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Dynamical chaos in the motion of fast charged particles in crystals
Nuclear Instruments
174
DYNAMICAL N.F. SHUL’GA,
CHAOS IN Tm N.V. LASKIN
and Methods
MOTION OF FAST CHARGED
in Physics Research
PARTICLES
B48 (1990) 174-180 North-Holland
IN CRYSTALS
and V.I. TRUTEN
Institute of Physics and Technology of the Ukrainian Acad. Sciences, 310108, Kharkov.
USSR
Rest&s of an investigation of the dynamical chaos phenomenon in the case of passage of fast charged particles through a crystat along one of the crystal axes are presented. Here we discuss the conditions for dynamical chaos arising in axial channeling and above-barrier motion of relativistic electrons and positrons in a crystal and some consequences of this in some of the physical processes that accompany the passage of the particle through the crystal, such as coherent radiation, multiple scattering by atomic chains, the fine structure of the radiation spectrum, the effect of thermal oscillations of atoms on the character of particle motion, and the effect of multiple scattering on the coherent radiation by fast particles. The results of this theory are compared with results of an experiment carried out at CERN on the effect of multiple scattering on the coherent radiation of ultrarelativistic electrons and positrons. The experiment is shown to confirm the main predictions of this theory.
1. Possibility of regular and chaotic motion of a fast charged particle in the field of a crystal atom chain Since a crystal has a periodic structure, it is natural to assume that any partic!e motion in this structure would be regular, quasi-periodic. In practice, however, this assumption is not valid and along with the regular motion there can occur a chaotic motion of the particle [l-3]. The essence of the matter is that even if the particle moves in the most simple fields dependent on two coordinates, an essentially irregular, chaotic motion of the particle is possible along with the regular one [4,5]. The character of the motion depends on the number of the motion integrals. Thus, for example, any finite motion of the particle will be quasi-periodic in a system with two degrees of freedom which has two motion integrals. If, however, the system has only one integral, then the motion may be stochastic. This is the case for a fast charged particle that moves in a crystal at a small angle to one of the crystal axes. Indeed, in this case the motion of the particle is mainly determined by the continuous potential U(x, y) of the crystal atom chains, i.e., by the lattice potential averaged over the coordinate z parallel to the crystal axis along which the particle moves [6]. In a field like this the particle momentum component parallel to the chain axis is conserved. The motion in the orthogonal plane is given by
pt=
-g/J(p),
(1)
where p = (x. y) and E is the particle energy. Note that a system of units with h = c = 1 is applied. Thus, we come to the problem of the two-dimensional motion of a particle in the periodic field U(x, y).
The potential energy U(x, y) has deep minima (for negatively charged particles, for example electrons) or considerable maxima (for positrons) at the coordinate values of x and y that determine the atomic-chain positions in the orthogonal plane, and it has saddle points lying on the straight lines that connect the neighbouring chains. The character of the particle motion in a field like this is determined by the number of motion integrals of eq. (1). One of the motion integrals is well known. It is the energy integral of the transverse motion [6], E,
= ;Erj2 + U(p),
lEll
KE.
(2)
Depending on the value of E, , the particle motion in the crystal can be both finite (channeling or, for positive particles, proper channeling) and infinite (above-barrier motion) with respect to the atomic chains. As regards the second motion integral, its existence is not obligatory in the problem under consideration, this conclusion holding true both for channeled and above-barrier particles [l]. In this report some results of an investigation of dynamical chaos phenomena during the passage of relativistic electrons and positrons through a crystal along one of the crystal axes are presented. Primary attention is paid to the condition of the dynamical chaos phenomenon arising when the motion of the particles in the crystal is above-barrier and to some consequences of the phenomenon for physical processes that accompany the passage of the particles through the crystal - such as multiple scattering by atomic chains, coherent radiation, influence of multiple scattering on the coherent radiation, and the fine structure of the radiation spectrum.
