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Calculation of the energy relaxation times of a two component plasma Jankulak, Francis Joseph 1963

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... CALCULATION OF THE ENERGY RELAXATION TIMES OF A TWO;COMPONENT PLASMA  "ay  FRANCIS JOSEPH JANKULAK B . A . S c , University of B r i t i s h Columbia, 1959  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR •' THE .DEGREE OF M.A.Sc. • i n the Department of PHYSICS. We.accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH.COLUMBIA A p r i l , 1963.  In presenting  t h i s thesis i n p a r t i a l f u l f i l m e n t of  the requirements f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study.  I further agree that permission  for extensive copying of t h i s thesis f o r scholarly purposes may granted by the Head of my Department or by his  be  representatives.  It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of The University of B r i t i s h Columbia, Vancouver 8, Canada. Date  ABSTRACT Employing v a r i a t i o n a l procedures on the transport equation we c a l c u l a t e ,  i n t h i s t h e s i s , three relaxation times of a t y p i c a l  shock tube plasma.  They are the electron-electron,  electron-ion relaxational times.  The results are c o m p a r ^ E t e to  those obtained by other authors, and are e s s e n t i a l l y on the work done by P h i l l i p s .  i o n - i o n , and  an . improvement  ii  ACKNOWLEDGEMENTS I wish t Q express my sincere appreciation t o Dr. L. Be Sobrino f o r suggesting t h i s problem-and f o r h i s continued guidance throughout the course of t h i s work. I wish also t o acknowledge:'-the help received from the s t a f f of the Computing Centre and i n p a r t i c u l a r H.D. Dempster.  iii TABLE OF - CONTENTS • INTRODUCTION  }  P a g e  1  -  !  2  CHAPTER I - ELASTIC AND INELASTIC PROCESSES 1. Cross Sections of Atomic Hydrogen 2. Energy Transfer Rates 3- E l a s t i c C o l l i s i o n Transfer Rates  2 3 5  CHAPTER 1. .2. 3-  2 - GENERAL THEORY The K i n e t i c Equation The C o l l i s i o n I n t e g r a l Evaluation of A ( E , E ) and B ( E , E 2 )  CHAPTER 1. 2. 3k.  3 - THE ELECTRON-ELECTRON RELAXATION TIME L i n e a r i z a t i o n of The Kin&tic Equation The E q u i l i b r i u m Condition Expansion of the Perturbation i n Modes' The V a r i a t i o n a l P r i n c i p l e  1  CHAPTER k -  2  7 . 7 9 12  1  19 19 -21 22 2k 28  THE ION-ION RELAXATION TIME  5 -THE ELECTRON-ION RELAXATION TIME L i n e a r i z a t i o n of the Ion and E l e c t r o n K i n e t i c Equations The Electron-Ion E q u i l i b r i u m Condition Vector Operator Representation .of the Ion and E l e c t r o n K i n e t i c Equations k. The V a r i a t i o n a l P r i n c i p l e  CHAPTER 1. 2J 3-  CHAPTER 6 - CONCLUSIONS  .  29 30 31 3^  -  36 1  +  0  APPENDIX A - THE ENERGY EXCHANGE IN..A BINARY COLLISION  >2  APPENDIX B - PROOF THAT THE OPERATOR"/^ IS SELF-ADJOINT  U6  APPENDIX C - PROOF THAT THE.OPERATOR Q IS SELF-ADJOINT  kQ  APPENDIX D  PROOF THAT; TO ORDER <£•  £f  k s  = 0; s / 1  =0;;r/ s ^ ii APPENDIX E - CONSEQUENCES 'OF THE CONSERVATION OF PARTICLES AND ENERGY ON THE TRIAL'FUNCTIONS AND  51  :  APPENDIX F - RECURRENCE RELATIONS  57 '  6l  APPENDIX G - TABULATION OF"RESULTS .  63  BIBLIOGRAPHY  65  ILLUSTRATIONS 1. Cross sections of Atomic Hydrogen.as a function of energy 2. V e l o c i t y r e l a t i o n s h i p s i n a binary c o l l i s i o n  3 kk-  .  INTRODUCTION  1.  We attempt ,in .this thesis to determine the rate at which,a plasma produced by a t y p i c a l shook tube i n our laboratory approaches equilibrium.  In a f i r s t  attempt to solve t h i s problem we neglect a l l s p a t i a l and anisotropic effects and, thus, reduce the problem to the study of the approach to equilibrium of" ah' i n f i n i t e , homogeneous,.isotropic  plasma.  In the f i r s t chapter we estimate the rate at which energy i s transferred between .the, components of the plasma due to the different interaction .. .processes.  We have found that,- for the conditions of interest to us, a  plasma having a number density, of IO ''' cm ^, a temperature of 35>000°K and 1  -  80$ i o n i z a t i o n , the rate of transfer of energy .'due to neutral-charged p a r t i c l e , interactions, i s n e g l i g i b l e compared with the rate of transfer of energy:due to electron-electron, and ion-ion ,interactions, and. i s two orders of magnitude smaller than the rate of transfer of energy produced by-electron-ion interactions.  The l a s t i s of course much smaller than.the rate due to the  electron-electron.and i o n - i o n interactions. In the following chapters we study the rate at which e q u i l i b r i u m ' s approached due to the charged p a r t i c l e interactions. divided into three parts. themselves.  The problem can be  F i r s t l y , . electrons will.approach equilibrium among  Secondly, ions w i l l approach equilbrium among themselves.  electrons and ions w i l l approach the same f i n a l equilibrium state.  Thirdly,  In the  three cases a v a r i a t i o n a l procedure to solve the transport equation and obtain relaxation constants for the different energy modes i s used. Several authors (Spitzer,. 1956; - Rosenbluth e t a l ,  1957;•Phillips, 1959)  have dealt with the problem and our work consists e s s e n t i a l l y of an .improve-. ment over the work of P h i l l i p s .  We find that the c h a r a c t e r i s t i c times  involved are about twice as large as those given b y / P h i l l i p s i n the case of electron-electron and ion-ion interactions and about t h i r t y times-as large for the case of electron-ion interactions.  2.  CHAPTER.1 ELASTIC AND INELASTIC PROCESSES In t h i s chapter we investigate elastic  the r e l a t i v e importance ©f various  and i n e l a s t i c processes as means of electron energy transfer.  The  cross sections for these processes for a hydrogen plasma are used to obtain orders of magnitude of the rates of energy transfer for a p a r t i c u l a r laboratory, plasma.  Under i n e l a s t i c processes we group the following:  . i o n i z a t i o n , excitation.and recombination. electron-electron, 1.  Under e l a s t i c processes we include  electron-ion and.electron^elastic  collisions.  Cross Sections of. Atomic Hydrogen R i p e l l e (19^+9) has shown that the i o n i z a t i o n cross section  C7^ ^for Gn  atomic hydrogen.can.be approximated quite well with the formula  0Tn  -  TC  Here e i s the charge of the.electron,  In E/E  E - i s the ionization.energy, of hydrogen Q  and-E i s the energy, of the incident electron. constants'XQ  and k  Q  (l.l)  ff  .is the following:  • The significance of the  under the influence of the charge  of the nucleus the incident electron acquires,.on the -average, an. a d d i t i o n a l k i n e t i c energy'X E -while the l i b e r a t i o n of an electron exacts,, on the average, O  an energy k E Q  0  0  from the incident electron.  jA formula of the type ( l . l ) . w a s fitted' to•experimental data obtained by Brackman and F i t e ( 1 9 5 ^ ) , o  n  i o n i z a t i o n cross sections of atomic hydrogen.  Care was taken to f i t the t h e o r e t i c a l curve to the-experimental curve as closely-as possible for the range of energies between : 13-6 and 100 ev. (electron v o l t s ) .  Figure 1 i l l u s t r a t e s the experimental-and t h e o r e t i c a l  2 cross sections i n units of Tfa , , where a 0  0  is. the f i r s t Bohr radius.  ls-2p excitation cross section i s shown for comparison.  