{"http:\/\/dx.doi.org\/10.14288\/1.0059943":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Chemistry, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Nielsen, Katherine Stephanie","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2011-06-05T22:59:49Z","type":"literal","lang":"en"},{"value":"1969","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"An expression for the spin-lattice relaxation time, T\u2081, 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 T\u2081\u02c9\u00b9. In this thesis, the relevant part of the transition operator, t\u2081, is estimated by a distorted-wave Born approximation (DWBA).\r\nThe monatomic gas is approximated by a specific model. In this model the collisions described by t\u2081 are governed by two potentials: one, the isotropic rigid sphere potential, V\u2080, and the other, the anisotropic dipole-dipole nuclear spin interaction potential, V\u2081. 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.\r\nAfter the DWBA transition operator is substituted into the equation for the relaxation time, the expression for T\u2081\u02c9\u00b9 breaks up into two terms, the \"diagonal\" and \"non-diagonal\" contributions. At this stage the explicit expression for T\u2081\u02c9 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\r\noverall relaxation time. The magnitude of a small parameter c\u00b2 , which appears in the exponential term of T\u2081\u02c9\u00b9 , is used as the basis for neglecting certain contributions to the integrals which arise in estimating T\u2081\u02c9\u00b9. 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.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/35129?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"COLLISION THEORY AS APPLIED TO THE CALCULATION OF A RELAXATION TIME by KATHERINE STEPHANIE NIELSEN Sc. (Hons.), Un i v e r s i t y of 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 in the Department of CHEMISTRY We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C olumbia, I ag r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f CHEMISTRY The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date MARCH, 1969 ABSTRACT An expression for the s p i n - l a t t i c e r e l a x a t i o n time, T^, o f a d i l u t e monatomic gas can be der ived s t a r t i n g from the quantum-mechanical Boltzmann equation. The r e a l d i f f i c u l t y 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 for a p a r t i c u l a r system l i e s i n the eva luat ion of the t r a n s i t i o n operator which appears i n the expression for T^ In t h i s t h e s i s , the re levant part of the t r a n s i t i o n operator , t^, i s estimated by a distorted-wave Born approximation (DWBA). The monatomic gas i s approximated by a s p e c i f i c model. In t h i s model the c o l l i s i o n s described by %^ are governed by two p o t e n t i a l s : one, the i s o t r o p i c r i g i d sphere p o t e n t i a l , V^, and the other , the a n i s o t r o p i c d i p o l e - d i p o l e nuclear sp in i n t e r a c t i o n p o t e n t i a l , V^. The l a t t e r i n t e r a c t i o n describes the coupl ing between the degenerate nuclear sp in states of the atoms and the t r a n s l a t i o n a l degrees of freedom i n the gas. The former ( i so t rop ic ) p o t e n t i a l governs the e x p l i c i t form of the r i g i d sphere d i s t o r t e d wave. A f t e r the DWBA t r a n s i t i o n operator i s subs t i tu ted into the equation for the r e l a x a t i o n time, the expression for T.. ^ breaks up in to two terms, the \"diagonal\" and \"non-diagonal\" contributions. At this stage the explicit expression for 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. i v TABLE OF CONTENTS Page Abstract i i Acknowledgment v CHAPTER I INTRODUCTION 1 CHAPTER II THEORY OF THE TRANSITION OPERATOR.. 5 CHAPTER III CALCULATION OF (t| +\" ) 30 1 Jg CHAPTER IV RELAXATION TIME - T \" 1 55 CHAPTER V A LINEAR APPROXIMATION TO T J 1 70 CHAPTER VI A QUADRATIC APPROXIMATION TO T \" 1 . . . 80 CHAPTER VII SUMMARY... 109 BIBLIOGRAPHY 115 V ACKNOWLEDGMENT I wish to thank Dr. R. F. Snider f or the stimulating and informative years that I spent as his research student. I also wish to acknowledge the f i n a n c i a l support which I received both from the National Research Council of Canada and the Chemistry Department of the Uni v e r s i t y of B r i t i s h Columbia. 