Comments on “separated high-energy electron beams using synchrotron radiation”

NUCLEAR INSTRUMENTS AND METHODS 118 (1974) 147-148 ; . CCs NORTH-HOLLAND PUBLISHING CO.

COMMENTS ON "SEPARATED HIGH-ENERGY ELECTRON BEAMS USING SYNCHROTRON RADIATION" D. WHITE and T. ERHEB Department of Physics, Illinois Institute of Technology, Chicago 60616, U.S.A. Received 7 January 1974 Inadequacies of a semi-classical estimate of energy straggling in synchrotron radiation are discussed . The lack of mass dependence in eq . (3) can be circumvented by taking into account the inelastic effects represented by eq. (1), i.e . making the replacement Eo-.Eo-AE. This can be exploited in spectrometers inserted downstream from the ch:catre, or indirectly,

In a recent article Farley et al.') proposed the utilization of a magnetic chicane to purify high-energy electron beams. The essential idea is to selectively strip offsome of the beam energy by inducing synchrotron radiation (magnetic bremsstrahlung), and then by direct or indirect means exploit the cumulative effects of radiation reaction to produce macroscopic shifts of theelectron trajectories . This possibility has also been examined in a parallel series of articlesz-1° ) . COm-

since eq . (1) gives rise to aclassical dampingincrement of the form")

SOcp = ~a

parison of Oese results with the corresponding formulas in ref. 1 indicates that there are some discrepancies in the estimates of the energy straggling which is associated with the bremsstrahlung. It is the purpose of this note to point out that Farley et al . have under-

or, in practical units, S®cD(mrad) = 1 .90x

AE = iamc z(Llxe) r z, li

--

nt - c3

i == N t+ 1t t+

Eo

HI(MG)Lt(mm) 0.95x 10 -3

1+1.33x l0-S Hz (1v1G)L(mm)E o (GeV)

The corresponding elastic deflection, to lowest urdce

e B o -HL.

(Sa)

dQQM (mrad) _

" +

chicane .

mc z/E, is

l0 -' H1 (MG) LJ(mm),

whereas refs. 8 and 9 indicate that

and this is the basis for the selective action of the

in

(4b)

then Farley et al. find (compare eqs (9), (11), and (16) of ref. I]

d0sc(mrad) = 0.57x

.

All the other quantities have their usual significance, and a -I = 137. It follows immediately that

a (dE)t ^' (dE)electr(metectrlnlt) :

10_ a H 3 (MG)Lz (mm).

measured in terms of the standard deviation,

(1)

r,- 4.414 x 101 3

(4a)

Within the framework of the classical relativistic theory, the radiated energy AE is a sharply defined quantity . However, quantum effects will induce a straggling, and this in turn is manifested in a spreading of the aeflected electron beam. If, as usual, this is

estimated the straggling effects in virtue of an inconsistent semi-classical approximation. All theessentialpoints can be conveniently illustrated by considering the following idealized situation: An electron with initial energy Eotraverses a path length L normal to a magnetic field H. The classical relativistic expression for the total energy loss is then

where

H_ s _L z , (A () CH~O

(3)

(Sb)

This discrepancy originates in the semi-classical arguments which lead to di)sc: Following Sokolov and Ternov") the energy dissipation corresponding to AE is discretized by introducing an equivalent number of quanta ; this in turn is assumed to be subject to (Gaussian) fluctuations . Although this heuristic introduction of temporal fluctuations is adequate to describe the

147

148

D. WHITE AND T. ERRER

TT--T~1-~yTT ~ NDRMAIZATION : AREA UNDER EACH CURVE : 56F-rt« tm°.tdI race trut -Cbssicd RMeti.k ps9 48 z 40 32

Gaussian character of the distribution is only approiimate. In any case a well-developed bremsstrahlun cascade is a prerequisite. It is easy to show that the lower limit, 1.617 L(mm) > H(MG)

14 ~ 16 0s

Fig. 1 . Simulation of beam spread due to energy straggling : the basic parameters are Eo =300GeV, H= 2 MG, L = 10 mm .

inception of betatron oscillations, it is not suq'lciently accurate for estimates of radiation-induced beam spreading. It is shown in detail in refs 8 and 9 that the magnetic bremsstrahlung process, in strictly quantummechanical terms, corresponds to a cascade over the Landau levels. Not only is the emission probability subject to fluctuations, but also the magnitude of the successive energy steps is liable to variation. Both of thc ,.e effects are responsible for the progressive time erolution of the bremsstrahlung spectrum . The difference of coefficients in the semi-classical and quantumme:hanical estimates (0 .57 versus 0.95) indicates that both sources of fluctuations must be taken into account --even in theweak-field limit . As a final caution, it should be remarked that the

is a necessary condition . Of course precise results require a computer-assisted evaluation of the exact theory"). Fig. I shows a representative scattering situation . We are grateful to Dr Baier for communicating the results in ref. 10 before publication. References

F. J. M. Farley, E. Picasso and L. Bracci, Nucl . Instr. and Meth. 103(1972) 325. T. Erber, Rev. Mod. Phys. 38 (1966) 626. T. Erber, Z. Angew. Phys . 24 (1968) 188. C. S. Shen, Phys. Rev. Letters 24 (1970) 410. H. G. Latal, Acta Phys. Austriaca34 (1971) 65. T. Erber, Schladming lectures, Acta Phys. Austriaca, Suppl. VIII (1971) 323. 7) C. S. Shen and D. White, Phys . Rev. Letters 28 (1972) 455. s) D. White, Phys . Rev. D5 (1972) 1930. o) C. S. Shen, Phys. Rev. D6 (1972) 2736. 10) V, N. Baier, V. M. Katkov and V. M. Strachovenko (Novosibirsk preprint, 1973). 11) A. A. Sokolov and 1 . M. Temov, Synchrotron radiation, (Engl. ed. ; Akademie-Vertag, Berlin, 1968). 1)

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