The  3-  electron energy, (electron v o l t s ) , FIGURE 1. CROSS SECTIONS OF ATOMIC HYDROGEN AS A FUNCTION OF ENERGY  2. Energy. Transfer Rates We now evaluate the energy, transfer rates due to the various processes and thus obtain an estimate of the importance of these processes-as means, of establishing equilibrium.  The energy transfer rate  ot .'= n <<Tv > • E  ,  ot ,.is (1.2)  where n i s the number density of p a r t i c l e s with which the electrons i n t e r a c t , E i s the average . energy l o s t by an electron .in ..one c o l l i s i o n , and <C0~v^  i s an  average of the product of the cross section (T for the process being considered and v. the speed of the p a r t i c l e .  We average  CTv such .that  co  <(T'v>  rc r v f  =  m  e \3/2 - mva/kT ;.vhv e  kv  ,  (l-3)  \2TTkT/  where m i s the electron mass,. k Boltzmann's constant and T .the temperature e  . m  o„ K.  ; -' r  We take the case of a plasma produced by-a t y p i c a l electromagnetic shock tube having a temperature of 35;000°K, and an. average density of p a r t i c l e s of 10-'-7 i r3.  I f we consider the plasma to be 80^-.ionized we obtain  C I  x IO ?  cm"3 ,'  = 7 x 1G"  cm /sec ,  n ,= ^ O P v>  2  1  11  . (l.U) •  3  E = l U ev = 2 x I O "  1 1  ergs ,  and for the energy transfer rate due to. i o n i z a t i o n we obtain ^ion  =  3 x IO" ergs/sec 5  •  ( l . 5)  Since the e x c i t a t i o n cross section never exceeds the i o n i z a t i o n cross section, of the two processes ionization, w i l l be the more important. cross section decreases rapidly-as a function of energy.  The recombination  We use an.approximation  which Bohm.and A l l e r (19^7) .applied to gaseous-nebulae.-.We have as .an estimate n ..=  2  x 10  l 6  cm"3 ,  <lCT^ v>. = 1+ x 1 0 ^ cm /sec 3  - 1  c  . E = 2 x 10"  1 1  ergs  ,  (l.6)  ,  and the energy transfer rate.due to recombination i s oL rec  = 7 x l O " ? ergs/sec . ' . .  (l-7)  :  3-  E l a s t i c C o l l i s i o n Energy:Transfer Rates We consider, an electron .having an.energy equal to the average energy of  the electrons i n the plasma,,that i s an energy of 3-kT, or approximately; 5 ev . The energy, l o s t per c o l l i s i o n ..by/an. electron i n t e r a c t i n g with a neutral p a r t i c l e of mass m^ i s at most 5 ev. but on the average i s m /mj_. times 5  e  v  e  •  If we use experimental data obtained by-'Brackman and F i t e (1958) we obtain n = 2 x 10  l 6  cm"3 ,  4CT _ v>.= 9 x 10-9 cm3/sec , e  n  .(1.8) '  .  i_—. ,ev.= k x I O 3 ' 2 x TO  .E = 5 x  -13  ergs  ,  and as a r e s u l t we get for the energy transfer rate due t o . e l e c t r o n - n e u t r a l elastic  collisions o C _ , = 7 . x 10" ergs/sec 7  e  n  .  (l-9)  . F o r the electron-electron e l a s t i c c o l l i s i o n s we use as an approximation the formula used by Bohm and A l l e r (19U7) *e-  e  =  30^n^v>e^ E  #  (  l  i  l  o  )  Substituting the appropriate values we.obtain f o r the energy transfer rate due to electron-electron e l a s t i c c o l l i s i o n s . od _ g  e  =  10 ergs/sec.  (l-ll)  Since the energy exchanged i n an electron-ion ..collision- i s m^/m- times the energy exchanged i n . an electron-electron .collision.we obtain.as an.estimate of the energy transfer rate due to electron-ion ..collisions oC _ •= 5 x 10" ergs/sec . 3  e  ±  • (1..1-2)  Comparing the energy transfer rates, due to the various'processes we see that e l e c t r o n - e l e c t r o n . e l a s t i c  c o l l i s i o n s . a r e of several.orders of magnitude  greater than ..both the i n e l a s t i c c o l l i s i o n s .and the electron-neutral and electron-ion.elastic collisions.  We see. also-that the electron-ion .energy  transfer rate i s s t i l l of at least two orders of magnitude greater than the i n e l a s t i c and electrpn-neutral energy transfer rates.  In the subsequent  development we s h a l l be concerned only with charged p a r t i c l e e l a s t i c interactions.  7CHAPTER 2 GENERAL THEORY . I n , t h i s chapter we. discuss the k i n e t i c equation.which ..describes- the evolution ef the one p a r t i c l e d i s t r i b u t i o n function.  Under certain assumptions  we obtain an , i n t e g r o - d i f f e r e n t i a l equation .which can .then .be -linearized-.and simplified to the point where numerical calculations.-can .be-performed. We s h a l l . r e s t r i c t ourselves t o : d i s t r i b u t i o n s functions-which are isotropic i n v e l o c i t y space and independant of position;/ We can ,define a one p a r t i c l e d i s t r i b u t i o n function f ( E - p t ) . 'By t h i s notation.we mean that f (E]_,t )dE-j_  denotes the number of p a r t i c l e s of type 1 whose energies l i e i n  the range between ;Ej_. and E j .+ dEj_ . -The normalization of f ( E ^ , t ) , is. such that the i n t e g r a l of f ( E ^ t )  , over a l l energies ' E i i s the number density  N ( t ) , o f p a r t i c l e s i n the system 1  co |f(E ,t)dE  ° ,1.  1  1  =  N (t) x  .  (2.1)  The Kinetic Equation If there are no external forces acting on the system the rate of change  of the d i s t r i b u t i o n function i s due entirely, to  interactions.-between.particles.  This can be expressed-by ^f(E ,t) ^t 1  ,=. J  ; L 1  + J  ±2  (2-2)  ,  :where <L_Q represents the contribution to.the rate of change of f(E^,t),due .to-interactions among p a r t i c l e s of type - 1 , and J - ^ <represents the contribution due to .interactions among p a r t i c l e s of type l w i t h type 2 . :  If we consider p a r t i c l e s of type 1 to be electrons and p a r t i c l e s of type 2 to be ions, we see that equation ( 2 . 2 ) states the following:  the rate of  change of the electron d i s t r i b u t i o n ..function i s due p a r t l y to interactions  among electrons themselves and partly.,to interactions among electrons and :  ions - a l l other.: interactions "being negligible., compared . to these two. However,.the e l e c t r o n - e l e c t r o n . i n t e r a c t i o n s are much more e f f i c i e n t  with  respect to .energy exchange between electrons than the electron-ion . interactions as we have seeniby the c a l c u l a t i o n s c a r r i e d out i n Chapter 1. would, expect the J  g e  term.  0  r  Therefore, we  ^ie "term to-be n e g l i g i b l e compared with  t h e ' o r  This then.would.describe an.approach to .equilibrium of the  electrons among themselves.  A similar reasoning holds for the i o n s . i f we  consider p a r t i c l e s of type 1 to be ions and p a r t i c l e s of type 2 to^be-electrons However, the electrons must f i n a l l y come to equilibrium with the.ions. t h i s case we w i l l consider the electrons, to be • i n equilibrium  For  .at a l l times  among themselves and slowly coming t o . e q u i l i b r i u m with the ions.  In. t h i s case  the J . t e r m - w i l l be zero since the electrons, are already . i n equilibrium among e e  themselves.  Thus we can .consider the evolution,of the plasma to o c c u r , i n  three stages each of which w i l l . b e described by an .equation .or equations.-of the type ( 2 . 2 ) . themselves."  In the f i r s t stage the electrons come to equilibrium among  In the second stage the ions come to: equilibrium .among, themselves.  F i n a l l y - i n the t h i r d stage the electrons•and ions w i l l approach a common equilibrium state. • In.discussing interactions between p a r t i c l e s we s h a l l - c a l l t h e - p a r t i c l e we are following as i t moves through the plasma.the""test p a r t i c l e " .  