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 quantum-2 3 mechanical Boltzmann equation of Waldmann and Snider . If a rigorous, analytical evaluation of t for the relevant interaction potential could be obtained, then it 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 strictly 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, T1 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 all 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 mathematically than, for example, t h i s more extensive treatment, the improvement obtained by a more exact method would have to be considerable to outweigh the time and e f f o r t involved i n evaluating t more p r e c i s e l y . In f a c t , i t was found that the simpler treatment gives 78% of the value f o r the r e l a x a t i o n time that i s calculated by the more extensive treatment. Because of the p a r t i c u l a r chosen i n t h i s t h e s i s , the d i s t o r t e d waves could be expressed exactly i n terms of a p a r t i a l wave expansion. A neater s o l u t i o n to t h i s p a r t i c u l a r s c a t t e r i n g problem would be a Cartesian evaluation of the r i g i d sphere wavefunction. More generally, the most useful s o l u t i o n to any c o l l i s i o n problem would be a good a n a l y t i c a l Cartesian approximation to the d i s t o r t e d wave for a general i s o t r o p i c p o t e n t i a l . In Chapter IV the relaxation time i s written down e x p l i c i t l y f o r the t discussed above. Unfortunately, the expression which i s thus obtained f o r TV ^ i s s u f f i c i e n t l y complicated that an exact evaluation cannot be c a r r i e d further. Consequently, i n Chapters V and VI a \" l i n e a r \" and a \"quadratic\" approximation re 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 order to estimate the remaining sums and int e g r a t i o n i n T ^ . 2 A c e r t a i n small parameter, c , which appears i n the expression 4 for TV* was used as a guide i n performing the in tegrat ions i n the \" l i n e a r \" and \"quadratic\" approximations. Because, for 12? 2 -4 example, f or Xe, c i s o f the order of 1.25 x 10 at \u00a9 300 K . , expansions 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 . In the l a s t chapter a summary o f the approximate evaluat ions to T 1 ^ i s g iven. CHAPTER II THEORY OF THE TRANSITION OPERATOR A b r i e f resume of sca t t e r i n g theory of molecules with degenerate i n t e r n a l states w i l l be given i n t h i s chapter. The general purpose of t h i s summary i s to introduce and discuss the t r a n s i t i o n operator, but the ultimate aim i s to exhibit the e x p l i c i t operator equation f o r the anisotropic t r a n s i t i o n operator 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 anisotropic p o t e n t i a l . In fa c t , i t i s not the whole anisotropic t r a n s i t i o n operator which i s of in t e r e s t here, but only a \" l i n e a r i n anisotropy\" approximation, or, equivalently a \"distorted-wave Born approximation\" (DWBA) to the whole anisotropic t r a n s i t i o n operator. A two body c o l l i s i o n problem involving a central p o t e n t i a l which i s also an operator i n i n t e r n a l state space, can always be reduced to a pseudo one-body scattering problem. Since the center of mass i s not affected by the c o l l i s i o n , the problem can be expressed e x c l u s i v e l y i n terms of the stationary state Schroedinger equation written i n r e l a t i v e coordinates, namely 6 (2-1) where jJL i s the reduced mass o f the two molecules, r. i s the r e l a t i v e coordinate r ^ - r ^ , and ^ i s the t o t a l r e l a t i v e energy of the system plus the energy o f the i n t e r n a l s ta tes . The Hamiltonians and 3\\, i n t ''int i n t e r n a l s tate space of molecules 1 and 2 r e s p e c t i v e l y , while . ' ao?e operators s t r i c t l y i n the V ^ n t (r) i s an operator i n both p o s i t i o n space and i n t e r n a l s tate space. The corresponding free p a r t i c l e Schroedinger equation for molecules with i n t e r n a l states i s A. j t 2. (2-2) where 'I 5 A (2-3) 7 and - 3 \/ 2 i ^ - A -A JL = Z> (2-4) -3\/* -jlk-A . .A JL U> = 