Noise in planar crossed-field electron guns—II: Numerical solution

1634 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 10, OCTOBER 1979

Noise in Planar Crossed-Field Electron Guns-II: Numericial Solution

Ahtract-Theory and numerical results for the shot effect in an I 111-

bounded, parallel-plate, crossed-field diode are presented. The lorv- frequency r6gime is studied, where transit-time effects are negligible. It is further assumed that short-circuit conditions prevail, and that the diode is operating in the space-charge-limited r6gime. Previous theory is first developed into a form suitable for automatic computaticm. Problems and approaches associated with the numerical solution ire then discussed. Finally, results of the computation are presentedl i n normalized form, giving the noise factor for a representative diode ;IS a function of magnetic field. The results show this factor rangimg through values less than unity. It follows that, in the present mold:1, feedback effects due to the potential minimum continue to smoo1;h the noise in the presence of magnetic field, just as in the case of 1110

magnetic field.

I I. INTRODUCTION

N AN EARLIER PAPER by Harker and Crawford [ l ] , r e - ferred to henceforth as [ I ] , a theory was presented for tlla

noise in crossed-field diodes. The fundamental mechanism of this theory was feedback arising from the reflection of electrons back to the cathode, due to the effective potential arising from the combined action of the electric and magnetk fields in the presence of beam space charge. The model for tl1e theory ignored transit-time effects, assumed an infinite planu geometry, assumed both RF and dc short-circuit conditions, and assumed operation in the space-charge-limited (as oppor;c!d to magnetic-field-limited) regime.

The purpose of this paper is to develop this theory into a form suitable for automatic computation, and then to presellt the results obtained for the variation of the noise factor r2 , 1,s

a function of magnetic field. We are unaware of any cas8:& where I" for an unmagnetized diode is ever greater than unity. Our results will show that r2 is still less than unity in 1jle presence of a magnetic field.

The reader is referred to [ l ] for an account of previo11;r work on noise in crossed-field diodes and general backgrour d material, and also for the details of the theory under discussion here. No attempt will be made to repeat here the equ3.- tions developed in [1] , and references both to definitions and equations in [ 11 will be freely made.

In Section I1 we present further developments in the theor,, in particular the conversion of the integro-differential equaticn for the perturbed potential into a more manageable integrd

Manuscript received April 4, 1979; revised June 13, 1979. This research was sponsored by the Air Force Office of Scientific Researcl~, Air Force Systems Command, USAF, under Grant AFOSR 77-3306.

The authors are with the Institute for Plasma Research, Stanfold University, Stanford, CA 94305.

equation. In Section 111 discussion is made of the numerical techniques used. Finally, in Section IV the numerical results are presented and their significance discussed.

11. FURTHER THEORY The planar crossed-field diode which we shall study is

sketched in [ I , fig. 11. It is immersed in a uniform magnetic field Bo, aligned parallel to the anode and cathode. A rectan- gular coordinate system ( X , Y, Z) is chosen with the X-axis perpendicular to the electrode surfaces and the Z-axis parallel to the static magnetic field. We shall use the normalized variables of [ l ] , the most basic of which are distance X , velocity w, and potential 7, defined in terms of their respective unnormalized counterparts, x, u, and @J, by the relations

where T, is the electron temperature, K is the Boltzmann con- stant, and w, is the cyclotron frequency.

Our purpose in this section will be to further develop the theory to a point where it is suitable for automatic computation. Our first step is to make the equations in [ l ] more con- venient by introducing new variables. Two of these, the potential perturbation T1 and the steady-state transmitted current I,, are related to the old variables by the relations

-

1 s 1 i s 0

The other two new variables, h and g, are related to the old normalized velocities at the cathode, wXb and wyb, by the relations

h = W y b -t (1 -t ,"xm

a2

where

vm is the minimum potential, and X , is the cathode-to- potential minimum distance. In terms of these variables, the

0018-9383/79/1000- lS34$00.75 0 1979 IEEE

HARKER AND CRAWFORD: NOISE IN PLANAR CROSSED-FIELD ELECTRON GUNS-I1 1635

Fig. 1. Integration region for I" (darkened) in cathode velocity space (Wxb, Wyb).

noise factor r2 is determined by the equation

y2 ( g , h ) exp (- wib - w$b)g dg d h (6 )

where

W X & = [a"h2 - g 2 ) ] WYb = a2h - (1 t a 2 ) X m . (7)

