2-34) i n t o V i i . to obtain (2-35) It i s v i t a l to n o t i c e t h a t the Green's f u n c t i o n i n Eq. contains o n l y the the t o t a l o f two other p3rt»c/e H a m i l t o n i a n , H^, as opposed to Hamiltonian. Now sum free (2-35) suppose t h a t the p o t e n t i a l V i s w r i t t e n as potentials: a n i s o t r o p i c and appropriate one i s o t r o p i c and denoted by V^. The denoted by V^, a and the Schroedinger equation f o r d e s c r i b i n g a system governed by these p o t e n t i a l s is The t e c h n i q u e and be employed a g a i n argument used e a r l i e r t o s o l v e Eq. (2-15) can to w r i t e a formal (2-36), s o l u t i o n f o r Eq. namely (2-37) 20 is where a solution to [ E - (H0-hV )\X w i t h the O = 0 (2-38) same k i n d o f boundary c o n d i t i o n imposed on A* ^ as was imposed on " 3 ^ ^ satisfies the i n Eq. (2-18). equation I £ - M +V )i] a where cf" i s The G r e e n ' s f u n c t i o n G^ Gr 0 the D i r a c d e l t a = 0 / (2-39) function. The p h y s i c a l l y r e l e v a n t s o l u t i o n to E q . (2-38), c o n s i s t s o f both a p l a n e wave and an outgoing spherical scattered wave. The s c a t t e r i n g due to the i s o t r o p i c p o t e n t i a l . is, Because the i n E q . (2-37) d i f f e r s from the 0 ^ ^ in the more common r e p r e s e n t a t i o n ^C-^ ^ \ l} of course, now i n t e r p r e t a t i o n of i n t e r p r e t a t i o n of o f 3?^^ as g i v e n in + Eq. It (2-25), is assumed t h a t the e f f e c t is often on 3?", c a l l e d the " d i s t o r t e d " wave. . due to the 1 s m a l l compared to the e f f e c t anisotropic K, potential, V^, is Consequently, To terms it i s reasonable of V^. to s o l v e E q . (2-37) by i t e r a t i o n . l i n e a r i n V ^ , the d i s t o r t e d - w a v e Born approximation 21 is obtained, namely -r(±) v ft) (±) J±) Parallel to the decomposition of the potential V into two contributions, the transition operator can be written as a sum of two parts, namely o i t = t +t = (V + Q + where G is given by Eq. (V Q + V p (2-17). G(t + tp (2-41) Upon expansion of Eq. (2-41) Q the "isotropic" transition operator, t^, and the "anisotropic" transition operator, t^, can be naturally identified. They are V il 3 Q (2-42) 0 where SLQ is defined as in Eq. (2-34) but with V replaced by V , and Q h = v i + v i G t o + v o G t i + v i G In the DWBA the last term in Eq. V ( 2 (2-43) on the basis that i t is quadratic in V . Eq. - 4 3 ) - is dropped (2-43) can then be rearranged into the form (1 - V G) Q = t l V H l (2-44) 0 which implies that t 1 (1 - V = G)" Q 1 V1_O. 0 (2-45) . The last equation can be made more useful by obtaining another expression for (1 - G) * or.; essentially, for V^G. If both sides- of Eq. (2-42) are multiplied from the right by G and the resulting terms rearranged, then VG Q = t G(l + t G ) Q (2-46) _ 1 0 and (1 - V G)" Q 1 =" [ 1 - t G(l + t G)" ] 1 Q 1 + t Q Q G 0 • (2-47) The superscript t on the "isotropic" Moelier wave operator denotes a transpose. At last, Eq. (2-45) can be written in the final form ti --ft-5 v i-fl- ( 2 0 - 4 8 ) 23 for the DWBA. An alternative derivation for Eq. (2-48) is given below. in The matrix element t ^ y (which is s t i l l an operator internal state space) can be written down from Eq. (2-33) as (-JIT/JLJS {jZ )^IJL>^ +S> In the term K, i terms of the distorted 1 J (2-49) can be rewritten in *- wave^^C^,-'! in the following manner. First, the solution to Eq. (2-38) can be found analogously to the solution of Eq. (2-15), namely - l p > + g£> V 0 X k & . (2-50) By means of the algebraic relation A" 1 - B" 1 = A" 1 (B-A) B 1 (2-51) the free particle Green's function, G^—\ can be replaced by 24 the expression (±) When Eq. (2-52) (t) (±) is substituted back into Eq. ( 2 - 5 0 ) , the distorted wave can be written exclusively in terms of I viz. Finally, the substitution of ^£''\ V l 3? £ ? + x ^ of Eq. I JJ^ (2-49) from Eq. yields (2-53) into , (2-54) where the superscript denotes an adjoint. In this case I the adjoint is taken in position space, but since j^G^^ ^ 26 is the identity in internal state space, the distinction of which space the adjoint is defined in, is pedagogical. step in Eq. ( 2 - 5 4 ) requires the recognition of Eq. The last (2-37), namely in the f i r s t and third terms of the second line of Eq. If Eq. ( 2 - 5 1 ) is used to express G (2-54). in terms of U G, u It i.e., G 0 = G H " H G V l0 <- ) G 2 56 where G^ is the Green's function for the total Hamiltonian HQ + VQ + Vj,. then l£'in exclusively in terms of s^C/ Eq. ( 2 - 3 7 ) can be written i ^ , viz. This equation (Eq. ( 2 - 4 6 ) ) allows X. in Eq. ( 2 - 5 4 ) to be K <§ re-expressed as 27 (2-58) The o p e r a t o r V^SL^ where t h e DWBA has been used. immediately i d e n t i f i e d can be as the " i s o t r o p i c " t r a n s i t i o n operator V A new t r a n s i t i o n o p e r a t o r t^ can be d e f i n e d Eq. (2-35) by u s i n g V^as the p o t e n t i a l function, as the G r e e n ' s namely H = where G and from v V i W + l + = V r = V 1 is s t i l l + + 2 V l ( G V l G t - H H 2 V G V 1 H G V l G ( ) - G (t H V l H )t 2 2 - V (2-59) 1 the G r e e n ' s f u n c t i o n f o r the t o t a l n Hamiltonian, H H H Q i n t h e term + V Q + V^. (?C^ / V E q . (2-59) appears i n Eq. (2-58) its p h y s i c a l i n t e r p r e t a t i o n as a m a t r i x element o f a t r a n s i t i o n ± + V j G ^ / ^ clarifying o p e r a t o r between two d i s t o r t e d waves, each governed by the isotropic potential, V ^ , but whose i n t e r a c t i o n i s governed by the a n i s o t r o p i c p o t e n t i a l , V . 1 28 The DWBA of this particular matrix element, namely (")C^ IvJ'X. can be further manipulated to yield (2-60) r - i where a natural identification of t ^ as 1-0. Q has been made. The adjoint of transpose o f _ Q . ^ + 3 f \ i"^o '' V can be re-expressed as the in the following way: + = Consequently t (2 becomes _Q.