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UBC Theses and Dissertations

Collision theory as applied to the calculation of a relaxation time Nielsen, Katherine Stephanie 1969

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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 © 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 = <AIJP>Z> (2-4) -3/* -jlk-A . .A JL U> = <jpyJL lA/\ (2-5) The label actually stands for the pair of quantum numbers i , d. of molecule 1. The former quantum number labels X l states with different energies and the latter one labels the degeneracy. The solution to Eq. (2-2), (P. . ( r ) , is a wave » k , I — function in position space, but s t i l l an abstract vector in internal state space. Finally, the ket 1 x^,1is an sig eigenfunction of the internal state Hamiltonian for a pair of molecules, i.e., is & JL. JL. fy « (2-6) With the substitution of Eq. (2-6) into Eq. (2-2), the latter 8 equation can be written as ' sL Jl* (2-7) Because the set of states l l i / 3 form a complete orthonormal set of vectors i n i n t e r n a l state space, ^? ( r ) , the s o l u t i o n to Eq. (2-1), can be expanded i n terms of them i n the manner (2-8) the p a i r of operators , 7^ act upon "jF(r) When i . mt **int as given i n Eq. (2-6), the r e s u l t i s Ufa = C i f . < i » € . / x „ 0 . (2-9) ?Cr) i« It can be seen from the l a s t equation that j f(r) i s not n e c e s s a r i l y an eigenfunction of 31 + 3~l i n t C i n t 9 Since the total energy of the system is conserved, can be expressed as ^ ~ -fl- + ^ ~ -+ ej/ (2-10) 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 is, 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_ is defined in terms of the relative linear momentum JD by means of the equation (2-13) where g is the relative velocity of the two molecules. For elastic scattering the wavenumber must be the same before and after the co l l i s i o n , i.e., /AI - IA'I -A. (2-14) If Eq. (2-11) is rewritten to look like an 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 it by the collision process. ^(j.) must essentially be the sum of an incoming plane wave plus an outgoing s p h e r i c a l scat tered wave. The e x p l i c i t boundary condi t ion i s thus A. (2-18) i ^ ^ (j_) designates that jT^(_r) cons i s t s o f a free p a r t i c l e inc ident wave plus an outgoing s p h e r i c a l scat tered wave. The subscr ipt i i s a reminder that the incoming plane wave has a p a r t i c u l a r i n t e r n a l s tate associated with i t . Analogously , j (j_) designates the sum of a free p a r t i c l e outgoing wave i n i n t e r n a l s tate i , plus an incoming s p h e r i c a l wave. The p a r t i c u l a r s o l u t i o n £\>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. •—- % (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 ^ —^ 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 € 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')£ 4<W A (2-30) where k_' — kr\ Comparison of Eq. (2 -30) with the boundary condition on ? (r), Eq. ( 2 - 1 8 ) , allows the scattering _K , 1 — 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/'fA . 7 ^ (2 -31) where Eq. (2 -5) was used without internal states. The bra K. p_'l represents the asymptotic final state of momentum p_' into which the particle has been scattered. Eq. ( 2 -21 ) can be written in the form 4^ Lk^ l ^ p ^ / (2 -32) 18 where SX.^ i s the Moeller wave operator. _ T L ( + ) has the property that i t takes the i n i t i a l incoming plane wave <r| £, i > 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 — 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»c/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 £ - 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(±) v f t ) (±) 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 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 • (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 £ > 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 2 4 the expression (±) (t) (±) 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 ^£''\ V x l 3? £+? ^  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£'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 <§ 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± + 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 •' 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 ) — 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 ><r (3-1) where 0* i s the diameter of the r i g i d spheres, and f o r a dipole-dipole nuclear spin 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 p o t e n t i a l i s given by 31 V6> = 3tf£* ^  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 ( ^ = <xfr)/y/xft)> (3-2) «"» ^- j t «%, / **" 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 £ z, the only angle dependence left in X /O- A A ^(r_) is C7 =• 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) — 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 « 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 — > j i £ (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 = (<A X ' % ) A: _ 3 JL ZL The notation I I I V?