beam brightness of patchy temperature-limited

8/2/2019 Beam Brightness of Patchy Temperature-Limited

http://slidepdf.com/reader/full/beam-brightness-of-patchy-temperature-limited 1/3

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-34, NO . 8, AUGUST 1987 1859

Beam Brightness of Patchy Temperature-Limited

Cathodes

RICHARD F. GREENE, SENIOR ME MB E R, IEEE, AND C . R . K . MARRIAN

Abstract-The beam brightness B of smooth thermionic cathodes is

examined in the temperature-limited regimeas affected by work-func-

tion patchiness. Beam brightness is expressed in terms of the autocor-

relation function of the work function nonuniformity in the laminar

approximation. The dependenceon extraction field strength and other

factors is discussed.

I. INTRODUCTION

I HAS LONG been recognized [l] hat the brightness

B of beams from thermionic cathodes is degraded by

thermal spread, surface roughness, work function non-

uniformity or “patchiness, ” etc. , and that this degrada-

tion may affect the performance and reliability of travel-ing waveubes (TWT’s) nearhe knee” between

temperature-limited (TL) and space-charge-limited (SCL)

regimes. More recently, the sensitivity of free electron

laser efficiency [2], [3] to B has increased this interest.

Recently, it has been shown [4] hat surface roughness

may strongly degrade B . In this paper, we examine the

effect of work function patchiness on B and show how this

effect depends on the patchiness autocorrelation function

and the extraction field in the T L regime.

Beam emittance E and beam brightness B are approxi-

mate invariants [5] only for paraxial unaccelerated beams.

These conditions do not obtain near the cathode, but the

effect of patch fields on trajectories nevertheless can beestimated in the laminar approximation. Emittance growth

due to space-charge forces in the nonuniform beam pro-

duced by a patchy cathode occurs largely outside the gun

region and is treated elsewhere [ 6 ] as an implicit beam

emittance.

11. LAMINAR PPROXIMATION

This involves the assumption that there is a single-val-

ued velocity field u at each point ( R , z ) = ( x , y , z ) , so

that we can transform the Lorentz (nonrelativistic) equa-

tion of motion

d v / d t = ( - e / m ) ( E

+u X B / c ) ( 1 )

from the moving (Eulerian) coordinate system to the (La-

grangean) coordinate system fixed relative to the cathode

Manuscript received August 11, 1986; revised May 4, 987.

The authors are with the Naval Research Laboratory, Washington, DC

IEEE Log Number 8715724.

20375-5000.

a t z = 0

( a / a t -t (v v)} u = ( - e / m > ( E + x ~ / c ) . 2)

In the steady-state case

E = -V + ( 3 )

so that

(v V ) v = - ( e / m ) E (4)

when magnetic forces are neglected. Writing Eo and vo

for the patch-free case, one alsohas

(vo V ) vo = - e / m )Eo. ( 5 )Then, writing

E = Eo + 6E, v = VO + 6~ (6)

so that

uo = (0, 0, 1 ) (-2ErJz/m) ( 7 )/2

and 6u , the patch velocity satisfies the field equation

( P o * V ) 6v + (6u 0)o + (6u - V ) 6u

= - e /m ) E . ( 8 )

In the laminar approximation the quadratic terms in 6v are

dropped.The transverse components of (8 ) then look like

zlo(z) (a/&) 6vx= ( - e / m ) 6Ex, etc. (9)

This can be integrated conveniently in terms of transverse

Fourier integrals, such as

M X Q , z ) = 1d2Rexp (i& - R ) 6ux( R , 2 ) / 2 ~ ,

Q = ( Q x , Q y 7 0 ) (10)

in view of Laplace’s equation (TL case!)

(a2 /az2 + Q 2 ) Sd(Q, z ) = 0 (11)

which gives

S & ( Q , z ) = 6d(Q,O)e-Qz, z > 0 (12a)

and

6& (Q,) = (Q, ) e-eZ, z > 0 . (12b)

Note that

- e 6 + ( R , 0 ) = 6 9 ( R ) (13 )

U S . Government work not protected by U. S . copyright,



1860 IEEERANSACTIONS ON ELECTRONEVICES,OL. ED-34, NO. 8, AUGUST 1987

where 6 9 ( R ) s the work function nonuniformity. One

then has

6 & (Q, ) = - e/2mEo)”2 6& (Q, )

ai

lo d.$ exp ( - E 2 ) . (14)

In the absence of long-range order in 6 9 the patch fields

evanesce for z >> R,, the work function autocorrelation

length. Then

6gx(Q, ) = ( r e / 4 m ~ ~ ) ” ~ Q, ) - 1 / 2 ,

z >> R,. (‘5)

111. TRANSVERSEENERGY

For z >> R,

( 6 ~ : ) = LIM- d2R 6 vi R ) 2 ,L - r m 4L2

which gives the mean transverse energy in terms of the

work function patchiness autocorrelation function.