^V JTiL 1 n which is Eq. (2-48). -6i) CHAPTER I I I CALCULATION OF (t 1 J g In t h e p r e v i o u s c h a p t e r t h e e x p r e s s i o n Eq. (2-48) was d e r i v e d Xi. f o r t h e DWBA t o t h e a n i s o t r o p i c t r a n s i t i o n o p e r a t o r f o r a system w i t h two p o t e n t i a l s . t h a t Eq. (2-48) o r , e q u i v a l e n t l y , Now Eq. (2-60) has been e s t a b l i s h e d , m a t r i x elements o f t h e a n i s o t r o p i c t r a n s i t i o n •' o p e r a t o r i n momentum space can be c a l c u l a t e d once t h e i s o t r o p i c and a n i s o t r o p i c p o t e n t i a l s a r e s p e c i f i e d . o v e r a l l object o f t h i s c h a p t e r i s t o e x h i b i t the c a l c u l a t i o n o f t h e p a r t i a l m a t r i x element potential V V (r) n 0* (t ) — for a rigid sphere namely = OO , 0 where The r r ><r i s the diameter o f t h e r i g i d dipole-dipole nuclear (3-1) spheres, and f o r a s p i n i n t e r a c t i o n p o t e n t i a l V^. l a t t e r p o t e n t i a l i s g i v e n by The 31 V6> = 3tf£* where I_ and ^ (3-2) the nuclear spins of molecules 1 and 2 respectively, are separated by the relative coordinate r_. (t^l" from Eq. (2-60) can be written with in position representation as ( ^ = <x /y/x > fr) «"» ^- ft) jt «%, / **" (3-3) jfc 32 where the equality X (A) -Jt' has been used. - (3-4) Eq. (3-4) can be derived starting from Eq. (2-53) in the following manner, namely A! (3-5) 33 It is clear from Eq. (3-1) that the distorted wave (r) is the solution to a Schroedinger equation with a rigid sphere potential. For r j,(r_) s a t i s f i es (3-6) Since the rigid sphere (and, in general, any central potential) problem has cylindrical symmetry about the direction of the A A incoming momentum k £ z, the only angle dependence left in /O- A A ^(r_) is C7 =• k-r. Therefore, for a solution in spherical X ^l) polar coordinates ^ Legendre, polynomials, Jt > c a n ^ e ex P-(cos(9), P ded i an terms of as A JL-o n A (3-7) Eq. (3-6) now becomes >E JL A Jt*o * J. A. A, ^A, 34 = o . (3-8) Since the Pj^'s are linearly independent, "^fV^fkr) must satisfy (3-9) which can be recognized as the differential equation for spherical Bessel functions^. In order to satisfy the usual asymptotic boundary condition imposed by a scattering problem (see Eq. (2-18)), i is convenient to write ^ ^ ( k r ) , the solution to Eq. (3-9), in the particular form ^JL JL JL J (3-10) 35 h fl) ( k r ) , the s p h e r i c a l Hankel f u n c t i o n o f the f i r s t behaves a s y m p t o t i c a l l y as an outgoing 0~i 7 * kind ' s p h e r i c a l wave, i . e., -I, UA) (3-11) while the s p h e r i c a l Hankel f u n c t i o n o f the second k i n d , (2) h^(kr), function r e p r e s e n t s an incoming s p h e r i c a l wave. is, The l a t t e r (1) — the complex conjugate o f h ^ (kr) rz in fact, *- The p l a n e wave p o r t i o n o f the asymptotic on ^ ^ ( k r ) boundary i s f u l f i l l e d by the p a r t i c u l a r l i n e a r o f s p h e r i c a l Hankel f u n c t i o n s h (kr) Z condition combination which i s r e g u l a r at the o r i g i n , namely 1 * * first Ji J it (3-12) Note t h a t the s p h e r i c a l Hankel f u n c t i o n s k i n d appear t o be d e f i n e d differently o f the i n Messiah i n Morse and Feshbach, V o l . I I , p . 1 5 7 3 , b u t , i n f a c t , and b^(z) r e s p e c t i v e l y , definition books are e x a c t l y the same. o f the s p h e r i c a l d i f f e r by a minus Neumann f u n c t i o n s sign. 7 and the It i s i n the t h a t the two (z) 36 where Jjg_(^ ) r obvious h^(kr) 1 S real, t n e Bessel l i n e a r l y independent * and h ^ ( k r ) , L spherical function. The o t h e r l i n e a r combination o f namely JL Z (3-13) d e f i n e s the s p h e r i c a l Neumann f u n c t i o n n ^ ( k r ) , which i r r e g u l a r at the o r i g i n . Eq. (3-10) are chosen to rigid at r The c o n s t a n t s satisfy in the boundary c o n d i t i o n s on the sphere r a d i a l w a v e f u n c t i o n , " ^ ^ ( k r ) , both at i n f i n i t y and 5 1 CT. The former c o n s t a n t , comparing the asymptotic ..exp > ( i k « r ) X. and is Cg, can be determined by form o f the p a r t i a l wave expansion in JL Ot) A J (3-14) w i t h the ^ ^ + asymptotic ^ (r,6). expansion o f j ^ ( k r ) from E q . (3-10) The p a r t i a l wave expansion o f a p l a n e wave in is of 37 J-JnZ-A. (3-15) and as r approaches i n f i n i t y , ^ J-Jt-JhL becomes o o (3-16) The term (kr) describes the asymptotic form of the spherical Bessel function jj^fkr). Thus the two specific terms to be compared are ft —> ji £ (3-17) 38 and ^? r r Of) -I (3-18) From these last two expressions C can be identified as 0 (3-19) The second constant is found from the condition that for a rigid sphere potential, the wavefunction must vanish at the molecular diameter, i.e., X Ji (kr) = O -3/* AkJ)-\d(kZ)JL(kr) Ji i. (3-20) 39 Consequently, W^fkO") is the quotient (3-22) 40 It i s convenient to write (t^ )-|- from Eqs. (3-3) and (3-2) a s * (3-23) f 1 ~ where D i s e x p l i c i t l y Eq. given (with the use o f Eq. (3-22) i n (3-3) by D = (<A X'%) A: _3 JL ZL The n o t a t i o n symmetrized tensor I I I V?^ means the completely and t r a c e l e s s p a r t o f the g e n e r a l l^l^' ^,—1—2^^'' ^ S w r ^ t t e n o u t second rank explicitly i n Eq. (3-2' For a g e n e r a l d i s c u s s i o n o f C a r t e s i a n t e n s o r s , see r e f e r e n c e 8. where r i s t h e u n i t v e c t o r i n the x_ d i r e c t i o n . c o l l i s i o n s are e l a s t i c a l l y energetic, the Because the t h e magnitudes (but not d i r e c t i o n s ) o f the wave v e c t o r s -iV and k_ a r e the same. g The second rank t e n s o r integrations: magnitude r . evaluated D breaks up n a t u r a l l y one, o v e r angles and t h e o t h e r , o v e r t h e The r a d i a l i n t e g r a l will bt first. C o n s i d e r the f o l l o w i n g spherical i n t o two Bessel functions, two d i f f e r e n t i a l equations f o r namely c m obi A A. JJX. (3-25a) and A " dn. 3 a(kC) 'YL. =0. A (3-25b) I f Eq. (3-25a) i s m u l t i p l i e d by f ^ f k ' r ) , through by g ( k r ) , n then a f t e r s u b t r a c t i o n e q u a t i o n s , a simple i n t e g r a t i o n and Eq. (3-25b) o f the two r e s u l t i n g yields 42 = A, (3-26) The prime denotes differentiation with respect to r, and the arguments of f respectively. and g^ are understood to be k'r and kr For elastic scattering (k' = k) this last equation becomes A, (3-27) The following equation JL+I (3-28) 43 is valid for a l l spherical Bessel functions and for those linear combinations of spherical Bessel functions whose coefficients are independent of both X <i kr. Thus from this an recursion relation, ^-j^(kr) in Eq. (3-22) can be rewritten as AZ(k^=(**iS'Ji[X A s L JL-I -tX +(w JL+l AH JL-i JL JL Mi JL-\ J (3-29) The form of ^ ^ ( k r ) as given in the last equation breaks up the radial integral djx. into four * A ~7C other integrals, each of which can be easily integrated by V means of Eq. (3-27). These four integrals are: oo UA*I) A A-l Jl! Ji' (3-30a) JL~l) ? 44 Ai ~~\ fx +0 V (.91 JL' (bd X (knddjx. JL-+I A Yz x'-xx'\L fat+iAJL'fo (3-30b) <=>© -—: L A J Al JL-I U x'-X (SJLU) \ £rl A' A' jLU'+^-jLa-i) (3-31a) 0£> M \ A-(J 45 It is to be understood f o r the remainder o f t h i s if , j Similarly, been w r i t t e n , W - w i l l be a shorthand n o t a t i o n f o r W fk0") . JL The f i r s t that there t h i n g to n o t i c e about E q s . (3-30 a) and (3-30 i s no c o n t r i b u t i o n t o the i n t e g r a l s at lower l i m i t because o f the boundary c o n d i t i o n on for a r i g i d same reason sphere p o t e n t i a l .