^  means the completely symmetrized and traceless part of the general second rank tensor l^l^' ^,—1—2^^'' ^ S w r ^ t t e n o u t e x p l i c i t l y i n Eq. (3-2' For a general discussion of Cartesian tensors, see reference 8. where r i s the unit vector i n the x_ d i r e c t i o n . Because the c o l l i s i o n s are e l a s t i c a l l y energetic, the magnitudes (but not the directions) of the wave vectors -iV and k_ are the same. g The second rank tensor D breaks up n a t u r a l l y into two integrations: one, over angles and the other, over the magnitude r. The r a d i a l i n t e g r a l w i l l bt evaluated f i r s t . Consider the following two d i f f e r e n t i a l equations f o r spherical Bessel functions, namely A obi JJX. c m A. (3-25a) and A3" dn. 'YL. A a(kC) =0. (3-25b) If Eq. (3-25a) i s m u l t i p l i e d through by g n ( k r ) , and Eq. (3-25b) by f ^ f k ' r ) , then a f t e r subtraction of the two r e s u l t i n g equations, a simple i n t e g r a t i o n y i e l d s 42 = A, (3-26) The prime denotes differentiation with respect to r, and the arguments of f and g^ are understood to be k'r and kr respectively. 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 an<i kr. Thus from this recursion relation, ^-j^(kr) in Eq. (3-22) can be rewritten as AZ(k^=(**iS'Ji[Xs -tX +(w A L JL-I JL+l JL-i JL JL-\ AH JL Mi J (3-29) The form of ^ ^ ( k r ) as given in the last equation breaks up the radial integral djx. into four V * A ~7C other integrals, each of which can be easily integrated by means of Eq. (3-27). These four integrals are: oo A UA*I) A-l Jl! Ji' JL~l)  ? (3-30a) 44 A (.91 i f x ~~\ JL' +0 V (bd X (knddjx. JL-+I fat+iAJL'fo A Yz x'-xx'\L (3-30b) - — : L A J <=>© Al JL-I (SJLU) jLU'+^-jLa-i) U x'-X M \ \ £rl A' A' A-(J 0£> (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= <T*. The evaluat ion o f the set of equations Eqs. (3-30a), (3-30b), (3-31a), and (3-31b) at i n f i n i t y requires more e f f o r t . The r i g h t hand s ide o f Eq.(3-30a) i n the l i m i t as r approaches i n f i n i t y w i l l be discussed f i r s t . Another recurs ion r e l a t i o n ^ , t h i s time for the d e r i v a t i v e of s p h e r i c a l Bessel funct ions and appropriate l i n e a r combinations thereof , i s given below d <t i \<r 0 3 2 ) Eq. (3-32) i s used to rewri te A. X -X X from JL-\ JL' JL' JL-\ Eq. (3-30a) so that the l i m i t can be evaluated, i . e . , 46 Jt /L"  X X' t-l JL' - X X' [X \jkL -wJ. )+ Jl X~ -2 <^/}X + AG -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 a (kr) as given in 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 H / -A, A.  X X'-x X ] _ -/ JL' JtH (3-35) where Nti\l) s JL,U'+l)-U+i)U+Zs). (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 - 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¥\ 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£) . - -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 „ ( z ) n (z) - j (z)n (z) Ji Ji-) £~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<r) - w (k<r)h (k<r) Q.+I A J*+J (1/2)[h (k<r) + h* (k(ril (i/2)^ h (k<r) + h*(k<r Jlh (k<r) l JC A J 0+1 h (k<T) A k2<r2h^(k<T) (3-45) Furthermore, the entire second term in Eq. (3-41) can be rewritten by means of Eq. (3-44), i . e., -Jk (T Jt'-I A' A'-l j (J -VJX \ + J [J -VJA - / J J A' MU'A) _ IN (A'A) Ji' A (3-46) where the arguments of j , W , and h are, of course, k<T. 52 Now i f the expression (2W^ - 1), which also appears in Eq. (3-41), i s re-expressed i n terms of s p h e r i c a l Hankel funct ions on ly , namely JL JL A/kr) - / JL  A / k i ) (3-47) where Eq.. (3-12) for ] has been used again , then Eq. (3-41) can be wr i t t en as a quotient o f s p h e r i c a l Hankel funct ions . F i n a l l y , from Eqs. (3-46) and (3-47), Eq. (3-41) becomes oo J* X (JtHSXik^J*. JL' JL T JL JL r a/ A j k r ) JL X' (3-48) 53 with the r e s t r i c t i o n s i n Eqs. (3-42a) and (3-42b) s t i l l v a l i d . The in tegrat ions over angles which remain i n D of Eq. (3-24) could 