To determine the region of space for the integration in (6 ) , it is first useful to find the integration region for (w,b, w y b ) space. Recalling that ( w X b , w y b ) are the normalized velocities at the cathode, it follows that the integration region is the darkened region wXb 3 0 shown in Fig. 1. That this is the appropriate region follows from the fact that we wish to integrate only over electrons emitted from the cathode. Re- flected electron trajectories are not considered separately in the integration, but are included with the corresponding emitted electrons. The equation for the boundary curves A , B, C1, and C2 in this figure are obtained by setting X = 0 in [ 1, eqs. (71) and (74.)]. The curves A , B, and C1 correspond to curves A , B, and C in [ l , fig. 2 1 . Curve A , which includes the segments Al and A 2 , is given by the equation

w;b = -x, [a", - 2(1 t a2)Xm - 2Wyb] (8)

WYb = -(1 + a ' ) X m +UW,b (9)

curves C2 and C1 by the equations

and curve B by the curve w x b = 0 extending from W y b = --oo to -(1 + a 2 ) X m , i.e., to h = 0. Curve D extends along w,b = 0 from wyb = -(1 t a 2 ) X m to 03.

The final integration variables chosen were (g, h). The corresponding integration region is shown in Fig. 2 . Curves B and D map into the curves g = -h and g = h, as can be easily seen by setting wXb to zero in (7). Curves C1 and C2 map into the curve g = 0. This can be seen by settingg = 0 in (3b), in which case (9) is obtained. Finally, curve A maps into the lines

h fg = X , . (10)

s t

Fig. 2. Integration region for r2 (darkened) in (h , g) and (h, G) space.

T g, and replacing h2 - g2 by w i b / a 2 to obtain

w;* = -X,[a'X, - 2hU21. (1 1)

Substituting (3a) then gives (8) for curve A . By setting g = 0 in (lo), it can be seen that the three curves

Al , A 2 , and C join at the points (g, h ) = (0, X,). Since only the curve h t g = x, can intersect the curve w,b = 0 [i.e., h = g ] , it is clear that this curve corresponds to curveA2, and, by the process of elimination, that h - g = X , corresponds to curve AI ,

From these considerations it also follows that regions 1,2, 3, and 4 in Fig. 1 map into the corresponding regions of Fig. 2 . Regions 1,2, and 4 all have g 2 > 0, so we take g as the positive ordinate in Fig. 2 . Region 3 has g 2 < 0, so we take G as the negative ordinate in Fig. 2 , where G is defined by the relation

G2 = -82 (1 2 ) In order to evaluate r2 we must determine I,, and y by

solving [ l , eqs. (79) and (SO)] , the latter variable for each value of ( g , h ) in Fig. 2 . In terms of the variables of this paper, these equations become

I,, = 1 { I - erf [(I + a 2 ) X m ] 2

exp [-a2(l t a?)(& -

' 7 I(Xc, wb) d X c + Q. -

(14)

The perturbed potential Fl in (14) is in turn determined by the equation

t [ T l ( X c ) - T l ( X ) ] -fB d X , + c ( X ) X ,

X- X,

This can be demonstrated by the squaring the equation h - X , = (1 5 )


(1 6)

( 1 ' 7 )

' exp (-w;AO) erf wxdO - 711/2 1: dwyBO

* exp (-W&o) erf WXBO - --= J'," dX,

1 * exp <-w;co) erfwxco

1 " a2(x) = - Jwys0 dWyA0

[exp ( - 4 4 0 - w&lo + 1)0(X))1 /%A0 (111;)

1 WY40 a J ( x ) = - -

71 I, dwyBO

' iexp ( -w2B0 - w;BO ' 1)0(x))1 /wxBO (1 9) a

P(XC) = ; exp [Vm - a2(1 +a"(X, - X,)'] (2CI)l

and

c(X) = S(X)/ {71% [(h - X)Z - g' ] 112). (21)

In (18) and (19) wy40 and wys0 are defined by the equations

w , , ~ ~ = X - (1 t a2)X ,

wy50 = (I + a')(XQ - X,) - (X, - X ) (22)

while w d o and W , B ~ are given by

W d O = - [(X - X,)(a z X , t ( 2 t a ) x - 2

* (1 + a')X, - 2wy&))] l'z (2:3)

W ~ O = [X((2 + a2)X - 2(1 + a2)X , - 2 w Y ~ o ) ] '1'. (24)

The evaluation of a l ( X ) is a specialized topic which is discussed in the Appendix.