(kr)h' o f the (kr) A' 1 A and (3-31 b) r e s p e c t i v e l y , set o f equations and (See (kr)h' A Eqs. (3-30a), i n f i n i t y w i l l be d i s c u s s e d relation^, this For the i n Eqs. (3-31a) ( 3 - 3 1 a ) , and i n the first. l i m i t as r Another r e c u r s i o n time f o r the d e r i v a t i v e o f s p h e r i c a l Bessel and a p p r o p r i a t e l i n e a r combinations t h e r e o f , is below d <t \<r i (3-32) i s used to r e w r i t e A. from JL-\ l i m i t can be e v a l u a t e d , i . e . , X JL-\ JL' Eq. ). at effort. approaches Eq. (kr) (3-30b), The r i g h t hand s i d e o f E q . ( 3 - 3 0 a ) given (kr) E q , (3-20) (3-30a) so t h a t the -X X JL' b) their JL' 4+1 v a n i s h at r= <T*. The e v a l u a t i o n (3-31b) at i n f i n i t y r e q u i r e s more functions as then the argument i s k r . JL r= Cf that , and h ^ appear w i t h no f u n c t i o n a l dependence, they have j u s t is chapter 0 3 2 ) 46 Jt X X' /L" t-l - X X' JL' [X \jkL -wJ. )+ Jl X~ -2 _ / ^ + AG <^/}X -w J. ) (5" (3-33) where M(jt',l) is defined as (3-34) In order to actually calculate the limit in Eq. (3-33) the asymptotic expressions for h.(kr) and j (kr) as given in a Eqs. (3-11) and (3-16) respectively, are needed. Now the 47 right hand side of Eq. (3-30b) can be written down by inspection, for r approaching infinity, from Eq. (3-33), namely -A, - ^ A H / _ X A. X'-x -/ X ] JtH JL' (3-35) where Nti\l) s JL U'+l)-U+i)U+Z ). , s (3-36) The recursion relation Eq. (3-32) is used again to find the limit at infinity of the right hand side of Eq. (3-31a), i . e., U \ x'-X JL' £ \ JL' JL-l 1 4 J£I JL' JL'-I JL-Z JL> 48 - (V) A - K / ") . (3-37) By i n s p e c t i o n o f E q . (3-37) the l a s t upper l i m i t , t h a t Eq. ( 3 - 3 1 b ) , can be w r i t t e n down. The r e s u l t in is u x x - r\ 9 NCI] A? A¥\ v JL' A' AH) .-A+A (3-38) The o n l y r e m a i n i n g e v a l u a t i o n s calculations (3-31b), are the two t r i v i a l on the n o n - z e r o lower l i m i t s o f E q s . (3-31a) and namely K A-/ Ji) JL x' &-i A f (3-39) 49 and JL N(A*j£) - -J?** SI X' . L- (hr>- W JL (V>L- X (kh-U (3-40) Eq. (3-32) has been used again in the above two equations. Finally the radial integral in Eq. (3-24) can be written down from Eqs. (3-33), (3-35), (3-37), (3-38) (3-39), and (3-40) as J cr at ^ (MN)~'[ JZrl A' J (3-41) (Vii. 50 w i t h the f o l l o w i n g two r e s t r i c t i o n s , l! namely A-1 (- ) 3 42a and A The reason t h a t * 1 Eqs. A+\ . (3-42a) and (3-42b) must be s a t i s f i e d t o keep the denominator i n E q s . Eq. (3-42b) (3-30a,b) and (3-31a,b) (3-41) can be f u r t h e r s i m p l i f i e d by l o o k i n g at Wronskian o f the s p h e r i c a l Bessel is finite. the f u n c t i o n with a s p h e r i c a l Neumann f u n c t i o n . The Wronskian i s j (z)n'(z) A A - j'(z)n A (z,) = -z (3-43 2 where the prime now denotes d i f f e r e n t i a t i o n w i t h r e s p e c t From the r e c u r s i o n r e l a t i o n E q . ( 3 - 3 2 ) , and (3-13) f o r j (z) and n (z) A be r e - e x p r e s s e d and from E q s . respectively, to (3-12) E q . (3-43) can A i n terms o f s p h e r i c a l Hankel f u n c t i o n s as 2 -z = j„(z)n Ji (z) - j Ji-) = i/2\h £~i (z)h* Li-/ A (z) (z)n (z) A - h* (z)h JL-\ A (z)"l. (3-44) J An example o f how the Wronskian from E q . (3-44) can be used i n E q . (3-41) is ) $z. 51 j Q.+I (k<r) - w (k<r)h (k<r) J*+J A (1/2)[h (k<r) + h* (k(ril (i/2)^h (k<r) + h*(k<r Jlh (k<r) l JC A h (k<T) J 0+1 A (3-45) k <r h^(k<T) 2 2 Furthermore, the entire second term in Eq. (3-41) can be rewritten by means of Eq. (3-44), i . e., -Jk j (T Jt'-I A' A'-l (J -VJX \ + [J -VJA J MU'A) _ IN(A'A) - / J J A' (3-46) Ji' A where the arguments of j , W , and h are, of course, k<T. 52 Now Eq. (3-41), i f the e x p r e s s i o n i s re-expressed functions only, (2W^ - 1 ) , which a l s o appears i n i n terms o f s p h e r i c a l Hankel namely JL JL - / A/kr) JL A where Eq.. (3-12) can / (3-47) k for ] i ) has been used a g a i n , then E q . (3-41) be w r i t t e n as a q u o t i e n t o f s p h e r i c a l Hankel Finally, functions. from E q s . (3-46) and ( 3 - 4 7 ) , E q . (3-41) becomes oo J* X JL' (JtHSXik^J*. JL T JL JL a/ r A JL j k r ) X' (3-48) 53 with the r e s t r i c t i o n s i n E q s . (3-42a) and (3-42b) The i n t e g r a t i o n s Eq. still valid. over a n g l e s which remain i n D o f (3-24) c o u l d be c a r r i e d out here e x p l i c i t l y . The c a l c u l a t i o n s would i n v o l v e the f o l l o w i n g e x p r e s s i o n , namely 3 <=2 (3-49) and then c o n t r a c t i n g each s i d e o f E q . (3-49) w i t h each o f t h r e e k, A, - k ' second rank t e n s o r s i n o r d e r to f i n d the the constants B, and'G- The t h r e e c o n t r a c t i o n s i n t o the r i g h t hand s i d e o f E q . (3-49) left are s t r a i g h t f o r w a r d . The c o n t r a c t i o n s i n t o hand s i d e o f the same e q u a t i o n r e q u i r e t h a t the the Addition Theorem f o r s p h e r i c a l harmonics be used to uncouple - k ' » r and k « r so t h a t the r - i n t e g r a t i o n can be done. However i t be shown i n the next c h a p t e r t h a t E q . (3-49) i s not will needed, and the angle i n t e g r a t i o n s can be c a r r i e d out i n a d i f f e r e n t , 54 and simpler, manner. Now Eq. (3-23) can be fully written out as oo JL (3-50) where ^I^I_2^|''^ denotes the completely symmetrized, traceless tensor. t' CHAPTER IV RELAXATION TIME Chen and S n i d e r have d e r i v e d an e x p r e s s i o n spin-lattice f o r T ^ contains possible t h i s t h e s i s the potential f o r the r e l a x a t i o n phenomenon to o c c u r . intermolecular potential dipole-dipole nuclear spin i n t e r a c t i o n . Chapter I I I w i l l be i n s e r t e d i n t o the In f o r the monatomic gas w i l l be approximated by a r i g i d sphere p o t e n t i a l Snider, gas. the p a r t i a l m a t r i x element o f the t r a n s i t i o n o p e r a t o r whose i n t e r a c t i o n makes i t the r e l a x a t i o n time T ^ f o r a d i l u t e monatomic This expression ts- for Thus, (t^ and a )— from e q u a t i o n o f Chen and namely X 56 •v. (4-1) and the relaxation time will be approximately evaluated by analytical methods. as follows: The symbols which appear in Eq. (4-1) are n i s the number density; m is the mass of one molecule; -J^£ is Boltzmann's constant (which is to be distinguished