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A theoretical study of space charge and hyrdomagnetic waves in solids Cook, John George 1962

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A THEORETICAL STUDY OF SPACE CHARGE /IND HYDROMAGNETIC WAVES IN SOLIDS by JOHN GEORGE COOK B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1960 A THESIS SUBMITTED IN PARTIHL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MaSTER OF 'SCIENCE i n the Department of PHYSICS accept t h i s t h e s i s as conforming t o the standard r e q u i r e d f o r candidates f o r the degree of MASTER OF SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA February, 1962 t In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics  The University of British Columbia, Vancouver 8, Canada. January 24, 1962 - i i -ABSTRACT T h i s t h e s i s i s a t h e o r e t i c a l study of some aspe c t s of space charge waves and hydromagnetic waves i n s o l i d s . D i s p e r s i o n equations obtained i n the hydrodynamic approximation are studied to g a i n Information concerning t r a n s v e r s e waves propagating along an a p p l i e d magnetic f i e l d , and the c o n d i t i o n s f o r which space charge waves may grow. For the hydromagnetic waves v a r i o u s assumptions are made as t o the r a t i o of the e l e c t r o n and hole masses and e l e c t r o n and h o l e number d e n s i t i e s . P a r t i c u l a r a t t e n t i o n i s paid t o the e x t r i n s i c and i n t r i n s i c c a s e s . I t i s shown that o f t e n waves which are apparently d i f f e r e n t from waves p r e v i o u s l y s t u d i e d , may be considered as simple extensions or s p e c i a l cases of the type of wave motion t h a t a r e w e l l e s t a b l i s h e d . In s t u d y i n g growing space charge waves i t i s assumed t h a t the s o l i d i s i n t r i n s i c , the hole mass equals the e l e c t r o n mass, and the plasma found i n the s o l i d i s c o l d . Recombination and damping of the c a r r i e r s i s taken i n t o account at a l l times. For t h i s model exact c o n d i t i o n s are gi v e n f o r which growth of space charge waves propagating along an a p p l i e d e l e c t r i c f i e l d may occur. - i i i -ACKNOWLEDGEMENTS I take pleasure in expressing my gratitude to my thesis supervisor, Professor R. E. Burgess, for advice and encouragement given during the preparation of the material in this report. I am indebted to the National Research Council of Canada for a Studentship. - i v -TABLE OF CONTENTS Page PART I INTRODUCTION 1 PART I I TRANSVERSE WAVES PROPAGATING ALONG 3 AN APPLIED MAGNETIC FIELD 1. I n t r o d u c t o r y Comments and O r g a n i z a t i o n 3 2. The D i s p e r s i o n E q u a t i o n 5 3. D i s c u s s i o n of the D i s p e r s i o n E q u a t i o n 7 ( i ) General A l f v e n Waves ' 7 ( i i ) The E x t r i n s i c Case 8 ( i i i ) The Near I n t r i n s i c Case 11 4. Some Notes on Watanabe's Approach 14 5. The Background Medium - 15 6. D i s c u s s i o n 17 PART I I I GROWING SPACE CHARGE WAVES 20 1. I n t r o d u c t o r y Comments and O r g a n i z a t i o n 20 2. D e r i v a t i o n and D i s c u s s i o n of the 21 D i s p e r s i o n Equation 3. A n a l y s i s of the Dispersion E q u a t i o n 23 ( i ) D i f f u s i o n Neglected 24 ( i i ) Zero D r i f t - V e l o c i t y 28 ( i i i ) The General Case 29 4. D i s c u s s i o n 30 V -Page APPENDIX A DERIVATION OF THE DISPERSION 34 EQUATION OF PART I B THE WATANABE EQUATIONS FOR ANY 36 VALUE OF M p/Ma C THE QUARTIC EQUATION WITH COMPLEX 39 COEFFICIENTS BIBLIOGRAPHY 41 1. I . INTRODUCTION. This thesis investigates some aspects of wave motion i n the gas of electrons and holes which may be found i n s o l i d s . Section I I i s devoted to waves propagating along a constant magnetic f i e l d ; section I I I studies space charge waves propagating i n the di r e c t i o n of an applied e l e c t r i c f i e l d . For the hydroroagnetic waves various assumptions are made as to the ra t i o s of electron and hole masses and number densities; t h i s discussion i s thus v a l i d also for gaseous plasmas. For the third section, which i s e s s e n t i a l l y indepen-dent of the second, i t i s assumed that the holes and electrons are i d e n t i c a l except f or the sign of the charge. We s h a l l f i r s t study plane transverse waves propagating along an applied magnetic f i e l d . Transverse unperturbed magnetic f i e l d s and longitudinal perturbations are assumed to be non-existent. The discussion was motivated by a recent paper by T. Watanabe (1961) which seeks the frequency ranges i n which undamped Alfven waves may propagate in a gas of any degree of i o n i z a t i o n , the mass of neutral and p o s i t i v e l y charged p a r t i c l e s being equal and large compared with the electron mass. In t h i s thesis we s h a l l discuss a gas which corresponds to the gas of electrons and holes in s o l i d s : neutral and charged p a r t i c l e s with both signs are present with a mass so heavy that at the frequencies considered these p a r t i c l e s may be considered to be immobile, and l i g h t p o s i t i v e l y charged pa r t i c l e s and negatively charged p a r t i c l e s are present i n such numbers that o v e r a l l charge n e u t r a l i t y i s maintained. We 2. s h a l l not r e s t r i c t ourselves to the damped or undamped Alfve'n waves as commonly known, but instead seek those conditions for which the magnetic f i e l d i s a dominant influence on the waves. Alfven waves are such waves, and the o s c i l l a t i o n s i n Sodium which have recently been found at low temperatures by Bowers et a l (1961) are included i n t h i s scheme. We s h a l l find that the magnetic f i e l d dependence i s not the same i n e x t r i n s i c (monopolar) and i n t r i n s i c (bipolar) cases. In the third section we study a paper by Groschwitz (1957) i n which a dispersion equation i s derived for space charge waves which propagate i n a s o l i d along the d i r e c t i o n of an applied e l e c t r i c f i e l d . The dispersion equation contains the d i f f u s i o n and recombination c o e f f i c i e n t s , important i n s o l i d s . In i t s derivation i t was assumed that electrons and holes have equal mass and are present i n equal numbers. The equation was solved f o r the complex wave number K. = K, . We show in the third section that the solutions given are not correct, and that thermal v e l o c i t y fluctuations are not taken into account c o r r e c t l y . The Gros-chwitz paper i s extended to a study of the conditions f o