Equation (15) is not suitable for integration because of tf11.l~:

singularity arising from the term c(X) , which goes to infinity at the point of reflection of the injected noise electrons. This situation can be remedied by transforming this integro. differential into an integral equation through use of Green's function. The equation

3 = f ( X ) , T,(O) =V,(X,) = 0 dx2

(25;l

can be written in the equivalent form

- v l ( Z ) = - J x a G ( Z , X ) f ( X ) d X ( 2 6 )

0

where G(Z, X ) is the Green's function given by the relation

1 G(Z, X ) = - X<(X, - X,)

X , where X < and X , are the greater and lesser of Z and X , respectively. Substituting the right-hand side of (1 5) into (26) yields

- Jr;" Z(XQ - X)a(X)Tl (X) dX X ,

If we reverse the order of integration in the double-integral terms, relabel the variables of integration so that only Tjl(X), and never ql(Xc) , appears, then (28) reduces to simply

X , - I K + ( Z , X ) V l ( X ) d X - - C(Z) (29)

1

X , xll where

K-(Z, X ) = X(X, - Z)a(X) t X(X, - Z ) - P ( X d dXC I x,-x

K+(Z, X ) = Z(X, - X ) a ( X ) + Z(X, - X ) P ( X d d X , I x,-x

+lxa Z(XQ - X ) c ( X ) dX

TABLE I

Region s(x) Range Type

0eXeh-g 2

1 Reflected

h-g<X9Xa 0

2 O<x<xa Transmitted 1

3

Transmitted 4

1 O<X\FXa Transmitted

OQX<Xa 1

x, Z(X, - X,) dX, +s, x , - x

Region 11

c = - { ( X , - Z ) [ [ ( h - Z ) ' ='/'a 1 -g']'/' -(h' -g')'/'

L

- (h - X,) ln h - X, - [(h - X,)' - g'] '"]} h - Z - [(h - 2)' - g'] '1'

C = -{(X, 1 - Z ) h In - h n'/'a h - Z

t Z(X, - h ) In -

at g = 0. (38)

(33) Region 111

The integration of the expression for A ( Z , X ) is straight- -g' =G ' > O (39) forward and yields the result

IZ - XI A(Z, X) = (X , - Z ) X In - X

C= (X, - Z ) [[(h - 2)' t G' ] '1' - (h' t G')'/' x, - x n'/'a IZ - XI'

t (X, - X ) Z In - t h l n

Z - h t [ (Z -h ) ' tG ' ] ' / ' for Z # X (34) (h' t G2)'/' - h I

The evaluation of C ( Z ) is more complicated because of the multiple circumstances which determine its value. These stem from the nature of the factor S(X) in the function c(X). The function S(X) is unity for transmitted noise electrons and two (zero) for reflected e1ectron.s with X less than (greater than) its value at the plane of reflection, X = h - g. The behavior of the function S ( X ) is summarized in Table I.

Substituting (21) into (32) and using the information in Table I , it can then be determined that C is given by the following formulas in the various regions of Fig. 2:

Region I with Z > h - g 2 C=- (X, - Z ) h In g

n'l'a { h - (h' - g')'/' 1 - (h' - g')'/'

I

C = - (X, - Z ) h lng , nearg= 0. n'/'a (36)

Region I with Z h - g

C = - {(X, - Z ) [ [(h - Z)' - g'] '1' - (h' - g2)'/' 2 n'/'a

(h - Z ) - [(h - 2)' - g'] '/' h - ('2 - g')'/' t h In 1

+ Z (X, - h)ln -- [ g h - Z - [(h - Z)' - g'] '/'

+ [(h - Z)' - g'] '1' 2 11

c=- - Z(X, - h ) Ing, near g = 0. (37) n'l'a

( X , - h ) + [(X, - h)' t G' ] '1' Z - h t [ (Z -h ) ' +G']'/ '

t [(Z - h)' t G'] '1' - [(X, - h)' + G'] '/'I} C = -{(X, 1 - Z ) h In - t Z(X, - h ) In h-~,

n'l'a h - Z

a t G = O f o r h > X ,

-2 c= - (X, - Z ) h In G, n'l'a

near G = 0 for 0 < h <X, and h <Z

2 c=- - (X, - h)Z In G, n'/'a

n e a r G = O f o r O < h < X , a n d h > Z .

