Nuclear Instruments and Methods in Physics Research A 331 (1993) 739-741 North-Holland

NUCLEAR INSTRUMENTS

& METHODS IN PHYSICS RESEARCH

Section A

Striped superconductor undulator scheme

A.V. Smirnov and A.A. Varfolomeev Russian Science Center "Kurchatov Institute'; Moscow 123182, Russian Federation

A new microundulator scheme is proposed. Superconductor (SC) material layers are interspaced with ferromagnetic layers composing a striped block (blocks). Electrons from a relativistic beam passing transversally to the layers direction near this striped surface will be deflected periodically due to the interaction with self-induced currents: SC provides repulsion and ferromagnetic layers provide attraction. A gas loading of the order of 1-0.1 Torr would be necessary for space charge neutralisation and for providing pure current-current interaction. Since the interaction strength is proportional to the current itself, rather high beam currents are needed for noticeable beam deflections. As it is shown kA currents would be effective enough.

One of the specific features of this scheme is that the undulator factor depends linearly on the beam current. The scheme is analysing from the microundulator design problem point of view.

1. Introduction

Conventional classical undulator schemes are based on electron beam interaction with an external magnetic field produced by permanent magnets or electromag- nets. These undulators can be treated as " far field" undulator designs since the distance between the elec- tron beam and the undulator pole surface can notice- ably exceed the beam size and can be comparable to the undulator gap. One of the disadvantages of the traditional schemes is the saturation of the undulator field leading to small K w values for small undulator periods.

We wish to consider an alternative interaction de- fined by the intense beam forces. Such nonl inear undu- lator schemes do not use any primary field and can be conditionally considered as "near - f ie ld" schemes simi- lar to the terminology used in particle acceleration physics.

In the suggested scheme electron wiggling is en- sured by its interaction with surface self-induced cur- rents. In what follows some estimates for the undulator strength and its dependence on the gap value is evalu- ated. The lower limit for the undulator period is esti- mated.

role of opposite walls spaced by a small gap. The gap area is filled with a gas.

The electron beam propagates parallel to the striped surfaces transversally to the layers direction. It induces image surface currents. Due to ionization the beam space is neutralizing.

For proper action of the undulator scheme the following two effects should be provided. First, a self- focusing effect should take place which allows high current beam propagation in the undulator without an external focusing solenoid. The second effect is the shielding of space charge. If the Coulomb interaction is neutralised, superconducting stripes provide repulsion while ferromagnetic stripes provide attraction.

Contrary to the self-focusing forces acting on an electron, the forces from the surface induced currents are periodical. They are axially asymmetric and depend more weakly on the electron position in the beam. These forces can appreciably exceed the self-focusing forces, being averaged on the beam cross section. This is true for the case when current beam instabilities can be neglected. As a result the whole beam is undergoing the periodic deflections.

3. Basic relat ions for the "near-f ield" undulator

2. The "near-f ie ld" undulator based on self - induced currents in superconductor

The schematic of the proposed undulator is shown in fig. 1. Ribbon superconductor stripes are inter- spaced with ferromagnet ic stripes composing a sand- wich-like block structure. Two such surfaces play the

Let us consider the results [1-7] which can be used for the approval of the scheme issues to determine the real conditions for the performance. It will allow us to determine the conditions for the realization of the scheme.

Bennet t [1,2] predicted that the self-focused high current relativistic electron beam (REB) can propagate

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740 A.V. Smirnov, A.A. Varfolomeev / Striped superconductor undulator scheme

in gas with its space charge being neutralised by plasma ions. Bennet t ' s condition of the self-focused beam propagation is

1 - f ~ - < 3 2 ( 1 - f r o ) , (1)

where f~,m are the coefficients of electric and magnet ic neutralisation respectively, and fl is the beam velocity related to the velocity of light c.

The absolute value and sign of imagine current I ' induced on a homogeneous surface by the R E B cur- rent I can be writ ten in the form

I ' = I (1 - f m ) ( ~ 2 - / $ 1 ) / ( ~ 2 4-/zi) , (2)

where /Xl, 2 are the permeabil i t ies of the gas (plasma) and the surface respectively.

For simplicity we will assume that the beam induced fields have no influence on the beam trajectory symme- try and on the cross section profile. In accordance with the refs. [3,7] the beam cross section size and the profile are not distorting if the following condition is satisfied along with condit ion (1):

r < d. (3)

Here r is the beam radius and d is the distance of the beam to the surface.

Some more information concerning REBs guiding can be found in refs. [4-6], including the case of guiding by the high tempera ture superconducting (HTSC) bulk materials [6]. Note that according to ref. [6] the H T S C compounds would have an advantage. For this case the surface current density of nanosecond range is not l imited by the quench current density and can be essentially larger than this value (by more than two orders).

