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Superlattice Crystal Accelerator: Acceleration Beyond GeV/mS.A. BogaczAccelerator Physics Department,Fermi National Accelerator LaboratoryP.O. Box 500. Batavia, IL 60510 USA

AbstractHere, an idea of using a visible light wave to accei-erate relativistic particles via the inverse FEL mecha-nism is explored. A strain modulated crystal structure -the superlattice, plays the role o f a microscopic undula-tor providing very strong ponderomotive coupling be-tween the beam and the light wave. Purely classicaltreatment of relativistic protons channeling through asuperlattice is perfo rmed in a self consistent fashion in-volving the Maxwell wave equation for the acceleratingelectromagnetic field and the relativistic Boltzmannequation for the protons. It yields the accelerating effi-ciency in terms of the negative gain coefficient for theamplitude of the electromagneticwave - the rate the en-ergy is extracted from the light by the beam. Presentedanalytic formalism allows one to find the accelerationrate in a simple closed form, which is further evaluatedfor a model beam - optical cavity system to verify fea-sibility of this scheme.I. INTRODUCTION

The main idea of using a modulated crystal struc-ture as an undulator is illustrated schematically inFigure 1. A beam of relativistic particles while channel-ing through the crystal follows well delined trajectories.The particles are periodically accelerated erpendicular otheir flight path as they traverse the channel. The undu-later wavelengths typically fall in the range 50-500 A,far shorter than those of any macroscopicundulator.. . . . . . l * . . .

. . . . . . l

. . . . . . l l

. . . . . l l

. . . . . l

. . . . . .

. . . .

. . .**

. . l

. . . . . . l l * . .

Figure 1 Center of the channeling trajectory in [110]direction in a strain-modulatedsuperlattice.O-7803-1203-1/93$03.001993 EEE 2587

Furthermore, the electrostatic crystal fields involve theline averagednuclear field and can be two or more ordersof magnitude larger than the equivalent fields of macro-scopic magnetic undulators. Both of these factors holdthe promise of greatly enhanced coupling between thebeamand the acceleratingelectromagnetic wave.II. SUPERLATTICE CHANNELINGOne can describe a high intensity proton beam interms of a classical distribution function, f(p, x, t),governed by the relativistic Boltzmann equation. Thetransverse dynamics of relativistic protons propagating

in a strain modulated superlattice is modeled by a har-monic crystal field potential and leads to generation ofa transverse current. This couples the Vlasov equationto the Maxwell wave equation. Therefore, presentedproblem reduces to a self consistent solution of theVlasov and the wave equations.Collective behavior of a particle beam channelingalong the z axis can be described in terms of the rela-tivistic Vlasov equation

pb-;,@ $- e% %-O1x a~Here A is a vector potential of an electromagnetic fieldand @ s a phenomenological harmonic crysta l-field po-tential*, which describesboth transverse ocusing of thebeam and longitudinal modulation of the minimum ofthe harmonic potential well.

$ = oo+~~l (x - x1 cos gz)*. (2)where g = 27cclI; is the strain modulation periodicityand x1, $,, $, are parametersof the potential.

Eq. (1) will be treated iteratively and only linearterms in the A-field will be retained. In the 0-th ordersolution A = 0, and the corresponding distributionfunction f = 6 is obtained from the solution ofab")at' J$a~-e&$i$z(). (3

A class of solutions, f(O), escribing a beam with asharply peaked nitial momentum distribution, A can beconstructedas followstie = no 6(x - x(O))F(p, - p(i) )h(pZ - p,), (4)

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where nos a concentration of particles per unit area ofthe channeling plane and the steady state trajectory isdcscrikd as followsX 0) = 1 Xlu2 cos (gz) (3

one can model the effect of coupling by adding a smallcomplex part ir? = a + ip to the k-vector; here a is again/loss coefficient and p describesa small shift in thephase velocity of the optical mode. Deamplification ofthe back traveling wave can be summarized by the fol-lowing expressionax(O)px0) p -z& .

Here U = g$, where, k is a focusing strength of thet .rystal channel given exp tcrtly below

00a-=-an 2 p, aAs dp, Q mr q Lr(v-) t (11)--mwhere

kb2 = f$-z z 03 v- = mykclpz + k - g. (12)

We have assumed hat only the transversecomponentofthe A-field is present and AX= A(z, t). We seek a per-turbed solution, t, in the following form Q= E Igx;, . (13)Here,6 = no 6(x - x(O))&p, - pf ) h(z, p, , t) , (7)where h describesbunching of particles due to the pres-ence of the A-field. Substituting Eqs.(4)-(7) and (2)into Eq.(l) leads to the following kinetic equation for h

(14)is the characteristic form occurring in diffraction theory,with the principal maximum at x = 0.

ah P ah eP gX1-L-+-.-L aAaA .at myaz c my 1 _ us z ap, singz = 0. (8)The inhomogeneous term in the above equationplays the role of a driving force representing accelera-tion of the particles by the ponderomotive force due tothe transverse motion (induced by the crystal field) inthe presenceof the A-field. The resulting transverse ur-

rent couples Eq(8) to the following wave equation

00

4rcne dp,h p, gxlC 2 sin gz,my 1-IJ (9)resulting in a closed system of equations for h and A.IIere the A-field can be identified as a sum of themacroscopic driving field anda self consistent electro-magnetic field propagating in the crystal structure. Westart with a single plane wave solution of arbitrary wand k propagating in free spacealong the z axis in bothdirections and use t as a 0th order iteration step