IV. AXIAL ENERGY SPREAD

For a discussion of beam brightness one wants to obtain

the axial energy spread from the transverse velocity

expression (23). To carry this out, we consider that, in

the laminar approximation, an electron passing through

( R , z ) must have come from a definite point (Ro, 0 ) on

the cathode. In steady state the energy is invariant along

this trajectory so that

( m / 2 ) v ( R ,? - e4(R, )

= ( m / 2 ) U(R0,o)2 - &(R, 0 ) . (24)

Similarly, in the absence of patchiness

( m / 2 )vo(zY - e4o(z ) = vo(O? - e40(0).

( 25 )

where 6 v i R )= 0 for R > L ( 1 6 ) Then, since v(Ro, 0)2= ~ ~ ( 0 ) ~kT , one finds that,

d2Q I 6v:(Q)I2

using Parseval’s theorem.(17) For he asymptotic region

Then, introducing (14)z >> R,, 64 = 0 (274

( 6 ~ : ) = 6Ek(Q,0)12/Q (18) whencene

4mEo

- ne- - LIM -!L I] d2Q d2Rd2R’4mEo 4L2

- exp iQ - ( R - R’ ) 6Ei (R, )

- GE:(R’, O ) / ~ ~ ~ Q

But

d 2 Q ( ~ X PQ * ( R - R‘ ) ) / Qm

= 2~ 1 dQ Ja(Q0R) = 2n/AR0

where AR = R - R‘ . Thus

(6v,2>=-8mEo 4L2

IM S d2R2AR

* SE,” ( R , 0 ) 6E : ( R + AR, O)/AR

e= -

8mEo 4L2IM- d2AR

‘ S

m v osv , ( ~ , ) + ( m / 2 )~ Z J ( R ,)’

= 69 ( R ) , z >> R,. (27b)

Since the RH S of (27b) s independent of z , and since vo

oc &, t follows that 6vz oc 1/& for z >> R,, and that

(19) mvo6v, ( R , ) + ( m / 2 )6vR(R,

= a+(&), z >> R, ( 2 81

and

( 2 8 ) mvo(6vz) -t (msvR(R,?/2)

= ( 6 + ) = 0 , z ,>>R, (29)

where 6vR s the transverse part of 6v . This makes it pos-

sible to find the axial energy spread

(21 1( > = (mv2/2 - m v ; /2 ) = (mv o~ vz m&v,2/2

= mvo(6vz) = - m 6 v i / 2 ) , z >> R,

n

= (1 16eEo) d2ARAR- ’ V2 ( ( R )- (6E$ (R ,0 ) 6Ek ( R + AR , 0 ) A R (X!)6+(R + A R ) ) . (30)

Using the result (A3) of the Appendix, this becomes

V. BEAMBRIGHTNESS

16eEoWe can obtain B from the conventional expression [5]

B = ( J / 2 r ) (AT, ,TI, where J is the current density.

* ( 6 9 ( R ) 6 9 ( R -A R ) ) ) / A R (23) Comparing this to the thermal value, one has, in the lam-

(m6v,2/2) =- d2AR ((a2/i3Ax2)



GREENE AND MARRIAN: BEAM BRIGHTNESS OF CATHODES

inar approximation

Bpatch 1

Bthemd 16eEokT-- - d 2 A R E V 2

- ( S O ( R ) S @ ( R+ A R ) ) . (31)

Thus, for example, for a dispenserathode with a smooth

surface (type A4 or CPD) the work function autocorrela-tion, in the unpoisoned condition, is determined by sur-

face diffusion and evaporation of Ba on the oxygenated

substrate, so that one might expect

( 6 @ ( R ) O ( R + A R ) )

= (6c2)a O / a c ) " (exp - A R / & ) (32)

where C is the Ba surface concentration. (The diffusion

length6 ight be expected to be about 10 pm on tung-

sten at 1050°C [ 7 ] . )On the other hand, for a ressed powder LaB6 cathode,

the work function is probably uniform over each micro-

crystal, and uncorrelated with the value on neighboring

microcrystals. In that case one might expect

( S O ( R ) S O ( R + A R ) ) = ( S O 2 ) , for A R < d (33a)

5: 0 , for A R > d (33b)

where d is the mean particle size (perhaps 0.5 pm).

