ON THE LINEAR THEORY OF ELECTRON PROCESSES IN THE COLLISIONLESS DIODE

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ON THE LINEAR THEORY OF ELECTRON PROCESSES IN THE COLLISIONLESS DIODE

A. Ender, V. Kuznetsov

To cite this version:

A. Ender, V. Kuznetsov. ON THE LINEAR THEORY OF ELECTRON PROCESSES IN THE COLLISIONLESS DIODE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-523-C7-524.

�10.1051/jphyscol:19797253�. �jpa-00219238�

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JOUIiNAL DE PHYSIQUE CoZZoque C7, s u p p t 8 m e n t a u n 0 7 , Tome 40, J u i Z Z e t 1979, page C7- 523

ON ME ^{L N A R}THEORY OF ELECTRON PROCESSES IN THE COLLISIONLESS DIODE

A. Ya. Ender and V.I. Kuznetsov.

I o f f e P h y s i e a Z - T e c h n i c a t I n s t i t u t e o f t h e U.S.S.B. Academy of S c i e n c e s , L e n i n g r a d , U.S.S.R.

Linear electron processes will be ana- lyzed assuming,that undisturbed potential is not negative in all the interelectrode spacing (cathode potential is taken to be zero) and the diode,being considered,has a plane-parallel geometry,with a planar emitting cathode and an absorbing anode.

For such one-dimensional case the electron concentration is defined by the formula from / I / :

HereTo,uo

-

time and velocity,at which an electron leaves the cathode;fz

-

^electron

distribution function on the cathode (uOrO);G and Q are calculated using the following formulas :

In these expressions the integrals are taken along the trajectory or the particle

'C(ZO,UO) ,which reached the point T at the

time e and had left the cathode at the velocity uo;the derivative with respect to C0 is calculated at the fixed values of t and uO. The function G is connected with the exchange of energy between electrons and the electric field,while the function Q expresses the change of the form of the particle number conservation law in the non-stationary electric field.

#hen a disturbance is imposed,the potential is 1(2,7)= I?,(T)+~(Y)~-~*'. Then GI

=G(T> e-iw*, Q=Q(T) e -iwz. On linearizing (2) with respect .to ,we get for and Q

Here u, ( ~ , u a ) = J m - velocity of the electron on the undisturbed trajectory and z(T,uO) - time of flight of the electron along the undisturbed trajectory from the cathode to the pointy

z(r,u") = T - T O

=d

f ^u;'(x.uO)^dx ^{( 5 )}

In order to solve E q . ( 4 ) ,we'll introduce a

new function to be determined ~ = a e - l ~ ~ / u ~ . The boundary conditions for F,nameLy,F(O)=

=0,u: (T)~F(T$/.~O d are derived from Eq. (4).

Twice differentiating the equation for F and having in mind,that the energy conservation law for the undisturbed trajectory is T~(T)-U? (T)/2=const ,we reduce this equation to &(ud$)=i~. e -iwz (TI d -(T) Hence,finally we get * 7

-

^.,

By linearizing the Poisson equation in the vicinity of the undisturbed state and taking into account the fact,that the ion

concentration is undisturbed,we obtain, using (3) and (6),the following equation:

Uith the boundary conditions ?(0)=0, ~ ( S ) = O ,

where 8 - the value of the interelectrode distance in terms of the Debye radius.

In the important particular case of the electron beam fi (U~)=N,~$(U~-V~ ) ,and (7) transforms into the equation

which after the introduction of the new independent variable (5) and the new function to be determined

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797253

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t e e s the form

idz

U o . ~ l l - u ; . F ' + N ~ . X . F = ruo2.e- ( 10

>

with the boundary conditions

F(0) = 0, U ; ' ( Z )

&

^F(Z⁾^/,=. ⁼^'.I ^{( 1 1 )}

Here I - amplitude of current disturbance.

As the measurement units we choose: for timew:= (45ie2 N~ /m) , and for spatial coordi- nat e hD =Vo /a.

~et's find the solution of Eq.(lO) for the case,when r 2 , ( ~ ) =

w~

(the case of the constant electric field),imposing no limi- tations whatever on the value 2 of the field. In the well-known work of Pierce

(see /2/) the solution is obtained for%=O.

In the case of the constant field Eq.(lO) becomes

( z + ~ , ~ ) F " - ~ t + N..%/Z.F = l r ( z + ~ / z ) ~ < ~ ( 12 ) After the replacement of the independent variable z=ktF-V, /a? and the unknown function ~=t'a (in the case, being considered, fr=V=2, ~=x/~N,.v,) ~ ~ ~ ( 1 2 ) reduces to the non-homogeneous Bessel equation

Here d =~v,Z/P, t = d - $ m .

Using (11) and taking into account (9) we get from (13) for q(Y)

t . ov.(xCd2)

- 2 c x , e ' "' dx+

Here ~~~(x,y)=J~(x).~(y)-~(y).~(x) and J L ( x ) , Ni(x) are Bessel and Neumann functions of the i-th order.

In the particular case of X=O Eq.(14) after using the asymptotic formulas for the cylindrical functions transforms into the well-known formula of Pierce

The solution so obtained can be used for analyzing stability of the diode with an arbitrary undisturbed distribution of the potential in the region 70(~)*0,since such a distribution can be well approxi- mated by portions with constant,but dif- fering values of electric field strength;

it should be notedsthat at the boundaries of these portions the condition of conti- nuity of the functions q(r)

,&i(n

and$&?(t) is satisfied. The necessity to satisfy the condition v(6)-0 leads to the dispersion relation.

In the case,being considered,i.e.r;l,(T)=

=*r,the dispersion relation is of the form:

04 5 10 fi 1.

The figure presents the regions (see hatched areas) of the aperiodic diode in- stability in the plane (6,r;), ) ,where

?,=*a

- anode potential. The boundaries of these regions are defined by the equation

At qa-O and r;),-wthe solution of (1 7) are

&xsJ(I-~/~?J and8~Csr~",where s=1,2,3,.., while Cs- constants,e.g. Cl =2.1593 /3/.

The intercepts (R(2s-I,52s) of the axis qa=O correspond to pierce 's instabilities.

As is shown by the numerical calculations, Eq.(16) at ReWfO has no solutions. Thus, in the casetE'f0,as well as in the case8=0 the in~tability~which develops,when the electron beam passes through the collisionless diode,can be only aperiodic.

References :

1.V.I.Kusnetsov, A.Ya.Ender, Ioffe Ph.T.1.

preprint, N 575, Leningrad, 1978.

2. J.R.Pierce, J.Appl.Phys.,~, 721, 1944.

3. E. ~ahnke,F.Bnde,~.~osch, Special functi~ns,~Nauka~, 1968.