A(o) = A+ 0

e - ictw * Ikz (10)

III. ACCELERATION RATEImposing resonant condition, v- = 0, in Eq.(ll)fixes the wave vector of the optical mode as follows

g = mykclpz + k (13One can notice that, apart from a slowly varying func-tion Q, the remaining functions occurring in the inte-grand in Eq.(ll), namely, A and F are sharply peakedfunctions of momentum characterizedby the respectivewidths:

(16)Now one can compare relative sharpnessof both func-tions; A and F. Typical value of the relative momen-tum spread s of the order of 10m4. ssuming superlat-lice modulation o f 5008, and crystal length of 5 cm al-lows one to evaluate the width of F. Both characteristicwidths can be summarizedas follows= 1o-4 (17)The integration in Fq.(l 1) is carried out assuming thatthe sharper function, namely F, is approximated by the

Putting A = A: (left and right propagating waves) inEq.(9), one can solve it analytically for h = h(f) byconstructing a Greens function with the appropriateboundary conditions built in it. The solutions forA(i)(z) can be written explicitly in terms of the Greensfunction for the I lelmholtz equation. On the other hand,

F-function. This reduces he gain/loss coefficient to thefollowing simple expressiona~-~ n2 . (18)

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The above final result will serve as a starting point fo rfurther fcasihility discussion.IV. THREE WAVE MIXING

Spontaneoushunching of the proton beamchannel-ing through a superlattice and interacting with the clec-tromagnetic wave results in energy flow from the waveto the beam. This particular kind o f particle densityfluctuation, h, has the form of a propagating planewave of the same requency, o, as the emitted electro-magnetic wave. The phase velocity of the movingbunch matches the velocity of protons in the beam.Therefore, the quantity ymolpZ = k, represents thewave vector of the propagating particle density bunch.Keeping in mind that the periodicity of the undulatorrepresentsa static wave with a wave vector g, and that kis the wave vector of the electromagnetic wave, we cananalyze our results in the language of three wave mix-ing3.

Furthermore momentum conservation of all threemodes yields the, v- = 0 conditions. The last condition,k, = g - k, is equivalent to a momentum recoil be-tween the particle density bunch and the electromag-netic wave (deamplification of the backward propagatingwave), where a four momentum (0, g) is transferredfrom the backward propagating wave to the forwardmoving proton bunch.V. FEASIBILITY ASSESSMENT

We will discuss the feasibility of the proposedscheme by considering (110) p lanar channeling in astrain modulated Si crysta14.We write the undulator pe-riod as C= Nd, where d = 1.92 8, is the spacing betweensuccessive attice planes and N is the number of suchplanes. The strain modulation, of course, requires a scc-ond component, such as Ge; however, we will use theparameters f Si for convenience.Relativistic particles while channeling along thepath undergo transverseharmonic oscillations from thecrystal field potential, an analog of the bctatro~lscilla-tions, with the characteristic frequency oB -+$i.

One can see rom Eq.( 13) that if the angular velocity ofa particle traversing the strain modulated path, o =27rv,,lf, approacheso,$G(Doppler shifted betatron frc-quency), the undulator parameter,Q, has a resonance U+ 1), which would enormously enhance the gain/losscoefticient. However, the excessivegrowth of the undu-later parameterwould soon result in a rapid dechanncl-ing of the particles. One can see this easily if Q isrewritten in the following form(1))

whcrc vI ,v,, are transverseand longitudinal componentsof the particle velocity, rcspcctively.For small values of y(IJ = l), the following sim-ple physical criterion allows one to estimate the maxi-mum value of Q. Decharmeling will occur if the trans-verse kinetic energy of the particle exceeds he bindingenergy of the harmonic potential (a particle leaves thechannel). If the maximum transverse velocity of achanneling particle is vI and a is the distance betweenadjacent channels (for (110) channeling in Si a = 5 A) ,the above condition can be written as follows:

e$ a2>1I-my 2 0 (20)The equality sign in Eq.(20) along with Fq.(19) fix themaximum allowed value of the undulator parameteras

Qrn"=S-\I- !l!L A-.--my2471 . (21)The above expression can be evaluated for relativisticprotons channeling through our model superlattice as

Qnax= 7.5 x 4 -.-L-x lo-l4 cm12g2 . (22)y2- 1Now, one can evaluate Eq.( 11) assuming only one pro-ton - by assigning n to be an inverse area of the chan-neling plane per one particle for typical values of thebeam concentration5, n = 1016cm-*. This way cx de-scribes the rate of optical amplitude depletion per oneparticle - the acceleration rate. Assuming y of 2, C=SOOA nd 42 -( >il - 1o-4 (2%yields the following value of the acceleration ate

a = 3.S3 x low4 cm- (24The nominal acceleration efficiency in units of cV/cmwill, obviously, depend on the energy density of the ac-tual optical cavity, which is left out for further discus-sion elsewhere.

REFERENCES1. B. L. Berman, et al, Nuclear Instruments andMethods in Physics ResearchB, 2, 90 (1984).2. J. A. Ellison, S. T. Picraux, W. Ii. A llen andW. K. Chu, Phys. Rev. B, 37, 7290 (1988).3. J. A. Ellison, Phys. Rev. B, 18, 5948 (1978).4. S. Datz, et al, Nuclear Instruments and Methodsin Physics ResearchB, 2, 74 (1984).5. M. A. Kumakhov, Sov. Phys. JEPT, 4, 781(1977).

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