175
N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
2. Scattering of fast particles by crystal atom chains A fast particle moving at a small angle J/ to the axis, but with transverse energy above the barrier between channels, successively collides with different crystal atom chains parallel to the z-axis. The collisions may or may not be correlated. The correlations take place, for example, in the case when J, >> J/, ($, is the critical angle for axial channeling) [6] and when in addition the angle a (a * $) between the incident beam and one of the close-packed lattice planes is small. In this case the relative variation of the impact parameter between collisions of the particle with nei~bou~ng atomic chains is small so that the particle trajectory will be changed smoothly with penetration depth. Besides, the trajectory will mainly be determined by the continuous potential U,(x) of the plane, to which the motion is nearly parallel,
.t=
-$$_r,(x),
@=2@
af w 4 3Z
z)-f(&
z)],
(4)
where Q, is the azimuthal angle of particle scattering by atomic chains in the orthogonal plane, f ( I&,z) is the particle-dist~bution function in the angle cp at the depth z, #(b) is a function of the particle deflection in the field of an atomic chain, n is the density of atoms and d is the atomic spacing along the axis z. The total angle of particle scattering 8, is related to + as 13= 211, sin ($1/2).
( 1-exp
[ - 2ndr$/_I
db sin2( ?)I).
(5) If, in addition, \I/ * J/,, we have
iP=
where x is the coordinate orthogonal to the plane. Thus, we have come across a one-dimensional problem of particle motion in the field U,(x). It is clear the motion in a field like this will be a regular quasi-periodic one (in the case under consideration the transverse-motion energy relative to the axis and to the plane will both be conserved). In the chaotic particle motion in the (x, v) plane the impact parameter varies strongly between successive collisions of the particle, thus collisions with different atomic chains can be treated as a random process. In this case one can consider multiple scattering of the particle by atomic chains. But in contrast to an amorphous medium, in the case under consideration multiple scattering occurs not on separate atoms but on clusters of atoms, i.e. crystal atom chains parallel to the crystal axis. To treat this process one usually uses the random chain model in which the process of the particle scattering in the periodic field of the atomic chains is assumed to be replaced by a process of scattering by the atomic chains located irregularly but parallel to each other [3,6,7]. In this case the multiple scattering can be considered using the kinetic equation
=nd$f” db[f(+++(b), --co
Note, however, that the statistics of random events (that is collisions with atomic chains) during the particle passage through the crystal can be different. Thus, it is important to know to what degree the above mentioned simplest model of multiple scattering is valid. With that end in view the values of the mean squares of the multiple-scattering angle of above-barrier electrons and positrons, obtained by computer simulation of particle passage through a crystal, were compared with results which were obtained using a theory based on the random chain model. As is generally known [3], in the random chain model we have
z/_“,
dh[-&/_m_
dyU(b,
Y)]2,
(6)
The computer simulations of the mean square value of the multiple-sitters angle in the periodic field of atomic chains was based on the formula p(z)
= $
,E 44’ I=1
sin2( 2),
(7)
where N is the number of particles in the beam and (pi is the azimuthal scattering angle of particle i at the depth z which is obtained bynumerical solution of eq. (1) with different initial conditions (that is, impact parameters). As an example we consider scattering of electrons and positrons which move with an energy of 30 GeV in a silicon crystal of 50 Pm thickness at a small angle to the (111) axis. The results of the calculations are given in fig. 1. The solid curve relates to the theory based on the random chain model. The results of computer simulation are given by the symbols A and 0 (the symbols relate to different orientations of the n- and y-axes against the beam). c is the mean square of the multiple-scattering angle in an amorphous medium. The results obtained demonstrate that for (c,5 \I/,the results of the theory based on the random chain model and computer calculation are the same. For $J > +, the results of the simulation are smaller than the corresponding results of the theory. The difference increases with 4, being greater for positrons than for electrons. As an analysis of Poincare cross sections demonstrates [l], this result can be explained by the fact that the regions of phase space that correspond to the regular motion expand as 4 increases. They expand considerably more for positrons than for electrons and affect the particle multiple scattering in the crystal. Indeed, if there are regions in phase space that correspond to III. CHANNELING,
DECHANNELING
N. F. Shui’ga et al. / Dynamical chaos in passage through crystals
176 e-
5, (444)
Fig. 1. Dependence of the mean scattering angles of electrons (a) and positrons (b) on orientation in a silicon crystal.
regular motion, then the particle moving chaotically cannot get into these regions. Thus, the phase space regions of regular motion have an effect on the multiple scattering of particles in a crystal, the influence increasing as the mentioned regions expand and the time during which the particle is near the regions becomes longer. The influence of thermal oscillations of atoms and the electron subsystem of the lattice on the scattering was not included in the results deduced above. Note that these factors have not only to give rise to multiple scattering on fluctuations of the lattice potential but also to decrease the phase space regions for regular motion, which in turn will increase the multiple scattering on atomic chains.