4d \" (-** \\ . w ^ 4 - A ' \/ ) * * * * .f-w (2-19) to Eq. (2-17) i s chosen because i t represents an outgoing s p h e r i c a l wave ( o r i g i n a t i n g at t^}. Since G ^ (r_,r-^) only p a r t i a l l y s a t i s f i e s the asymptotic boundary condit ions on 13 \"^(r_), Eq. (2-16) cannot represent the en t i re s o l u t i o n o f the inhomogeneous d i f f e r e n t i a l equation. Consequently, i f another term were added to the r i g h t hand s ide of Eq. (2-16) i n such a way as to f u l f i l l the plane wave p o r t i o n of the boundary c o n d i t i o n , Eq. (2-18), then the expression fov*$^+} (r) as a-s o l u t i o n to Eq. (2-15) would be complete. The appropriate term i s the p a r t i c u l a r free p a r t i c l e wave funct ion ^(r) which s a t i s f i e s the equation. AjJL 6 M = o. \u2022\u2014- % (2-20) Eq. (2-20) i s , of course, jus t Eq. (2-^) for an e l a s t i c s c a t t e r i n g process . Now the complete formal s o l u t i o n to Eq. (2-15) i s -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: -4. 5 and (2-22) (2-23) (2-24) (2-25) The free particle Hamiltonian H^ which acts on wave functions in position representation is 4 \" < (2-26) 15 Formally, the Green's function G i s just the inverse of the appropriate d i f f e r e n t i a l operator. For example, with respect to Eq. (2-17) G can be written as where E, an eigenvalue of H^, i s understood to be a multiple of the i d e n t i t y operator. However, when (physical) boundary conditions are incorporated into the Green's function, as they are i n Eq. (2-25), then the sol u t i o n of Eq. (2-17) i s chosen to be i n the form of Eq. (2-19) and G becomes G ^ . The operator expression for G ^ \u2014^ i s away from the influence of the i n t e r a c t i o n p o t e n t i a l V ( r ' ) , G = (E-HJ -1 (2-27) = ( E-H n + i \u20ac r 1 . (2-28) can be expanded i n the following manner: 16 A. - A. (2-29) If this asymptotic expansion is used in Eq. (2-21), then - V * * 7\/> -4^ A V4')\u00a3 4 into the f u l l scattered wave _Z\" ^ . ( r ) . Eq. (2-32) can be used i n Eq. (2-31) i n order to express the sca t t e r i n g amplitude as a matrix element between the asymptotic incoming and asymptotic outgoing plane wave states, namely (2-33) The operator VJfL. i s s u f f i c i e n t l y important to be given the symbol, t , denoting the t r a n s i t i o n operator. Note that t i s an operator i n both momentum space and in t e r n a l state space while t \u2014 i s a matrix element i n momentum space \"but s t i l l an g operator i n i n t e r n a l state space. I f Eq. (2-25) i s rewritten i n terms of the Moeller wave operator, then the 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 i n t e g r a l equation f o r t can be written down by s u b s t i t u t i n g Eq. (2-34) into Vii. to obtain (2-35) It i s v i t a l to notice that the Green's function i n Eq. (2-35) contains only the free p3rt\u00bbc\/e Hamiltonian, H^, as opposed to the t o t a l Hamiltonian. Now suppose that the p o t e n t i a l V i s written as a sum of two p o t e n t i a l s : one i s o t r o p i c and denoted by V^ , and the other anisotropic and denoted by V^. The Schroedinger equation appropriate f o r describing a system governed by these p o t e n t i a l s i s The technique and argument used e a r l i e r to solve Eq. (2-15) can be employed again to write a formal s o l u t i o n f o r Eq. (2-36), namely (2-37) 20 where i s a s o l u t i o n to [ E - (H0-hV0)\\X = O (2-38) with the same k ind o f boundary condi t ion imposed on A* ^ as was imposed on \" 3 ^ ^ i n Eq. (2-18). The Green's funct ion G^ s a t i s f i e s the equation I \u00a3 - Ma+V0)i] Gr0 = \/ (2-39) where cf\" i s the Dirac d e l t a func t ion . The p h y s i c a l l y re levant s o l u t i o n to Eq. (2-38), cons i s t s of both a plane wave and an outgoing s p h e r i c a l scat tered wave. The s c a t t e r i n g i s , o f course, now due to the i s o t r o p i c p o t e n t i a l . Because the i n t e r p r e t a t i o n of i n Eq. (2-37) d i f f e r s from the i n t e r p r e t a t i o n of 0 ^ ^ in the more common representat ion of 3?^^ as given i n Eq. (2-25), ^ C-^+^ \\ l} i s often c a l l e d the \"dis tor ted\" wave. It i s assumed that the e f fec t on 3?