The  p a r t i c l e s with which the test p a r t i c l e interacts w i l l be c a l l e d . " f i e l d p a r t i c l e s If  c e r t a i n conditions are s a t i s f i e d the term.J^g , i n equation,(2.2)  resembles the c o l l i s i o n i n t e g r a l of the Boltzmann equation.  One c o n d i t i o n , i s  that .the d i s t r i b u t i o n . f u n c t i o n must, not change appreciably i n distances of the order of the Debye length where the Debye length  i s given by  •9Since we are considering a homogeneous plasma t h i s c o n d i t i o n , i s f u l f i l l e d . second condition requires that the d i s t r i b u t i o n function changes l i t t l e  A  in  times of the order of the time taken by.-a particle.,-to cross the-correlation' .sphere,.-i.e.a  sphere of radius  \^ .  This holds true for times..t,, of the  order of the product of the Debye length.and the inverse of the average velocity,  that .-.is  t  ., kT  ~  m=,  :.2  kT m  (2.U)  e  i+TTN  ,e  OOp  2  where C i ^ i s the plasma frequency.. . A - t h i r d and. fourth condition requires that the number of particles.. i n the c o r r e l a t i o n sphere be large and that the p a r t i c l e s are uncorrelated before they enter the correlation. ..sphere. plasma i s close to equilibrium and satisfied.  i  T ^ ^ l , these conditions, w i l l be  hTIN  F i n a l l y , . t h e . d i m e n s i o n s of.the  Debye length.  • If the'  system must be greater than the  Since the Debye length f o r - a plasma such as was considered i n  •"  " '  ' ' ' '  6  Chapter. 1, Is of the order..of k -x -10  u  cm-and since t h e ' l i n e a r dimensions of  the plasma i n which, we are interested are of the., order of several .centimeters, this condition.is f u l f i l l e d .  We s h a l l now go .on. to write an expression for  the c o l l i s i o n . i n t e g r a l . 2.  The C o l l i s i o n Integral Using the results of Bohm and A l l e r (19^7) and assuming the preceeding  conditions to. be s a t i s f i e-•.00 d we can write i n general M2  _  f(E„,t)dE, "'^2 ' 'o  f(E -  / ^"2>  1  w  —  f(E ,t) x  ^ E , 0 (T( ^ E , E 1  1  1  AE E ) 1}  2  CO  (T ( A E , E , E ) D  1  1  2  V d( A E j )  ,  (2.5) .  10. where we desnigate by: E-j_ , the energy ef the test p a r t i c l e ; E  2  , the energy of the f i e l d p a r t i c l e ;  AE-j_ , the energy gained by, the test .particle, as a r e s u l t of a c o l l i s i o n ; V  , . the magnitude of the r e l a t i v e velocity, of the c o l l i d i n g p a r t i c l e s ; .the cross-section calculated using the Bebye potential  <J>-Q  The Debye p o t e n t i a l i s given by ^  where  J?L i  s  =  (2.6)  e" / ^ D , r  the Coulomb, p o t e n t i a l ; a t - a d i s t a n c e . r . . • Since the Debye  p o t e n t i a l - i s small. for distances greater then the Debye length we can ..use-a • Coulomb potential:which i s cut .off at the Bebye length as a good approximation to.the Debye p o t e n t i a l , (Rose and C l a r k , , 1 9 6 l ) .  The cross section corresponding  to. t h i s p o t e n t i a l ' w i l l be denoted by (T , and i s given by R  z^z e  2  (2-7)  2  jUY  2  win  2 where.Zj_ and Z - a r e the charge numbers of the f i e l d and t e s t . p a r t i c l e s 2  respectively and ^M. i s the reduced mass  yU  ..=  m  m  l  _  l?2 + m  2  m-j_ and m being the mass of the test .particle and f i e l d p a r t i c l e • respectively, 2  and *X i s the scattering angle in-the center of mass system.  11'. (fT^C  If we expand f(E-j_-AEpt) E-j_ i n terms of  x  - AE ^_ X  E  E 2  )  ^  n  a:l  ? a y l o r series about  AEj_ we obtain  f(E - AE ,t) x  ^ l>  0r ( A E ^ E ^ A E E ) - f ( E , t ) ; C^( A E , E , E ) 3  1 ;  rf(E!,t)  2  x  01)( A E i , E , E ) 1  2  ,.  1  + I(  1  2  f^ f ( E i , t ) l T i ( A E 2  AK ) X  )  ^ E.2  1>  E ,E )l 1  2  (2.8) If we take the expansion up to terms i n AE-j. • only, , then .the • equilibrium condition.is satisfied.  That i s i f  f°(E^,t)  denotes the equilibrium  d i s t r i b u t i o n , we have  (2-9)  ^ _ f ° ( E , t ) .= 0 .. •at ' '  Another j u s t i f i c a t i o n i s seen ..from an order, of magnitude c a l c u l a t i o n . ;  The  integrand of the second.integral of equation (2.5),is  co  (-1)  i l  °_ f(E ,t) r ( ^E  (AE )  1  n  n=l  AE ,E ,E )  D  1  1  V .  2  n  And we have,,near equilibrium  f(E ,t) x  ^f(  6^(AE ,E ,E ) 1  1  2  E  l  ,t)0^  2  V^ i E  -It follows then that  c \  V d(A  AE-,_d f (E-, , t )  ($"nf.AEi >E-i ,Ep)  )=  ^  f(  V d( A E ) -  ^  f(E ,t)  E l  X  f A i ^ f ( E , t ) CTC( A E , E , E ) | £ E £ <L_f ( E ^ t ) CT ( A E , E , E ) E  2  2  1  J.  2.32 3S iE.;F.] -co  —co  C  1  1  1  1  2  2  E l  ,t)  r  Q(^  ,  0(^E-^) ;  Vd(AE ).,tf(E ,t)0(^-).(2.10) 1  1  12. The r a t i o  A E ^ / E ^  i s of the order of the p o t e n t i a l . energy of the • test p a r t i c l e  at the c o r r e l a t i o n sphere t o • i t s k i n e t i c energy. d i s t r i b u t i o n function .in terms of  Since we are expanding the  AE^.we must have  A E I / E I < J < 1  hence j u s t i f i e d .in .retaining .only terms that are f i r s t o r d e r - i n  and we are A E ^ / E ^  .  This i s consistent with the Debye approximation. Using equation (2.8) we can rewrite equation (2..5) as  follows:  f(E ,t)dE j- | _ A(E ,E )f(E /t)]  '12  2  1  2  2  1  B(E ,E )f(E ,t)J 1  2^ 2  2  (2.11)  1  E  -,+eo  where •A(E ,E )..= 1  /(AE )  2  (T ( A E E , E ^ )  Vd(AE )  / ( A E ) 0~ ( A E , E , E )  Vd(AE )  i  1  2  1  J  3.  1 >  1  x  2  r°° B ( E , E ) .=  c  1  c  1  2  (2.12)  1  -<x>  Evaluation of A ( E , E ) . a n d B ( E , E ) 1  2  1  2  We f i n d i t convenient to i n t e g r a t e - A ( E , E ) and B(E]_,E ) over a l l possible 1  2  ;  2  scattering angles rather than over a l l possible energy exchanges ^E-|_.  We  therefore write A E ^ . a s . a function of angles and i n i t i a l v e l o c i t i e s of the two i n t e r a c t i n g p a r t i c l e s .  2^2 J 1 m +mj ?  A E ,  s ± n  From Appendix A we see that we can.write  2  % 2  2  Vo -vV . + ^ 2  2  + siruA-.cos.CLv v sintOcos 0  2  2  2  1  l  V  n^+m^ (2.13)  where we denote by: .V-L , the magnitude of the v e l o c i t y of.the test p a r t i c l e i n the laboratory system; v  2  , the magnitude of t h e - v e l o c i t y of the f i e l d p a r t i c l e - i n the laboratory system;  •13-  C 0 , . t h e angle between the v e l o c i t i e s ©,  of the two p a r t i c l e s ;  the angle between .the plane containing v , 2  v-^ and  and the plane containing the o r b i t s i n the centre of mass system.  If we choose our frame-df reference to be the one i n which-the v e l o c i t y of  the centre of mass i s constant,  and i f we f i x the orientation of the  initial  r e l a t i v e v e l o c i t y of the two p a r t i c l e s , we must integrate over, a l l s o l i d angles  cL-^  into which the tw© p a r t i c l e s are scattered. dil  i  We have then  ..= siiftcftaQ .  (2.1k)  We have now to average our result over, a l l possible directions of the i n i t i a l relative velocity.  Since these are i s o t r o p i c a l l y d i s t r i b u t e d , • the  p r o b a b i l i t y that they l i e i n a s o l i d angle . d -fLgATT  i n the laboratory'frame i s  , and the average w i l l be achieved by m u l t i p l i c a t i o n by •d-O.^/k'K  and integration over  &fl  2  'where  ...= sinWdoOdoC ,  g>  (2.15)  o^being the azimuthal angle describing the orientation of v-^ to v  2  Therefore equation. (2.12) w i l l now.appear as  A(E ,E ) = 1  2  B(E ,E ).