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

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