\", . due to the a n i s o t r o p i c K, 1 p o t e n t i a l , V^, i s small compared to the e f fect o f V^. Consequently, i t i s reasonable to solve Eq. (2-37) by i t e r a t i o n . To terms l i n e a r i n V^ , the distorted-wave Born approximation 21 is obtained, namely -r(\u00b1) v f t ) (\u00b1) J\u00b1) Parallel to the decomposition of the potential V into two contributions, the transition operator can be written as a sum of two parts, namely t = to + t i = ( V Q + + ( V Q + V p G(t Q + t p ( 2 - 4 1 ) where G is given by Eq. ( 2 - 1 7 ) . Upon expansion of Eq. ( 2 - 4 1 ) the \"isotropic\" transition operator, t^, and the \"anisotropic\" transition operator, t^, can be naturally identified. They are 3 V Q i l 0 ( 2 - 4 2 ) where SLQ is defined as in Eq. ( 2 - 3 4 ) but with V replaced by VQ, and h = v i + v i G t o + v o G t i + v i G V ( 2 - 4 3 ) -In the DWBA the last term in Eq. ( 2 - 4 3 ) is dropped on the basis that it is quadratic in V . Eq. ( 2 - 4 3 ) can then be rearranged into the form (1 - VQ G ) t l = V l H 0 (2-44) which implies that t1 = (1 - VQ G)\"1 V 1 _ 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 VQG = t Q G(l + t 0 G ) _ 1 (2-46) and (1 - VQ G)\"1 =\" [ 1 - t Q G(l + t Q G)\" 1] 1 + t Q G 0 \u2022 (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 t i --ft-5 v i - f l - 0 ( 2 - 4 8 ) 23 for the DWBA. An alternative derivation for Eq. (2-48) is given below. The matrix element t y ^ (which is s t i l l an operator in internal state space) can be written down from Eq. (2-33) as {jZ+S>)^IJL>^ (-JIT\/JLJS J (2-49) In the term can be rewritten in i K , 1 *-terms of the distorted 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 \u00a3 > 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^\u2014\\ can be replaced by 2 4 the expression (\u00b1) (t) (\u00b1) When Eq. ( 2 - 5 2 ) is substituted back into Eq. ( 2 - 5 0 ) , the distorted wave can be written exclusively in terms of I , viz. Finally, the substitution of I J J ^ from Eq. ( 2 - 5 3 ) into ^\u00a3''\\ V x l 3? \u00a3+? ^ of Eq. ( 2 - 4 9 ) yields ( 2 - 5 4 ) 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. The last step in Eq. (2-54) requires the recognition of Eq. ( 2 - 3 7 ) , namely in the first and third terms of the second line of Eq. ( 2 - 5 4 ) . i . e . , If Eq. (2-51) is used to express G in terms of Gu, U It G0 = GH \" GH V l G0 <2-56) where G^ is the Green's function for the total Hamiltonian HQ + VQ + Vj,. then l\u00a3'in Eq. (2-37) can be written exclusively in terms of s^C\/ i ^ , viz. This equation (Eq. ( 2 - 4 6 ) ) allows X.K in Eq. (2-54) to be <\u00a7 re-expressed as 27 (2-58) where the DWBA has been used. The operator V^SL^ can be immediately i d e n t i f i e d 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 operator t^ can be defined from Eq. (2-35) by us ing V^as the p o t e n t i a l and as the Green's func t ion , namely H - vi + W 2 = V l + V l ( G H - G H V l G ( ) ) t 2 = V r + V l G H t 2 - V l G H ( t 2 - V = V 1 + V 1 G H V 1 (2-59) where G i s s t i l l the Green's funct ion for the t o t a l n Hamil tonian, H H H Q + V Q + V^. Eq. (2-59) appears i n Eq. (2-58) i n the term (?C^ \/ V\u00b1 + V j G ^ \/ ^ c l a r i f y i n g i t s p h y s i c a l i n t e r p r e t a t i o n as a matrix element of a t r a n s i t i o n operator between two d i s t o r t e d waves, each governed by the i s o t r o p i c p o t e n t i a l , 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 - 6 0 ) r - i f where a natural identification of t ^ as 1-0. Q \\ V i \" ^ o + ' ' has been made. The adjoint of can be re-expressed as the transpose o f _ Q . ^ + 3 in the following way: = ( 2 - 6 i ) Consequently t becomes _Q.^V 1 JTiL n which is Eq. (2-48). CHAPTER III CALCULATION OF (t 1 J g In the previous chapter the expression Xi. Eq. (2-48) was derived for the DWBA to the anisotropic t r a n s i t i o n operator for a system with two p o t e n t i a l s . Now that Eq. (2-48) or, equivalently, Eq. (2-60) has been established, matrix elements of the anisotropic t r a n s i t i o n \u2022' operator i n momentum space can be calculated once the i s o t r o p i c and anisotropic p o t e n t i a l s are s p e c i f i e d . The o v e r a l l object of t h i s chapter i s to exhibit the c a l c u l a t i o n of the p a r t i a l matrix element (t ) \u2014 for a r i g i d sphere p o t e n t i a l V namely V n ( r ) = O O , r 0 r > = 3tf\u00a3* ^ where I_ and 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 ( ^ = (3-2) \u00ab\"\u00bb ^- j t \u00ab%, \/ **\" j f c (3-3) 32 where the equality X (A) -Jt' - (3-4) has been used. 