= 1  2  Ul^  AE (<9 1  ,E ,E ,0C, CO) (T^t Y,\^ 2 } 1  2  °° )V >'  | ^ E ( @ , E , E , 7 . , £ 0 ' ) [ O J O C E ^ E g , u))V 1  1  2  In performing the integration over  .  we must remember our. requirement  that the scattering cross section be cut off at the Debye length. w i l l impose a l i m i t on the minimum value w h i c h ^ can take. t h i s minimum value of 06 as \>. /  mm We define  A( IO) and B ( LO)  as  follows:  (2.16)  This  We designate  lh.  A(60)  ,=y"a% J^ ^l  =  B(CO)  AE (©,E.  V  d  [ A E  V  1  ( @  E , y,u))  1 >  J  E  r  2  1  We have,.then, using equations (2.7),and  , E  G  A ^ )  2  2  1 ;  (T (  + 2 sin  ^  sm  2  v ,  1^  2 2  d©  V  c o s l v .  2  ,  a)) .  n  r smY<d%  E2,Oj)  (2.18)  (2.13)  r  A(oo)  (T ( X,E  /  1  -2  2  0  1  m -m, 2 1 m + m-j^ s m 9  2  2  •2  v.  sina)cos0, .*ir  2  2  V  2 " l V  +  "i - iy m  2  m  2  + m  ZI_l_2_  e  2yUV3  l  sm in sm  (2.19) 2  ^  X, min  Now we have  ,7T  rsm^dQc  2~^T  J sm rvur\  =  1+ I n sm  2  2, l  n  f  with the angle 7 m i n corresponding to \ P.67)  0  being given by (e.g. S p i t z e r ,  3  1956,  ,15-  tan  2  A-  m,l 2 V  2(7TN)2ZiZ  We have replaced  m  22  varying logarithm. we find that  'XD  2  3/2  kT I 2_  2  (2.20)  e  t>y 3M since i t . appears i n the argument of a slowly  V  For the values of temperature and.density, used i n Chapter 1  In A 25 6.  Equation (2.19) now becomes  A(UJ) , = Z2S£AJL A2 y3  . 2 " i -2. Im +m m  m  2  l n  2  2  (2.21)  1  S i m i l a r l y , i n evaluating B ( u ) ) , i f we keep as dominant terms those containing In A i s large, we obtain, to order -7=—.  In A . , since  B(o0 ) ,= U 7 T v  2 ; L  v Z Z e 2  2  In A  s  integrate.  2 2TlZ,Z, e Jn& A(y y ).= ^2 2  2  1  .We now multiply by d-^g/^"^  i ;  2  2  2 v  i  n  (2.22)  ^  We obtain 2  +  i  sin COdu)  1  v  v3  r  zir  doC  o  2_. ,2 , m  .where  sin^du?  V  v  l  .(2.23)  (2.25)  -  .a  =  + m  (2.2U)  sinoJduo V and  2  2 2- 1 "2v-^v cosuJ + T  2  16.  The integration .of  I^-  and  I  i s straight-forward.  2  However, , since the  magnitude of the r e l a t i v e v e l o c i t y V,.must always be positive,,we have to consider separately, the cases v > v . a n d 1  T  2 •<?_ 2V  f  >  v (v 2-v ) 2  2  —  I  '  l 2  _•' 2 *T  lo  v  2  ..=  - Y2 2  >. V]_..  2  '  Y  ,  ,v  1  2  >  V  The r e s u l t  2  > y  2  ^  A(V V ) 1 ?  2  Z  l  Z  m  ^  ^  •  l  Y  I  n  A  '  V  ,  V  >  : m  Similarly, for B ( v , v ) 1  x  2  hTe  InA  k  27TZ Z e^ 1 2 2  where  2  1> 2 V  >V  :  .  sinuOdtO  |  2 ' 1-  (2.27)  2  2 2 2 sinC0v -y 1  2  ~V3"  2  d  0  ^  r3  2 V l 1  s i n 3 cO.dtO 3-  L  1  we have  2  / B(v ,v ) =  A(v ,v )  U  v^  2  ^  2' , 2 e -InA  (2.26)  1  Equations (2.26) together with (2.23) give us for (  is  v  2 ?2  ln/^'I,  :  (2.28)  (2.29)  17-  As before we obtain two different expressions for  I 3  Io = 3  (2.30)  3  3V  Equations ( 2 . 3 0 ) a n d equation ( 2 . 2 8 ) give us for Bty-^Vg) :  ^ B(v , V ) = 1  2  2 4  87TZ! Z .  e  2  2  A 2 W\v . 2  V  3 v.  2 2 1+ A £ 8TTZ-J_ Z . e In A v-j_ 2  1> 2 V  V  3 v-o  2  (2.310  > 1 V  We can now write equations ( 2 . 3 1 ) and equations ( 2 . 2 7 ) . i n terms of E . and Eg ±  as followsJ , i> 2. E  A ( E .  I  :  , E  2  B ( E - ^ , E  )  2  ) . -  m  E  l  m  2  -<  hG S 2 1 .  G -m 2  2 E  2  3m E 2 1  m  l 2 ^  2  ,  E  2  E  x  m^m^  E  (2.32) where  G •= 2 ^ TTZ-^Z^eVn A . Summarizing the theory, of t h i s chapter,,we have the k i n e t i c equation  given by f(E,,t)  , =• -J ;.+ J N  1  2  (2-33)  18. where the general c o l l i s i o n ..integral- J^g • i s given ."by CO  r  f(E,..,t).dE  '2-1  +  and  A(E ,E ) 1  2  1 ~d  Z  J^;  A(E ,E )f(E ,t)j 1  2  1  (BCE^EgJfCE^t)!!- . ,  , ^(E-pEg)  (2-3*0  .are defined by -eqaution -.(2. 3 2 ) . . In the following  chapters the k i n e t i c equation w i l l be discussed ..in d e t a i l , for the three stages • Jt  by which we represent the approach to equilibrium .of a plasma.  .19CHAPTER-3 THE ELECTRON-ELECTRON RELAXATION TIME  In this chapter we ofctain.a l i n e a r i z e d equation describing the of the electron d i s t r i b u t i o n function.due to the electron-electron  evolution interactions.  The relaxation time t o . e q u i l i b r i u m among e l e c t r o n s - i s obtained from a numerical c a l c u l a t i o n using the v a r i a t i o n a l p r i n c i p l e . We apply equation (2.33).to the electron d i s t r i b u t i o n . f u n c t i o n for reasons given i n Chapter 2, the electron-ion ..collision i n t e g r a l .  neglecting, We make  the following changes to equations (2.32) and (2.33) m  = m ,= m  1  2  e  ;  ^  = E  .;  1  E.^ = E  ,  Z  ±  = Z. = 1  Equations (2.32) and (2.3.3) become now ;  G  A ( E ' ') > E  B(E,E*)  —r~f  UG  ;E*  ,  E>E«  3m 2E2 P  m 2E2 E  kG E  G  —T—a •  m2 e  3m 2E'2  E ' 2  j .E'>.E (3.1)  and  -co  ^ f ( E , t ) .=  f(E',t)dE' « - | [ A ( E , E ' ) f ( E , t ) ] f  • 1-  +  2^ 2  B(E>E')f(E,t)  (3-2)  E  L i n e a r i z a t i o n of The Kinetic Equation We l i n e a r i z e equation (3.2) by assuming that.the d i s t r i b u t i o n .function  equal to the. equilibrium d i s t r i b u t i o n plus a small p e r t u r b a t i o n . . That i s  f(E,t).,=  f°(E)  >(E,t)]  ,  where f ( ' E ) „ . i s the Maxwell Boltzmann distribution..function Q  (3-3)  is  20... 1  LE  f°(E)  2  _E e" kT  (3-10  2 NF  L =  V7f(kT)3/2  and  J^(E,t)  i s a small deviation from equilibrium.  Substituting equations  ( 3 . 3 ) and(3-^) into equation ( 3 . 2 ) and neglecting products of the perturbation as being second order terms we get  f°(E) < ^ ( E , t )  ..=  J^f°(E'  )dE'  |^-.^_JA(E,E')f°(E)] -co  •+  ^[B(E,E')f (E)j  2  0  / f ° ( E ' )dE'  | ^ [ A ( E > E ' ) f ° ( E ) ^>(E,t3  •'o ^(E>E')f°(E) 2  ^ E  y(E,t)]l  |f°(E>))^E;t)jE'L^  +  [A(E,E«  )f°(E)]j  2  + 1^ 'If we l e t C(E)  (3-5)  [B(E,E')f°(E)]|  -co =.0/ /f°(E')A(E>E')dE'  =  4  -co  and D(E)  =  / f°(E')B(E,E*)dE'  ,  (3.6)  we can.rewrite equation ( 3 - 5 ) as follows:  f ° ( E ) ^J^(E,t)  .=  - £-[c(E)f°(E)l .+ 3 . 2 L  1  -^[c(E)f°(E)  Jf°(E«)  y(E,t)j  .+  I  0  c^E  J  E  .+  2 ^.[l)(E)f (E)l 2  L  ^_§(E)f°(E) E  >(E,t)j  jL [A(E,E> )f (E)]  ^ ( E , 't)dE'  e  •  i | L _ [B(K, .)^(E)] E  (3-7)  21. 2.  The Equilibrium • Condition The. r i g h t hand side of eqaution (3-?) should be zero i f f (.E,t). i s  to the equilibrium d i s t r i b u t i o n function.  equated  This provides a check on the ;  consistency of the equation.and on. the-approximations made. ^/>(E, t)f0  r i g h t hand side of equation . (3-7). "to be zero for  In o r d e r . f o r the 0, we; must show  only that 2  - ^ jc(E)f°(E)j. BELJ  =|^L_ji)(E)f (E)] Z% 'E  +  = 0  0  (3-8)  Straight forward integration ..of expressions - (3-6), gives GL  C(E)  e  2  -  E  •D(E)  ^  (*T)  E_  W  2  e"  m^2  e  W(x)  »JkT  "U  where =  (3-9)  E_ kT  m2  e  2  z  dt  i s the error function.  We f i n d that  -C(E)f°(E) identically,  .+  i<L 2  3E  rB(E)f (E)l • ,= 0 0  L  J  (3-10)  and hence the condition (3-8) for equilibrium i s f u l f i l l e d .  We  can now make use of equation (3-10) to simplify equation (3.7) and we obtain f°(E)|^(E,t)  .=  ||_|>(E)D(E)^(E,t^  °(E' ) ^ ( E ' ,t)dE'  < - < L . J A ( E , E » )f°(E)j  B(E,E» 2  ^E  2  )f°(.E)J  (3-11)  ,22. Using the Heayiside function H(x) defined by  c  1,  H(x) ,=  (x > 0)  ;  (x <-0) ;j  .0,  (3-12) S(x)  and i t s derivatives given i n terms of the Dirac d e l t a function d|(x)  8(x)  ; =  dfl(x),=  dx  (3.13)  ;  S'u)  .;  .  2  we can simplify the i n t e g r a l which occurs i n equation  (3.11)as  follows:  2 f (E') >(E',t)dE' |--jL.