IL NUOVO CIMENT0 Vet . 61 B, N. 1 l l Gennaio 1981

Theory o f a Pinched Electron B e a m in a Diode.

T. SAITO

Department o/ General Education, Soka University . Tokyo 192, Japan

(ricevuto il 5 Giugno 1980)

S u m m a r y . - - A finite-temperature relativistic fluid theory is used to analyse the z-dependent equilibrium state of a charge-neutralized electron beam in a diode with flat electrodes. Equil ibrium is considered to be supported when the radial gradients of particle pressure are balanced against selLmagnetic forces and the thermal energy of an electron is assumed to be much smaller than its rest mass energy. Under these conditions analytical expressions for the distr ibution of beam electrons are given in the pinched region. The density profile shows sharper pinching than a Bennett profile, as well as the enhancement of pinching resulting from an externally applied electric field.

l . - I n t r o d u c t i o n .

Studies of h i g h - c u r r e n t r e la t iv i s t i c e lec t ron b e a m s h a v e b e e n de ve l ope d f ron i

the app l i ca t i on for the pu lsed h e a t i n g of a solid t~ rge t to fus ion t e m p e r a t u r e .

The ,~nalysis of the re la t iv i s t i c e lec t ron b e a m in g diode is of p a r t i c u l a r im-

pm ' t ance in the scheme of self-focusing the e l ec t ron flow on a pel le t . I n i ts

i nves t iga t ion , it is necessa ry to der ive a n a l y t i c a l express ions for t h e d e n s i t y

profiles of the beams a n d for t he conf igura t ions of se l f -m~gnet ie fields (~).

(1) J. CHANG, M. J. CLAUSER, J. R. FREEMAN, G. R. HADLEY, J. A. HALBLEIB, D. L. JOHNSON, J. G. KELLY, G. W. KUSWA, T. H. MARTIN, ~). ,A. MILLER, L. t ). MIX, F. C. P~RRY, J. W. POU~EY, K. R. PRESTWICII, S. L, SIIOPE, D. W. SWAIN, A. J. TOEeFE~, ~V. H. VANDEVENDER, M. M. WIDNER, W. P . WRIGHT and G. YONAS : Relativistic electron beam induced ]usion, presented at the _Fi/th Co~t]erence on Plasma Physics and Controlled _~ruclear F,usion Research {Tokyo, 1974).

99

10~ T. SAITO

A few theoret ica l models for relat ivist ic e lect ron beams in diodes have been proposed. A ( (parapotent ia l flow model)> has been cons t ruc ted by DE PACKtt (2) and fu r the r s tudied b y several au thors (s,4). The model is con- s t ruc ted on the basis of ad hoc assumpt ions wi th ques t ionable va l id i ty (~,6). POUKEY and TOEPFEtr (7) have appl ied f in i te - tempera ture relat ivist ic fluid theo ry to the p rob lem of b e a m pinching in h igh-current diodes. They have der ived an envelope equa t ion for a s t reamline for the cases of the beams with par t icu lar t ypes of profiles. The s t reamlines they have obta ined b y numer ica l calculat ion show b e a m pinching. However , their s tudy has been based on the a s sumpt ions t h a t the b e a m profile remains cons tant in the z-direction (the direct ion of the s y m m e t r i c axis of a cylindrically symmet r ica l diode) and tha t the z -componen t of b e a m veloci ty is independent of the radial distance f rom the axis. F r o m a theoret ica l point of view, these assumpt ions abou t beam profile and abou t the d is t r ibut ion of the z -component of veloci ty should be model led af ter more physica l considerations.

Under these c i rcumstances , it is worth-whi le developing a theory t h a t fur- nishes ana ly t ica l expressions for the dens i ty profiles of beams and also for the d is t r ibut ion of the z -componen t of ve loci ty wi thout unfounded assumpt ions concerning these quant i t ies . Fo r this purpose, the electron flow in a diode has been t r e a t ed as a Vlasov equi l ibr ium state. Equi l ibr ium is assumed to be suppor t ed when radia l gradients of par t ic le pressure are ba lanced against self- magne t i c forces. Moreover , the case in which no external magnet ic field is appl ied is considered. The mot ion of ions in the diode is neglected and their role is a s sumed only to neutral ize the charge of electrons. We shall restrict ourselves to the case in which the t he rm a l energy of an electron is much smaller t h a n its res t mass energy. F in i t e - t empera tu re fluid theory is here extended so as to be appl icable to such cases. Analy t ica l expressions for the densi ty profile of the beam, for the dis t r ibut ion of the z -component of veloci ty and for the se l f -magnet ic field, respect ively, are ob ta inab le in the region where the b e a m is p inched in the diode. The densi ty profile in this case shows sharper b e a m pinching t h a n a B enne t t profile.