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_) satisfi 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 \u00a3 z, the only angle dependence left in X \/O- A A ^(r_) is C7 =\u2022 k-r. Therefore, for a solution in spherical polar coordinates ^ ^l) c a n ^ e e xP a nded i n terms of Legendre, polynomials, P-(cos(9), as Jt > JL-o A A (3-7) Eq. (3-6) now becomes A> E JL * 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 f l ) 7 * h ( k r ) , the s p h e r i c a l Hankel funct ion o f the f i r s t k ind ' behaves a sympto t i ca l ly as an outgoing s p h e r i c a l wave, i . e . , 0~i -I, UA) (3-11) while the s p h e r i c a l Hankel funct ion of the second k i n d , (2) h ^ ( k r ) , represents an incoming s p h e r i c a l wave. The l a t t e r (1) \u2014 funct ion i s , i n f a c t , the complex conjugate o f h ^ (kr) rz h (kr) *- Z The plane wave p o r t i o n o f the asymptotic boundary condi t ion on ^ ^ ( k r ) 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 combination of s p h e r i c a l Hankel funct ions which i s regu lar at the o r i g i n , namely 1 * Ji J it (3-12) * Note that the s p h e r i c a l Hankel funct ions o f the 7 f i r s t kind appear to be defined d i f f e r e n t l y i n Messiah and i n Morse and Feshbach, V o l . I I , p.1573, but , i n f a c t , the (z) and b^(z) r e s p e c t i v e l y , are exact ly the same. It i s i n the d e f i n i t i o n o f the s p h e r i c a l Neumann funct ions that the two books d i f f e r by a minus s i g n . 36 where Jjg_(^r) 1 S t n e s p h e r i c a l Bessel f u n c t i o n . The other obvious r e a l , l i n e a r l y independent l i n e a r combination of * h^(kr) and h^(kr) , namely L JL Z (3-13) defines the s p h e r i c a l Neumann funct ion n^(kr) , which i s i r r e g u l a r at the o r i g i n . The constants and i n Eq. (3-10) are chosen to s a t i s f y the boundary condit ions on the r i g i d sphere r a d i a l wavefunction, \" ^ ^ ( k r ) , both at i n f i n i t y and at r 5 1 CT. The former constant, Cg, can be determined by comparing the asymptotic form of the p a r t i a l wave expansion o f ..exp > ( i k \u00ab r ) i n X . Ot) JL A J (3-14) with the asymptotic expansion of j^ (kr ) from Eq. (3-10) i n ^ ^ + ^ (r , 6 ) . The p a r t i a l wave expansion o f a plane wave i s 37 J-JnZ-A. (3-15) and as r approaches i n f i n i t y , ^ becomes J-Jt-JhL 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 \u2014 > j i \u00a3 (3-17) 38 and Of) ^? r r -I (3-18) From these last two expressions C0 can be identified as (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 (kr) Ji = 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) as* 1 f ~ (3-23) where D i s e x p l i c i t l y given (with the use of Eq. (3-22) i n Eq. (3-3) by D = (\u00a9 Al JL-I (SJLU) jLU'+^-jLa-i) U x'-X M \\ \\ \u00a3rl A' A' A-(J 0\u00a3> (3-31a) 45 It i s to be understood for the remainder o f t h i s chapter that i f , j , and h ^ appear with no func t iona l dependence, as they have jus t been w r i t t e n , then the argument i s k r . S i m i l a r l y , W - w i l l be a shorthand notat ion for W fk0\") . JL JL The f i r s t th ing to not i ce about Eqs. (3-30 a) and (3-30 b) i s that there i s no c o n t r i b u t i o n to the i n t e g r a l s at t h e i r lower l i m i t because o f the boundary condi t ion on (kr) at r= Cf for a r i g i d sphere p o t e n t i a l (See E q , (3-20) ) . For the same reason . (kr )h ' (kr) and A (kr)h' (kr) i n Eqs. (3-31a) A A'1 JL' 4+1 and (3-31 b) r e s p e c t i v e l y , vanish at r= 48 - A (V) - K\/ \") . (3-37) By inspec t ion o f Eq. (3-37) the la s t upper l i m i t , that i n Eq. (3-31b), can be wr i t t en down. The re su l t i s NCI] A? u x9 -v A\u00a5\\ JL' x r\\ A' AH) .-A+A (3-38) The only remaining evaluations are the two t r i v i a l c a l c u l a t i o n s on the non-zero lower l i m i t s o f Eqs. (3-31a) and (3-31b), namely K A-\/ Ji) JL x' &-i Af (3-39) 49 and JL SI X' N(A*j\u00a3) . - -J?