^(E,E')f°(E)|  r dE' ^ ~  where  2  (E',t)K(E)E' ) "  E_ K ( E , E ' ) ; = e" kT  e  E_ " kT  1 | i ( B , E ' )f°(E)]  +  G  .•+•£-£ m2 e  (E,t)  ^  ^  2  H ( E - E ' ) . - 11  kT 2  3  E  ' -H(E-E') (kT) 3  /  (3.1U) H(E'-E)  kT .+  2  E  2  2  3  /  2  3 "(kTF  HfE'-E)  It i s apparent that the kernal K(E,E' ). i s symmetric, K(E,E') = K(E',E)  3-  Expansion  ofIthe Peturbation  E'  i  1  II  2'E  e" k r  that-.is  (3.15)  . i n Modes  If we make the following change of v a r i a b l e s :  (3.16) Kf>,r)  = (kf)*  k(E,E')  j  we can r e w r i t e equation ( 3 - l l ) w i t h the a i d of equations (3-1*0 and (3-*0 as x.  4  1 -tf-  (3-17)  We define two operators  and 771  a s  follows  (3-18) -2*  With the above n o t a t i o n eqaution (3•IT) becomes simply  (3-19)  at  LG  . The s o l u t i o n of equations of t h i s type which behave p r o p e r l y at i n f i n i t y can be w r i t t e n as  (3-20) n= o where the ^(y)  are the eigenfunctions of the eigenvalue equation  (3-21)  and the C  n  are determined by the i n i t i a l c o n d i t i o n s .  2k. k.  • The V a r i a t i o n a l P r i n c i p l e The following v a r i a t i o n a l p r i n c i p l e (Morse & Feshbach, 1953> p-1118)  applies for the eigenvalue equation (3.21)..  If ^ and 7?^, are s e l f - a d j o i n t  then  r>  —  o (3-22) LG  where the following boundary condition holdsj;!  e  T  Z  —> o  as  * —>  00  We mean by .-equation (3.22) that i f J = J(06^063'• • o ^ ) then  aw/  1,2...K  The self-adjoin-fe condition for operators  (3-23)  £ and7T|_ requires that  (3-24)  7?[ From the d e f i n i t i o n of adjoint.  i t i s immediately apparent that e ^ i s s e l f -  In.Appendix B i t i s further shown .that  i s also  Equation (3.22) i s obviously s a t i s f i e d for the exact { ^)  self-adjoint.  ^ ^  i f the function  i s known. - However,,since the modes are unknown we s h a l l attempt to  represent them by means of t r i a l functions.  These t r i a l functions w i l l be  expressed as a series of orthogonal polynomials and we select the Laguerre polynomials of order one-half.  associated  These have been chosen for two reasons.  These polynomials a r e , , f i r s t of a l l , exact solutions of.equations describing the behaviour of a Maxwellian gas.  .And secondly t h e i r normalization i n t e g r a l has  25the same weighting factor.as  .  Our t r i a l functions w i l l therefore he  represented by K  (3-25)  2 7  where  (^)  i s the jth.order-associated Laguerre polynomial defined'"by  a t  (3-26)  Using the Rayleigh-Ritz method of successive approximations (Morse and: Feshbach,.1953), we strive to obtain,a consistent value for the smallest non zero  ,  since the mode possessing t h i s relaxation rate w i l l be prolonged  for the longest time.  Substituting equation (3.25).into equation (3-22) we  can write  (3-27)  where  (3.28)  and  _e  y  0 -'o  ^  J  J  (3.29)  26. The v a r i a t i o n a l p r i n c i p l e . a s expressed b y e q u a t i o n  jL. Applying equation  J  = 0 •, f o r a l l  (3-23).requires t h a t  i  "  (3-30)  (3. 3^) , t o equation- (3^ 27). we obtain:  where each a d d i t i o n a l j. g i v e s a f u r t h e r , approximation.  Equations f»3-3l)  w i l l have a n o n - t r i v i a l s o l u t i o n ^ o n l y when, the,.determinant-oif the c o e f f i c i e n t s is  zero, t h a t i s  Recursion r e l a t i o n s f o r i n t e g r a l s appearing found  inati' J  and TH,;,. J  w i l l be  in.Appendix F. I f our approximation  polynomials •equation .  to  ^M^Y)  c o n s i s t e d o f the f i r s t  j  Laguerre  we f i n d t h a t t h e s o l u t i o n o f the determinant-as expressed by  (3-32) w i l l . g i v e  each c o r r e s p o n d i n g  us approximations  t o the f i r s t  j  relaxation rates,  t o . a d i f f e r e n t , mode. -The f i r s t .few p o l y n o m i a l s a r e :  0  (i)-fv^  -  1 X  2  2  2  In Appendix E i t i s shown t h a t . t h e c o n s e r v a t i o n o f p a r t i c l e s and the c o n s e r v a t i o n of  energy impose t h e c o n d i t i o n t h a t A Q and A]_ be i d e n t i c a l l y zero.  first  t r i a l f u n c t i o n must :be  >cm  =. 4  2)  (t)  }  Hence our  27and we obtain from equation .(3• 32), the following f i r s t approximation for  15T2 or 1)  =  .k  °K  0  >  LG  I5m i {2.  l&/KlnA  e  15me^ (kT)  3  Me  k  /  ' (3.33)  2  e  Equation (3-32) was programmed for.an'IBM 1620 computer.  The output  consisted of a symmetric matrix of order i which represented a system .of i homogeneous equations, i n the i unknown coefficients  m  .  The matrix was used  as input to a program written by.'H. Dempster for the Alwac'III-E computer which gave as output the-eigenvalues  and eigenvectors of the matrix.  The  eigenvalues.jcorrespond to the relaxation rates and the eigenvectors correspond to the modes.  The result for the lowest non-zero relaxation rate after nine  successive approximations was  •  '  (3.3M  The numerical c a l c u l a t i o n of the coefficient.;0.262; i s accurate to three decimal places.  28.  . CHAPTER k. THE ION-ION RELAXATION TTME  If we apply .equation -(2. 3 3 ) , "to the ion. distribution'.function and neglect • the ion-electron c o l l i s i o n i n t e g r a l , for the reasons given..in Chapter-2, we find that.the c a l c u l a t i o n of the ion-ion relaxation time has exactly the same form as. the electron-electron relaxation tigae with the: exception that the mass and charge of the electrons must he replaced by the mass and. charge of the i n t e r a c t i n g ion. . .We therefore obtain for the relaxation time of,the f i r s t mode  '  LL  where the coefficient  • ' .  o.2GZ • \G nx Z >  In A  0 . 2 6 2 , as before i s accurate to three decimal place's. .  29CHAPTER 5 THE ELECTRON-ION RELAXATION TIME In general the solution to the electron-ion problem follows the same pattern as that for. the electron-electron.and ion-ion case.  That i s , we  f i r s t l i n e a r i z e the kinetic equation and then apply, v a r i a t i o n a l techniques to obtain the smallest relaxation rate.  There i s , -however,,an added complexity  i n that we are now considering a system .containing two species of p a r t i c l e s We therefore must write the k i n e t i c equation for each species.  We r e s t r i c t  ourselves to singly charged ions and we s h a l l designate by:  f^.(E,t) , the ion d i s t r i b u t i o n function; f (E,t) g  We have from equation  ,:the electron d i s t r i b u t i o n function.  (2.33) J  St  °  f. ( E , t ) = i ' ' • v  .+ ee -  ei  J. . + ii  St If  (5-1)  J. I ie  for-reassas? -gkven„±&*Cfa8ptef -'2, we .consider "the electrons and the ions i  r  to be i n equilibrium among themselves and slowly coming to • equilibrium' :£e:ge~tb.er, ~ ^11=0 >  J„„ ee  w  e  c  a  n  w r  l t e equation ( 5 . l )  as  1 2  ^  ,c© 91  X (5.2)  30.  where  ( -  S rr\  r t  me, E t ;  'il 11 i  1.  "Mi  1111~ ~h~ifr~ T i n m l — r  nf t h " inrr .  - L i n e a r i z a t i o n of The Ion and E l e c t r o n - K i n e t i c Equations. . We l i n e a r i z e equations ( 5 - 3 )  equilibrium we can write  f (E,t)  ,=  f (E,t)  =  g  ±  before; b y -assuming small deviations from  s  f (E-,t) and f ^ ( E , t ) i n the-following way.: '.  :+  ^ (E,t)J  .,  f ° ( E ) [ l ,+  ]  .  f ° ( E ) j~l  I f we define Q(E)  a  e  ^ .=  /f°(V,- ) A ( - E E ) d E ; ' e l  >  i  i  "Jo C (E)  .=  o^(E)  =  L  /  M- (E )A. (E E )dE 0  e  e  r  /f°(E )B ,(E,E )dE i  6 i  i  e  l  i  to  oOJs) =  /^ (2 )B (E,S )dE 0  e  u  £  (5-5)  e  we can rewrite equations (5-2) as  5t  2  S  £ E  31-  3E  L  1  ^  rcE) B CE-E-)]L eL  , (5.6)  and a s i m i l a r equation r e s u l t i n g from the interchange of the subscripts e and i . We have once again neglected products of the perturbations  as'being  q u a n t i t i e s of second order. 