Region IV

C = -{(X, 1 - Z ) [ [(h - Z)' - g'] '1' - (h2 - g')"' +/'a

t [(Z - h)' - g'] '1' - [(X, - h)' - g'] '/']} . (40)

Near curve C, the function C i s singular and varies as In g for g2 > 0 and as ln G for G' > 0. The variation of function C near curve C, has been given along with the general formulas for C. For h >X,, the logarithmic singularities in function C


cancel out. A limiting process must still be taken, however, and so the value of function C is also given here for this case. For h < 0, function Cis well behaved along the h-axis.

111. NUMERICAL METHOD All integrations in which the integral was known analytically

were carried out by an adaptive method [2]. Equatim (17) was evaluated by this method. Integrations in which the integrands were only known numerically, such as t.:iose in (14), were carried out by Simpson's composite integration formula [3]. Double integrations, as in (6), were carried out by successive applications of this formula.

Integrations, such as in (1 8) and (19), in which the lomain extended from finite values to +-=J were carried out b y break- ing the integration region into two domains, one fmm the initial point to some arbitrary intermediate point, and the second from the intermediate point to infinity. In region I the integration was carried out with the original vlriable, while in 11, the variable of integration was replaced by its reciprocal so that the limits of integration become finite. Cauchy principal value integrals, such as those in (,30) and (31), were evaluated by the adaptive method, except for a small region symmetrically located about the singular point. In this region, the integrand was expanded about the singular point and integrated term by term.

The problem requiring the closest attention was the r;oI;ution of (29). This was accomplished by the quadrature method [4]. In this method, the interval between X = 0 and .Y' = X a is divided into m , intervals of equal length h with (n+ - 1) interior nodal points. Quadrature formulas (namely, the trapezoidal rule [3]) are used to express the integrals over X in (29) as linear functions of the values of Tl a t the (m, - 1) interior nodal points. If we then set Z in (29) to its values at the (ml - 1) interior nodal points, we abtain the following set of ( m , - 1) linear equations for the (mi - 1) values ofTl :

1 -7 C(Zn), n = 1, , m l - 1

A a

where

Zn =nh X, = m h X , = mlh . (42)

The end points at X = 0 and X , are excluded, sinceT1 1; zero at those points and the equations are identically satisfied The ml - 1 linear equations can then be solved by the Gamsian elimination method [5] for the (mt - 1) values of Tj1. It is those values of 'ql which are used in the integration by !;imp- son's rule of (14) referred to above.

and y in the vicinity of curve C2 in Fig. 2. As g approaclies 0, y varies as In g. As far as (14) and (29) are concerned, this

Another difficulty i s created by the singular behavior

problem is solved by writing r), y, and C as a coefficient multiplied by In g. Terms in (14) and (29) not containing In g as a factor (namely, the term Q in (14)) are then dropped, and the factor lng in the remaining terms cancelled out. The re- sulting equations can then be solved along curve C2 to find the coefficients for In g. Equations (1 4) and (29) are employed in the normal fashion to find q and y for points not on curve Cz .

In order t o deal with the problem of the In g singularity as it relates to the integration in (6), we rewrite this equation as

I?' = -hi exp {-a2h2 - [a'h - (1 t a2)X,] ' } d h 2a4

?+/'Ixo

(43)

where

Since the singularity is restricted to the integration variable g, we can confine our attention here to (44). Referring to Fig. 3, which is an enlargement of Fig. 2, we see that the integration in (44) corresponds to a column in the figure. The integration along the dotted points is far enough removed from curve C2 as to allow Simpson's composite rule to be applied. We confine our attention to the integral over the triplet of crossed points in each column of Fig. 3 lying closest to curve C2. Considering first the upper half plane, this integral is given by

(45)

We first expand 7 in the series - y = a , t a 2 g t a3 lng. (46)

Substituting this equation into (45) and integrating gives the result

\ (47)

Equation (46) shows that, as we let g approach zero, "J varies as cy3 lng. The coefficient a3 is, th'erefore, the coefficient of ;;J whose determination has just been described above. In order to determine the formulas for al and a2, we first set g in (46) to the values e/2 and E to obtain the pair of equations

where ;;Jl andT2 are the values o f r on rows 1 and 2 of Fig. 3.


s t

4 Fig. 3. Mesh points for numerical integration near curve Cz.

0.21 I I 0.05 0.10 0.15

wc IWpT

Fig. 4. Noise factor I" as a function of normalized magnetic field W , / W ~ T for E, = 80, E, = 14, and q, = -5.

