Kinetic equation for relativistic charged particles channeling in a bent crystal

Nuclear Instruments & Methods in Physics Research

Nuclear Instruments and Methods in Physics Research B67 (1992) 228-230 North-Holland

Section B

Kinetic equation for relativistic charged particles channeling in a bent crystal A.V. Vel'ko

Department of Physics, State Unil'ersityof Grodno, Grodno230023, USSR

Kinetic equations are suggested which describe the planar dechanneling of relativistic charged particles in a bent crystal with the inclusion of multiple scattering and radiative energy losses. The equations transform into equations of the Fokker-Planck type valid for linear crystals in the limit of infinite bending radius of the crystal.

The dynamics and kinetics of the distribution of relativistic charged particles in bent crystals have been under thorough investigation during recent years theoretically as well as experimentally [1-6]. The interest is caused by the possibility to use the effects of orientation change and radiative self-polarization of the charged particles' spin in a bent crystal for measuring magnetic particle characteristics. However, we must know the exact distribution of particles channeling in the bent crystal in order to solve these problems. The distribution function of incident particles changes with penetration depth in the crystal due to multiple scattering on electrons and on atoms of the crystal and, for light projectiles, also due to radiative energy losses. Multiple scattering predominates over radiative processes in the energy region E < El, where E= is the energy determined from the correlation between the dechanneling length and the characteristic length of energy losses of the particles [5]). With increasing particle energy, the energy losses increase too and the radiation becomes dominant. The ;tim of the present work is to examine the kinetics of relativistic charged particles in a bent crystal, taking into account multiple scattering and radiative energy losses. We consider the case of relativistic positively charged particles (such as positrons) entering a bent monocrystal under planar channeling conditions. The evolution of the local distribution function f ---f(t, p, r ) in phase space (p, r) is de~ribed by the Boltzmann equation [7]

af+ p af

-~t

~

~r --

vv

Of

ett~-pp=/'; + [2,

(1)

where t is the time, m the particle mass, and 7 = E/mc 2 the Lorentz factor, corresponding to particle energy E.

The motion of a charged particle in curved channels with radius of curvature R ( R > > h , where h is the half-width of the planar channel) is determined by the effective potential, which also includes the centrifugal part Veff---~Vcff(E, y) = V(y) - y E / R ,

(2)

where V(y) is the continuum planar potential of the straight crystal. The transverse (radial) coordinate of the particle y = r - R is read from the channel centre in the direction opposite to R. The collision integrals /i and [2 in eq. (l) describe the changes of f per unit length due to the influence of multiple scattering and radiation, respectively, i.e.

i, = fdq

If(t, p - q , r)w,(q, p - q )

- f ( t , p, r)w,(q, p ) ] ,

(3)

[2= f d k [f(t, p - k , r)w2(k, p - k ) - f ( t , p, r)we(k, p ) ] ,

(4)

where wt(q, p) is the local probability of increase of the particle's momentum due to multiple scattering. In the uniform field approximation the local probability, w2(k, p) of radiation is described by expression (10.24) in ref. [8]. Eq. (1) describes the distribution of channeled particles at all depths, i.e., both at small depths (near the surface of a crystal) and at large depths, where a statistical equilibrium is reached. We can simplify cq. (i) at large depths: As the mass of a particle is very small compared to the mass of the crystal atom, the collision integral i I may be written as the sum of two parts. The first describes the change of the absolute value of the momentum, and the second the change of direction. With the use of new local variables (t, Ey, E)

0168-583X/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

A.V. Verko / Relativisticchargedparticlesin a bentcrystal (where Ey is the local transverse energy of a channeled particle), the distribution function f(t, p, r) converts into a function u(t, Ey, E) as follows:

f(t, p, r) =~u( t, E., E),

(5)

where c3 ~: = ~ ~/2( Er-Veu ) . Introducing the mean distribution function of channeled particles in a bent crystal plane,