r "double-stream a m p l i f i c a t i o n " i n i n t r i n s i c semiconductors for which recombination i s not ignored but the velocity d i s t r i b u -tion functions are delta functions centred on the d r i f t v e l o c i t i e s of the electrons and holes. Throughout t h i s thesis dispersion equations are studied which are obtained from hydrodynamic equations using a l i n e a r i z a t i o n method. I t i s well to consider b r i e f l y the l i m i t s to the V a l i d i t y of this derivation. We consider the holes and electrons i n the eff e c t i v e mass approximation. The charac-t e r i s t i c times and distances over which macroscopic variables change (such as the period and wavelength of any wave motion present) must be large compared with the mean free time - not mean c o l l i s i o n time - and mean free path of the c o l l i s i o n mechan-isms: the p a r t i c l e s are "locked" together and l o c a l equilibrium i s attained. The system may then be described by such macro-scopic variables as mean density, mean mass f l o w 1 and a parameter such as temperature describing the fluctuations about the mean. F i n a l l y a l i n e a r i z a t i o n method i s used to make the equations more amenable to study and a superposition theorem i s ca l l e d upon to make the approach worthwhile. Again, i t must be kept i n mind that the l i n e a r approximation loses meaning i f higher order terms become important as the waves grow. I I . TRANSVERSE WAVES PROPAGATING ALONG AN APPLIED MAGNETIC FIELD. 1. Introductory Comments and Organization. On the following pages we study transverse hydro-dynamic waves propagating along an external magnetic f i e l d . Two charged c a r r i e r species, electrons of charge -e and posi t i v e ions or holes as we s h a l l c a l l them of charge +e , and a neutral or charged background are present. Electron-hole, electron-background and hole-background scattering i s taken into account, and no r e s t r i c t i o n s are placed on the hole mass nor on the electron and hole number den s i t i e s . For most 1. I t should be noted that a mean veloc i t y may not be defined for low mobility semiconductors, as the mean free path may be of the order of or smaller than the interatomic spacing or the electron wavelength. of the discussion the background medium i s assumed to be immobile. Much of the work i s thus v a l i d for such plasmas as semiconductor plasmas and gaseous plasmas; the p r i n c i p a l assumption i s the v a l i d i t y of the hydrodynamic approximation. The discussion is an extension of several published papers. Hines i n 1953 treated the case of an unspecified number of c a r r i e r species inter a c t i n g only with a mobile neutral medium. Oster i n 1960 discussed wave motion i n general, devoting a few pages to the type of wave we are interested i n but assumes no background i s present. Watanabe in 1961 studies these waves with our assumptions except that in h i s treatment the background medium i s neutral and mobile and the ion mass is large. Tonenbaum (1961) has discussed plane waves for any angle between the d i r e c t i o n of propagation and the magnetic f i e l d but again the ion mass i s taken to be much larger than the electron mass, AS t h i s author i s interested in s o l i d state plasmas t h i s assumption is too r e s t r i c t i v e and hence not made. In section II.2. the dispersion equation i s derived and b r i e f l y discussed. Section II.3. is devoted to a discussion of "general Alfven waves", and a study of the dispersion equation f o r the e x t r i n s i c and near i n t r i n s i c cases The approach we use i s compared with Watanabe's approach i n II and a recipe f o r including the effects of a mobile neutral background i s given in section II.5. F i n a l l y , matters f o r future Investigation are suggested i n II.6. 5. 2. The D i s p e r s i o n E q u a t i o n . Consider an i o n i z e d gas c o n s i s t i n g of e l e c t r o n s of mean v e l o c i t y U«., number d e n s i t y mass and charge -e; ions (t£ ,Np, r i P , e.) , and a t h i r d s p e c i e s of p a r t i c l e s which are immobile (more exact c r i t e r i a are g i v e n i n s e c t i o n II.5.) but may be n e u t r a l or charged. In the presence of an e l e c t r i c f i e l d "E and magnetic f i e l d ~B the equations of motion then are (Watanabe (1961) and Tannenbaum (1961)): ot + j v i ^ p 1^71TP (u,-U P) ^ ^ U ^ - e M j E t U , ^ ) 2.1 and J>r ^ * TTTTPFp ("r-U^+J>rM e WP ( E + tTP x 6 ) 2-2 The c o l l i s i o n frequency of an e l e c t r o n w i t h the holes i s denoted by »^e> ; i t i s assumed to be constant. ^•«.,»,,^pt> are d e f i n e d s i m i l a r l y ; J ^ N ^ M ^ , JV=KipM p . The Lagrangian o time d e r i v a t i v e i s denoted by ^ . Due to c o n s e r v a t i o n of momentum ^»p= . Define (i = ^ , and °t = ^ . Let plane waves propagate along the 3 - a x i s which l i e s along the primary magnetic f i e l d S D . Let no primary e l e c t r i c f i e l d be p r e s e n t , and assume 8 - 8 0 and a l l other v e c t o r s l i e i n the x-y plane and v a r y as exp ( i ( " t " K 5 ) ) . By standard p e r t u r b a t i o n techniques i t f o l l o w s (see Appendix A) from equations 2.1, 2.2, and the Maxwell equations that 6. »» and k obey 5.5 The signs are coupled. The lower signs give the Ordinary (0) wave (the f i e l d vectors rotate with the holes), the upper signs give the Extraordinary (E) wave (rotation with the elec-trons). This convention w i l l be followed throughout the following discussion. These waves are very f a m i l i a r from 2 ionospheric studies. The plasma frequency ^ec for the com-plete ionized gas has been defined as J +•p£)-|-The plasma frequency U t A for the electrons has been defined ) N A 1 a s j ^ - , and s i m i l a r l y f or O t p . The p e r m i t t i v i t y £ and permeability/*, w i l l be assumed to be r e a l . The gyrofrequencies CJ* and are defined as - and ^pj^ respectively. The magnetic f i e l d affects both numerator and denominator of the right-hand side. The denominator and c o e f f i c i e n t of v„.f i n the numerator involve a term i n B G Kip which vanishes as fb= jfc tends to one. Generally the numerator Includes a term i n &„ , but f o r fixed and large enough (i, (or small enough^ ) the numerator and denominator have a common factor and only a a o dependence remains. Varying «x for £ diff e r e n t from one has a s i m i l a r e f f e c t . 