The r ema inde r of this pape r gives f u n d a m e n t a l equations in sect. 2, deals wi th me thods of a p p r o x i m a t i o n and contains the ma in results in sect. 3, and sect. 4 is devo ted to concluding remarks .

(~) I). DE PACKI[: Naval Research Laboratory, Washington, D.C., Rep. RPIR 7, 1968 (unpublished). (3) J . M. CREEDON: J . Appl. Phys., 46, 2946 (1975). (a) B. N. BREIZMAN and D. D. RYUTOV: Soy. Phys. Dokl., 20, 857 (1976). (5) ~k. E. •LAUGICUND, G. COOPERSTEIN and A. S. GOLDSTEIN: Phys. Fluids, 20, 1185 (1977). (6) G. YONAS and A. J. TOEPFER : in Gaseous Electronics, Vol. l , edited by M. N. HIRSH and H. J. OSKAM (New York, N. Y., San Francisco, Cal., and London, 1978), p. 399. (7) J. W. POUKEY and A. J. TOEPFER: Phys..Fluids, 17, 1582 (1974).

T H E O R Y O F A P I N C H E D E L E C T R O N B E A M I N A D I O D E 1 0 1

2 . - F u n d a m e n t a l e q u a t i o n s .

A relat ivist ic electron b e a m which is a s t eady - s t a t e one and axia l ly sym- metr ic wi thou t macroscopic az imutha l mot ion will be s tudied in a diode wi th flat electrodes. I n the diode, comple te space-charge neut ra l iza t ion is a s sumed to t ake place. The cylindricM co-ordinate s y s t em (r, 0, z) is used in which the z-axis is t a k e n along the diode's axis of s y m m e t r y . The s t eady- s t a t e relat ivis t ic Vlasov equa t ion for the d is t r ibut ion funct ion /(r, p) is g iven b y

qp Ho (1) Po ~r ~- Po ~c--z ~- ~- qE, \rpo Po / ~P,

ei ( p_rHo

where Pr, PO and p~ are the radial , az imuthM and axial componen t s of mo- m e n t u m , respect ively, and Cpo is the energy c(m2c 2-~p~)'-'. Here Er and E~ are the radial and axial componen t s of electric field and Ho represents the self- magnet ic field produced by the electron beam. Fur the r , c, q and m are the speed of light, the electric charge and the rest mass of an electron, respect ively. The fluid equat ions are ob ta ined b y t ak ing m o m e n t s of eq. (1) in m o m e n t u m space over a relativist ic Maxwell ian dis t r ibut ion, ] = A exp [ - - ~oPo -~ ~,P,

~p~], as follows (s):

1 ~ ~ u~ (2) r ~r [ru,N(~)] -~- ~zz[ N(~)] = 0 ,

~ q (E,u, j - E~u~) = 0 (3) u,~[Uot;(~)] + U~Uz[UoG(~)] c

q (-- E, uo + Hour) = 0 + u~ Uzz [u,~(~)] + c

(5) mcN-l(~) ~zz [~-IN(~)] + u, [u~G(~)] +

q (Ezuo "~- Hour) + u~ ~ [u~c,(~)]-- c ~--0.

I n eqs. (2)-(5), u, and u~ are the radia l and axial componen t s of m e a n too-

(s) Fo r detai ls see A. J . TOEPFER: Phys. Rev..4, 3, 1444 (1971).

