particle bunching in a traveling-wave linear accelerator

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Page 1: Particle bunching in a traveling-wave linear accelerator

LETTERS TO THEEDITOR

PARTICLE BUNCHING IN A T R A V E L I N G - W A V E LINEAR ACCELERATOR

0 . I . Z h l i e i k o

A traveling-wave linear accelerator can be used to obtain extremely short pulses of accelerated e lec- trons. In a number of cases [1] the bunching capacity of the accelerator Itself Is found to be [mufficIent so that a system of lxettmlnary bunching is introduced. The presence of a current of accelerated particles leads to a reduction In the ampUtude of the electric field; this effect tends to Inhibit electron beaching in the a c -

c e l e r a t o r .

Below we consider eIectron bunching in an accelerator under the followln 8 assumptions: 1) an electron bunch of angular Iength 2Ar < 30" where emax = ~a + A r and emln = r 1 6 2 Is introduced Into the accelerator at equilibrium phase r 2) the amplitude of the l:hase oscillations Is small r162 < 15"): 3) accelerator parameters vary slowly along Its length ( inn distance equal to one oscillation wavelength In the apertared(disc-loaded) wave guide the phase velocity Is not increased by mcxe than 10% and the amplitude of theaecelera t ingf ie ld isnot reducedbymorethanS- l~ . Using these assumptions we analyze the longitudinal electrc,a oscillations about the equilibrium phase [2]. [3]; assuming that r = const we obtain the equation"

-~+out -- r j inJ ink . inj EzInI

v where Ar t Is the angular duration of the particle bunch at output of the acceleratc~; 8 = ~ - ; _v ts the patti-

ms cle velocity: c Is the velocity of light; mcorr - ( ~ .

Using Eq. (1) we can determine the angular duration required at injection In order to achieve a given bunch length at the output Ar

In Eq. (1) the quantity ~ can be given by the expression

z / r , o - lu ( l...+. r 4'2) H, I , ' e r

where a is the attenuation factor in the ;~r wave guide; _z is the longitudinal coordinate; Ps lathe power flux at the input of the wave guide (z = 0): I is the c ~ e n t of the.pattlcle beam; U Is the'kinetie energy

'S of the particles; p = a t + al~Z; r is a con~ant; Y= ~--~-~ E'r[l t ds is a f u l~ t t~ which depel~]$ OU the II

wave guide geometry and the phase velocity (~ is the area of the cross section over which there it an i n . a c - tion in the wave gulde). -.

Eq, (2) Is obtained by Irgegating the power-balance differential equation f ~ t ~ accelerator wave guide. In carrying out tl~s l~ocedure It :Is ;tss ~med L~'at eZ < 1.

' [Here and elsewhere "k" in subscript g "cc t rec ted ." - ed.]

Page 2: Particle bunching in a traveling-wave linear accelerator

Using Eq. (2) we find

= . - . : I I ,.__ ,,'o., <, + P.'-inl i out k ,% +(a)

Eq, (I) now assumes the form

~ al i "

.a_~.L = ~- ~ !~" ~outt~out + U o ) . %,,t (+'o.t

m ~ (4)

where 1" = I (1 + p); U, Is the rest energy of the electron.

On the basis of Eq. (4) the following conclusions can be reached= r

I. When the wave guide b loaded by the beam current the Istio is reduced, In other word~ & t o u t

if A r Is fixed. A tou t increases with Increasing particle cunen~. This factor manifests 1he effect of the current on particle bunching in the accelerator.

A ~in_. L increases with the ratio 2. The quantity A 9out Pout "

3. There Is an optimum value of the final electr0n energy flout for which the maximum possible bunch- Ing It obtalned with a given particle current and pc~er Pe.

4. The allowable value of Ar Lsreduced as the wall losses in ~e wave guide Increase (the term a z

Pe = UP--~e I(1 + az +cxtz}and the factor e" 4 ) .

Using Eq. (4) we can determine the required angular bunch length which must be achieved by the pre- llmlna~/b uncher.

It can aLso be shown that It is necessary that the quantity F vary In a specific way If optimum pa~,icle bunching Is to be achieved. It should be noted, however, that even a comlderable Increase In the ratio

r ln j does not yield a large gain In bunching [cf. Eq, (4)]. Thin, in the Stanford University accelerator (U.S.A.} rou t

=rln-~L- n 132, All other things being equal, as compared with = I this value Increases FlnJ [2] the quanti~ "out Four

the ratio A~r by only a factor of 1.85. To achieve this value of r--~-~- ~ great number of engineering dlf- A tou t Four

flculties must be overcc+ne.