Solving gives a1 and a2 in the form

al = 2Tl - Tz - cy3 In (e/4)

CY2 = -(-TI t rz - a3 In 2). 2 f

(49)

Results for the lower half-plane of Fig. 3 are obtained from the above formulas by merely replacingg2 by - G 2 , yielding in particular

Iv. RESULTS AND DISCUSSION Using the above method, calculations were made of the noise

factor for a range of magnetic field and a fixed diode configuration. The results are shown in Fig. 4, in which the noise factor is plotted as a function of w , / o ~ T . Previously defined in [ l ] , W ~ T is the plasma frequency of the emitted beam at the cathode. These results are given for E, = 80 and 4, = 14, where is the distance measured in units of 2ll2 (Debye wavelength) and is related to normalized and unnormalized distances X and x , respectively, by the relations

Fig. 4 uses the units [ and uc /up~ , instead of the units used thus far in this paper, because in the latter the magnetic field is included in the normalization. As a matter of interest, however, Fig. 4 plotted in these units would have the abscissa plotted in units of X, varying linearly from 0 to 12.8 under the constraints

It is easily shown that these are equivalent to the plot in Fig. 4 with and O , / O ~ T as the variables. In terms of a representative diode with an emission current density Js equal to 3000 A/mZ and a temperature Te equal to 1073 K, Fig. 4 in unnormalized units would correspond to

x, = 0.1 mm, x, = 0.6 mm, $, = - 0.46 V (53)

and Bo varying from 0 to 220 G. The upper value 0.1 6 for W , / O ~ T in Fig. 4 divides the space-

charge-limited regime (lying below 0.16) from the magnetic- field-limited regime (lying above 0.16). This upper value is determined by the condition [see 1, eq. (68)]

Ivml =x& (54)

or its equivalent in terms of E ,

Let us now turn to the interpretation of Fig. 4. We note that I?', starting from its value at W , / W ~ T = 0, rises gradually to a value of unity at 0.1 6. The physical basis for values of r2 less than unity in the space-charge-limited r6gime follows from the negative feedback character of the interaction between the emitted electron beam and the effective potential minima. For example, if there is an increase in the current emitted from the cathode, this causes the effective potential minima to deepen, which, in turn, reflects some of the electrons back to the cathode which might otherwise have reached the anode. Thus less than the full amount of the electrons emitted from the cathode reaches the anode, and smoothing of the noise (r2 < 1) results.

When W , / W ~ ~ passes through the upper limit given by (55) into the magnetic-field-limited region, there are no longer any effective potential minima, and, therefore, no sorting effect by the potential on the transmission of emitted electrons to the anode. The anode noise current for magnetic fields above the upper limit of (55) is the temperature-limited value corresponding to the transmitted current, and the noise factor has the corresponding value I? = 1.

Finally, this work will be compared with the work of Ho and van Duzer 171. Using a different model than ours, these workers studied the case of a cathode with finite length along a direction perpendicular to Bo and used open-circuit boundary conditions. They observed instability when the magnetic field exceeded a threshold which was inversely proportional to the length of the cathode. This implies instability for an unbounded cathode with any magnetic field in excess of zero. Because of the differences in the two models and the attendant assumptions, it is hard to compare the two theories.

1640

TABLE I1 LIMITS OF 119 I X R A T I O N FOR I,!l AND W

However, if one ignores these differences, there appears t c , be disagreement, because our theory shows that the unbounle,d cathode is stable (F2 < 1 ) for all values of the magnetic field.