2.  The Electron-Ion E q u i l i b r i u m -Condition The e q u i l i b r i u m c o n d i t i o n f o r equations (^-6), "which i s analagous to  equation (3-8) f o r the e l e c t r o n - e l e c t r o n case, requires that  o E  — I  —I  (5-7)  . Integration of expressions ( 5 - 5 ) , i s s t r a i g h t forward and the r e s u l t s are as f o l l o w s :  €k.T mi  LL  IT  S  32. sE "ET  -+- & e  matt)'  -  e  (5-8)  kr 6 = With the values (5.8) we f i n d , as was the case in- Chapter.3/ that  2  identically.  "o^Ce) f °CE)  =.  ©  =  o  Hence the equilibrium .condition i s f u l f i l l e d .  ^_f (E) G  =  0  That  (5.9) is  .  We can simplify equations (5-6) with the use of r e l a t i o n s ' ( 5 . 9 ) as follows:  • 5t  c  ^  E  33-  "1- ^1  B  & i  CE, E O f°CeO (5-10)  and a similar equation obtained by interchanging the subscripts;e and i . The integrals which occur i n .equation .(5•10).can be simplified using the Heaveside function defined by equation (3-12).  We get  3^ 3-  Vector Operator Representation .of the Ion and. Electron Kiqetic Equations :  If we perform the following change of variables  and uke•equations (5• ll),we. can write equations- (5-10),in vector operator notation as follows:  ~  -P  at  (5-12)  r  By t h i s notation,we mean that  > ( * , t ) . i s a column vector  (5-13)  —-*  who se adjoint  ^ / ( ^f^t)  . i s - a row vector, that  is  (5.1^) The vector operator "pcan.be represented by the matrix Tee  ©  TP  (5-15) Li  where  "Pu = Tee  =  (5.15b)  e ^  35S i m i l a r l y the vector operator £7 can.be represented by the matrix  & =  (5.16a)  6 whose components are defined-as  ii  follows:  _ e  . - Y C i + e) Ife. '/a  -OO  1  1  i .  J i f  2  — e (5.16b)  36. h.  The Variational- P r i n c i p l e The solution to equation  (5.I2),can be written.as previously  •(5.17) 1 where the  yes)  are the eigenfunctions  of the eigenvalue  equation  (5.18) where  The v a r i a t i o n a l p r i n c i p l e which was applied for the electron-electron, case can be applied i n the same manner to the electron-ion case.  The c a l c u l a t i o n w i l l  be taken to f i r s t order i n £- , the r a t i o of the mass of the electron.to mass of the ion.  the  If "tr and & are s e l f - a d j o i n t then the following i s a v a r i a t i o n a l  principle.  (5-19) where  (5.20)  The s e l f - a d j o i n f condition for operators [P and 0 ig 5  and  ,00  (5-2ffi)  The operator  i s s e l f - a d j o i n t because i t i s a diagonal matrix whose non  zero elements are numbers.  The proof that ft* i s self-adjoint w i l l be found i n  Appendix C. —^m We represent the unknown modes functions.  1^/ (o),  as before, by means of t r i a l  For reasons explained i n Chapter 3 e choose the"Associated w  Laguerre polynomials of order one-half.tp represent the components of our unknown vector  /  yf (tf)i  We have (5.22)  Wco  =  T  B ; L  (  ; V )  •  Substituting relations (5-2£) into equation (5•2d),and using,results of Appendix C, equation (C5 ), we obtain  38.  It  ^  J  1  fc •  -  'fee  Recurrence relations  involving the above integrals can be found i n  Appendix F. In can be seen, from Appendix B , t h a t to f i r s t order i n ;  ^ei  =  °.  >  s  6  r 1 ; (5-25)  With these r e s u l t s and applying the v a r i a t i o n a l p r i n c i p l e to  •-v  -  J , . namely, that  0 , , for a l l k ;  and  •=_ 0 , . f o r a l l s ;  (5-26)  s we obtain the following sets of homogeneous equations i n .the A  and  coefficients  B  (5-27) ]  2&:  oi l  V  where K i s the order of the - approximation.  o  5  S = Ojl.wfc  39Equations (5'27).will have a solution .only when the determinant o f the c o e f f i c i e n t s i s zero.  In- Appendix E i t .is shown that the conservation of  p a r t i c l e s imposes the condition that . A^ of energy demands that :A^  =  -B,m  =  0 , . B™.  •=  0 ..  The conservation  Our f i r s t t r i a l function w i l l then be  J  (5-28)  -Li  C*)  :We can obtain,. i n much.the same manner as f o r the electron-electron case, the following r e s u l t ' f o r the f i r s t approximation to the lowest non zero relaxation rate.  z  or a l t e r n a t e l y  rt  j£  1_<S  _  =  (?.29)  'The computer r e s u l t obtained-after twelve successive approximations was.  .OZ85  16  Me e In A.  me.  3 me Ck.t) ..- mi fe  3/t  (5-30)  and was found, to be smaller, than the co^ospondingquair>,ity. f o r .the second mode .'by. a factor of 1.21. . Complete tables of r e s u l t s appear i n , Appendix F. -  • ,'oOe e y p a c t *h=> fc>e  The numerical c a l c u l a t i o n of the coefficient- • ©2B5 -'iB accurate to % s i g n i f i c a n t r  figures.  ko. • CHAPTER 6  v  CONCLUSIONS  In t h i s thesis we have employed v a r i a t i o n a l techniques to obtain expressions which-give the rate at which a plasma approaches equai-llirium. We have attempted to solve the k i n e t i c equation by simplifying and l i n e a r i z i n g i t to the point where numerical c a l c u l a t i o n s could be performed and the relaxation constants  f o r the d i f f e r e n t energy modes obtained.  We summarize our assumptions and r e s u l t s : . (i)  We are dealing with.an i n f i n i t e , bimbgeneous i s o t r o p i c plasma,  (ii)  The plasma i s i n i t i a l l y hot f a r from equilibrium,  ( i l l ) . .The evolution-.of the plasma occurs i n three stages : a) ,the electrons approach equilibrium among themselves . with a relaxation time  ~T~ •  , b) ,the ions approach equilibrium among ;themselves with" a-relaxation.time . c)  ~U..;  the ions and electrons approach equilibrium with a relaxation time  ~C ie  The.results using an IBM 1620 and an Alwac' ;III-E computer are: 1/2.  \  -  V = f» • \ »R > J2l  2  3 / 2  .^v:  C6.D  hi.  P h i l l i p s (1959) calculated the relaxation rate by applying v a r i a t i o n a l p r i n c i p l e s to the Boltzman equation. on h i s work.  Tee  Our c a l c u l a t i o n s are 8Ln improvement  He obtained  -  1.68  Ik-f)^  .  ,  M  This r e s u l t - i s approximately a factor two smaller than.the similar quantity obtained i n t h i s thesis.  P h i l l i p s also derived a f i r s t approximation to the  electron-ion relaxation rate.  T ; e  He got  Q.132 -gU.  * c • QcT)  ..  (6.5)  This i s approximately thirty, times smaller, than the electron-ion relaxation rate obtained i n t h i s thesis. The r e s u l t s obtained f o r the - electron-electron relaxation time i n t h i s thesis i s greater by a factor of seven than times obtained by Spitzer • (1956) and Rosenbluth.et a l  (1957)-  V  e  have included in:Appendix F the eigenvalues  and eigenvectors obtained from the  "tvjel-fiVj  approximation f o r the electron-  ion case and the corresponding eigenvalues and eigenvectors from the ninth approximation f o r the electron-electron,case. : No estimate of the higher order eigenvalues i s given. .  .  1+2. . APPENDIX A THE ENERGY.EXCHANGE IN [A-BINARY COLLISION I n . t h i s appendix-we study the dynamics ef the c o l l i s i o n ..©f two ^particles. We l e t  v-]_, . V2. denote the i n i t i a l v e l o c i t i e s of p a r t i c l e l a n d p a r t i c l e 2.  Let v^'.v^'denote the" f i n a l v e l o c i t i e s of the same t w o . p a r t i c l e s • a f t e r a collision.  Let  m^-andiiv, he the masses'of the two-particles.  The energy-  gained by p a r t i c l e 1.as a result of the c o l l i s i o n .is AE.  = | m ( v^ ,- v 2  L  x  2 x  ) .  .  (Al)  We f i n d i t convenient-to .express t h i s r e l a t i © n s h i p - i n t e r m s of the ;  r e l a t i v e v e l o c i t y ©f the two p a r t i c l e s mass  V , and the v e l o c i t y of the centre of  V Q defined as follows:  1 'i2 1 n  +  (m v^. + m-^v^),..  (A2)  2  ^1  .. It can.be shewn (Chapman. and Cowling, 1952),< that | V | = j^l ;  •  Using equations (A2) we can ..rewrite the I n i t i a l v e l o c i t i e s of the two . p a r t i c l e s as  ^3and the f i n a l " v e l o c i t y ef p a r t i c l e 1 as  Y  •Let  ^  i  ••=  v  m2 , .  r ~  (A4)  - — • -v-  "be the • angle 'between V „ and V ,  and  the angle between •V  |  and V . • We then obtain.the following relations m_ 2 -2 22 V Q V cos <§ .- + l = G "~ " m-]_':+ 2 - ™1 m  v  v  V  m  v  2  .= V  ~,m 1 G  T iu  r  V COS  1  £  1  "2  V  G  <p +2 m  +  m  ^  u  v. =  m.  T  + "2 . :— 2_ V - m~ + m-,  +  (A5)  l  -r/  O •2m,  cos 9 + •  ± — : V .. V  m + m-|_ G 2  V m  2 - ". +m  ;  Using the above equations we can rewrite equation (Al) :m  g  m rru V  V  n  cos £  - cos  ,- v  .+  (A6)  We also obtain the r e l a t i o n  cos  2V .V  v  0  n  G  m  2 -  m  l  m  2  m  l  : +  (A?)  In order to . simplify the expression for A E ^ we w i l l make use o f . f i g u r e 2 which i l l u s t r a t e s the c o l l i s i o n process.  kk.  FIGURE 2  VELOCITY RELATIONSHIPS.MN;;A BINARY COLLISION .  With reference te f i g u r e 2, which i s drawn.in the centre o f mass system, the c o l l i d i n g p a r t i c l e s describe hyperbolae i n the o r b i t a l plane. i n t e r a c t i o n plane contains the i n i t i a l v e l o c i t i e s i s the p r o j e c t i o n of V Q on the o r b i t a l plane.  v-]_ and v  In general ©  2  The  • -Point-A , the angle  between the i n t e r a c t i o n and o r b i t a l planes can.have any value and i s not r e s t r i c t e d to 90° (Chandrasekar, of mass co-ordinates i s Jc •  19^l).  The s c a t t e r i n g angle i n the centre  h5-  From f i g u r e 1 we o b t a i n :  cos  CD  COS  and thence  Now  • =  COS:  1  COS  COS  1  COS  p  cos <&' •*•  =  cos ^  - cos ^  c  ^ cos  cos £ x  -  /  (A8)  cos cos ^ cos ft  | l  cos( ^ cos 1•-  cos  2(sin^X  L  ,  +  )  OS ^  ,  + siny_ tan ^TJ cos ^ + s i n ^ coslC.  £  2.  ,  t a n <£„ Ices $ .  e  •-j  x  (A9)  S u b s t i t u t i n g equation . (A^) , i n t o . e q u a t i o n . ( M ) we. have .m  m,  9  1  Tig . +  s i n 34 + s i n % -  cosOL tan qb ,ces $  9  Q  ra-^  I f the angle between v ^ and VgVis U  o  '  4-  .  (Al©)  then by t a k i n g the vector.-products o f  equations (A3).we f i n d that v-^. Vg. s i n LO  i=  VQ'V  sin ^  .  and we can w r i t e by/inspection of f i g u r e - 2 . • v^ , s i n oO cos ©  tan c/> = V  G  "V  (All)  cos ^  From equations ( A l l ) , A(l©).and (A7) we obtain the r e s u l t :  m  AE.  • J-  2  : r ;  iji  .= 2 —= -5,f. m _ • + m. 1 2 l m  2  ^  s l n  +  2  v  <i  :,+ m  ,_- '  •2  . -" 2'- i.; 2 m  m  +  1  s i n 2p cos ^  l r  m.--+ m-2 - .. l. 0  Uf  Vg. V]_. sinoJ .cos ® J" •  (A12)  k6.  • APPENDIX B '.PROOF..'THAT. THE OPERATOR Til, I& SELF—ADJOINT In t h i s appendix we shew that  -  &c~7ri  [if a6\<lv  5  (BI)  where  TTt a c o = t f  e  —  £_&(tf)  e  (B2)  -*tf y ^ , z  -+, and  -tf -tf -tf'  e  .  — ^ H Cfr-r)'  H (tf-tf'D  e  3. -|-tf H6MO  -2  ,a  3  3  tf^Ctf'-tf) (B3)  •For-the l e f t hand.side of equation (.Bl) we have  -co  -tt  _2tfx  I t i s c l e a r that hy interchanging  rp  D^Ctf") £tf  '  0 0  •  the dummy v a r i a b l e o f i n t e g r a t i o n  tf.by  tf and v i c e versa we get  Zit) since  K(tf,  K( tftf )  (/><*')  dt'df  5  (B ) 5  V7-  F u r t h e r , i n t e g r a t i n g by.parts,.we see t h a t ;  A*  5*  2*  1  •PO  -2#  r  V* e  — e  (B6)  5^  The f i r s t , t e r m on the r i g h t hand side i s zero because of the boundary.condition  tpO)e  ^  — a s  V-  C O  .  We have therefore OO  .CD  L: Ho S. o  r i—  2^  which.is p r e c i s e l y the r i g h t hand side of 'equation ( B l ) . s e l f - a d j o i n t operator.  (B7)  J Therefore  is. a  APPENDIX G .PROOF THAT THE OPERATOR & IS SELF-ADJOINT  In this appendix we shew that  U  i s s e l f adjoint i n the sense ef  (Cl)  tp CO  where operator  i s a column vector and  V-p  CO  i s a row vector.  &• i s defined.by relations (5.16a),and (5.16b).  The  Expansion of  equation (Cl) gives us the following:  CO  -CO  1-  PiC^ft*[^^  -t  p ^) ^ e i [ ^ C ^ ] ^ e  +  Jo  ^ ^ [ ^ ^ ( 0 2 )  J  0  I t ; i s immediately-apparent that i n order to prove equation (Cl) ; i t w i l l be sufficient-to  show only that the following relations are true:  ^°  r°°  T-eOO . ( 3 ^ [^C^dr  "°° lf (tf) €  r  -  i  &d [JiCOJcW  =  A.  r  1  IjSeOr) &e ^ COj^ e  f^*  ..  . (c ) 3  1  (C5)  (3«C*> ^ i [ ^ i C ^ ] ^  ( c 6 )  ©f.the four-above r e l a t i o n s i t i s apparent .that • (C5),and (c6),are the same i f we interchange the , s u b s c r i p t s e-andi.. Hence i t w i l l be s u f f i c i e n t t o prove r e l a t i o n s (G3), (Ch) and (C5). We see that r e l a t i o n (C3) .-is true since we have  —e .  ^  Jo  ^  oo _  I  (C7) ^0  ay\  Jo  >  since the term i n the 2nd l i n e of the equation.is zero because of the boundary condition  e  t  o  as  V" —>•  0 0  .  A s i m i l a r c a l c u l a t i o n can be•done to.show that r e l a t i o n (Ch) \is t r u e . In.order t o prove r e l a t i o n . (G5) we f i r s t - w r i t e - i t ••in , i t s complete form  •(cs)  mi 6i/^)t(>«C^«  •  y*dfr  50. If we perform .a change of variables i n the second i n t e g r a l , on the r i g h t hand ;  side of equation (c8) defined by :  we obtain  IT)  2,  By u s i n g e q u a t i o n . (C9) . the d e f i n i t i o n ©f (9^ , and t h e ' f a c t t h a t T ^ V " , ^ ) i s symmetric,  we see a t once t h a t the r i g h t hand s i d e o f e q u a t i o n . ( C 8 ) : i s j u s t  Thus we have proved that  sa  s e l f - a d j o i n t operator.  51. APPENDIX -D ks rs PROOF THAT TO ORDER e &. , ,= 0 j S / l i AND £/. . e i  ks T h e . d e f i n i t i o n ef  U  e i  .  =0;  jk and of  0 . . i s taken from, equations (5-2'H)' i i  With the d e f i n i t i o n of " ^ j ^ ^ i ) as - expressed beneath - equation ( 5 . l l ) we have the following:  mi  4tfx  x r ci)  3/2.  00  x  1 4  2, m i 3 nn  ^ (£>  L>e)  e  2.  °°  v7.  e  -*e % 4 d4  r  00  L  (yf)e d V i  Jtfe  .CO  + mi  (Bl)  52. I f we make use of the f a c t that  L ^ • (x) - 1 ,.and 0 we'-can-rewrite equation • (Dl) , i n the f o l l o w i n g form: J  - 2,  ^ 3 rn.i  e.L  J i  2  n^i  3  rru  T  1  (x)  •3 2 — x  (B2)  me  -Z L  where we l e t  A  ,0b  c$  C  co  -tfi  1  03  -Ye  ft 2&  (B3)  '  i f we consider  s e r i e s about  =  la)  'I  .as-a f u n c t i o n of 6 we can .expand, i t .in.a Taylor  £ = 0 .as-follows: 2.  I|C^) 2  .  ^  x  <v ks R e c a l l i n g that we are only, considering  •ei  to f i r s t order, i n &  •  we need only concern ourselves with I ( 6r) to zeroth order.  (B5)  I f we reverse the order-of i n t e g r a t i o n of :  I  2  1^ we get  -  Wc must include terms up. to-order order i n € .  3/2  p/Ks 'ks i n I,-, t o - r e t a i n O „:£© -ei  first  53I f we l e t  we have  v7  • and -r  6  2>? where we must keep', terms up to order Now,  55'  5=0  '  i: (o) = 0 2  =  0  =v-0 •  (B8) hence * 6 3  3/  ^  V^.  C-^  <L  C-7)  2  •We must Include terms up t o order &  3//2  in..I  3  as,well.  We have  5h.  