from the wavenumber k); (.1^/ is the expectation 2 value of the spin operator I ; T is the absolute temperature; Q = Qj = Q2 molecule; 1 S t n e and internal state partition function for one is the reduced relative velocity. The last quantity is appropriately defined in terms of the relative velocity g as It- = 57 w h i l e the (L , i s i n t e r n a l s t a t e p a r t i t i o n f u n c t i o n f o r molecule 1, g i v e n by (4-3) H is the e x t e r n a l magnetic field, and d e f i n e d p r e v i o u s l y i n Chapter I I . internal s t a t e s o f molecule ^H. t and I. The t r a c e T r ^ i s (4-4) exp f .ffi. H » I ) , can be used to j u s t i f y 1 from E q . (4-3) and keeping o n l y the f i r s t internal with i t H*I., , the term exp series. (4-4) expanding the field. exponential, i n a power s e r i e s about term. state Hamiltonian associated the e., AT « the magnitude o f the a p p l i e d magnetic Eq. over Chen and S n i d e r used the f o l l o w i n g h i g h temperature a p p r o x i m a t i o n , i . where H i s been 1. Now i n o r d e r to d e r i v e E q . (4-1) t£n have Because, in general, has a much l a r g e r zero the energy than does the n u c l e a r Zeeman H a m i l t o n i a n , (-J^/J&T) Consequently, is i s not expanded i n a power 58 (4-5) where Q ^ is the electronic partition function and (21^ + 1) is the degeneracy of the nuclear spin space. 2 The factor Q ^ also arises from the exponential term exp ( \ in the numerator of Eq. (4-1). Consequently, Eq. (4-1) becomes -Yi-yyy. (if<-v*t ^t~Tj xtx+i) (-ax+i}'*' JL v' ~,~2. > £ I g (4-6) 59 g' ' is the adjoint of (t^ f+) )^g' in spin space. (t^f+) )j^ For the particular case of Eq. (3-50), D acts as a coefficient in front of the operator ^ I ^ I ^ " ^ ^ . Since both and are Hermitian operators, they are unaffected by the adjoint */*' while the coefficient D is changed to D*. The traces over the spin states of molecules 1 and 2 (+) g' f+) g'"^" in Eq. (4-6) can be carried out by writing (t^ Jj, * ^\ )g in terms of D and as in Eq. (3-23). Then the result anc is •» Z- '-/'-'X.J -v I t ~2- l L ~ / '-• X. J Vr"'"-^-! Z ) ~/ j ~-7. 2 (4-7) 60 Consequently, Eq. (4-6) is further reduced to T - 3 -rum. IO - " 4 c x < A ^ y W ^ / ^ (4-8) 61 where Eq. (4-2) has been used to write the reduced relative velocities in terms of the wave vectors. As was mentioned at the end of the previous chapter, A i t is possible to perform the r-integration in D in more than one way. The alternative method to that associated with Eq. (3-49) uses the property of the isotropic, symmetric traceless tensor which appears in Eq. (4-8), i . e., (4-9) or, equivalently, that JJ « « ^ f=- a x: tz (4-10) 62 Eq. (4-10) will be integrated out completely in the following paragraphs. 7 The Addition Theorem for spherical harmonics is used to separate the wave vector angle dependence from the relative coordinate angle dependence in the Legendre functions in the JL LL ' JL X / ,* JL JL + JL' •a? X L Jl+JL -AA. JL JtL i X vi-J.' -rn-X b In It- 63 r . M-L « ~ a III 7 It 1/5 (4-11) Now i f r is chosen as the fixed reference axis, then Eq. (4-11)can be further simplified to ( 2.i &J > > LL'-O <-» A A x + 6/) A 64 LA I— » (-1) X L + 4 f <r-0 J L ^ ^ L+L. _ " i t ^ u j In order to complete the next step the following two results are needed, namely (4-13) 65 and JL 1 4T J K ?T JL (4-14) Use of Eqs. (4-13) and (4-14) further facilitates the calculation in Eq, (4-12), i . e., M'*-L' M=-L \ L °W AL'YLOH °Jl L 1 A! X C-i) X * JL t* J- * «•'> ^ , - -A, / -2: -^JL — X [ A' A x C&JL'+LX^JL+I) /jL 1 6 1 JL 6 OA o) C 67 -JL -4 (4-16) where the argument, k ( T , o f t h e s p h e r i c a l Hankel functions w i l l be denoted by z from now o n , v i z . cr s The f u n c t i o n A , Az\ JL JL 4£ (4-17) i s defined as tPJt+3YJl«Wt-r> 3 (4-18) and w i l l h e n c e f o r t h to E q . ( 4 - 1 6 ) . defined be r e f e r r e d to as a " d i a g o n a l " c o n t r i b u t i o n The " n o n - d i a g o n a l " c o n t r i b u t i o n A . , (z) is as 3L A 1 JL+Z &A+3)(jZ+Z)(6+h X 1*1 (4-19) JL MS. ) 68 The r e s t r i c t i o n on JL - v a l u e s as s t a t e d i n Eqs. (3-42a) and (3-42b) i s a u t o m a t i c a l l y accounted f o r i n t h e A ^ ( z ) ' s because 7 of the 3-j symbol which appears i n Eq. (4-16). £' JL A O o o A±l A o o o That i s , f o r z \ - o (4-20) because i s odd f o r every v a l u e Now the o n l y i n t e g r a t i o n l e f t o^X- i n the e q u a t i o n f o r the r e l a x a t i o n time i s t h a t over the energy parameter z, namely V* = ± yixtt+b (JL -CZ 3 djt X \*JL (4-21) where the c o n s t a n t c _g i s d e f i n e d as (4-22) 69 In o r d e r to complete summed o v e r and the the calculation of t h e j ^ ' s must be z-parameter must be i n t e g r a t e d out explicitly. It c 2 is which i s a p p r o p r i a t e at t h i s p o i n t defined i n Eq. ( 4 - 2 2 ) . to d i s c u s s the I f an e s t i m a t e o f parameter its 2 magnitude as a guide c o u l d be made, then the magnitude o f c i n choosing to c a r r y out the the o energy the type o f approximation with which integration (i. e., z-integration) ,in 129 for a Xe atom at r e l a x a t i o n time. Thus, f o r example, 2 -4 i s a p p r o x i m a t e l y 1.25 )C 10 , i. 300 K . , c could serve e., (4-23) CHAPTER V A LINEAR APPROXIMATION TO T " The central i s s u e i n t h i s and the next c h a p t e r i s f i n d a decent a p p r o x i m a t i o n to the q u o t i e n t functions (4-19). Eq. which appears Originally, i n the A ^ ( z ) ' s an attempt JL-value (4-21.)" f o r each from 0 to f o r t e r m i n a t i n g the came from the fact that like^" namely the Eq. first entire became apparent t h a t 3 quotient fruitless investigations infinite sum. Unfortunately, the infinite sums by i n t e g r a t i o n s same t i m e , 9 a further decision was made. % of it soon in each the further solving to approximate The remainder o f w i l l d e a l with a " l i n e a r - i n - w " a p p r o x i m a t i o n 2 z h (z)h*(z) (4-19) r e s p e c t i v e l y , formula was made a f t e r i n t o r i g o r o u s methods f o r At the to the p r o d u c t s series after The d e c i s i o n to approximate (4-21). and term i n each n i n e terms c o n s t i t u t e 99.5 Eq. then, 10 and f o r the Riemann-Zeta f u n c t i o n , o f s p h e r i c a l Hankel f u n c t i o n s t h i s chapter, The the p a r t i c u l a r q u o t i e n t s i n v o l v e d integration. integrate s e r i e s at (4-21) were d i v e r g i n g , r a t h e r than c o n v e r g i n g , succeeding Hankel (4-18) and 10 i n c l u s i v e . the g e n e r a l > £ ' t h behaves b a s i c a l l y the v a l u e o f the of Eqs. infinite to of spherical was made to r i g o r o u s l y justification Zl l / * £ ^ , 1 2 and z h (z)h*(z) i n E q . (4-18) and w i t h the E u l e r - M a c l a u r i n sum i n o r d e r to f a c i l i t a t e -1 an approximate e v a l u a t i o n o f T^ . 