an even multiple of the radiation formation length I, = 2E( E - o)/m2u (the coherence length) multiplied by 27. If the particle motion in the transverse plane is chaotic, then its collisions with atomic chains can be considered as random. In this case the spectrum and polarization of the radiation are found to be smooth functions of the frequency (see fig. 2), the radiation being determined only by peculiarities of the particle interaction with the field of a separate atomic chain in the frequency region for which the length 1, is small compared to the mean free path between successive collisions with chains. In this case the peculiar behaviour of the particle motion in the field of a chain affects its radiation. Thus, correlations between electron collisions with atoms in a chain increase the mean value of the angle of its scattering in the crystal as compared to the scattering angles in an amorphous medium, and this in turn intensifies the electron emission in the range of low frequencies as compared to emission in an amorphous medium. But the asymmetry of the electron scattering in the field of an atomic chain (the scattering takes place along the azimuthal angle cp) gives rise to linear polarization of the radiation. In particular, it is easy to verify that the polarization degree is 50% in the range of frequencies satisfying the condition I, > 2 R/J, (R is the screening radius of the atom potential). It is worth noting an analogous result can be obtained if one uses the exact formulas of the theory of coherent radiation in which all simplifications connected with dynamical chaos are avoided. However, it requires tiresome computer calculations.
I
3. Coherent radiation in the case of regular and chaotic motions of fast particles in a crystal In the theory of coherent radiation by ultrarelativistic electrons one usually considers the case of the electron motion at a small angle 4 (but 4 S- JI,) to one of the crystal axes and, besides, the angle a (a =Z 4) between the particle momentum and a crystallographic plane is small too [8,9]. In this case the electron successively collides with atomic chains located in the plane, the collisions being regular. The regularity of the electron interactions with the atomic chains gives rise to sharp maxima in the spectral density and the radiation polarization (fig. 2). The positions of the maxima correspond to the frequencies w, for which the particle free path length I = a/# between collisions with the neighbouring chains (a is the chain spacing) is equal to
Fig. 2. The emission spectrum of electrons with an energy of 1 GeV which move at the angle J, = 2 mrad to the (100) axis [9]. The solid curve corresponds to motion in the vicinity of the (100) plane; the dash-dotted line correspond to chaotic motion in the plane orthogonal to the (100) axis; the dotted line is the emission spectrum in an amorphous medium.
N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
Thus, under dynamical chaos conditions the formulas that determine the electron radiation in a crystal are greatly simplified. In this case, one can find a relation between the quantities that determine the scattering of and the radiation from a fast particle in a crystal.
4. Effect of multiple scattering on the coherent radiation The coherence length increases sharply as the particle energy grows and the frequency of the emitted photon gets lower. If the length exceeds the free path length of the particle between successive collisions with chains in a crystal and if the motion in the transverse plane is chaotic, then one has to take into account the effect of multiple scattering on the radiation. This problem has been solved in ref. [lo] where a path integration method is used. In particular, using the random chain model, the effect of multiple scattering on the coherent radiation spectrum is shown to be determined by the formula
where J’Ln is the electron radiation spectrum in an amorphous medium in the range of low frequencies, N is the ratio of the mean square angle values of particle scattering in the crystal and the amorphous medium and a(x) is a function that determines the multiple scattering effect on the coherent radiation (see eq. (6.4) in ref. [3]). The quantity w, is determined by the expression oF = 2 No,,, where wLP is the frequency at which the multiple scattering affects the bremsstrahlung in an amorphous medium [8,11]. For a wide range of angles E+=ZOGd,
St,
111
+(R/d