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 involve the fo l lowing express ion, namely <=2 3 (3-49) and then contrac t ing each s ide o f Eq. (3-49) with each of the three k, -k ' second rank tensors i n order to f ind the constants A, B, and'G- The three contract ions into the r i g h t hand s ide of Eq. (3-49) are s t ra ight forward . The contract ions into the l e f t hand side o f the same equation requ ire that the Addi t ion Theorem f o r s p h e r i c a l harmonics be used to uncouple - k ' » r and k « r so that the r - i n t e g r a t i o n can be done. However i t w i l l be shown i n the next chapter that Eq. (3-49) i s not needed, and the angle in tegra t ions 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 o o JL t' (3-50) where ^ I^I_2^|''^ denotes the completely symmetrized, traceless tensor. CHAPTER IV RELAXATION TIME Chen and Snider have der ived 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 ^ f o r a d i l u t e monatomic gas. This expression f o r T ^ contains the p a r t i a l matrix element ts- of the t r a n s i t i o n operator whose i n t e r a c t i o n p o t e n t i a l makes i t pos s ib l e for the r e l a x a t i o n phenomenon to occur. In t h i s thes i s the intermolecular p o t e n t i a l for 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 and a d i p o l e - d i p o l e nuc lear spin i n t e r a c t i o n . Thus, (t^ )— from Chapter III w i l l be inser ted into the equation of Chen and S n i d e r , namely X 56 •v. (4-1) and the relaxation time will be approximately evaluated by analytical methods. The symbols which appear in Eq. (4-1) are as follows: n is 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 1 S t n e internal state partition function for one molecule; and is the reduced relative velocity. The last quantity is appropriately defined in terms of the relative velocity g as It- = 57 while the i n t e r n a l s tate p a r t i t i o n funct ion f o r molecule 1, (L , i s given by (4-3) H i s the external magnetic f i e l d , and ^H.t and I. have been defined p r e v i o u s l y i n Chapter I I . The trace Tr^ i s over the i n t e r n a l s tates of molecule 1. Now i n order to der ive Eq. (4-1) Chen and Snider used the fo l lowing high temperature approximation, i . e . , t£n « AT (4-4) where H i s the magnitude o f the appl i ed magnetic f i e l d . Eq. (4-4) can be used to j u s t i f y expanding the exponent ia l , exp f .ffi. H » I 1 ) , from Eq. (4-3) i n a power ser ies about zero and keeping only the f i r s t term. Because, i n genera l , the i n t e r n a l s tate Hamiltonian has a much larger energy assoc iated with i t than does the nuclear Zeeman Hamil tonian, H*I., , the term exp ( - J ^ / J & T ) i s not expanded i n a power s e r i e s . Consequently, i s 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 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 •» Z- ' - / ' - ' X . J L ~ / '-• 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 « « ^ 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 •a? -A- A. i JL JtL X X L vi-J.' -rn-X b In It-63 r . « ~ 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 > > <-» Ax + 6/) A A 64 I— » LA L+L. (-1) XL + <r-0 J L ^ _ 4 f ^ " i t ^ u j In order to complete the next step the following two results are needed, namely (4-13) 65 and JL1 JL 4T J K ?T (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 X A! * C-i) X JL J- t* * «•'> ^ , --^JL -2: -A, / X — [ 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 £ 3 tPJt+3YJl«Wt-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 £' JL A O o o A±l 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* = ± 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 > £ ' 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 / * £ ^ , 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 ° 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 £ju ( * ) £ ( £ 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<r) . (5-6) 73 The approximation which i s exhib i ted i n Eq. (5-5) w i l l be c a l l e d the " l inear" approximation to the diagonal term A . Now, of course, Eq. (5-5) can be subst i tu ted into the whole diagonal term A , (w) which, upon keeping only the main term i n the E u l e r - M a c l a u r i n sum formula, becomes f J U L Jl o o JLlL - - (5-7) 74 where x i s def ined by x — MI+t). (5-8) The r e s u l t i n Eq. (5-7) can be put back into the diagonal term o f Eq , (4-21) i n order that the w- integrat ion be completed. Consequently, the 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 