71 The E u l e r - M a c l a u r i n sum formula i s g i v e n by t ° ~*~o more c o r r e c t i o n terms (5-1) where the summation i s is r e p l a c e d by an i n t e g r a t i o n . v a l i d f o r n approaching i n f i n i t y and, i n f a c t , Eq. (5-1) in this t h e s i s n can be d i r e c t l y r e p l a c e d by i n f i n i t y . The constant is the equal t o one s i n c e c o n s e c u t i v e v a l u e s o f ^ , parameter, d i f f e r by one; v a l u e o f JL It term ^g( ) z i n the is and the constant is h summation the lowest sum. convenient at t h i s p o i n t to d e f i n e a new d i a g o n a l ^ JL JL £ju ( * ) £ ( £ JL JL (5-2) 72 and, similarly, a new non-diagonal term A , (z) by (5-3) An approximation to ^g( ) will be considered f i r s t . z The product of the power series expansions for zh (z) 7 and zh*(z), namely (5-4) up to terms quadratic in (1/z), allows A^( ) z t 0 D e expressed as A^vi) * [Q/*)jLU+d . (5 5) where w is defined by IV = ? 2 = (A<r) ^ . (5-6) 73 The a p p r o x i m a t i o n which i s c a l l e d the i n E q . (5-5) " l i n e a r " a p p r o x i m a t i o n to the d i a g o n a l Now, o f c o u r s e , diagonal exhibited E q . (5-5) term A , can be s u b s t i t u t e d will be term A . i n t o the whole (w) which, upon keeping o n l y the main term i n the E u l e r - M a c l a u r i n sum f o r m u l a , becomes f J U L Jl o o JLlL - - (5-7) 74 where x i s d e f i n e d by x — The r e s u l t i n E q . (5-7) term o f E q , (4-21) (5-8) can be put back i n t o the d i a g o n a l i n o r d e r t h a t the w - i n t e g r a t i o n be completed. Consequently, time MI+t). the d i a g o n a l c o n t r i b u t i o n to the r e l a x a t i o n is o o oO I 3 3 L C (5-9) The e x p o n e n t i a l integral £ - E i ( - y ) \ ^ is o o d e f i n e d by T V (5-10) 75 where 0 fr., the Euler-Mascheroni• constant , has the numerical ±u 1 0 value 0.577215. . . . The linear approximation to the non-diagonal term A^,^ ^z) ^ P ^ S term: e r o r m e ^ analogously to that o£ the diagonal that i s , the power series of zh„(z)zh* JL JL+Z ^ - £ E (-/) I zA&JL A (z), namely ^ U+A+A\V.U+&1£ (5-11) is cut off at terms quadratic in (1/z) . Similarly, the power series expression for Re| z h (z)h* (z)| (z)| is cut off at 1/z . 2 2 - _ L Consequently, A , JL JL+Z JL (w) becomes JL+£ J (5-12) _ 2 where again w = z and the "linear" approximation actually refers to the parameter w rather than z. Now the f u l l nondiagonal term A^, JLJL+& sum formula, becomes , with the f i r s t term of the Euler-Maclaurin ' . 76 n A j M ML 2 J o L ^H+jt+A JVI+A + 5 A + 6 (I-%H)' 4 (5-13) From Eq. (5-13), t h e non-diagonal c o n t r i b u t i o n t o the r e l a x a t i o n time, Eq, (4-21), i s oO /Lit* 0 [w J C (5-14) 77 Thus the complete e v a l u a t i o n l i n e a r - i n - w approximation o f T^ from E q . (4-21) f o r the is _/ T j_ (f£^-ni(T+i) f-rr AT) Lc (5-15) For the d i a g o n a l term, namely contribution, the f i r s t correction l/2^g(oo) + g ( x ) ] , from E q . (5-1) o f the E u l e r Q M a c l a u r i n sum formula is A Ji Jw) + A/w) /o(vi+iy. L (5-16) And s i m i l a r l y t h e f i r s t c o n t r i b u t i o n as g i v e n Eq. (5-13) is c o r r e c t i o n term o f the n o n - d i a g o n a l i n the i n t e g r a n d o f the f i r s t line of 78 Ao o 6v/) J3& f t * <X -f AJ\J) . 3 J_ O + (5-17) e2 Eqs. (5-16) and (5-17) the r e l a x a t i o n time, c o r r e c t i o n terms. can be put back i n t o t h e e x p r e s s i o n E q . ( 4 - 2 1 ) , to f i n d the magnitude The r e s u l t for o f the is -cot*. •/& +('/5Y/+c*)j. l-£z(-c*)\ C (5-18) None o f the terms i n E q . (5-18) c o n t r i b u t e to the 2 dominant 1/c term i n the main c o n t r i b u t i o n , E q . (5-15), to the relaxation time. This fact is considered to be sufficient justification for using the Euler-Maclaurin sum formula to change the infinite sums in the original problem to integrations. CHAPTER VI A QUADRATIC APPROXIMATION TO T In -1 1 t h i s c h a p t e r t h e p o l y n o m i a l s o f z h ( z ) h * ( z ) and JL 2. z h(z)h* JL JL+Z (z) i n Eqs. (5-4) and (5-11) r e s p e c t i v e l y , w i l l be -4 -2 t e r m i n a t e d at z analogous JL r a t h e r than a t z t o t h a t used . Then a procedure i n the previous chapter w i l l out i n o r d e r t o o b t a i n a b e t t e r approximation be c a r r i e d to the -1 r e l a x a t i o n time T^ The Eq. from Eq. (4-21). q u a d r a t i c - i n - w d i a g o n a l term can be found from (5-4), namely X (6-1) I f Eq. (6-1) i s put i n t o t h e f u l l d i a g o n a l term A^ ^ ( ) w a n i f o n l y t h e main term o f t h e E u l e r - M a c l a u r i n sum formula, Eq. ( 5 r l ) , i s kept, then t h e r e s u l t i s d 81 C A _s a s fa) /f ^ -f...» 4X CLIL W+(*M)(TL-£) J - 3W(W%L) / S(w+i) /6<Vf/V5W -f 3 W z /*tefeW^3W-%)?'* (6-2) 82 where x i s d e f i n e d second l a s t integral as i n Eq. (5-8). l i n e to the l a s t i s o f the general The i n t e g r a t i o n from t h e l i n e o f Eq. (6-2) i s e x a c t . The form (6-3) where S i s d e f i n e d as (6-4) Integral (6-3) can be found i n any book of i n t e g r a l s . i n the l a s t 11 which l i s t s The i n t e g r a t i o n over energy o f the f i r s t l i n e o f Eq. (6-2) i s s t r a i g h t f o r w a r d . i n t e g r a t i o n o f the r e m a i n i n g two terms, however, more e f f o r t . discussed namely term The requires The aforementioned energy i n t e g r a t i o n w i l l be i n the f o l l o w i n g For tables paragraphs. convenience the f o l l o w i n g definition w i l l be made, 83 9^ (6-5) The energy i n t e g r a t i o n o v e r f(w) s t r a i g h t f o r w a r d manner. Because the f a c t o r L-(5w Thus f(w) f(w), y Thus f(w) 3 w + / 2 (w + 3/2) to expand f(w) = a i n an in 3/2 (6-6) can be w r i t t e n i n terms o f y i n the f(y) appears l o g a r i t h m and i n the o t h e r term i t was found to be convenient i n v e r s e power s e r i e s in a must be approximated. + 3w - 9 / 4 ) J both i n the argument f o r the of cannot be done e x a c t l y + Where the remainder F(y) b/y is, + d/y + obviously, F(y) form (6-7) 84 F(y) = f(y) - a - b/y - d/y (6-8) 2 Consequently the complete energy i n t e g r a l o f Eq. (6-5) i n terms o f Eqs. (6-7) and (6-8) i s /> 3 0 0 -3- . (6-9) In t h e i n t e g r a l Jexp I exp (-c w) F(w)dw 2 the exponential i s unnecessary f o r convergence s i n c e F (w) behaves, at l e a s t , as _2 y . T h e r