It is clear rom these results that the beam brightness issignificantly degraded by work function patchiness only

forsmaller extraction fields, close o he space-charge

knee.

VI. IMPLICITEAMEMITTANCE

The nonuniform beam extracted from a patchy cathode

will, of course, experience emittance growth by space-

charge forces. Although such an emittance growth will

occur mainly beyond the gun region, such a growth is

implicit in the work function nonuniformity via Richard-

son's equation. A treatment of this implicit beam emit-

tance, based on excess Coulomb energy, will be pre-

sented separately.

APPENDIX

The autocorrelation function for the patch field at the

cathode is (choosing SEX( ) = 0 for R > L , and using

the Wiener-Khinchine theorem)

(6Ex ( R ) 6Ex ( R + A R ) )* "

d 2 R SE:(R) 6E'I;(R+ A R ) ( A l )

= LIM 4 ~ 2 2Q 6E: (Q)SE:(Q>* exp i Q A R

= LIM 4 ~ 2 2 Q Q z 6 $ " ( Q )S$ " (Q ) * exp iQ - A R

1861

12 r P

- --

a h 2LIM -& 3 d 2 Q S$" (Q ) @"(e)*xp iQ

- A R

=

e 2 a A x22 ( S O ( R )S O ( R + A R ) ) .

REFERENCES

[l ] C. Herring and M.H. Nichols, Rev.Mod.Phys. , vol. 21,p. 185,

[2] R. M. Phillips, IEEE Trans. Electron Devices, vol. 7, p. 231, 1960.

[3] P. Sprangle, R. A. Smith, and V. Granatstein, in Infrared and M illi-meter Waves, Val . I-Sources of Radiation, K . J. Button, Ed. New

York Academic, 1979.

1949.

[4] Y. Y . Lau, J . Appl . Phys. , vol. 61, p. 36, 1987.

[5] J. D. Lawson, The Physics of Charged Particle Beams. New York:

[6] R. F. Greene and C. R. K . Marrian, to be published (1987).

[7] R. E . Thomas, private communication (1986).

Oxford, 1977.

*

Richard F. Greene (SM'79) was born in New

York, NY, in 1925, received the B.S. degree in

physics from Lehigh University, Bethlehem, PA,

in 1946, and received the Ph.D. degree in theo-

retical physics from the University of Pennsylva-

nia, Philadelphia, in 1951. He was a postdoctoral

fellow at the University of Illinois, Urbana, in

He served in the U.S. Army and was dis-

charged in 1947. He worked at theWestinghouse

Research Laboratory from 1953 to 1958. He joined

the Naval Ordnance Laboratory in 1958 and transfemd to the Naval Re-

search Laboratory in 1975, where he has since been Head of the Surface

Physics Branch. He has published papers in statistical mechanics, nonlin-ear mechanics, chemical physics, surface physics, semiconductor trans-

port, and device theory. He has taught undergraduate and graduate physics

at several universities. He holds patents in solid state devices, infrared

devices, field emission devices, lithographic pattern transfer processes, and

thermionic emitters.

Dr. Greene has served on theEditorial Board of Applications ofSurfaceScience.

1951-1953.

*

C . R. K. Marrian was born in Cambridge, En-

gland, in 1951. He studied at Cambridge Univer-

sity and received the B.A. degree in engineering

in 1973. He remained at the Electrical Engineer-

ing Department at Cambridge and was awardedthe Ph.D. degree in 1978 for his dissertation onthe Auger spectroscopy of thermionic cathodes.

He subsequently spent nearly three years at

CERN in Switzerland working on an Avalanche

chamber used in a quark search experiment. In

1980, he crossed the Atlantic and joined the Sur-

face Physics Branch at the Naval Research Laboratory. At NRL he has

continued his studies of tungsten-based thermionic emitters, particularly

under conditions of nonideal vacuum. More recently, he has become in-

terested in the limits of lithographic techniques for microfabrication.

beam brightness of patchy temperature-limited

Documents