102 T. SAITO

m e n t u m , respect ively, and CUo is the m e a n energy c(m2c~d-u~ d- ~ ~. There N(~) is the electron n u m b e r densi ty in the rest f r ame of a fluid e lement and defined b y

N(~) : 4 ~ m e A ~ - l K2(mc~) ,

where K~(x) is the modif ied Bessel funct ion of order n. The var iable ~ is defined b y (~0 2 - - ~ - ~)�89 and re la ted to the t e m p e r a t u r e of the electron gas in the rest f r ame of a fluid e lement , :To, b y the relat ion ~ = c(l~To) -~ p). The funct ion G(~) is defined by

G(~) = K3(mc~)[ K2(mc~) ] -~

and re la ted to the internnl energy of the electron be,~m. The m,~gnetic field Ho is g iven b y the 3[axwell equa t ion

(6) ] ~ (rHo) 4~q N(~)u~ r Dr me 2

B y v i r tue of the spa(.e.-ehargc neutral izat ion, E~---- 0 and E~ represents the axial electric ficld appl ied external ly . I f we define the electric potentia.1 r as a func- t ion of z only, t hen eq. (3) can be rewr i t ten as

(7) U ~ r ~ uoG(~)d-cqeP d- Uz~z-- U o ( I ( ~ ) § c r - O .

El imina t ing the force t e rms with Ez and Ho f rom eqs. (4) and (5) by the use of eq. (3) and ,~pplying the following ident i ty to the resul tant :

Ka(x) 2 1 dK2(x ) (8)

K2(x) x K2(x) d x

we find t h a t

(9) [N(~)L(~) ] - [ - u~ ~-z[N(~)L(~) ] = 0 , U ,~r

where the funct ion L(~) is defined b y

L($) = mc~[K~(mc~) ] -1 cxp [ - - mc~G($) ] .

E q u a t i o n (7) shows t h a t the to ta l energy including the internal energy of a fluid is conserved a long a s t reamline and also eq. (9) demons t ra tes t h a t the

(9) W. ISRAEL: J. Math. Phys. (N. ](.), 4, 1163 (1963).

TI tEORY OF A P I N C H E D ELECTB.ObI BEAM IN A DIODF~ 103

e n t r o p y per un i t mass is conse rved a long a s t r eaml ine (~o). As a b o u n d a r y

condi t ion , i t is a s s u m e d tha t , on t he surface of t h e ca thode , i n t e r n a l energy , m e a n k ine t ic energy, po t en t i a l ene rgy a n d also t h e e n t r o p y per u n i t mass of

t he fluid e l emen t t ake definite values, respec t ive ly , w i t h o u t r e spec t t o t he r~di~l

posi t ion. T h e n the fluid e lement on each s t r eaml ine has t he same t o t a l e n e r g y per un i t mass . The same a r g u m e n t is appl icable t o t he e n t r o p y pe r u n i t mass .

Thus the t o t a l ene rgy ~nd e n t r o p y pe r un i t mass of t he fluid e l emen t t a k e con- s tun t values , so a n d So, respect ive ly , i.e.

(lO) uoG(#) + -q r = So, r

(11) ~{~)L(~) = So.

Let us define the (( modif ied m o m e n t u m ~) ~ a n d t he (( modi f ied dens i ty ~) v as follows:

(12) ~ - - u ~ u ~ i e o - - q - q s ~ , for / = r a n d Z, \ ] C

q r (13) ~, ~ N ( ~ ) U o e o - - c .

T h e n the con t inu i t y e q u a t i o n (2) becomes

(14) 1 D (rO:rV) ~- D r Dr ~T (~zv) = o .

Different ia t ing b o t h sides of eq. (11) a nd us ing eq. (8), we can show t h a t

(15) V[~-~N(~)] = mcN(~) VG(~),

where V represents the g rad ien t o p e r a t o r in a 2 -d imens iona l space (r, z). F r o m eq. (15) a.nd defini t ions (12) a n d (13), t he fo l lowing e q u a t i o n is de r ived :

- ~ r - - ~ r ~ ~ ~ 2 o DzDr e0 ~b 2 ~ - - So-- q} - - ~ , - - ~ , = �9

I f we m a k e use of eqs. (10) a n d (15), i t can be seen t h a t each of eqs. (4) a n d (5) leads to the same equa t ion , wh ich is

(17) D~r Do~ q Ho = 0 Dz Dr @ c

(10) J. L. SYNGE: The Relativistic Gas, Chap. VI (Amsterdam, 1957).

104 T. SAITO

In tegra t ing formally eq. (6) over r and rewrit ing the resul tant in te rms of ~ and v, we have

r

( i s ) H0 = 4~_qmc fdr'~(r' , ~)~(r', ~1. o

Equat ions (14), (16) and (17) together with eq. (18) const i tute the closed system of equat ions for the unknown quanti t ies ~r, ~* and v.