e f o r e t h e e x p o n e n t i a l can be expanded 2 series. Because t h e parameter c i s so s m a l l (see Eq. (4-23) ), t h e power s e r i e s 2 at terms l i n e a r i n c . i n a power can be cut o f f e x p l i c i t l y That i s , (6-10) since [.exp (-c w) 85 1 2 i + c wj is 1 1 of the order of c . Thus Eq. (6-9) becomes 3c o _ c (6-11) 86 The constants a, b, and d must be calculated explicitly, and the magnitude of the two integrals, namely oo (6-12) and (6-13) must be estimated. The calculation of a, b, and d will be exhibited f i r s t . a, b, and d are found from the asymptotic form of f(y) in Eq. (6-7) , i . e. , f(y) y -> -> a + b/y + d/y 2 (6-14) In order to identify a, b, and d i t is useful to notice that various powers of the quotient X= .c 5 - /a (6-15) 87 appear in the expression for f(w) in Eq. (6-5). Now for w approaching infinity, x i t s e l f can be approximated in the following manner: Oh •7 ^ J51 ( I + A J L ) (6-16) where u is given by u = -6 / 5(w + 3/2) (6-17) This approximation for x can be used to simplify the argument of the logarithm in Eq. (6-5), namely (6-18) L/-X\/5J 3 it 88 where the fact that u approaches zero as w approaches infinity is used in the expansion of the logarithmic terms. Thus the whole second term in Eq. (6-5) can be approximated by J+7L 33L % < W / a ) 3 : I - X ~9w 3 ^ 9 /* /_ 3 - 5 / 5 + 0(AA?) 89 - 9 3 . ^5)l(^Va^^^l n . 0(l (6-19) 90 By p r e c i s e l y the same t e c h n i q u e s , Eq. (6-5), the f i r s t term o f f(w) f o r w a p p r o a c h i n g i n f i n i t y , can be expressed in as 731V = Z / 1 _ 3 /_ xL/v- S(w+*/si) + / Mto+v^y- +0(~j-*)s (6-20) The two r e s u l t s i n E q s . (6-19) and (6-20) c o n s t i t u t e explicit e v a l u a t i o n o f the c o n s t a n t s for f(y) i n Eq. (6-14). identified as the i n the asymptotic T h e r e f o r e , a , b , and d can be expansion 91 a = 6(arctan (3/160) (5) (3/800) 11_. + (5) ) 0.0391, (6-21a) 2 27(arctan 2 (5) (5) ) 2 % (6-21b) -0.0726, and d = 7 (3/2000) - 243(arctan ( 5 ) ) 2 4 (5) 2 •0.0363. (6-21c) In o r d e r t o complete the e v a l u a t i o n o f the r i g h t hand s i d e o f Eq. (6-11) the magnitudes o f t h e i n t e g r a l s 1^ and I i n Eqs. (6-12) and It was concluded and (6-13) r e s p e c t i v e l y , must be a f t e r several numerical investigated. i n t e g r a t i o n s on 1^, a f t e r s e v e r a l graphs o f the exact i n t e g r a l were drawn, t h a t t h e c o n t r i b u t i o n o f 1^ t o Eq. (6-11) i s n e g l i g i b l e (of the o r d e r o f 10 2 (a/c ) ^ ) compared t o , f o r example, the main term 2 3 x 10 . the i n t e g r a n d F i r s t l y , because o f the s i m i l a r i t y o f i n 1^ t o t h a t i n 1^ and s e c o n d l y , because o f the 2 fact that c m u l t i p l i e s I„, i t was s i m i l a r l y concluded that 92 the integral 1^ contributes negligibly to Eq. (6-11). Therefore, the complete energy integration for the quadratic-in-w approximation to the diagonal term, Eq. (6-2), in the relaxation time is i.e. J © O 0 J -X. 1 a <> 3 3 tic (6-22) In order to complete the calculation for the relaxation time, the quadratic-in-w approximation to the non-diagonal term 93 A (z) must also be obtained. From Eq. (5-11), A (w) becomes A (W) (6-23) ^ ('/*>>(Jl+zfu+if The denominator in Eq<. (6-23) is such a complicated function of JL that the main term of the Euler-Maclaurin sum formula cannot be rigorously integrated. Consequently both factors in the denominator were approximated by the same expression, namely Cw + (3/8) (jl+ 2) ( JL + lj\ . Js 2 Now the whole non-diagonal term can be further approximated by the principal term in the Euler-Maclaurin sum formula to become 94 4 ^M[f/-tJf)*-t] , [6-fJl)y-fl* (6-24) In Eq. (6-24) the three integrals with the factor (2 Jl + 3) have straightforward JL- and w-integrations. Each of these three integrals is of the form I given in Eq. (6-3). The ^-integrations and their subsequent w-integrations yield the following results: 95 9 X V/- lyp _ y£ 74 V 3 (6-25) 96 The t h r e e remaining i n t e g r a l s , preceded by t h e f a c t o r (2 JL + 3) \ a l s o have j£.-integrations straightforward. Eq. (6-3). integral which are The i n t e g r a l s a r e o f t h e type I g i v e n i n In each o f the t h r e e succeeding w - i n t e g r a t i o n s an i n the form o f (6-26) appears. E q . (6-26) s t i l l must be estimated i n o r d e r to complete t h e e v a l u a t i o n o f t h e q u a d r a t i c a p p r o x i m a t i o n to the non-diagonal c o n t r i b u t i o n to the r e l a x a t i o n t i m e . A brief clarify d i s c u s s i o n o f the i n t e g r a n d i n E q . (6-26) the method used to c a r r y out the i n t e g r a t i o n . will First o f a l l , t h e r e i s no problem at the o r i g i n because, by l'Hopital's rule, i. the l i m i t .as y approaches zero i s finite, e., (6-27) 97 Secondly, as long as the exponential exp (-c y) is left alone 2 and not expanded in a power series in c , the integral in Eq. (6is finite. Finally, i t would facilitate the integration i f the logarithm, ln (1 + series expansion in x. 2 x), could be replaced by a power Unfortunately, with such an expansion the powers of (1 / c ) would increase after the w-integration, and from the linear-in-w approximation in Chapter V, the main contribution to the relaxation time should be no larger than CI / c ). 2 A method which avoids the aforementioned difficulties 2 entails splitting the integral I (c ) into two parts, namely oo 7 ^ ^ s •4 (6-28) and expanding each integral in a different manner, u is defined by 2u 5 <^y. (6-29) The f i r s t integral which appears on the right hand side of 98 Eq. (6-28) can,now be evaluated by expanding the exponential , 2 . in powers of c , I . e., JL CJJUL. Jt^(l-t^AA^) 2. (h^ J)JU(MjJ)MJ- + V£ ^''+3JJ^JUO+J^CU. (6-30) where q 3 ( c /c< 2 ). (6-31) Because the f i r s t coefficient in front of the second 4 -8 integral on the right hand side of Eq. (6-30) is c ( <5S' 10 ), and because the succeeding coefficients increase in powers of c 2 6 starting from c , Eq. (6-30) can be reduced to '(6-32) 99 12 The dilogarithm function L ^ C y ) is defined by S L* (S) Z = -^JLrL. O-JUL) aLLL. . (6-33) Now the second integral which appears on the right hand side of Eq. (6-28) is more difficult to evaluate. If the logarithm is rewritten, then the integral under consideration can be split into two contributions, viz., o o (6-34) 100 Thus the problem has been shifted to evaluating the integral oo oo (q) = |[ e x P (-2qu)"^ (In u)du . (6-35) / The next few paragraphs will deal with the evaluation of I(q). 