2. Any book on the physics of the ionosphere may be consulted. A very detailed study i s given by R a t c l i f f e i n his book on the magneto-ionic theory (1959). In much of the discussion he assumes the ion mass i s large compared to the electron mass. 7. These matters are discussed i n greater d e t a i l on pages following. 3. Discussion of the dispersion equation, (i) General alfven Waves. Usually Alfve'n waves are considered to be trans-verse waves propagating through a two-Carrier gas along the d i r e c t i o n of an applied magnetic f i e l d such that the equations and E + 7jex*Bo = o 3.2 are v a l i d (Watanabe (1961)). The mean mass density and mean vel o c i t y are defined as P«. = p P » T£ = ui> For these waves the phase vel o c i t y i s given by a. i f t h i s v e l o c i t y is small compared to the vel o c i t y of l i g h t . In general to* In the following sections the descriptions "Alfven Waves" s h a l l be r e s t r i c t e d to these waves. They are one form of a more general type of wave which i s e s s e n t i a l l y governed by the applied magnetic f i e l d . For example, consider those waves which S a t i s f y the dispersion equation (III.2.): o1- ^ These waves are not e s s e n t i a l l y dependent on the magnetic f i e l d f o r o>|u« l or ^v, > , and even i f «o«l«»\ and VH.l»«lOM.l the r i g h t hand s i d e v a r i e s not w i t h , but B<. . However, i n t h i s d i s c u s s i o n a s u f f i c i e n t c o n d i t i o n f o r these waves to be i n a c l a s s c a l l e d "General Alfve'n Waves" s h a l l be that CJ-C<1"H.I . These waves may be h e a v i l y damped. This convention i s suggested as plasma p h y s i c s i s overburdened w i t h t e c h n i c a l terms and d e s c r i p t i o n s . ( i i ) The E x t r i n s i c Case. I f (&<<! , o < / b « i , ^ < < o t , -^^ W <•< " > V , and AiW > > ^pi. the holes cease to exert any i n f l u e n c e 3 and the d i s p e r s i o n equation becomes the f a m i l i a r appleton-Hartree formula f o r a p u r e l y l o n g i t u d i n a l magnetic f i e l d : I f the i n e q u a l i t i e s g i v e n are reversed a s i m i l a r equation f o r the holes i s of course found. T h i s equation has been s t u d i e d i n g r e a t d e t a i l (see R a t c l i f f e (1959) and Oster (1960), f o r example). We b r i e f l y d i s c u s s some of the important f e a t u r e s . (a) High Frequency Waves. I f (J » | U , J , e q u a t i o n 3.4 y i e l d s the f a m i l i a r e quation p u t t i n g c x= l/u£ . The group v e l o c i t y i s l e s s than C a n d the product of the group and phase v e l o c i t i e s i s e x a c t l y equal 3. These i n e q u a l i t i e s may be u n n e c e s s a r i l y r e s t r i c t i v e . At h i g h f r e q u e n c i e s , f o r example, the c o l l i s i o n f r e q u e n c i e s are not important. 9. to C l , a c h a r a c t e r i s t i c of high-frequency radio waves. (b) General X-vlfve'n Waves. For we have _ _ ato(± to*.-iS* _ u£. Oe» which describes general Alfve'n waves. I f „^v. ^ ° > should also be ignored. We may demand that tJ be r e a l , thus r e s t r i c t i n g K x , or we may, depending on the problem, demand that K be r e a l . I f ^ i s r e a l and K.- +• i Ka. , then Kikj. and have opposite sign und 2-kiki. _ u t > Ai»  Thus the waves are damped and K. tends to be r e a l or imagin-ary as \Li*\/^>H\o becomes large. This may be readily ascertained from 3.5; ignoring i t becomes Let w be positive and r e a l . Then for the E-<-^ue K is positive and hence k. i s r e a l ; for the o- «J« Me. ^ i s r e a l for to> U e ~ A i t v l and imaginary for U < U*"/£|u>J. One of the l a t t e r Cases may not be possible. I f Oa^^O^\o and for example, we Cannot have Hence K. must be r e a l . For low f r e q u e n c i e s ( s p e c i f i c a l l y , I o(±u»*.;V»b)| -«i ) equation 3.4 y i e l d s and, f o r r e a l u) , I f ' l >* u/iu l.| » -1 then K i s imaginary f o r the 0- . For r e a l K and u,+,ux , t J l j , s p o s i t i v e and = - iu„l 3.6 , Eq u a t i o n 3.7 has r e c e n t l y been d e r i v e d by Bowers et a l (1961) who f i n d that i t g i v e s good agreement w i t h a new o s c i l l a t o r y e f f e c t i n Sodium which they found e x p e r i m e n t a l l y at low temperatures. These authors do not d i s c u s s damping t h e o r e t i c a l l y . E quation 3.6 p r e d i c t s damping f o r the Bowers experiment which i s s m a l l e r by a f a c t o r of ten than the damping a c t u a l l y observed. The d i s c r e p a n c y i s probably due to the f a c t that we have t r e a t e d media i n f i n i t e i n e x t e n t . (c) H e a v i l y damped waves. For 1 , O - « V > K I O , I f w i s r e a l and u * * X ) ^ b < < ^ , and both tend to 11. Let us consider what we have found for the e x t r i n s i c case. In general, the ^ppleton-Hartree formula i s v a l i d . We have the f a m i l i a r behaviour of the high frequency-waves and, f o r general Alfve'n waves fo«t<J«.l ) the o-u<wt nuy only propagate under severely r e s t r i c t e d conditions. For very low frequencies the Bowers formula i s found. Thus the o s c i l l a t o r y effect i n Sodium i s a s p e c i a l Case of the type of wave o r i g i n a l l y discussed f o r the rare ionospheric gases; i t may also be described as a low frequency general Alfve'n wave. ( i i i ) The Near I n t r i n s i c Case. For £=.1 and ^Zl^L ^ p << I "-iv>P<° + i (o-iv>„ t) I , the dispersion equation i s again, as in the following, the signs are coupled. For <=> 7^ > 1 we must c e r t a i n l y have f o r the w^ves to be general nlfve'n waves. I f o v>-^v,,^ PU, i u r u w.(o±£J P) *»_ - o*t L 1^ 1" o J 3.9 which are general ulfven waves without further r e s t r i c t i o n s . In equation 3.9 and 3.8 as elsewhere, the lower signs corres-pond to tW o . u a j e , the upper signs to the E - u ^ e . I f we have not high frequencies but « J « ^ t > v , P W > These are general Alfve'n waves i£ ve » CJ Thus, approximately u«oft • then or l«Jw.|V lii>»uV»b< = ^ 1 v . w ^ u v p W . i ^ ' J 3 ' 1 0 The real part of the right-hand side of this equation -which i s always smaller in magnitude than the imaginary part - V a r i e s as A„ while the imaginary part may vary as &J~ i f I n t e r p a r t i c l e c o l l i s i o n s play an important role i f | L)±CJP-\^\0\ < ;^s^,.p and I u ± u v _ | < ^ p I f i t i s completely dominant, _ = i an equation s i m i l a r to that obtained for heavily damped waves in the e x t r i n s i c case. i n t e r p a r t i c l e scattering may be ignored and oJi <o-;-»Vb*-£Cu-;iJ*0 3.11 Let us treat t h i s equation i n more d e t a i l . For frequencies higher than the c o l l i s i o n and gyro-frequencies, a s i m i l a r equation was obtained under the same conditions f o r the e x t r i n s i c case. Thus at high frequencies the gas behaves e s s e n t i a l l y as a monopolar gas, the p a r t i c l e mass being equal 13. to the reduced mass. Consider 3.11 for the case of general Alfven waves. Then from 3.9 for the high frequency waves ( *J» At.