V. SUMMARY The theory for noise in a planar crossed-field diode has lbrlen

brought to a form suitable for numerical computation. Tf.e:se computations have been carried out, yielding the variatior of the noise factor as a function of magnetic field for one 11ar- ticular diode configuration. This solution for the noise factor shows values always less than unity in the space-charge-liml lied regime. If excess noise is associated with noise factors gre;rler than unity, then it is clear that it must stem from an effect t ~ o t included in the model of this paper. If it is assumed that rhe noise still results from the effective potential minima, then possible candidates for these effects would include transit time, bounded geometry, and magnetic fields with a csm- ponent normal to the electrode surfaces.

APPENDIX As pointed out by Lindsay [ 8 ] , the evaluation of a l ( x : ~ by

(1 7) is best avoided, inasmuch as it involves the numerically 'un- stable process of multiplying a large number[exp (qo - w j j O ) ] by a small number (1 - erf wx0). This can be overcomr: by going back to the original definition of a,

al(x) = - exp qo(x)J exp (-wi - w;) dw, dw,. (,ctl)

The integration area for a l for X < X , and X 3 X , is given by the shaded regions in Figs. 5 and 6, respectively. 111' we transform to polar coordinates (w, I)), then (Al) can b t : integrated over w to give the result

2 n A

_ _ _ "- I_

0

0

0

0

0

0

__

"B+

wC

"C

WA- -

- W

- "C

"C

"B-

m

"A+

"C -

CS

- "A+

wA+ -

Fig. 5. Integration ratio for a l ( X ) in velocity space for 0 < X Xrn.

Fie. 6. Integration region for ai ( X ) in velocitv mace for X - =? X X,,.


where $- and I)+ are the lower and upper limits of $ and w- and w+ are the minimum and maximum values of w for each value of $. In determining the integration limits, w+ and w-, we need first to know the intersection of the radius vector with curves A , B , and C. If we express [l, eqs. (23), (24), and (74)] in terms of the polar coordinates (w, I)), then these limits are found to be given by

( X , - X ) sin $ f [(X, - + (v - qa) cos2 $1 ‘1’ cos2 * wA+_ =

(1 + a”(X - X , ) w c =

sin * + a cos * * ( ‘45)

Also of importance are the angles $, and $ b , which correspond to the coalescence of w+ and w-. Those are given by setting the square-root term in (A3) and (A4) to zero, and are given, respectively, by

Finally, we must know the angles of tangency $4 and $ s of curve C to curves B and A . The ordinates of these points are obtained by setting X , = X , and 0 in [ l , eq. (72)] , in which case we obtain

wx4 = ax wx5 = a(X - X a ) . 647)

Combining these with (21) and (22) gives these angles as

X - (1 -t 2 ) X m tan J/4 = ax

From inspection of Figs. 5 or 6, it can be seen that the area of integration can be broken up into a series of subregions. These are tabulated in Table I1 for the two cases. In examin- ing the table it should be remembered that I)- and I)+ are

constants, while w- and w+ are functions of $. Substituting these into (A2), setting terms to zero when w+ = 00, and to unity when w- = 0, yields the final formulas for a l ( X ) :

CaseI: X < X ,

Case 11: X > X ,

[31

[41

181

(‘41 0)

REFERENCES K. J. Harker and F. W. Crawford, “Noise in planar crossed-field electron guns-I: Theory,” this issue, pp. 1623-1633. C. de Boor, “Cadre: An algorithm for numerical quadrature,” in Mathematical Software, J. R. Rice, Ed. New York: Academic PIess, 1971, ch. 7. R. W. Hamming, Numerical Methods for Scientists and Engineers. New York: McCraw-Hill, 1962, p. 121, p. 153. D. F. Mayers, “Quadrature methods for Fredholm equations of the second kind,” in L. M. Delves and J. Walsh, Numerical Solution of Integral Equations. Oxford: Clarenden, 1974, ch. 6. G. Forsythe and C. B. Moler, Computer Solution of Linear Alge- braic Systems. Englewood Cliffs, NJ: Prentice Hall, 1967. H. Margenau, The Mathematics of Physics and Chemistry. New York: van Nostrand, 1943, p. 299. R. Y. C. Ho and T. van Duzer, “The effect of space charge on shot noise in crossed-field electron guns,” IEEE Trans. E6ectron. Devices, vol. ED-15, pp. 75-84, 1968. P. A. Lindsay, “Exact solutions for the electron distribution in plane magnetrons,” Proc. Royal SOC. (London), vol. A287, pp. 183-200, 1965.

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Noise in planar crossed-field electron guns—II: Numerical solution