In general, it is ve ry difficult to solve the system of equations (14)-(18) exact ly , since t he y are grossly nonlinear. Hence the case is considered in which the self- magnet ic forces (q/c)Hou, N(~) are balanced against the radial gradients of part icle pressure mcD/Dr[N(~)/~]. Applying this condition to eq. (4), then we get

D D (19a) ur D-r [urG(~)] @ Uz ~Z [urG(~)] = 0 .

This equat ion can be re-expressed in terms of ~r and ~= as

(19b) D0~r D ~ r ~ - ~ - + ~o Dz- = o �9

For la ter convenience, let us introduce the quan t i ty R(r, z):

(20)

r

R(r, z) = f dr' r' ~z(r', z) v(r', z). o

I n te rms of R(r, z), the cont inui ty equa t ion (14) becomes

DR DR (21) ~" gr- + ~ -~z = o .

B y the el iminat ion of at/a, f rom eqs. (19b) and (21), it follows t h a t

(22) DR &or DR D~ Dr Dz Dz D r - - 0 "

The lef t -hand side of eq. (22) is just the Jacobian of [R(r, z), at(r, z)] with respect to (r, z). Therefore, eq. (22) means t h a t ~r(r, z) must be a funct ion of /~(r, z). I t m a y t hen be supposed t ha t zoo(r, z) = AI[R(r, z)]~, where A1 and j are constants . Subs t i tu t ing this relat ion into the equat ion which is obta ined by the el imination of 8otr/Dz f rom eqs. (17) and (19b) and using eqs. (15) and (20),

T t I E O R Y OF A P I N C H E D E L E C T R O N B E A M I N A D I O D E 105

we have

(23) ~0~ z

A~jR2~-~(r, z)rv(r, z) + - ~ - - aor- l R(r , z) = 0 .

Here ao is 4~zq~/mc 2 (the classical radius of electron). sume t h a t

(25) l imv(r , z) = v(0, z) , r~*0

F u r t h e r we m a y as-

where v(O, z) represents the modif ied densi ty on the axis of s y m m e t r y . F r o m eqs. (20) and (25), R(r, z) is found to be propor t iona l to r 2 in the ne ighbourhood of r = 0. Therefore, the first t e r m of eq. (23) is p ropor t iona l to r ~j-1 and the th i rd one to r in t h a t region of r. I n order t h a t every t e r m on the l e f t -hand side of eq. (23) has the same dependence on r, the cons tan t j m u s t be t a k e n as j = �89 Then eq. (23) is reduced to

(26) :2- A~rv(r, z) + ~c-~- + a ~ z) = 0 .

Equa t i on (16) also implies t h a t the quan t i ty ( C o - - q q b / c ) - ~ - - ~ has to be a funct ion of v(r, z), i.e.

(27) ( e o - - ~ r ~ - ~ - - ~ : = F[v(r, z ) ] .

To deterniine the form of F[v(r, z)], rewri t ing the le f t -hand side of eq. (27) b y using eq. (12), we get

�9 2 ~ = m ~ c "2 e o - - (/) uo "~ (28) Co-- r ~ , - - ,

(29) eo ~ q5 2 2 m2e2[G(~)]2

Here eq. (10) has been used to ob ta in express ion (29). Supposing t h a t t h e t h e r m a l energy of an electron, 3kTo/2, is much smaller t h a n its rest mass energy, i.e. t h a t mc~ = mcZ/kTo >> l , we can a p p r o x i m a t e the expressions for G(~) and L(~) as follows (11):

5 (30) a(~) = I @ :2mc~'

(31) L(~) = e - -~ (mc~)~.

(11) See, for example, Chap. V and VI in res (10).

106 T. SAITO

:Eliminating ~ and L(~) f rom eqs. (11), (30) and (31) leads to

(32) G(~) = 1 § 2--mcc,

where c I is a constant . Subs t i tu t ing eq. (32) into eq. (29) and using eq. (13) yield

_ 5

I f we recall t h a t (Co-- qffP/C) Uo ~ is equal to G(~) and compare the r igh t -hand side of eq. (30) wi th t h a t of eq. (33), it can be seen t h a t the second t e rm on the r igh t -hand side of eq. (33) is of the same order of magni tude as 1/mc~ = = kTo/mc 2.