13 By means of the following tabulated integral , namely oo OO (6-36) I(q) can be expressed in terms of the exponential integral 101 J fexp (-2qu)\ ln (1 - (2u) *)du . £-Ei (-q)\ and the integral t J u The latter quantity is explicitly calculated as follows: oo JUL oo _ = 0 0 -2.A.AA. -/ \JL MA, / + JL atu. * J. J JUC / 00 + JL T AA, (6-37) The last integral in Eq. (6-37) can be approximately evaluated by the usual trick of adding and subtracting (1 - 2qu) to the integrand, namely 102 Muu-_ JUL. oO ( I ' ^ j J ) A*, (l " <£juu) -j. I i JUU. (6-38) so that one (the last one on the right hand side of Eq. (6-38) ) of the integrals can be eliminated on the basis of the argument that the leading coefficient of this integral is of the order of c . There is no problem about the integrand in V (1 - (2u) ) + (2u) ]du at 1 (exp (-2qu) - 1 + 2qu) u / _1 infinity since the factor (u) (exp (-2qu) - 1 + 2qu) _± -1 multiplied by the term-by-term expansion of the logarithm _3 behaves at least as (u) (exp (-2qu) - 1 + 2qu) at infinity. The integral Eq. (6-38) remains to be evaluated. namely I I aioL from It can be done exactly, 103 o o _ -I ^ J '/z o jt <3L i I ©2 J vx^ " 2 (6-39) Now I(q) from Eq. (6-35) can be expressed in terms of Eqs. (6-36), (6-37), and (6-39) as * ->L + u A ( ' £ > + ± i - £ j t f - f $ (6-40) 104 Finally, Eqs. 2 I ( c ) from E q . (6-26) can be w r i t t e n down from (6-32), ( 6 - 3 4 ) , and (6-40) as oo -2 (6-41) 2 where, o f c o u r s e , q =• c / o ^ Now the energy i n t e g r a t i o n s o f the set o f t h r e e preceded by the f a c t o r ( 2 j £ , +-3) integrals * i n E q . (6-24) a r e g i v e n by 105 (6-42) 106 It is interesting to notice that there is no (1 / c ) contribution to the relaxation time from Eq. (6-42). The numerical value of the relaxation time for the quadratic-in-w approximation can be obtained by using Eqs. (6-22), (6-25), and (6-42) i n E q . (4-21). of the quadratic evaluation of T ^ The portion in Eq. (4-21) that is of principal interest is the (1 / c^) contribution which comes from Eqs. (6-22) and (6-25), namely 7*r _ / (6-43) The only calculations which remain to be done are those pertaining to the f i r s t correction term of the Euler-Maclaurin sum formula, Eq,' (5-1), for.the diagonal and non-diagonal contributions to the relaxation time. The correction term for A„ .(w) in Eq. (6-2) is 4>JL — L eo / (6-44) 107 which is exactly the same as that for the correction term to the linear diagonal contribution to the relaxation time, namely Eq. (5-16). The correction term for A (w) in Eq. (6-24) is (6-45) After the energy integrations are performed on the last two equations and the results are multiplied by the appropriate factors, then the correction to the relaxation time, (T^) , can be written down as ^ 1 'corr ^ Vf) ~J^T \7Zk~r) (6-46) Use of the Euler-Maclaurin sum formula in Eqs. (6-2) and (6-24) has thus been justified by the fact that the f i r s t correction terms do not yield a contribution in the form 2 of (1/c ) to the relaxation time. CHAPTER V I I SUMMARY A comparison o f t h e r e s u l t s f o r t h e p r i n c i p a l term 1/c between t h e l i n e a r and q u a d r a t i c approximations t o the r e l a x a t i o n time, i s made below. In t h e l i n e a r d i s c u s s e d i n Chapter V, t h e energy c o n t r i b u t i o n , Eq. i n t e g r a t i o n o f the diagonal (5-9), y i e l d s a n u m e r i c a l f a c t o r o f 4 (8/3) T r approximation 2 f o r t h e 1/c term. The analogous r e s u l t from t h e 4 non-diagonal c o n t r i b u t i o n , Eq. (5-14), i s 87T complete e x p r e s s i o n f o r T . From t h e : Eq. (4-21), i t i s seen t h a t t h e r e e x i s t s a f a c t o r o f 2 i n f r o n t o f t h e non-diagonal p o r t i o n o f the r e l a x a t i o n time. t h a t 8 IT 4 This fact, i s a l r e a d y l a r g e r than combined w i t h t h e evidence (8/3)7}* , 4 means t h a t any weakness i n t h e approximations used t o e v a l u a t e t h e nond i a g o n a l term i s t o be r e f l e c t e d i n t h e s i g n i f i c a n c e t h a t can be a t t a c h e d t o t h e e s t i m a t e d v a l u e o f T ^ . In c o n t r a s t t o the l i n e a r approximation, t h e r e s u l t s from t h e q u a d r a t i c e v a l u a t i o n g i v e n i n Chapter VI a r e (8/3) "ft" (1.625) and 8 /?" (0.583) f o r the d i a g o n a l , Eq. (6-22), 4 < 4 and n o n - d i a g o n a l , Eq. (6-25), terms r e s p e c t i v e l y . Again, the l a t t e r number must be m u l t i p l i e d by 2 b e f o r e t h e r e l a t i v e c o n t r i b u t i o n s t o t h e r e l a x a t i o n time can be compared. Now w i t h 110 t h i s f a c t o r , the non-diagonal term c o n t r i b u t e s t w i c e as much as the d i a g o n a l one t o T^. With r e s p e c t t o the q u a d r a t i c a p p r o x i m a t i o n , i t i s f e l t t h a t more c o n f i d e n c e can be a t t a c h e d t o the v a l u e from the d i a g o n a l term than t o the one..from the non-diagonal term o f the way because i n which c e r t a i n approximations are made i n the expressions. two In the E u l e r - M a c l a u r i n approximation t o the d i a g o n a l c o n t r i b u t i o n , the f a c t o r I 4 - 3 i s expanded f o r l a r g e JL 1 \ Eq. (6-2), i n the obvious manner. However, i n the non-diagonal term, the approximation t o the denominators i n Eq. (6-23) i s , i n a- sense, f o r c e d on the problem by c o n s i d e r a t i o n s r a t h e r than a r i s i n g manner. Thus the two practical i n an obvious, n a t u r a l r i g o r o u s , d i s t i n c t denominators were r e p l a c e d by a s i n g l e e x p r e s s i o n Lw + ^(3/8) (j£ + 1 ) ( j£ + 2 ) ] l which was 2 s e l e c t e d a f t e r several other equally reasonable c h o i c e s had been i n v e s t i g a t e d . U n f o r t u n a t e l y , the integrations -4 with these o t h e r denominators e i t h e r d i v e r g e d or gave c contributions. 2 Because t h e r e e x i s t/ zs a f a c t o r o f c i n the c o e f f i c i e n t ch appears i n Eq. (4-21) f o r T \ 3. /y£\?irft+fr /V llo\<r?r) J^T the 1/c 2 \ whi \^J£r) terms from the d i a g o n a l and non-diagonal p o r t i o n s o f -1 0 T^ are e f f e c t i v e l y c terms. Now for c 2 = 0, i t i s obvious Ill that the whole contribution to comes from these terms. 2 Thus, again for c = 0, i t was found that the value of the numerical factor in front of the expression for the total relaxation time decreased from . (7/2.