^k) M l 2.1(Up±1-0 17^  " * — - r ~ 3.12 - - l i t Utc Li 1 -For o r e a l , K. i s r e a l . I f we obtain the f a m i l i a r disper-sion equation for Alfve'n waves: , 3,13 For low frequency waves such that i n addition to o«^b,>? Pb and o « o p s o ( l u j u p + v„u^Pw) «cV M. CVpW + s j ^ O , 3.10 and 3.11 show that for r e a l O , For r e a l ^ equals the same expression. Ignoring the t r i v i a l Case v . t ^ ^  v P i . the right-hand side i s seen to be large but decreasing with increasing magnetic f i e l d for |tJju p < V„u v^u , reach a minimum f o r I ur> ~ ^bVpU , and then increase with increasing magnetic f i e l d . Thus for very high frequencies or c o l l i s i o n frequencies larger than the wave frequency and gyro frequencies the behaviour of the bipolar gas i s not more complicated than the behaviour of the monopolar gas. In general the behaviour is more complex, the magnetic f i e l d not only appearing i n the 14. r e a l part of the r i g h t - h a n d si d e of the d i s p e r s i o n equation, as i n the case of Alfve'n waves, but a l s o i n the imaginary p a r t . The r e a l p a r t may vary both as G>oX (3.13) but a l s o as fi>o (3 .10 , or 3.12) . 4. Some Notes on Watanabe's approach. In s e c t i o n I I . 3 . the d i s p e r s i o n equation has been used to study the behaviour of the waves. As Watanabe (1961) has shown, the equations of motion may be used d i r e c t l y t o shed some l i g h t on at l e a s t the A l f v e n waves. From 2 .1 and 2 .2 , Qui _ — _ _ ot + -«?npT + e.*L (pVpk tV-V^wLO-the equation of motion f o r the plasma and the g e n e r a l i z e d form of Ohm's law. The c o n d i t i o n s may be sought f o r which we o b t a i n 3.1 and 3.2, the equations c h a r a c t e r i z i n g the undamped Alfve'n waves. These are obtained approximately f o r example i f <* ~ l . , c= 1 , u ^ ^ . b , and I iu>+ V w p \ <*. -jy T h i s agrees w i t h I I . 3 . T h i s method i s more d i f f i c u l t to use f o r g e n e r a l 4 ' o< and , however. Furthermore, a study of g e n e r a l Alfven waves would seem to be e a s i e r with' the d i s p e r s i o n e q u a t i o n . 4. See a l s o s e c t i o n I I . 5 . and Appendix B. 5. The Background Medium. In the equations of motion 2.1 and 2.2 a damping force per unit mass of the form «^v> or ^ .kUp has been used. The cause of t h i s damping force may be quite general; i t may be due to neutral or ionized impurity scat-tering or phonon-scattering in s o l i d s , or i t may be due to scattering between ions and heavy molecules i n gaseous plas-mas. Even though the model we use Is somewhat i d e a l , such a damping term i s used for a great variety of scattering mechanisms. I t i s evident from the papers of Hines (1953) and Watanabe (1961) that f o r Alfven waves, which are a sp e c i a l Case of the waves we study, the mass of the background p a r t i c l e s i s very important. From Watanabe's table I , f or example, i t may be seen that many frequency ranges exi s t for which Alfven waves may propagate in highly ionized gas ( large ) . Only for high frequencies may the effect of the f i n i t e mass of the background p a r t i c l e s be ignored. This behaviour could be studied i n solids also. The mass of neutral impurities could be introduced, or also the mass of ionized impurities, although the l a t t e r would create more d i f f i c u l t i e s as the charge of the background p a r t i c l e s would have to be introduced. Phonon scattering seems to present d i f f i c u l t i e s . Shockley (1951) has shown that as far as energy and momentum transfer are concerned the scattering between electrons and acoustical phonons i s equivalent to the scattering between two gases of hard spheres, 16. KT one species of mass M*. , the other species of mass , S the velocity of sound f o r the mode considered. The number density of the heavy spheres may not be determined however, as the mean free path for hard sphere scattering - which must be compared to the actual mean free path f o r electron-acoustical phonon scattering - i s a function of both this density and the cross-sectional area of the spheres. It i s a simple matter to introduce the mass of neutral background p a r t i c l e s (density Nb , mass M b ) into the dispersion equation. We only substitute the expressions K r ^ ~£p7^» -tor Mp H rVpb uj, in the equations of motion, and in addition use the equation of motion f o r the neutral p a r t i c l e s : pUw Hb ru — _ . npnb — — \ J>b Dt ^ b ^ n k + nrt (U b-U^K ^b^bp n t + ri p (lib- • U^ , may then be eliminated from 3.1 and 3.2. It i s found that OC 3wAb ^ pl» . ~ , i+i v** +• , . • — r — r - must be substituted for i+*vn» and iJk^Vm for %>Pb . J^fb «• J--v>.b •Pb-v'fb and v>Pb/^wp obey a r e l a t i o n similar to that given for V » P and • From these substitutions c r i t e r i a f or the v a l i d i t y of ignoring effects due to the f i n i t e value of J\* may be obtained. The frequency i s of prime importance as is to be expected. I t must be noted that tip must equal rt*. . These s u b s t i t u t i o n s were c a r r i e d out and the r e s u l t a n t equation compared w i t h equation 3.11 of Watanabe (1961) and equation 19 of Hines (1953) f o r t h i s c o n d i t i o n under which these equations are v a l i d . Complete agreement w i t h Watanabe was found; as the a l g e b r a became i n v o l v e d c o n s i s t e n c y with Hines was t e s t e d only f o r s e v e r a l l i m i t i n g cases. The equations which Watanabe uses to d i s c u s s A l f v e n waves may be r e a d i l y extended so t h a t they are v a l i d f o r any v a l u e of the r«><o . The r e s u l t a n t equations are g i v e n f o r completeness i n Appendix B. These equations may then be s t u d i e d d i r e c t l y , f o l l o w i n g Watanabe 1s example c l o s e l y . T h i s was not done i n g e n e r a l ; i t was only found that f o r Hp~«V the c o n d i t i o n s f o r Alfve'n waves to e x i s t i n weakly i o n i z e d plasmas are almost e x a c t l y those g i v e n by Watanabe f or ">"> n*.. 