As (co-- qfP/c) u ; ' can be a p p r o x i m a t e d to uni ty , then the fac tor 5/2mcc~. �9 v~(r, z) also becomes of the same order of magn i tude as kTo/mC 2. Equa t ion (33) m a y be regarded as a cubic equa t ion with respect to [(e0--qq)/C)Uo~] ~ and m a y be solved up to the first order of 5~2receive(r, z). The solution is

(34) ( c o - - q ~ b ) u o ~ = ] @ Ao.vl(r, z)

with the abbrev ia t ion A 2 ~ 5 / 2 m c c l . F r o m eqs. (28), (34) and the relat ion ~ 2 A~R(r, z), the following can be obta ined:

(35) Co-- qb - - A ~ R ( r , z ) - - ~ ( r , z) = m~c'[1 @ 2A2vl(r, z)] .

I n der iving eq. (35), we have neglected the t e rms which are higher t h a n the second power of A2vi(r, z).

I t should be r e m a r k e d t h a t the cont inui ty equa t ion (21), the m o m e n t u m equa t ion (26) and the ene rgy-en t ropy equa t ion (35) are more appropr ia te equat ions in cases in which self-magnet ic forces are ba lanced against gradients of par t ic le pressure and the t he rm a l energy of an electron is much smaller t h a n its rest mass energy.

3 . - A n a l y t i c s o l u t i o n s i n t h e p i n c h e d r e g i o n .

I n this section, the behav iour of the electron beam in the region where the b e a m is p inched in the diode is invest igated. I n this region, the radia l c o m p o n e n t of the b e a m ' s m e a n veloci ty is m u c h less t h a n its axial one. Such

t e n d e n c y of the b e a m ' s veloci ty in this p inched region has been shown b y

T I I E O R Y OF A P I N C I ] [ E D E L E C T R O N B E A M I N A D I O D E 107

numerical simuI~tion in the work by POUKEY ,~n(:[ TO:EPFEtr (75. Fol lowing their analysis, we assume t h a t lu,(r, z)]<< u~(r, z), i .e. I~,(r, z)]<< ~ ( r , z). Ree- ognizing these facts and tha t ~ : A~Rt ( r , z), we neglect the terms which contain the f,~ctor Aa in eqs. (26) and (355. We then have from eqs. (26) and (35)

(36)

~nd

(37)

r ~--~ aoR(r , z) : 0

W(z ) -- ~ ( r , z) - - Aav~(r, z) = 0,

respectively, where we have defined W(z) as

(38) W ( z ) ~ eo - - ~ - - m ~c "~

~md defined A3 as equal to 2m~-c"-A2. For l~ter cot~venienee, we write 4own the equat ion which is der ive4 from eq. (37) by sett ing r ~ 0:

(39) W(z) -- ~2(0, z) - - A~v~(O, z) = O .

])iffcrentiating eq. (36) with respect to r and postul~ting tha t

~c~ << cr t ' (4o)

we get

(4J) ~r ao r ~ ( r , z) v(r, z) = 0 .

Eliminatil lg r(r, z) from eqs. (375 ~md (41) le~ds to n differcntial equa t ion of ~Jr, z):

(42) c~-~- gz i ( r , z ) [ W(z) - - ~( r , z)]-~ = A ;~ ao r .

Equa t ion (42) can be easily solved to give

C W ( z ) ] (43) CW(z) - - ~':(r, z ) + In 2W(z)

A3~z(r, z) I ~-

+ 2 V / W ( z ) [ W ( z ) - - ~( r , z)] I

where Dl(z) is an arb i t rary funct ion of z. I f we set the logari thmic t e rm of

1 0 8 T. SAITO

the above equa t iml to - - D~(r, z) t en ta t ive ly , then we can solve it fo rmal ly as

(44) ~(r , z) ~-- W(z) -- W(z)(D~(r, z) + A~iW~(z)[�89 r2 + Ddz)]} - : .

At r----0, this equa t ion becomes

(45) a~(0, z) : W ( z ) - - W(z)[D2(O, z) - ~ A ~ W ~ ( z ) D I ( Z ) ] -2 .