- in the linear approximation r 1 to 2.56 in the quadratic approximation. No such corresponding decrease is exhibited from the linear to the quadratic approximations in the diagonal contribution i t s e l f . In fact, the latter approximation to the diagonal term yields a result which is 1.625 times larger than that from the linear approximation. An explanation for this increase is given below. JL In the exact expression for A (z), Eq, (5-2), the quotient can be written down as JL / (7-1) where i t has been assumed that a, b, etc. are a l l positive. Now i f i t is true that a l l the coefficients in the f i r s t line of Eq. (7-1) are non-negative, then the result given by the linear approximation to the diagonal term is a lower bound for the diagonal contribution to T^*. This substantiates the aforementioned increase from the linear to the quadratic evaluations of the diagonal term. Although no proof has been exhibited here to show that a l l the coefficients are in fact positive, investigation has shown that, at least for the lower 2 powers of 1/z , the assumption in Eq,.'' (7-1) .is valid. In contrast to the diagonal term, the decrease in the numerical factor multiplying is reflected in the values obtained from the linear and quadratic approximations to the non-diagonal term. This decrease is almost by a factor of 1/2. Because of the nature of the approximations to the quadratic non-diagonal term as discussed above, there exists an uncertainty as to whether the numerical factor should have increased or decreased. At this point, the obvious question to ask is how the results in this thesis can be improved or further substantiated. It seems that rather than calculating more correction terms 2 and/or rather than keeping more powers of 1/z in both 2 2 z h fz)h*(z) and z h (z)h* (z), i t would be more useful to A JL JL £+z 113 find an entirely different analytical approximation to the quotient of polynomials in Eqs. (5-2) and (5-3). Then perhaps a good estimate of the upper and lower bounds to the relaxation time could be found. As they now stand, the results for T * from this thesis can be compared with the result of Chen and Snider. First, 2 c must be put equal to zero. Then from Eqs. (5-15) and (6-43), the relaxation times corresponding to the linear and quadratic approximations are (7-2) and respectively. Chen and Snider obtain (7-4) 114 where their d is to be identified with (JT here. It would appear from Eqs, (7-2) and (7-3) that the value obtained by the other workers is good, especially considering their crude approximation which consisted of replacing the distorted wave by a plane wave. In conclusion, i t can be said that this thesis exhibits the feasibility of using analytical methods to estimate the relaxation time for a specific system. 115 BIBLIOGRAPHY 1 F . M. Chen and R. F . S n i d e r , J . Chem. Phys. 46, 3937 (1967). 2 L . Waldmann, Z . N a t u r f o r s c h . 12a, 660 (1957). 3 R. F . S n i d e r , J . Chem. Phys. 32, 1051 (1960). 4 L . S. Rodberg and R. M. T h a l e r , I n t r o d u c t i o n to the Quantum Theory o f S c a t t e r i n g , (Academic P r e s s , New Y o r k , 1 9 6 7 ) . . 5 G. A r f k e n , Mathematical Methods f o r P h y s i c i s t s , P r e s s , New Y o r k , 1966), p . 608. 6 P. M. Morse and H. Feshbach, Methods o f T h e o r e t i c a l P h y s i c s , (Mc G r a w - H i l l Book Company, I n c . , New Y o r k , V o l . I , p . 622, and V o l . I I , p . 1574. (Academic 1953), 7 A . M e s s i a h , Quantum M e c h a n i c s , (John Wiley and Sons, I n c . , New Y o r k , 1966), V o l . I , p . 489, p . 496, and V o l . I I , p . 1054. 8 J . A . R. Coope, R. F . S n i d e r , and F . R. Mc C o u r t , J . Chem. Phys. 43 2269 (1965). 9 A . R a l s t o n , A F i r s t Course i n Numerical A n a l y s i s , (Mc G r a w - H i l l Book Company, I n c . , New Y o r k , 1965), p . 133. 10 E . Jahnke and F . Emde, T a b l e s o f F u n c t i o n s , P u b l i c a t i o n s , New Y o r k , 1945), p . 1, 2. 11 See, f o r example, B. 0. P i e r c e , A Short T a b l e o f (Ginn and Company, New Y o r k , 1929), p . 13. 12 L . Lewin, D i l o g a r i t h m s and A s s o c i a t e d (Mac Donald, London, 1958). 13 B i e r e n s de Haan, N o u v e l l e s T a b l e s D ' I n t e g r a l e s D e f i n i e s , (G. E . S t e c h e r t and C o . , New Y o r k , 1939), T a b l e 359. (Dover Integrals, Functions,
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Collision theory as applied to the calculation of a relaxation time Nielsen, Katherine Stephanie 1969
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Title | Collision theory as applied to the calculation of a relaxation time |
Creator |
Nielsen, Katherine Stephanie |
Publisher | University of British Columbia |
Date Issued | 1969 |
Description | An expression for the spin-lattice relaxation time, T₁, of a dilute monatomic gas can be derived starting from the quantum-mechanical Boltzmann equation. The real difficulty in calculating the relaxation time for a particular system lies in the evaluation of the transition operator which appears in the expression for T₁ˉ¹. In this thesis, the relevant part of the transition operator, t₁, is estimated by a distorted-wave Born approximation (DWBA). The monatomic gas is approximated by a specific model. In this model the collisions described by t₁ are governed by two potentials: one, the isotropic rigid sphere potential, V₀, and the other, the anisotropic dipole-dipole nuclear spin interaction potential, V₁. The latter interaction describes the coupling between the degenerate nuclear spin states of the atoms and the translational degrees of freedom in the gas. The former (isotropic) potential governs the explicit form of the rigid sphere distorted wave. After the DWBA transition operator is substituted into the equation for the relaxation time, the expression for T₁ˉ¹ breaks up into two terms, the "diagonal" and "non-diagonal" contributions. At this stage the explicit expression for T₁ˉ is sufficiently complicated that, in order to finish the calculation, analytical approximations to the diagonal and non-diagonal terms are made. These approximations may be succinctly described by stating that they result in two separate evaluations, a linear and a quadratic one, for the overall relaxation time. The magnitude of a small parameter c² , which appears in the exponential term of T₁ˉ¹ , is used as the basis for neglecting certain contributions to the integrals which arise in estimating T₁ˉ¹. The linear and quadratic approximations yield numerical factors of 3,50 and 2.56 respectively, in the expression for the relaxation time. These values are to be compared with the factor of 2 obtained elsewhere. |
Subject |
Spin-latting relaxation Collisions (Nuclear physics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0059943 |
URI | http://hdl.handle.net/2429/35129 |
Degree |
Master of Science - MSc |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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