6. D i s c u s s i o n . L e t us summarize the r e s u l t s of the p r e c e d i n g pages. Although a d i s p e r s i o n equation f o r t r a n s v e r s e waves propagating along an a p p l i e d magnetic f i e l d i s r a r e l y g i v e n f o r g e n e r a l values of o ( . p j ^ and C* = [the paper by Hines (1953) i s an e x c e p t i o n ; he ignores e l e c t r o n - i o n s c a t t e r i n g ) , i t was found that t h i s d i s p e r s i o n equation i s r e a d i l y d e r i v e d . V/e then t r e a t e d t h i s equation f o r the two cases N p * ^ * . and tip. << r / ^ w i t h oi 1* i . The e x t r i n s i c case (rtp«*lw) has been p r e v i o u s l y 18. studied i n the l i t e r a t u r e . It i s i n t e r e s t i n g that the c l a s s i c Appleton-Hartree dispersion equation which has been used extensively in Ionospheric studies was seen to agree c l o s e l y with the o s c i l l a t o r y e f f e c t that Bowers and his fellow workers discovered recently in sodium at low temperatures, AS the ionospheric gaseous plasma and the plasma found i n sodium are at f i r s t sight quite d i f f e r e n t , this agreement i s s t r i k i n g . It was shown that although the sodium o s c i l l a t i o n s are not Alfven waves as these are usually thought of, the s i m i l a r i -t i e s between these two wave motions are such that i t i s too r e s t r i c t i v e to not describe the sodium waves as "general Alfve'n waves". The i n t r i n s i c case was found to be i n general more complex. The i n c l u s i o n of electron-hole scattering did not lead to any d i f f e r e n t type of wave motion. The dependence of the waves on the magnetic f i e l d was found to be quite d i f f e r e n t for d i f f e r e n t cases; for one frequency range i t could be as that of the Alfven waves ( &o dependence) while at other frequencies the v a r i a t i o n could be with as for the sodium o s c i l l a t i o n s . There are several topics which could be studied in more d e t a i l . Boundary conditions could be introduced, although Sturrock (1958) points out that l i t t l e new informa-tion would be introduced in so doing. The effect of the l a t t i c e upon the waves could be investigated. One method would be to use the dispersion equation with the mass of the back-ground p a r t i c l e s included, or the equations of Appendix C. A second method would be to treat the e l e c t r o n - l a t t i c e i n t e r -action as the neutron-lattice i n t e r a c t i o n has been dealt with in scattering or thermalization studies. F i n a l l y , the behaviour of the Ordinary and Extraordinary waves could be studied for the waves in the i n t r i n s i c case (with <*~± ) as i t has been studied in the l i t e r a t u r e f o r the e x t r i n s i c case. 20. I I I . ON GROWING SPACE CHARGE WAVES IN SOLIDS. 1. Introductory Comments and Organization. The following pages s h a l l deal with unstable longitudinal space charge waves in s o l i d s . The dispersion equation we use, which has been previously derived by Groschwitz (1957), includes the effects of damping, recombi-nation, and, supposedly, d i f f u s i o n . The word "unstable" requires c a r e f u l d e f i n i -t i o n . Sturrock: (1960) has recently shown that the d i s t i n c t i o n between amplifying and evanescent waves (the l a t t e r are fami l i a r from optics) requires some attention. In thi s thesis monochromatic waves are considered, and the term "growing wave" s h a l l be used f o r a wave which has r e a l frequency C J but complex wave number K such that the r e a l and imaginary parts of K are of the same sign. Thus, putting K=iCi + >Ki. in ** r ( i ^ t - k ^ ) ) a n d l=~-tt ^ w e obtain «*p ^ £ t which shows an i n s t a b i l i t y f or >o or kxkit-'>o. This c r i t e r i o n avoids the p i t f a l l of c a l l i n g an attenuating wave a growing wave from the reverse d i r e c t i o n . We s h a l l assume throughout the discussion with only few exceptions that the frequency i s r e a l , i . e . the waves are excited with a constant amplitude at any given p o s i t i o n . We f i r s t determine the conditions f o r which waves obey this c r i t e r i o n while d i f f u s i o n may be ignored. We s h a l l f i n d some errors in the Groschwitz analysis which, although only involving a sign, indicate that growing waves e x i s t at a l l times. We then d i s c u s s the Case f o r which d i f f u s i o n i s supposedly taken care of, and show that growing waves are i n d i c a t e d i n the absence of any a p p l i e d f i e l d s . The reason f o r t h i s c o n t r a d i c t o r y r e s u l t i s that s p a t i a l d e n s i t y g r a d i e n t s are not taken account of c o r r e c t l y i n the Gro3chwitz e q u a t i o n s . T h i s matter i s b r i e f l y d e a l t w i t h and f i n a l l y a paper d i s c u s s i n g i n s t a b i l i t i e s i n semiconductors u s i n g the Boltzmann equation i s d i s c u s s e d . 2. D e r i v a t i o n of the D i s p e r s i o n E q u a t i o n . The d i s p e r s i o n equation we s h a l l analyze has been d e r i v e d by Groschwitz i n 1957. He proceeds from the equations of motion Qui ^ _ _ _ . Ot+^tf«.=:-ri(E+UR.fcB*) 2.1 o-t • o u r = = -p; ( E + ur K Q> ) 2.2 i n which i t has been assumed that the damping c o e f f i c i e n t s and masses of the e l e c t r o n s and holes are the same. The e l e c t r i c f i e l d has been d e f i n e d as a macroscopic average of the f i e l d a c t i n g on the p a r t i c l e s , and hence a pressure term i s not used e x p l i c i t l y . We s h a l l see that t h i s matter r e q u i r e s f u r t h e r d i s c u s s i o n . The equations of c o n t i n u i t y are Ship i —• «\ ^ i _^ . <l being the recombination c o e f f i c i e n t , Jp = N l P e ^ r - eOVNjp } T^ = - ^ e M » +cov n . . o i s the d i f f u s i o n c o e f f i c i e n t which is l i k e R, assumed to be the same fo r electrons and holes. Let plane space charge waves propagate along the 3-axis , the d i r e c t i o n of an applied e l e c t r i c f i e l d E» ; no applied magnetic f i e l d i s present. Thus ^ x - = o , ^  = -iK. A perturbation method i s used as in section II to obtain the dispersion equation. The unperturbed values of M*. and rtf are assumed to equal N ; (the i n t r i n s i c case); the unper-turbed d r i f t v e l o c i t i e s are of equal magnitude u o and opposite in d i r e c t i o n as MM= H P= M and &«,=<?. Defining as the vector sum of the perturbations in and u r , u * s a t i s f i e s t. 5 u«- i x > u * -xVu* _ u x o » 3*> • o x j r ^ % *a (N>OX*U0XO) 5vJ-t - * OaTJt* +• £v*ox*V *i«0W;Uox) This equation d i f f e r s from that given by Groschwitz in two c o e f f i c i e n t s . L e t t i n g vary a s exp (i C^x-ic^ ) ) t the dispersion equation follows: c ^ ^ * ( c < . ; ^ ) K \ (C, + i «0 ic* * r 0 +;cka _ o 2 . 