... 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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Calculation of the energy relaxation times of a two component plasma Jankulak, Francis Joseph 1963
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Title | Calculation of the energy relaxation times of a two component plasma |
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Jankulak, Francis Joseph |
Publisher | University of British Columbia |
Date Issued | 1963 |
Description | Employing variational procedures on the transport equation we calculate, in this thesis, three relaxation times of a typical shock tube plasma. They are the electron-electron, ion-ion, and electron-ion relaxational times. The results are compared to those obtained by other authors, and are essentially an improvement on the work done by Phillips. |
Subject |
Ionization of gases |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-11-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085876 |
URI | http://hdl.handle.net/2429/39307 |
Degree |
Master of Applied Science - MASc |
Program |
Physics |
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Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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