** L- (hr>- W JL (V>L- (kh-U X (Vii. (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 at cr ^ (MN)~'[ JZrl A' J (3-41) 50 with the fo l lowing two r e s t r i c t i o n s , namely l! A-1 (3-42a) and A 1 * A+\\ . (3-42b) The reason that Eqs. (3-42a) and (3-42b) must be s a t i s f i e d i s to keep the denominator i n Eqs. (3-30a,b) and (3-31a,b) f i n i t e . Eq. (3-41) can be fur ther s i m p l i f i e d by looking at the Wronskian of the s p h e r i c a l Bessel funct ion 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) - j ' ( z ) n (z,) = - z 2 (3-43 ) A A A $-where the prime now denotes d i f f e r e n t i a t i o n with respect to z. From the r e c u r s i o n r e l a t i o n Eq. (3-32), and from Eqs. (3-12) and (3-13) for j (z) and n (z) r e s p e c t i v e l y , Eq. (3-43) can A A be re-expressed i n terms of s p h e r i c a l Hankel funct ions as 2 -z = j \u201e ( z ) n (z) - j (z)n (z) Ji Ji-) \u00a3~i A = i \/ 2 \\ h (z)h* (z) - h* (z)h (z)\"l. (3-44) Li-\/ A JL-\\ A J An example of how the Wronskian from Eq. (3-44) can be used i n Eq. (3-41) i s 51 j (k \u00a3 I g (4-6) 59 f+) g' ' f+) g' (t^ )j^ is the adjoint of (t^ )^ - in spin space. 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, a n c* ^\\ )g in terms of D and i s as in Eq. (3-23). Then the result \u2022\u00bb Z- ' - \/ ' - ' X . J L ~ \/ '-\u2022 X. J -v ~2- V r \" ' \" - ^ - ! ) ~\/ ~-7. j I t l Z 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 it 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 \u00ab \u00ab ^ 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 , * JL JL + JL' X Jl+JL \u2022a? -A- A. i JL JtL X X L vi-J.' -rn-X b In It-63 r . \u00ab ~ a M-L 7 I I I 1\/5 It (4-11) Now i f r is chosen as the fixed reference axis, then Eq. (4-11)-can be further simplified to ( 2. &J LL'-O i > > <-\u00bb Ax + 6\/) A A 64 I\u2014 \u00bb LA L+L. (-1) XL + ^ , --^JL -2: -A, \/ X \u2014 [ A A ' x C&JL'+LX^JL+I) \/jL1 JL O A C 1 6 6 o) 67 - J L -4 (4-16) where the argument, k ( T , of the s p h e r i c a l Hankel funct ions w i l l be denoted by z from now on, v i z . c r s (4-17) The func t ion A , Az\\ i s defined as JL JL 4 \u00a3 3 tPJt+3YJl\u00abWt-r> (4-18) and w i l l henceforth be r e f e r r e d to as a \"diagonal\" c o n t r i b u t i o n to Eq. (4-16). The \"non-diagonal\" c o n t r i b u t i o n A . , (z) i s defined as 1 &A+3)(jZ+Z)(6+h A JL+Z 3L X 1*1 JL MS. ) (4-19) 68 The r e s t r i c t i o n on JL -values as stated i n Eqs. (3-42a) and (3-42b) i s automatically accounted for i n t h e A ^ ( z ) ' s because 7 of the 3-j symbol which appears i n Eq. (4-16). That i s , for \u00a3' JL A O o o A\u00b1l A z \\ - o o o o (4-20) because i s odd f o r every value o^X-Now the only in t e g r a t i o n l e f t i n the equation f o r the r e l a x a t i o n time i s that over the energy parameter z, namely V* = \u00b1 yixtt+b (JL 3 - C Z X \\*JL djt (4-21) where the constant c i s defined as _g (4-22) 69 In order to complete the c a l c u l a t i o n o f thej^'s must be summed over and the z-parameter must be integrated out e x p l i c i t l y . It i s appropriate at t h i s point to discuss the parameter 2 c which i s defined i n Eq. (4-22). I f an estimate o f i t s 2 magnitude could be made, then the magnitude of c could serve as a guide i n choosing the type o f approximation with which to carry out the energy i n t e g r a t i o n ( i . e . , z - in t egra t ion ) , i n 129 the r e l a x a t i o n t ime. Thus, for example, f o r a Xe atom at o 2 -4 300 K . , c i s approximately 1.25 )C 10 , i . e . , (4-23) CHAPTER V A LINEAR APPROXIMATION TO T \" 1 The c e n t r a l issue i n t h i s and the next chapter i s to f i n d a decent approximation to the quotient o f s p h e r i c a l Hankel funct ions which appears i n the A^(z) 's o f Eqs. (4-18) and (4-19). O r i g i n a l l y , an attempt was made to r i g o r o u s l y in tegrate Eq. (4-21.)