Now we can consider the features of the beam wiggling between the two striped surfaces made of block arrays with interchanging the materials H T S C

superconductor ferroma,qnetic Z

I L ~ . N N ~ ~/2 l

. . . . . . . . . . . . . . . . . . . . . . , ~ " ' ~ - " " - ~ t

e- be g./2 [

Fig. 1. The schematic of the "near-field" undulator.

K value w

°1

0 0.1 0,2 0,3 0,4 0 5 0,6

gap/period ratio

Fig. 2. Calculated "near-field" undulator strength K w as a function of the relative gap gw for a beam current I = 2 kA.

and ferromagnetic (see fig. 1). Now we impose more strict conditions than condition (1):

)re >> Y 2, fm << 1, (4)

where y = ( 1 - / 3 2 ) -1/2. For ferromagnetic material with permeabili ty /.% we have /x 3 >>/xu For this case the absolute values of the currents induced in the ferromagnetic and the H T S C stripes are equal to the primary one: l I ' I ~ I. We use Biot -Savar t ' s law for an estimate of the amplitude of the induced fields in the median plane Box for picewize constant induced cur- rents. In this way we derive for the undulator factor K w = qeBoxFhw/2"rrmo c2 the following expression:

I F 1 l= + 1/4 Kw-=

IA rrgw ~ = -1 /4

oo l l=(2n+1)/4]

+ 2 E (-I)~ ~gw~+/2 (5) n = 1 l = ( 2 n - 1 ) / 4 J '

where gw=g/Aw is the relative gap, IA=moC3/qe, and F is the relative amplitude for the fundamental harmonic of the field.

The undulator strength dependence on the gap calculated with the help of formula (5) for a current I = 2 kA and F = 1 is shown in fig. 2. It is also seen that K w = 1 is obtained for gw = 0.25. It follows from eq. (5) that the constant K w value is provided for the condition I/g2w = const, and g2 >> 1/16. For small gaps (g~ << 1 /16) the other relation, I / g w .~ const., is valid.

If the minimum undulator period value is est imated from condition (3), the effect of wiggling motion should be taken into account:

A w = 4 r / ( g w - Kw/'rry ). (6)

A.V. Smirnov, A.A. Varfolomeev / Striped superconductor undulator scheme 741

4. Numerical example

Let us consider two numer ica l examples. 1) If we take b e a m p a r a m e t e r s tha t are cor respond-

ing to those of ref. [6] ( I = 3.8 kA, r = 1.5 ram, e lec t ron energy E = 2 M e V ) t hen f rom fig. 2 and eq. (6) it follows tha t A w = 22 mm, g = 7.3 mm, and K w = 0.85.

2) In the second example we use the following b e a m p a r a m e t e r s r e ached in a r e sonance e lec t ron linac [8]: I = 1.5 kA, E = 24 MeV, r = 0.3 ram, and pulse dura t ion tp = 20 ps. U n d e r these condi t ions we get f rom fig. 2 and eq. (6): A w = 5.6 mm; g = 1.2 mm, and

K w = 0.85. For shor t e r b u n c h lengths the "nea r - f i e ld" undula -

tor factor sharply decreases because of de le ter ious re ta rd ing of the b e a m induced fields.

5. Conclusion

The undu la to r scheme based on the in te rac t ion wi th self- induced fields is able to ensure the undu la to r effect for h igh b e a m currents at r a the r small undu la to r periods. The scheme has some poten t ia l advantages : 1) it seems to be simple in manufac tu r ing ; 2) it does not

need in a power supply; and it can be used with mul t i -kA beams.

O n the o the r hand, a h igh th resho ld cu r ren t can be r ega rded as the main disadvantages. In addi t ion the m i n i m u m b e a m pulse length is l imited by the r e t a rded wake-f ield effect. However, the b e a m repuls ion f rom supe rconduc to r makes the scheme effective and facili- ta tes the p rob lem of b e a m t r anspor t t h rough the undu- lator.

References

[1] W. Bennett, Phys. Rev. 45 (1934) 890. [2] W. Bennett, Phys. Rev. 98 (1934) 1584. [3] W. Link, IEEE Trans. Nucl. Sci. NS-14 (3) (1967) 777. [4] A:N. Didenko, A.I. Riabschikov, V.A. Tuzov and Y.P.

Ysov, Soy. Phys.-Techn. Phys. 19 (12) (1975) 1625. [5] A.N. Didenko, V.P. Grigoriev and Y.P. Ysov, Powerful

Electron Beams and their Applications (Atomizdat, 1977) in Russian.

[6] H. Matsuzawa, Y. Ishibashi, T. Osada et al., Jpn. J. Appl. Phys. 29 (1990) 785.

[7] K.V. Hodataev, Nucl. Energy 23 (5) (1972) 381, in Rus- sian.

[8] S. Takeda, K. Tsumori, N. Kimura et al., IEEE Trans. Nucl. Sci. NS-32 (5) (1985) 3219.

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