4 where c , = ^ -o^aaoM. + i O w o r o^-uoM - U ^ ^ o S « j j c ) We have used the plasma frequency Oct = J-pTe' instead of o t r v =» £J«.p =J'TTT which Groschwitz uses. AS this equation involves only powers of K x i t is reversi b l e i n space. It i s r e v e r s i b l e in time only i f 0<=o and •P = o . As a r e s u l t of the exponential v a r i a t i o n assumed for the perturbations the complex conjugates of ^ and K may be inserted f o r ^ and K simultaneously but not separately. I f and only i f 0=/o and ^Wo is the equation a cubic in K*" ; i f one of D and i s zero i t i s reduced to a quadratic equation. I f U0 and 0 equal zero only c„ and d„ are non-zero and Thus the f i e l d vectors rotate over the whole space with uCo-\?) = u£l , which for o » v y i e l d s the f a m i l i a r Tonics and Langmuir r e s u l t w x = ^t.c (Oster (I960)). It should be stressed that not even in this simple equation may damping be taken into account by replacing u in the c o l l i s i o n - f r e e dispersion equation by0-'"^ . 3. Analysis of the Dispersion Equation. Let us seek the conditions such that at least one root of the dispersion equation i s growing, or the boundary between growing and attenuating waves. We s h a l l f i r s t study the dispersion equation when one of E 0 and O vanish, and then b r i e f l y when neither vanishes. (i) Non-Zero D r i f t V e l o c i t y , Diffusion Neglected. Consider the case Eo-to but 0 = 0 . Let us i n addition put V=o and R = £> for a preliminary study. For these undamped waves the dispersion equation i s u e M K * - C ^ } U0L K1 * C u x _ t J c c ) = o or, equivalently, k> + u«,tO x C « - U « < V " 3 .1 This equation i s f a m i l i a r from the theory of double stream a m p l i f i c a t i o n . The frequency and wavelength may be r e a d i l y solved f o r ; we obtain Ke*feX= i ^ e c * O x ± i J8uxO?c + U«c From the f i r s t equation It follows that for " > " e c , K i s r e a l for a l l four waves, but for o<JJc<. K i s r e a l f o r two waves and imaginary f o r the other two. I f i t i s desired to keep K r e a l i t follows from the second equation that for U o < l ~ l l M t w o s o l u t i o n s i& u are r e a l and two are Imaginary; IU«C I a l l four waves are r e a l . This type of evanescent but undamped wave is also found in l o s s l e s s waveguides (for which the cut-off frequency is a function of the waveguide dimensions). The analogy may not be carried too f a r as we have an applied e l e c t r i c f i e l d and longitudinal waves. 25. These four waves have been the subject of much discussion in the l i t e r a t u r e . The dispersion equation has been used to study such devices as the t r a v e l l i n g wave tube, but several errors have been made as to the exact waves which exhibit a m p l i f i c a t i o n . The s i t u a t i o n has recently been c l a r i f i e d by Swift-Hook (1960) who finds the pair of solutions in w for r e a l K which may lead to am p l i f i c a t i o n . I f the effects of damping but not recombination are included in 3.1 i t becomes Let us discuss this equation more f u l l y with i n addition the effects of recombination included. I f only d i f f u s i o n i s ignored - the plasma i s thus "cold" - we have a quadratic in X2". Thus two roots in e x i s t , and f o r each root Kx two roots in K. . The values of k,2"-^1, and ki.Kx occur i n pairs; growth or attenuation are displayed by an even number of roots. The roots i n K niay rea d i l y be found, but as we s h a l l obtain results d i f f e r e n t from those of Groschwitz the method for solving 3.3 is given in appendix C. The equation of interest i s equation 4 of appendix C: - dt ± j=T J- •o«, 1 + M C0CX + JU?-A?'- UCeC,.yV^C4fl<1 - H^d^Y 3.5 Define the right-hand side as - « t , * "Z . As ol, has the same sign as t> and c% and o are p o s i t i v e , we may without loss of generality assume o > o (henceforth assume *J r e a l ) . Hence two modes are attenuating at a l l times, and two modes are growing i f and only i f 2 > « l | . Or, Consequently two modes are growing i f M ^ C i V C , 1 * ^ 3.6 or, i f 3.6 does not hold, c± Jo * Co cx o(,x > c, e t d„ ol, 3.7 Substituting the values the c's and d*s take on, Up** L (\>x- 0^"c)*+ H o1-\>x + 8 <J10^"«. + 3.8 and 3.9 As an example l e t 2 = o ; then 3.8 cannot hold and 3.9 becomes - i. / a * * i \ Consequently a l l four modes are attenuating i f uo +\? > ^  o^, t and for MO +^ X< X "ec two modes are attenuating and two are growing. I f no damping occurs but recombination does occur, 3.8 cannot hold and 3.9 becomes c j A < « J * « . . Iff?, tends to zero we obtain the evanescent waves previously discussed. The conditions f o r which K" i s r e a l or imaginary may also be obtained from equation 3.5. I f x«k x = -dt-2. , Kx i s r e a l i f and only i f d, + 2 = o . Thus and 2 must both equal zero, as both are non-negative. As H i s of the form _V4-/v*+w* ,H. = o i f "VWo and W=o ; for V<o,z4:o . I f M £». K.l<a, = — <s4| J: , we must have d, = z. , or the inequality of 3.7 must become an equality. Let & again equal zero, as an example. Then 2 may equal zero,as V^>o . Upon simplifying the in e q u a l i t i e s given in the preceding paragraph i t is found that two modes have K*" r e a l i f Hu>* + vx=, jt u<JL , and K.** i s r e a l f o r four modes i f cj—o or -0- o . It follows that a l l four modes have K r e a l or imaginary on the axis of the ) coordinate system, but two of these modes are of this nature on the axis only and are attenuating elsewhere, while two modes are r e a l or imaginary on the axis and on the e l l i p s e 4 o l + O l = \ t~jc<. , are attenuating on the exterior of this e l l i p s e and growing on the i n t e r i o r . The s i t u a t i o n for non-negligible recombination is s i m i l a r . Two modes are always attenuating, and two modes are growing under the conditions given by 3.8 or 3.9 and the converse of 3.8. Growth in bands i s not possible as 3.8 and 3.9 are both low frequency i n e q u a l i t i e s . These re s u l t s d i f f e r from those of Groschwitz, who finds that four modes are always growing. Our expressions for K, and K x (see equation 5 of appendix C) are almost i d e n t i c a l to those of Groschwitz ( 9-t in his equation 11 should be changed to - Pj ) but we disagree with the signs of k| and K x. The actual signs ma7 be obtained as i n appendix C. ( i i ) Zero D r i f t V e l o c i t y , D i f f u s i o n Not Neglected. A somewhat disturbing r e s u l t i s obtained i f the e l e c t r i c f i e l d i s assumed to be completely absent, but d i f f u s i o n i s not ignored. We then f i n d that, i f in addition recombination may be ignored, the dispersion equation after some manipulation to make the leading c o e f f i c i e n t r e a l becomes 1.1 x ^ *. 