To es t imate the dependence of Ddr, z) on r, we compare 8Ddr, ~)/cr with the der iva t ive of the first t e r m of the lef t -hand side in eq. (43). Then by di rec t calculat ion the ra t io of the fo rmer to the la t te r turns out to be [W(z) - - -- ~(r , z)]o:~(r, z). B y using eq. (37), we can express this quan t i ty as follows:

(46) W(z) - - ~ ( r , z) A3v~(r, z)

a~(r, z) a~(r, z)

The r igh t -hand side of eq. (46) is smaller t h a n A3vi(0, z)a~-2(0, z), if ~:(r, z ) > > a~((), z) and v(r, z ) < v(O,z). After the same a rgumen t as the one under eq. (33) and b y the definit ion of A3, we can es t imate the quan t i ty Aav~((), z)-

�9 ~z~(0, z) as

kTo (47) A3vi(() , z)~[~(O, z) - -

m-l~(O, z)

The r igh t -hand side of the above equat ion is near ly equal to thernml energy/ /modified kinet ic energy. Thus the the quan t i t y Aavi(0, z)a-2(0, z) is consid- ered to be smaller t h a n un i ty and hence SDdr, z)/~r is smaller t h a n the de- r iva t ive of the first t e r m on the lef t -hand side of eq. (43). Thus we can neglect the dependence of Ddr, z) on r and a p p r o x i m a t e D~(r, z) to D~(0, z). Replacing D2(r, z) in eq. (44) b y Dd0, z), which is given b y eq. (45), we find t ha t

(48) ~(r , z) = W ( z ) - [ W ( z ) - ~ (0 , z)]-

l a A "~r~W(z)V'W(z) ' zd(Oi ~ ) -~ . �9 ( 1 + ~ o 3

The above equa t ion shows t h a t a~(r, z) is an increasing funct ion as a funct ion of r. This p r o p e r t y of az(r, z) is consistent wi th the assumpt ion made jus t unde r eq. (46). Once the ana ly t ic expression of the modified m o m e n t u m is ob ta ined as eq. (48), the modif ied densi ty v(r, z) can be given fi 'om eqs. (37) and (48) as follows:

(49) v(r, z) ~-- A[:~[W(z) -- a~(0, z)] :~ (1 + �89 aoA; ~ r ~ W ( z ) % / W ( z ) - ~ ( 0 , z)) a.

F u r t h e r subs t i tu t ing eqs. (48) and (49) into eq. (18) and in tegra t ing the r e su l t an t

"I~III~]OIr O F A P I N C I I E D ~ L E C T I ~ O N B ~ , A M I N A D I O D ~ 1 0 0

over r, we obtain for the self-magnetic field

(50) Ho(r, z) - - 3qrW(z )

[~( r , z) - - zoO(O, z)] .

Finally we calculate ~,(0, z) which appears in eqs. (48)-(50). For this pur- pose, we subst i tute eqs. (48) and (49) into eq. (20) and consider the result in the limiting case in which r --~ 0. This yields

(51) R(r, z) 1A~a~(0, z)[W(z) ~ 0 , = _ ~ ( , z ) ] ~ r 2

for a small value of r. The combinat ion of eqs. (21) and (51) furnishes the fol- lowing differential equation, if the t e rm including ~ is neglected in eq. (21):

(52) ~-~ {~(O, z ) [W(z ) - - ~:(O, z)]~} = O .

This equat ion enables us to write an integral

(53) ~(0 , z)[W(z) -- ~2~(0, z)] ~ = B ,

where B is a constant . Recognizing t h a t a~(0, z) is near ly equal to W(z) f rom eq. (39) and the ,~rgument under eq. (47), we can solve eq. (53) with respect to ~(0 , z) as being approximate ly

(54) 2 0 ~ - - , ~ ( , z) W(z) B~W-~(z)

Subst i tu t ing eq. (54) into eqs. (48)-(50) yields concrete expressions for the distr ibution of the z-component of modified m o m e n t u m ~ ( r , z ) , for the modified density v(r, z) and for the self-magnetic field Ho(r, z), respectively, as functions of W(z) , r and z.