i s o o oO I 3 3 L C The exponential i n t e g r a l £ - E i ( - y ) \ ^ i s def ined (5-9) by o o T V (5-10) fr. • 1 0 75 where 0 , the Euler-Mascheroni constant ± u, has the numerical value 0.577215. . . . The linear approximation to the non-diagonal term A^,^ ^ z) ^ S P e r^ o r m e^ analogously to that o£ the diagonal term: that is, the power series of zh„(z)zh* (z), namely JL JL+Z ^ A ^ zA&JL - £ E (-/) I U+A+A\V.U+&1£ ' . (5-11) is cut off at terms quadratic in (1/z) . Similarly, the power (z)| is cut off at 1/z2. series expression for Re| z 2h -(z)h* (z  _ L JL JL+£ J Consequently, A , (w) becomes JL JL+Z (5-12) _ 2 where again w = z and the "linear" approximation actually refers to the parameter w rather than z. Now the full non-diagonal term A^ , , with the first term of the Euler-Maclaurin JLJL+& sum formula, becomes 76 j n A M ML 2 Jo L ^H+jt+A JVI+A + 5 A + 6 (I-%H)'4 (5-13) From Eq. (5-13), the non-diagonal contribution to the relaxation time, Eq, (4-21), i s oO /Lit* 0 [ w J C (5-14) 77 Thus the complete evaluat ion of T^ from Eq. (4-21) for the l i n e a r - i n - w approximation i s T _ / j_ (f£^-ni(T+i) f-rr AT) L c (5-15) For the diagonal c o n t r i b u t i o n , the f i r s t c o r r e c t i o n term, namely l/2^g(oo) + g ( x Q ) ] , from Eq. (5-1) of the Eul Maclaur in sum formula i s er -A Jw) + A/w) Ji L /o(vi+iy. (5-16) And s i m i l a r l y the f i r s t c o r r e c t i o n term o f the non-diagonal c o n t r i b u t i o n as given i n the integrand o f the f i r s t l i n e of Eq. (5-13) i s 78 J3& f t * <X . A 6v/) -f AJ\J) o o J_ e2 O + 3 (5-17) Eqs. (5-16) and (5-17) can be put back in to the expression for the r e l a x a t i o n t ime, Eq. (4-21), to f i n d the magnitude o f the c o r r e c t i o n terms. The r e s u l t i s -cot*. •/& +('/5Y/+c*)j.Cl-£z(-c*)\ (5-18) None of the terms i n Eq. (5-18) contr ibute to the 2 dominant 1/c term i n the main c o n t r i b u t i o n , Eq. (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 -1 1 In t h i s chapter the polynomials of z h (z)h*(z) and JL JL 2. z h ( z ) h * (z) i n Eqs. (5-4) and (5-11) re s p e c t i v e l y , w i l l be JL JL+Z -4 -2 terminated at z rather than at z . Then a procedure analogous to that used i n the previous chapter w i l l be c a r r i e d out i n order to obtain a better approximation to the -1 r e l a x a t i o n time T^ from Eq. (4-21). The quadratic-in-w diagonal term can be found from Eq. (5-4), namely X (6-1) I f Eq. (6-1) i s put into the f u l l diagonal term A^ ^ ( w ) a n d i f only the main term of the Euler-Maclaurin sum formula, Eq. (5rl), i s kept, then the r e s u l t i s 81 _s C Aa s fa) -/ f ^ - f . . . » 4 X CLIL J W+(*M)(TL-£) - / 3W ( W %L) S(w+i) / 6 < V f / V 5 W z - f 3 W -/ * t e f e W ^ 3 W - % ) ? ' * (6-2) 82 where x i s defined as i n Eq. (5-8). The inte g r a t i o n from the second l a s t l i n e to the l a s t l i n e of Eq. (6-2) i s exact. The i n t e g r a l i s of the general form Integral (6-3) can be found i n any book which l i s t s tables of i n t e g r a l s . The in t e g r a t i o n over energy of the f i r s t term in the l a s t l i n e of Eq. (6-2) i s straightforward. The int e g r a t i o n of the remaining two terms, however, requires more e f f o r t . The aforementioned energy int e g r a t i o n w i l l be discussed i n the following paragraphs. For convenience the following d e f i n i t i o n w i l l be made, namely (6-3) where S i s defined as (6-4) 11 83 9^ (6-5) The energy i n t e g r a t i o n over f(w) cannot be done exact ly i n a s tra ight forward manner. Thus f(w) must be approximated. Because the f a c t o r L-(5w + 3w - 9/4)J 2 / (w + 3/2) appears both i n the argument for the logarithm and i n the other term of f(w), i t was found to be convenient to expand f(w) i n an inverse power ser ies i n y 3 w + 3/2 (6-6) Thus f(w) can be wr i t t en i n terms of y i n the form f(y) = a + b /y + d/y + F(y) (6-7) Where the remainder F(y) i s , obv ious ly , F(y) = f(y) - a - b/y - d/y 2 84 (6-8) Consequently the complete energy i n t e g r a l of Eq. (6-5) i n terms of Eqs. (6-7) and (6-8) i s 3/> 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 • 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 % <W 3 / a ) : J+7L I - X ~9w 3 / * ^ 9 /_ 3 - 5 / 5 + 0(AA?) 89 - 9 3 n . ^ 5 ) l ( ^ V a ^ ^ ^ l . 