3.10 Using the methods of section (i) , The two roots i n \C~ correspond to the two signs possible; the two roots in K corresponding to the upper sign display attenu-ation for o« x c • > a n d growth for <\>x + <->*• . The lower sign gives «• - Vo at a l l times. This r e s u l t is contrary to expectations a s in the absence of a primary e l e c t r i c f i e l d no growing waves are possible. I t is suggested that the dispersion equations involv ing D (such as 3.10) are i n error. This matter i s discussed in III .4; we f i r s t treat the general case b r i e f l y f o r completeness ( i i i ) The General Case. Let us consider equation 2.2, making no r e s t r i c -tions on fl.,0, or e c . Although this equation could be studied to f i n d the exact conditions for which Hi ^ u > o , we s h a l l not do so. The main reason as we have seen i s that there i s some doubt as to the correctness of this equation for 04 o ; a second reason i s that the a l g e b r a becomes quite involved. Instead of this we s h a l l b r i e f l y discuss the condi-tions when K i s re a l or imaginary. As the left-hand side of equation 2.2 equals the product of a l l the terms k, , where K s is any solution, we f i n d upon expanding this product that i f not a l l three of do%<&\ , and Mx are zero at least one root k* is complex. I f d0 = <Ji = dj.=o and i n addition a l l roots are r e a l i t must be true that (Burnside and Panton (1904), p.84) (c„- s c , c x + a c i ) 1 + MCC 1-C 2 X) 3 <O , a necessary condition. If at least one root i s r e a l , C. 5 yi e l d s two equations from which we have and again necessary conditions. 4. Discussion. It is apparent from section 3 . ( i i ) that there is reason to doubt the correctness of the complete dispersion equation. It i s suggested that the error i s due to incomplete equations of motion. It has been assumed that the pressure term, which is of course very important for longitudinal waves, may be absorbed into the alternating e l e c t r i c f i e l d by virtue of a c o l l e c t i v e e f f e c t or pseudo force. The effect of a concentration gradient has been at least p a r t i a l l y taken into account through the d i f f u s i o n current. It i s apparent that we may not without some j u s t i f i c a t i o n group the effect of a c a r r i e r concentration gradient or pressure under the e l e c t r i c f i e l d in the equation of motion, but take the effect of this gradient e x p l i c i t l y into account in the equation of continuity as a d r i f t current. Equivalently, the temperature of the plasma may not be taken i n t o account through the d i f f u s i o n c o e f f i c i e n t while i t i s a r b i t r a r i l y l e f t out through the" pressure. A more rigorous procedure would be to use a s u i t -able pressure term and then determining from the dispersion equation or the equations of motion themselves those condi-tions f o r which'the pressure term may be ignored. Oster has shown that the pressure term may cause the electron acoustic v e l o c i t y to appear in the equations, and increase the number of modes possible. The complexity of the dispersion equation would thus be increased; six or more modes may exist i n general. This matter i s further discussed by Fried and Gould (1961). The r e s u l t s of section 2.(i) are val i d for those conditions f o r which the ve l o c i t y fluctuations about the mean .may be neglected; i . e . , the plasma is "cold". This is further borne out by comparing our r e s u l t s and other published papers. It may be seen from equation 3.S and 3.3 that the roots K vary inversely as U© : i f u0 decreases the wavelength decreases, let but the amplification or attenuation f a c t o r TcJ" remains f i x e d . The d r i f t v e l o c i t y may be reduced to any non-zero value and s t i l l amplifying waves may e x i s t . Other authors have shown (see e.g. Pines and S c h r i e f f e r (1961)and their references) that no amplifying waves are possible for d r i f t v e l o c i t i e s less than a certain c r i t i c a l value which i s a function of the mean thermal v e l o c i t y . This c r i t i c a l v e l o c i t y tends to zero as the momentum di s t r i b u t i o n s of the c a r r i e r s tend to delta functions. Recombination may s t i l l occur, as instead of averaging the effect of recombination over the d i s t r i b u t i o n function the recombination c o e f f i c i e n t has that value which corresponds to the d r i f t v e l o c i t y at which the delta function has i t s sharp peak. We have found the conditions f o r which growing waves may exist in a cold plasma. It remains to be determined which of the growing waves may serve for the amplification of an injected s i g n a l . This would perhaps be done using the c r i t e r i o n of Buneman which is given in J. E. Drummond's book (1960). Buneman determines those conditions for which power may be transferred to an external load by studying a dispersion equation obtained by matching admittances between the plasma and the f i e l d perturbations. This method has also been used by Swift-Hook (1960). I t w i l l not be used by this author. The Boltzmann equation has recently been used by Pines and Schrieffer (1961) to study the p o s s i b i l i t y of observing high-frequency i n s t a b i l i t i e s i n InSb plasmas. The Boltzmann approach i s more s t r i c t l y v a l i d , more elegant, and more powerful than the hydrodynamic approximation. These authors postulate a displaced Maxwellian d i s t r i b u t i o n for both carri e r s and from derived dispersion equations find conditions such that the imaginary part of the frequency i s p o s i t i v e , for several different electron and hole concentrations. The conditions are found for which the energy and momentum exchange for electron-electron interactions dominate the exchange for e l e c t r o n - l a t t i c e interactions and hence the pos-tulate of a displaced Maxwellian d i s t r i b u t i o n for the electrons is v a l i d . Similar calculations are carried out f o r the hole-l a t t i c e i n t e r a c t i o n . The electron-hole interaction i s not mentioned; the electron and hole gases probably exist i n separate q u a s i - e q u i l i b r i a because of the large ("*»m) hole to electron mass r a t i o . This fact also makes InSb more suitable for observing the d r i f t i n s t a b i l i t y . I t may be added that the displaced Maxwellian c a r r i e r d i s t r i b u t i o n i s often postulated when i n s t a b i l i t i e s or negative