\" for each JL-value from 0 to 10 i n c l u s i v e . The j u s t i f i c a t i o n f o r terminat ing the i n f i n i t e ser ies at 10 came from the fact that the general > \u00a3 ' t h term i n each ser ies behaves b a s i c a l l y l i k e ^ \" 3 and for the Riemann-Zeta func t ion , namely Zl l \/ * \u00a3 ^ , the f i r s t nine terms cons t i tu te 99.5 % o f the value of the e n t i r e i n f i n i t e sum. Unfortunate ly , i t soon became apparent that the p a r t i c u l a r quotients involved i n Eq. (4-21) were d i v e r g i n g , ra ther than converging, a f t er each succeeding i n t e g r a t i o n . The d e c i s i o n to approximate the quotient o f s p h e r i c a l Hankel funct ions was made a f t e r further f r u i t l e s s inves t iga t ions into r igorous methods for so lv ing Eq. (4-21). At the same t ime, a fur ther dec i s i on to approximate the i n f i n i t e sums by in tegrat ions was made. The remainder o f t h i s chapter , then, w i l l deal with a \" l i n e a r - i n - w \" approximation 2 2 to the products z h (z)h*(z) and z h (z)h*(z) i n Eq. (4-18) and (4-19) r e s p e c t i v e l y , and with the E u l e r - M a c l a u r i n sum 9 -1 formula i n order to f a c i l i t a t e an approximate evaluat ion of T^ . 71 The E u l e r - M a c l a u r i n sum formula i s given by ~*~o t \u00b0 more c o r r e c t i o n terms (5-1) where the summation i s replaced by an i n t e g r a t i o n . Eq. (5-1) i s v a l i d for n approaching i n f i n i t y and, i n f a c t , i n t h i s thes i s n can be d i r e c t l y replaced by i n f i n i t y . The constant h i s equal to one s ince consecutive values o f ^ , the summation parameter, d i f f e r by one; and the constant i s the lowest value o f JL i n the sum. It i s convenient at t h i s point to def ine a new diagonal term ^g( z) ^ JL JL \u00a3ju ( * ) \u00a3 ( \u00a3 JL JL (5-2) 72 and, similarly, a new non-diagonal term A, (z) by (5-3) An approximation to ^g( z) will be considered first. 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 2 ^ IV = ? = (A 0 0 - 3 - . Jexp (6-9) In the i n t e g r a l I exp (-c2w) F(w)dw the exponential i s unnecessary f o r convergence since F (w) behaves, at l e a s t , as _2 y . Therefore the exponential can be expanded i n a power 2 se r i e s . Because the parameter c i s so small (see Eq. (4-23) ), the power series can be cut o f f e x p l i c i t l y 2 at terms l i n e a r i n c . That i s , (6-10) 85 since [.exp (-c w) Eq. (6-9) becomes i 21 1 + c wj 1 is of the order of c . Thus 3 c o _ c (6-11) 86 The constants a, b, and d must be calculated explicitly, and the magnitude of the two integrals, namely o o (6-12) and (6-13) must be estimated. The calculation of a, b, and d will be exhibited first. a, b, and d are found from the asymptotic form of f(y) in Eq. (6-7) , i . e. , 2 f(y) y -> -> a + b\/y + d\/y (6-14) In order to identify a, b, and d it 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 itself can be approximated in the following manner: Oh \u2022 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 33L % 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) J s(jl+ 2) ( JL + lj\\ 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\u00a3 74 V 3 (6-25) 96 The three remaining i n t e g r a l s , preceded by the f a c t o r (2 JL + 3) \\ a l so have j\u00a3.-integrations which are s t ra ight forward . The i n t e g r a l s are of the type I given i n Eq. (6-3). In each of the three succeeding w- integrat ions an i n t e g r a l i n the form of appears. Eq. (6-26) s t i l l must be estimated i n order to complete the eva luat ion of the quadrat ic approximation 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 time. A b r i e f d i s cuss ion o f the integrand i n Eq. (6-26) w i l l c l a r i f y the method used to carry out the i n t e g r a t i o n . F i r s t o f a l l , there i s no problem at the o r i g i n because, by l ' H o p i t a l ' s r u l e , the l i m i t .as y approaches zero i s f i n i t e , i . e . , (6-26) (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. (6-is finite. Finally, it would facilitate the integration i f the logarithm, ln (1 + x), could be replaced by a power series expansion in x. Unfortunately, with such an expansion 2 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 o o ^ 7s ^ \u20224 (6-28) and expanding each integral in a different manner, u is defined by 2u 5 <^y. (6-29) The first 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 Jt^(l-t^AA^) CJJUL. 