The quantity ~ can be very different In different types of at~ertured wave guldeJ wldch are suit~ rou t

able for electron acceleration. For Imtance. the value of ~ can be estimated using the fosmmta) givea In Reg. [4], r o u t

If tt Is assumed that the energy of the accelerated particles IS not ]e• than l Mev. while 81nj TM 0.S rUin j u 80 key). E~ (4) can be ~ m ~ e ~

(5)

1034

Page 3: Particle bunching in a traveling-wave linear accelerator

A@inJ .

l,a~ AWInl _

~ Mev

.n:i. ,,. ,.v

Oout. Mev 0 05 " L~

Fig- I. The ratio AO!nJ has a function of the A~out

final elec~n energy Uou t for various beam etmrents ~ (Pt = 2 megawau~, az = 0.25).

e~

6

!o.,

Fig. 3. Th~ optimum value of the final electron energy Uou t opt asa function of a beam current I a t varlouspowers Po (az = 0.~5).

Fig, 2. The ratio A~lnJ as a func~on of beam current A ~out

I fc~ various final elecuon energies (Pc 2 megawatu c~z = 0.25).

we now comlder ceztain exx~ples In which A~ou~

is determined for various given val~es of% z . r , I and U.

If lhe phase velocity in ~ e wave guide is increased from 0 .5c t o c by increasing the ~ r n e m r of ~e apert~. , the function r rcmaim virtually co~tant. We will assume az = 0.25. Then Eq. (5) can be wtiz~-n in the form

a~out V \ e . out)~,, V. l

Now it Is easy to determlne~ magnitude of Uou ~

A~n for which a maximum value of ~e ratio ~ l---~j- is obtalned~

Aqout i/ 6P. . , ~ "

Uout op~ = T L . ~ , ~ - " ' / " {7)

m Fig. 1.are shown the t e l a ~ between A =~in|~ and ~oul

Uout f~ Pe -- ~ me~awat~ as c a l c ~ d from F.q. (~ . The curves in Fig. 2 show the depender~cr of ~inJ- on the magnitude of the beam current f~ ~iffcrent final ~an~cle

~'out energies. The curves in Fig. 3 show the dependence of the optimum final energy on bea~ current. By op~mtTn

. ; " ,A ~nj �9 . final particle ener.~y we are ~ und~standthe e n e t ~ at w~ch the maximum value of ~'~ ~ a ~ o ~ t~ ach~ev~l for a given particle curren~ A qtout

LITERATURE C I T E D

[I] H. Moil, Trans. IRE AP-4, Ho. 3, 3 " / 4 ( I ~ ,

1035

Page 4: Particle bunching in a traveling-wave linear accelerator

[2] I .C. Slater. Rev. Mod. Phys.. 20. 473(1949).

[3] M. Chodomw cr al.. Rev. Scl. Instr, 26, 134 (1955).

[4] P,. G. Mlrlmanov and G, L Zhilelko. Elecv~lcs and Elec~on Physics 2. 137 (1957).

Received March 19. 1957

RADIOMETER-ANALYZER FOR FIELD US E

G. R. T o l b e k . V. V. M a t t e e v . and R. S. S h l y a p n i k o v

In carrying cut geological surveys at radioactive-ore sites it Is extremely de~irable to be able to auay the radioacti~ ores directly at the deposit,

In a number of papers [1] - [3] it has been shown that the nature of 0:-, energy s~ecCa of 7 -radiation from radioactive elements of the uranium and thorium series makes it possible to carry out such �9 determina- tlon.

Acc~xdingto the theory given in Ref, [3] the equation for. ~,-rays. emitted by a rock which contains e le- menm of the radioactive families of uranium and t h o ~ r a Is of the form:

~ r = a R R a + bRTh

fro" ~ e ~Ul y-ray spectra and

= a,~aaRe. . + ~ r h R ~

fix 7"rays with energies above a given value, where a and b are con~tafi~ which characterize the rock and the m

Inscument. l~a and RTh are the radium and thor~m content of the rock. N r' is the counting rate for all y - quanta. I~ is the counting rate for 7-rays with energies higher than a given value, and g = HT/~I'.

t t follows from these equations that

Nrx,/ ,a- NT R.,,=

(I)

The relative thorium and uranium con~ent of radioactive ores can be characterized by t ~ quantity

R a . + Rzm " Ca)

where RTh Is the thorium content and RRa is the radium content, whlc~h is pwportional to the uranium content a t radioactlve equilibrium between uranium and radium. $ubstitutlng Eq. (I) In Eq. (2) weobtain an expres- s/o,. for In terms of me~surable qu~n~id~

T , m ~ R l ~ = n b . . - o (s )