F -~F(t, Ey, E ) = f a2u(t, Ey, E) dNy,

(6)

al

where dNy is the probability that the particle has the coordinate in the interval [y, y + dy] with transverse energy Ey, one can obtain the following kinetic equation from eq. (1) after a few transformations: 0t

0E~

&

"~

(Ey-Veu

F -

\ At [

J

--~+ atl/ l

+ A 2 ( E y , E, F ) + / 2 ( E r , E, F ) ,

(7)

where 1 [ ~-E

~'Ey~\

AI(Ey, E , F ) = ~ " [2-~-T + 3-'~-7" ) / F

(

° 0 E~-~ + i~-'E

-~-

(8)

A2( E,, E, F) = " ~ "-~ +---~ ) EbVeff] F a "}-~-E

E

229

The turning points a i (i = 1, 2) are determined by the equation U(ai) = Ey. The kinetic eq. (7) describes the change of the distribution function of channeled particles in the field of the curved crystal plane (taking into account multiple scattering and radiation). Eq. (7) without the terms A1(Ey, E, F), A2(Ey, E, F) and ]2(Ey, E, F) is of the Fokker-Planck type for a straight crystal [9]. In the approximation E >> AE, which corresponds to the region of particle energy E > yE/R, this term is small too. It corresponds to the case of particle channeling in a potential well with a depth smaller than or of the order of the quantity Vl -- V(b), where b is the screening constant, b ~ Z - i/3, Z is the atomic number of a crystal's atom. In particular, the equation is transformed into the diffusion-type equation proposed by the authors [3] when the redistribution of the flux and the damping are neglected. Note that the diffusion approach does not give sufficiently correct particle distributions in the crystal channel (it gives a larger fraction of dechanneling). The error in the results obtained with the kinetic Fokker-Planck-type equation is determined by the approximations used for numerical simulation (such as approximation of continuum potential, electron density distribution, etc.). Note that we do not consider the beam smearing along the atomic plane. This smearing may be simply included in the kinetic equation as additional terms. Besides we do not have to take into account variations of the Lorentz factor 3' with transverse energy Ey (which corresponds to the dipole approximation). If this is taken into consideration, then the kinetic equation is more complicated.

y

"Ey 2 Kff n "~

F ,

(9) Acknowledgement

i2(E~, E, F) = f d( ho,)[ F( t, e. - A e . e + ho, ) X (w2(hto, E + h t o ) ) - F ( t , E r, E)

× ],

with ~"E/At and -A-'Ey/At the mean changes of the total and transverse energy, respectively, per unit length [4,8,9]. The brackets ( . . . ) designate averaging over the period of particle oscillations T = T(Ey, E), viz. ( x ) = "~c a, ~ 2 ( E y - Verf)/E "

The author is grateful to A.O. Grubich for useful discussions.

00) References

[1] O.I. Sumbaev, Soy. Phys. JTP 57 (1987) 2067. [2] A.M. Taratin and S.A. Vorobiev, Soy. Phys. JTP 58 (1988) 403. [3] N.A. Kudryashov, S.V. Petrovsky and M.N. Strikhanov, Soy. Nucl. Phys. 48 (1988) 666. II1. CHANNELING, DECHANNELING

230

A.V. Vel'ko / Relatit,istic charged particles in a bent crystal

[4] V.G. Baryshevskii, Channelling, Radiation and Reactions under High Energy in Crystals, Belgosuniversitet, Minsk (1982). [5] V.G. Baryshevskii and A.O. Grubich, Soy. Nucl. Phys. 37 (1983) 1093. [6] V.G. Baryshevskii and V.V. Tikhomirov, Soy. Phys. Usp. Fis. Nauk. 159 (1989) 529.

[7] V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Relativistic Quantum Theory, part 1 (Pergamon, New York, 1971). [8] V.N. Baier, V.M. Katkov and V.S. Fadin, Radiation from Relativistic Electrons (Atomizdat, Moscow, 1973). [9] V.V. Beloshitsky and M.A. Kumakhov, Docl. Akad. Nauk. USSR 212 (1973) 846.