0 ( l (6-19) 90 By p r e c i s e l y the same techniques , the f i r s t term of f(w) i n Eq. (6-5) , for w approaching i n f i n i t y , can be expressed as 3 1 V Z = / 1 _ 3 7-/_ / -xL/v- S(w+*/si) + Mto+v^y- +0(~j-*)s (6-20) The two r e s u l t s i n Eqs. (6-19) and (6-20) cons t i tu te the e x p l i c i t eva luat ion o f the constants i n the asymptotic expansion for f(y) i n Eq. (6-14). Therefore , a, b , and d can be i d e n t i f i e d as 91 a = (3/160) 6(arctan (5) ) (5) 2 0.0391, (6-21a) (3/800) -0.0726, 11_. + 27(arctan (5) 2) 2 ( 5 ) % (6-21b) and d = (3/2000) 7 - 243(arctan (5) 2) 4 (5) 2 •0.0363. (6-21c) In order to complete the evaluation of the r i g h t hand side of Eq. (6-11) the magnitudes of the in t e g r a l s 1^ and I in Eqs. (6-12) and (6-13) r e s p e c t i v e l y , must be investigated. It was concluded a f t e r several numerical integrations on 1^, and a f t e r several graphs of the exact i n t e g r a l were drawn, that the contribution of 1^ to Eq. (6-11) i s n e g l i g i b l e (of the order of 10 ) compared to, f o r example, the main term 2 2 (a/c ) ^ 3 x 10 . F i r s t l y , because of the s i m i l a r i t y of the integrand i n 1^ to that i n 1^ and secondly, because of 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 J i.e. J © O 0 -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) 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£ 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£.-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 ^ •4 (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£ ^ ''+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 £-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 " <£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 1 (exp (-2qu) - 1 + 2qu) / _1 infinity since the factor (u) (exp (-2qu) - 1 + 2qu) V (1 - (2u)_ ±) + (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 ©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 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 =• 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£ , +-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„ .(w) in Eq. (6-2) is 4>JL — 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£ + 1)( j£ + 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£\?irft+fr /V \ / z whi llo\<r?r) J^T \^J£r) ch appears i n Eq. (4-21) for T \ 2 the 1/c terms from the diagonal and non-diagonal portions of -1 0 2 T^ are e f f e c t i v e l y c terms. Now for c = 0, i t i s obvious I l l that the whole contribution to comes from these terms. 2 Thus, again for c = 0, it was found that the value of the numerical factor in front of the expression for the total relaxation time decreased from r. (7/2.-1 in the linear approximation to 2.56 in the quadratic approximation. No such corresponding decrease is exhibited from the linear to the quadratic approximations in the diagonal contribution itself. 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. In the exact expression for A (z), Eq, (5-2), the quotient JL can be written down as JL / (7-1) where it has been assumed that a, b, etc. are all positive. Now i f it is true that all the coefficients in the first 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 all 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), it 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, it 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 . Snider , J . Chem. Phys. 46, 3937 (1967). 2 L . Waldmann, Z . Naturforsch . 12a, 660 (1957). 3 R. F . Sn ider , J . Chem. Phys. 32, 1051 (1960). 4 L . S. Rodberg and R. M. T h a l e r , Introduct ion to the Quantum  Theory o f S c a t t e r i n g , (Academic Press , New York, 1967).. 5 G. Arfken , Mathematical Methods for P h y s i c i s t s , (Academic Press , New York, 1966), p . 608. 6 P. M. Morse and H. Feshbach, Methods of T h e o r e t i c a l  Phys ics , (Mc G r a w - H i l l Book Company, I n c . , New York, 1953), V o l . I , p . 622, and V o l . I I , p . 1574. 7 A. Messiah, Quantum Mechanics, (John Wiley and Sons, I n c . , New York, 1966), V o l . I , p . 489, p . 496, and V o l . I I , p . 1054. 8 J . A. R. Coope, R. F . Snider , and F . R. Mc Court , J . Chem. Phys. 43 2269 (1965). 9 A. Rals ton , 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 York, 1965), p . 133. 10 E . Jahnke and F . Emde, Tables of Funct ions , (Dover P u b l i c a t i o n s , New York, 1945), p . 1, 2. 11 See, f o r example, B. 0. P i e r c e , A Short Table of In tegra l s , (Ginn and Company, New York, 1929), p . 13. 12 L . Lewin, Di logarithms and Associated Funct ions , (Mac Donald, London, 1958). 13 Bierens de Haan, Nouvelles Tables D'Integrales D e f i n i e s , (G. E . Stechert and C o . , New York, 1939), Table 359. 

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