In conclusion, the explicit expressions are presented below for the dis- t r ibut ion of the z-component of veloci ty of the beam, v~(r, z), and for the dens i ty in the labora tory frame, n(r, z), respect ively:

(55)

and

( )1{ [ 1 ]} q cI) W(z) - - B i W - ~ ( z ) 1 + ~ aoAdW~(z ) r 2 -2 v~(r, z) : SO - - e

- - so- - r W-�89 1 -r ~ a o A d W ~ ( z ) r ~ m e ' '

where B1 and A4 are constants defined by B1 = B ~ alld A4 ~ B i A [ ~, respec- tively.

110 T. SAITO

4 . - C o n c l u d i n g r e m a r k s .

With in the f r a m e w o r k of f in i t e - tempera ture relat ivist ic fluid theory, an- a ly t ica l expressions have been ob ta ined for the densi ty of an electron beam, for the z -componen t of the m e a n veloci ty of the beam and also for the self- m a g n e t i c field, respect ively , in the p inched region of a diode with fiat electrodes unde r the following three assumpt ions : i) the t he rma l energy of all electron is m u c h smaller t h a n its rest mass energy, ii) lug(r, z)l<< u~(r, z), iii) the |nag- n i tude of the second par t ia l der iva t ive of ~ ( r , z) with respect to r is smaller t h a n the first par t ia l der ivat ive.

The first a s sumpt ion means t h a t 5.7.10~T01>>1, where To is the tem- Fcra ture of the electron fluid in kelvin. This condition seems to be sat- isfied in prac t ica l exFeri lnents (~2). The physical meaning of the second ,~s- sumpt ion has been given in sect. 3. The th i rd assumpt ion restricts the do- ma in of (r, z) in which our theory is applicable. Calculating the inequal i ty ]r~"-~/~r:l < c~:/~r b y the use of cqs. (48) and (5t), we find th ' t t the ine-

qual i ty is replaced b y

t21(z) r r ~ 3t)o(z) r~ (57) [~ + tA(z)r~] ~ + i + t2,_,(z)r ~ < ~

with the "~bbrcvi~tions

) 2 - 4 a Ql(z) = 1 ~ W ~(z) and (&(z) - ~ a o A ~ W ~ ( z ) .

For a large va lue of r, the second t e r m is dominan t on the lef t -hand side of eq. (57). Hence we can infer t h a t our solutions in sect. 3 are reasonable results in the domain l imited b y Q~(z) r2< �89

Final ly , i t should be r e m a r k e d t h a t eq. (56) shows the enhancement of b e a m pinching b y the ex te lna l ly appl ied electric field and shows sharper beam pinching t h a n the B e n n e t t profile (la) used in ref. (7).

This work was pa r t l y suppor ted b y the Scientific Research F u n d of the

Educa t ion , Science and Culture in J a p a n . The au tho r wishes to express his t hanks for Profs. S. GOTO, T. KATO and

:N. MISttIMA for va luable discussions and for reading the manuscr ip t .

(12) See ref. (e) and references on experiments therein. (13) W. BENNETT" Phys. Rev., 45, 890 (1934); 08, 1584 (1955).

TItEORY OF A PINCHED ELECTI~ON BEAM IN A DIODE 111

�9 R I A S S U N T O (*)

Si usa m m teor in per i fluidi, r e la t iv i s t i ea e a t e m p e r a t u r a f in i ta per ana l izzare lo s t a t o di equi l ibr io d i p e n d e n t e d~ z di un fascio e le t t ron ico n eu t r a l i z za t o nel la car ica in un diodo con e le t t rod i p ia t t i . Si cons ide ra che l ' equi l ibr io sia m a n t e n u t o q u a n d o i g r a d i c n t i r~dial i del la p ress ione delle pa r t i ee l l e sono b i l anc ia t i r i s p e t t o alle forze a u t o m a g n e t i e h e e si suppone che l ' energ ia t e r m i e a di un e l e t t r one sia mo l to pifi p iccola de l l ' ene rg ia del la sua massa in riposo. In ques te condiz ioni si d a n n o espress ion i ana l i t i che p e r la d is t r i - buz ione degli e le t t ron i del fascio nel la regione di cost r iz ionc. I1 profilo di dens i t s m o s t r a cos t r i z ione pill n e t t ~ di un profilo di B e n n e t , cosi come l ' a u m e n t o del la c o s t r i z i o n e d o v u t a a ua eampo e le t t r ico app l iea to da l l ' e s t e rno .

(*) Traduzione a eura della Redazione.

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