resistance are sought. Pines and Schrieff e r have postulated i t to study the twin-stream i n s t a b i l i t y , which may occur when the d i s t r i b u t i o n function f or the composite gas departs s u f f i c i e n t l y from the unperturbed s i n g l e peaked d i s t r i b u t i o n f u n c t i o n . Adawi (1961) has shown that n e gative r e s i s t a n c e cannot occur f o r e x t r i n s i c semiconductors f o r which the c a r r i e r d i s t r i b u t i o n i s d i s p l a c e d Maxwellian. T h i s r e s u l t may be i n t e r p r e t e d as f o l l o w s , a c c o r d i n g to the f a m i l i a r low f i e l d theory, the c o n d u c t i v i t y v a r i e s as where t C O i s the energy dependent r e l a x a t i o n time of the p e r t u r b a t i o n s of the unperturbed c a r r i e r d i s t r i b u t i o n U n l i k e other suggested forms f o r C O - such as the Davydov or c r u y v e s t e i n d i s t r i b u t i o n s - the d i s p l a c e d Maxwellian d i s t r i -b u t i o n has the p r o p e r t y that " 5 T > ° f o r a f i n i t e energy range. Thus Adawi has shown that ^ c O v a r i e s w i t h energy i n Wo such a manner t h a t the p o s i t i v e c o n t r i b u t i o n of TE i s not l a r g e enough to make the i n t e g r a l p o s i t i v e . APPENDIX A. Derivation of the Dispersion Equation of Section II Let & = 8 E + 6 enp ( i ( ^ t - ^ ) ), 1 =T«fcp(i(ot-Kj)V and s i m i l a r l y for U» and <^p . The perturbation symbol ( v ) is used only i n this appendix. Elsewhere i t s omission w i l l cause no confusion. As the waves are purely trans-V V V v verse 6 , E , d p and U» l i e i n the x-y plane and are perpendicu-l a r to &o . It is well-known (Oster (i960)) that under these conditions Np and rt* are constant in time and space. For example, from the equations of continuity f o r semiconductors as given In section III i t follows that (;o *• o H <u * o K . x C v KJ^)] with R the recombination c o e f f i c i e n t , O the d i f f u s i o n c o e f f i c i e n t and tl*o , *ipo the unperturbed c a r r i e r d e n s i t i e s . This dispersion equation i s completely independent. The equations 3.1 and 3.2 may be l i n e a r i z e d : rip Z~ ^- 1. — x J V y f «\P»"*.f n « t « P (U**-Ur ) + _p„\\,b = - « - » i R ^ E + * 6 „ ) j>p JF+-j> p ^7d^rK-uJ^Vr b U r = e r J P ( E + a p , B j The assumption that G 0 = O may be equivalently stated: I U M *~5 U< |U*x Bo I and Otr as. From Maxwell's equations i t follows that 1 ^. t _v J « J^+Jp = - i (eu -k.^u) E . Hence the dispersion equation becomes (over) i"+->>..(. +• T7J *V - u „ -£>Y- |Ti v>M f o where Y= l M ( v C £ ( J _ , - M . , <oP = M P = - ^ O . Expanding the determinant leads to The dispersion equation given i s found upon taking the square root (which gives the double sign) and solving f o r f 36 .-JPENDIX B. The Watanabe Equations f o r any Vnlue of v . V/atanabe's formula numbers are giv e n followed by the c o r r e c t e d e q u a t i o n s . The n o t a t i o n i s changed to conform to the n o t a t i o n of the present paper, which uses the n o t a t i o n common to s o l i d s t a t e s t u d i e s . The s u b s c r i p t 'b' i s used to r e f e r t o v a r i a b l e s of the background medium. ,iS <s- i , 3 nc. Define M f t * ; k , u » ~~pI7yl— • U s i n g M.K.S. u n i t s , W2 .6: P» ~- =M c M b U p b + <x„ - u l + t7 ) '52 . 9 : />c or , -j>b "ox J * . o Z O / J L ^  — _ _ , or ot t ? o t ( u 4 1 c k J = c r(E" li . i < 1 ) where W2.4: o r P < , = W2.5: ^ ( . t r Kb M ^ t M b W2.6: oCpb = M p ) - M k E = E 4- - x ou 6 — -!=>p O t Most of these equations are g i v e n i n the r e f e r e n c e s i n .Vatanabe's paper. ,7e add one equation and d i s a g r e e i n some minor p o i n t s . For M P = O<M*., M » << Hp but not M P < < H B , W 3 - 3 : u l _ , f t e - ^ T " " \ ^ ' ? * , T - x • , where T >^ W2.22 A = ^ L ( i w T r J ^ - V p t ] W2.23 >>x = [d-t-oi T H , ) + ^ v>pl, ] Watanabe's equations f o l l o w from these f o r 38. For m P < < N b , l o t A _ \ - G 0 IT* u\ ( U c - U u J no »c 7i ft —— .2.22 ^ = "k ^ - W -v>P-') ',Y2 .23 V l = ^ (v^b +• oi Vpw) ,'2.24 v , i+rf ^ P«. v (1- 77i +- ( i w i 1 -These equations are of the same form as those of Watanabe's paper. APPENDIX C The Quartic Equation With Complex C o e f f i c i e n t s . Consider the equation Hence = - e . - . o l , ±iv4 . i W C.2 Let X+-«Y= ±/v+ i W ; as 2 XY=W ,we must have _ . C.3 W<o : ± y V + i W - ± jf* The outer sign (which we s h a l l c a l l £, ) gives the two solutions K*" . I f is the same for both signs of W" , then as W becomes zero and changes sign the Value of changes smoothly for V>o , but to obtain a smooth t r a n s i t i o n withV<o, S, must change sign at W=o. AS i t is reasonable that a solution in K changes smoothly as such parameters as t*t<L,\> vary, we adopt the following: V W S, 5 X F i r s t solution i n K.x >o >o +i +i >0 <0 4-1 _ i <o -CO +| - I < o >0 — I + | V W i, >o -1 • I >o <o -1 -1 <o <o - I -1 <o >o -1 The inner sign i n /vTrtT has been defined as X t . AS V changes from -IVl to *|Vl with U>o , the f i r s t s o lution must become the second and the second the f i r s t , and s i m i l a r l y for the reverse change. Thus, . — -— - — — 't From this equation K,kx may be obtained d i r e c t l y . The sign of qw.kx equals the sign of g ^  ~ f ' -^i4- |=r J - c , l + d , t 4 - u c 0 c 1 . + y (c 1 x _c4, l -Mc 0 O x +MCc 1 ol 1 -ac a c/ 0 ^ Our convention does not a l t e r the fact that i f t h i s expression with o ^ s + i i s p o s i t i v e , at least two roots have Ca.k,kx p o s i t i v e . From equation C . 4 , which may be put i n the form K X=P+JQ , i t follows that C.5 C.6 where Sx. and equal ±1 and $\%H has the same sign as Q . 41. BIBLIOGRAPHY I. Adawi, J. Appl. Phys. 32, 1101 (1961) R. Bowers et a l , Phys. Rev. L e t t . 7, 339 (1961) 0. Buneman, Plasma Physics ed. J". E. Drummond (McGraw-Hill, New York, 1960) Ch. 4 V/. S. Burns ide and A. W. Pan ton, The Theory of Equations (Dublin, Hodges, and F i g g i s , London, 1904) B. 0. Fried and R. //. Gould, Phys. Fluids 4, 139 (1960) E. Groschwitz, Z. Naturforsch. 12a, 529 (1957) C. 0. Hines, Proc. Cambr. P h i l . Soc. 49, 299 (1953) L. Oster, Revs. Mod. Phys. 32, 141 (1960) D. Pines and I. R. S c h r i e f f e r , Phys. Rev. 124, 1387 (1961) J", A. R a t c l i f f e , The Magneto-Ionic Theory (Cambridge Univ. Press, 1959) W. Shockley, B e l l System Tech. J. 30, 990 (1951) P. A. Sturrock, Phys. Rev. 112, 1488 (1958) D. T. Swift-Hook, Phys. Rev. 118, 1 (1960) B. S. Tanenbaum (to be published) T. Watanabe, Can. J". Phys. 39, 1197 (1961) 

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