2. (h^ J)JU(MjJ)MJ- + V\u00a3 ^ ''+3JJ^JUO+J^CU. (6-30) where q 3 (c 2 \/c< ). (6-31) Because the first 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 2 6 c 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*Z(S) = -^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 o o \/ o o (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 o o O O (6-36) I(q) can be expressed in terms of the exponential integral 101 \u00a3-Ei (-q)\\ and the integral J fexp (-2qu)\\ ln (1 - (2u) *)du . Jt u The latter quantity is explicitly calculated as follows: o o JUL o o _ -2.A.AA. = -\/ \\JL \/ MA, + 0 0 JL * J. J\/ JUC atu. + 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. o O ( I ' ^ j J ) A*, (l \" <\u00a3juu) -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 1 (exp (-2qu) - 1 + 2qu) \/ _1 infinity since the factor (u) (exp (-2qu) - 1 + 2qu) V (1 - (2u)_ \u00b1) + (2u) - 1] du at u multiplied by the term-by-term expansion of the logarithm _3 behaves at least as (u) (exp (-2qu) - 1 + 2qu) at infinity. The integral I I aioL from Eq. (6-38) remains to be evaluated. It can be done exactly, namely 103 o o _ -I ^ J '\/z o jt <3L i I \u00a92 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 ( ' \u00a3 > + \u00b1 i - \u00a3 j t f - f $ (6-40) 104 2 F i n a l l y , I(c ) from Eq. (6-26) can be wri t ten down from Eqs. (6-32), (6-34), and (6-40) as o o - 2 (6-41) 2 where, o f course , q =\u2022 c \/ o ^ Now the energy in tegrat ions o f the set of three i n t e g r a l s preceded by the fac tor (2 j\u00a3 , +-3) * i n Eq. (6-24) are given 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) inEq. (4-21). The portion of the quadratic evaluation of T^ 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 first 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\u201e .(w) in Eq. (6-2) is 4>JL \u2014 L e o \/ (6-44) 1 0 7 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 first correction terms do not yield a contribution in the form 2 of (1\/c ) to the relaxation time. CHAPTER VII SUMMARY A comparison of the r e s u l t s for the p r i n c i p a l term 1\/c between the l i n e a r and quadratic approximations to the rela x a t i o n time, i s made below. In the l i n e a r approximation discussed i n Chapter V, the energy int e g r a t i o n of the diagonal contribution, Eq. (5-9), y i e l d s a numerical f a c t o r of 4 2 (8\/3) T r f o r the 1\/c term. The analogous r e s u l t from the 4 non-diagonal contribution, Eq. (5-14), i s 87T . From the : complete expression f o r T Eq. (4-21), i t i s seen that there exists a fa c t o r of 2 i n front of the non-diagonal portion of the re l a x a t i o n time. This f a c t , combined with the evidence that 8 IT4 i s already larger than (8\/3)7}* 4 , means that any weakness i n the approximations used to evaluate the non-diagonal term i s to be r e f l e c t e d i n the s i g n i f i c a n c e that can be attached to the estimated value of T ^ . In contrast to the l i n e a r approximation, the re s u l t s from the quadratic evaluation given i n Chapter VI are (8\/3) \"ft\" 4 (1.625) and 8 <\/?\"4 (0.583) for the diagonal, Eq. (6-22), and non-diagonal, Eq. (6-25), terms re 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 before the r e l a t i v e contributions to the re l a x a t i o n time can be compared. Now with 110 t h i s f a c t o r , the non-diagonal term contributes twice as much as the diagonal one to T ^ . With respect to the quadratic approximation, i t i s f e l t that more confidence can be attached to the value from the diagonal term than to the one..from the non-diagonal term because of the way i n which c e r t a i n approximations are made i n the two expressions. In the Euler-Maclaurin approximation to the diagonal contribution, the factor I 4 - 3 1 \\ Eq. (6-2), i s expanded f o r large JL i n the obvious manner. However, i n the non-diagonal term, the approximation to the denominators i n Eq. (6-23) i s , i n a- sense, forced on the problem by p r a c t i c a l considerations rather than a r i s i n g i n an obvious, natural manner. Thus the two rigorous, d i s t i n c t denominators were replaced by a si n g l e expression Lw + ^ (3\/8)l(j\u00a3 + 1)( j\u00a3 + 2 ) ] 2 which was selected a f t e r several other equally reasonable choices had been investigated. Unfortunately, the integrations -4 with these other denominators e i t h e r diverged or gave c contributions. 2 Because there ex i s t s a factor of c i n the c o e f f i c i e n t 3. \/y\u00a3\\?irft+fr \/V \\ \/ z whi llo\\