If we perf©rm the f o l l o w i n g change of v a r i a b l e s (DIG)  we obtain -_€:X -X  J_ X  Expanding I^ about  dX  <D11)  €r = 0 we get  3/  a  (D12)  From ^equations (DI2), (B9) and (Bp) we have  r  ks  CO  U C^)e X  ei.  1  ' LTOfc  e  n  d^  d*e  C ° 0 0 L <Q e x*dx s  S = 1;  m  /z e  (B13)  55-  where  -  £  j fA )  e"*d*«  L  -  4  ^  L  "  O  rs Taking the d e f i n i t i o n of & ii  ^XX  from equations (5-2:4-) we have  f£tt  5tf  ^  5  (Dlif)  which i f we integrate "by parts gives -6*  tf.e  2tf  &  £tf  For small x we have the expansion  W(x)  .=  .3  J  x  >  10  Hence we get  2 2  6tf  e • •=  1  3  .  - €tf = 3  2 5  io  ^2 £2tf 2 s»2 2  (P15)  To f i r s t order i n £ we have for  &  rs  .00  3 toe'fe-  2  €-  3  m~J  :  (r (s -1)!  = S} ;  since  r  (x)  .=  s  APPENDIX E CONSEQUENCES OF THE CONSERVATION OF PARTICLES AND ENERGY' ON THE TRIAL FUNCTIONS .  In t h i s appendix we show that i n the expansion.of the modes of the perturbation i a the;electron-electron case, where the modes are given by  where K is. the order of the approximation^a consequence of the conservation of p a r t i c l e s - a n d the conservation of energy i s that  A^. .= 4-j_- = 0-  For the  electron-ion case where there are two modes, one for the electron and one for  the ions, given by K 1  ^  i"<£>  we have the same conservation.laws giving the requirement:that A^.= and  = '-B  •= 0,  .  From the conservation of p a r t i c l e s we get :  <L dt  r  •fCtft)dtf  - O ,'  ( E  3  )  From the conservation of energy we have  dt where f (t) -is the Boltzmann d i s t r i b u t i o n function.  (EU)  From the normalization  58. i n t e g r a l of Associated Laguerre polynomials we get  n.  o We express the d i s t r i b u t i o n function f(tf,t) as  )  (35)  3 m-^ n follows:  (E6)  4nt  where  (E7)  Using equation-(E3).we get  (E8) -Since t h i s equation must hold true for any. i n i t i a l conditions i t must hold true for any set of -  , p a r t i c u l a r l y - for  That i s  or  However, since I^ ^ 2  (E9)  J=0  O  ( t f ) , = 1; we can-write, equation . (E9) as  i f f C*>  ^  L > l  V*e J * - o  ,  (E10)  and-using equation (E5) we get A  Q  ..=  0  ,  for a l l n.  (Ell)  ..59From equation (~Ek) we obtain  r  -Vt  CO  CO  [4J e  C  n  Yet) At  ~  O  (E12)  .For the same reasons as before  j tf f ^ t f ) > c t f ) d *  « o  n  (—) L ^ (tf)-  Using the fact that  2  -  .  (E13)  3 ^  v\ — t f , . a n d using equation ( E l ) we can  Z  i  write equation (EI3).as  -  J  1  J J  =• o  Using equation (E5) we see.that A.  •=  0  , , f o r a l l n.  .(B15)  For the elctron-ion,case the conservation of p a r t i c l e s fan be written as  —  =  Ur\  o  o  (E16)  since the number of electrons, and the number, of ions remains, constant.  The  conservation of energy does allow energy to be exchanged.between a l l p a r t i c l e s and hence we..have  d j _ jtffeCtf^citf  at  r  (E17)  60. The proof that  A  0  = B  Q  =0  follows the same argument as for the electron-  electron case. We-can rewrite equation (E17).as  r n=\  L  J  x  ft—\ ...  0  from which for the same reasons.as given previously we obtain  Ui  J  U l  6  which becomes  4r  .2,  Lo OO  *  E  a )  /-> A, L O A J *  (E20)  k-=.i .with-the f i n a l r e s u l t that  4  B:  ,  for a l l VT.  (E2l)  61. APPENDIX F RECURRENCE RELATIONS Below are given recurrence r e l a t i o n s used to evaluate the ;  integrals  which appear i n the numerical c a l c u l a t i o n of the v a r i a t i o n a l procedure.  They  are written for the electron-ion case,, however, the same r e l a t i o n s hold for the electron-electron case simply by equating  6  to unity.  62.  _L  -CO  (3D  00  -tfe  n  i-tf,  r  CO  -tf.  nu  I  rv\+l  mi  CO  /~ m _L *J  . f -tfJtf  (Fk)  oo K\4l  -tfi  r irv\  -tf  tf tf^"edtf  33  tfj. e  63APPENDIX G TABULATION OF RESULTS  In table 1 ve tabulate the eigenvalues associated with the ninth approximation f o r the electron-electron case.  The eigenvectors associated  with t h i s ninth approximation are i n table 2 . . In table 3 and k we have the corresponding values f o r the electron-ion case which involves  twelve  -3j>£> VoX.i mat i o ns.  ZzK'S  V ^ 3  1  .2666  2  .11+97  •559  3  .1115  .258  .920  1+  .093*+  .165  .1+03  1.32  5  .0824  .125  •235  • 597  1.76  6  . -071+9  . 101+  .165  • 323  •779"  2.23  7  .0702  • 093  .132  .216  .1+26  1.00  2.71  8  . 0698  .087  .107  • 153  .262  .529  1.22  3.19  9  .0698  .087 :  .106  .153  .262  .502  .861  1.52  ;  3-61+  Table 1 A2  3-  &  10  H  10  -.063  3 .160  -.316  .1+81+  -571  .1+88  -.259  -.179  •339  -.1+32  .326  -.000  -.1+03  • .566  -.275  -.021+  .363  -.1+92  .312  .136  -.1+11  .081+  .1+59  -31+9  -.033  .577  -.301  -.308  .1+08  .185  -.361+  -.206  • 327  -.561+  -.253  .385  •335  - V 224  -•397  -.021+  .381+  -.026  -.369  -•501+  -.195  .21+6  .1+02  .172.  -.208  -.1+81  .211+  -.177  -•353  -•359  -.177  .096  .251+  . .291  .316  -.651.  -.108  -.280  -.1+23  -.1+52  -•31+9  -.116  .108  .196  .581+  .021  .075  .158  .260  •357  .1+1+0  .1+68  .1+18  .1+30  A  '  Table 2  A  • 063  -.005  .018  •  ' • 1  1  1. 333  2  . 2527  2.81  3  . 1152  . 510  5. 53  4  . 0814  . 206  . 806  9.76  5  . 0673  . 122  . 323  1. 17  15.6  6  0 590  . 091  176  .459  1.62  ?.3.4  7  .0532  . 075  . 121  . 243  .612  2. 19  33.2  8  . 0487  . 065  .094  . 159  . 322  .786  2. 89  45. 6  9  .0453  .0 58  .079  . 117  . 203  .411  . 983-  3.72  59. 3  10  . 0424  .0 54  .069  . 095  . 145  .2 54  . 508  1 .21  4.71  76,0  11  .0400  .0 50  . 062  .081  . 114  . 177  .311  .616  1.46  5.85  95.2  12  .0379  .046  .058  .071  .094  . 135  .213  . 373  .733  1.76  71.14  TABLE 3  117.  ink.  in  Tf  m  cr-  CO  co  o  CO  ro  00  in  ro  CO ro  O -£> ro  O  CO  O  ro CT-  in o ro  o o  oo  in  in m o  Tf  CO Tf  co  in m ro  O  00  oo  O co ro  O ro  ro O O  O  o  00  ro  ro  ro 00 ro  ro Tf  CT 00  ro CTro  in m ro  ro  Tf  in  ro rO  Tf  •tf  Tf  CT- 00  in  Tf  Tf  r-  ro ro  o  in in ro  CTTf  CO co  o  •<f  ro  Tf  O O  oo  co co  O  o co  ro co ro  CTCO  vO r—I  CTo  00  ro oo ro  CO  ro  m  co ro  m co  ro ro  O ro  in  ro  ro  CO  vO CTro  in o o  Tf  00  m oo ro  Tf  o  O  oo CO ro  o  I—I  CT-  r-  o  rro  in ro  in ro  CT-  co CO ro  rTf  ro  in co  o o  CToo ro  W PQ  <  ro  CTCO ro  oo o co  co  o oo ro  ro CTco  Tf  in co  co  ll —»  Tf Tf  O  in  o  0  s  o o  ro o o  CO  CT-  00 00  Tf  i—i  CT00  ro  ro  00 ro  Tf  CO  r-  Tf  o  Tf  CO  in co  Tf i—I  o o  CT-  00 ro  ro  rro O  Tf  r—t  ro  Tf  ro  oo in ro  O ro ro  m  CTCT-  o  ro  in  Tf  ro  Tf  Tf  o  ro  CT-  ro in co  Tf  o  ro  CO  O  o  CO  o  00  00  I  O O O  Tf  o o  O in co  o CO o  in  vD  Tf Tf  Tf  ro  CT  00  CT-  ^  oo o ro  O  a.  co in  vO  ro O  BIBLIOGRAPHY D. Bohm and L. A l l e x , Astrophys.J. 105, R.T.  Brackman and W.L- Fite,,Phys;Rev.  S. Chandrasekhar, Astrophys.J.  131(19^7)  112, llkl  (1958)  93, 285 (19U1)  S. Chapman and T . G . Cowling, The Mathematical Theory of Non-Uniform Gases, section 3-^lj (Cambridge University Press, i960) ' • •. M. Fabre De La Rdp-elle, J.Phys.Radium G. Knorr,) Zerts.Naturforsch.  10, 319 (19^9)  139, 9^1 (1958)  W.M. MacDonald, M.N. Rosenbluth and W. Chuck, Phys.Rev.  107, 35O (.1957)  P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part Chapter 9, (McGraw-Hill, 1953) 73, 800 (1959)  N.J.  Phillips,  D.J.  Rose and M . C Clark, Plasmas and Controlled Fusion, Chapter 8,. (M.I.T. Press and John Wiley and Sons I n c . , 19-61)  L. Spitzer, York,  Proc.Phys.Soc.  II,  Physics of F u l l y Ionized Gases, 1953)  Chapter 5, (interscience, New  

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