B # 2 #,) the quantity N is large compared to unity (N - R/4d#, see also fig. I), which is why multiple scattering should affect the radiation in a crystal more than in an amorphous medium. Lately the effect under consideration has been investigated at CERN for the passage of electrons and positrons with 10 and 20 GeV energy through a silicon crystal of 100 urn thickness in the vicinity of a (110) axis. Fig. 3 shows the results of the experiment relating to the angular range # 2 #, where comparison of theoretical and experimental data is possible. The results obtained in ref. [12] confirm the main predictions of the coherent radiation theory based on the random chain mode: for a wide range of frequencies o, the spectral density of radiation in the crystal greatly exceeds the spectral density of radiation in an amorphous medium and depends weakly on w; but at low frequencies, w 5 200 MeV, the intensity of coherent radiation decreases. However, in order to make a quantitative comparison between theory and experiment some additional data are needed. Indeed, according to the coherent radiation theory, in the case considered and for an ideal lattice, N = 121/-‘. Such a value of N, however, leads to values which exceed the experimental data by a factor of 2. The difference can be caused by a decrease of the mean scattering angles in the crystal as compared to the values given by the random chain model (see fig. 1) and by crystal imperfections, i.e. lattice defects or impurities. The mean scattering angles were not determined in the experiment considered, therefore the quantity N in eq. (8) can be treated as a phenomenological normalizing parameter. If one has found the value of N comparing the theoretical and experimental data obtained at some definite angle in the frequency range, where the
+ - 10Gd,
si,<1/O)
o (MeV) Fig. 3. The emission spectral density of positrons small angles to the (110) axis in a silicon crystal
(left) and electrons (right) with 20 and 10 GeV energy, respectively, which move at the thickness of which is 100 pm. The dots correspond to experiment [12], the solid lines to theory [lo]. III. CHANNELING,
DECHANNELING
178
N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
multiple scattering does not affect the radiation, then the radiation spectrum can in accordance with eq. (8) be determined for the total frequency range both at this angle value and other values of q, and at other energy values as well. As a result we obtain NeXp= 5.3V’. The radiation spectra corresponding to this value of N are shown with solid lines in fig. 3. The results obtained demonstrate attenuation of the coherent radiation intensity in the low-frequency range, which has been observed experimentally, to be due to the effect on the radiation of multiple scattering of particles by atomic chains. The quantity N in eq. (8) is the ratio of the mean squares of the particle scattering angles in the crystal and in an amorphous medium. Therefore the value of N can be independently determined under the concrete conditions of the experiment from analysis of the angular distribution of particles scattered by the crystal. Realization of the “complete” experiment, in which the radiation spectra as well as the angular distributions of the scattered particles are measured, provides an opportunity not only to determine the experimental value of N but to verify independently the dynamical chaos hypothesis that is the basis of the coherent radiation suppression effect.
5. Emission by relativistic positrons and electrons in a crystalline medium during their regular and chaotic motion Now we consider the radiation by an ultrarelativistic positron beam passing through a crystal along one of the crystallographic axes and show the possibility of obtaining information on the character of the particle motion in the crystal from the spectrum of the radiation emission. To specify the conditions we consider the beam to fall on a silicon crystal along the (111) axis. Fig. 4 shows the equipotential surfaces of the continuous potential energy U(x, y) in the case considered. Firstly let us note that in the case of positron beam incidence on a crystal along its axis, the particles are separated into two groups corresponding to finite and infinite motion in the orthogonal plane. The particles will have finite trajectories in the field shown in fig. 4 if the points of their penetration into the crystal are inside the region confined by the dotted line. The analysis based on the Poincare cross-section technique [l] shows that the positron trajectory in the field under consideration, both in the case of channeling (that is finite motion) and for above-barrier motion, can be regular as well as chaotic. We want to find out how the behaviour of particle motion affects the emission of radiation. To simplify the problem we consider the case in which the emission can be treated in the dipole approximation of classic electrodynamics.
Fig. 4. Equipotential contours of the continuous potential energy of interaction of positrons with a silicon crystal, for particles moving along the (111) axis. Energy values are in eV.
The spectral energy density emitted by a separate particle in a crystal in the case of present interest is given by the expression [9]
dr@(t)
W(q)=/_mw
ei@,
where S = m2w/2 E2 and p(t) is the positron trajectory in the transverse plane. If the motion in the transverse plane is regular and periodic with period T, then eq. (9) can be written as ~?;=Nr;;*+[l-2%(1-i)] 2
X
=dtji(t)
eig’
,
where Nr is the number of oscillations and g = 2an/T, n=l,2, . . . . The formula shows that the emission spectrum of a separate particle which moves regularly has a sharp maximum the position of which is given by the condition 6 = 2a/T. At first we consider emission of radiation during above-barrier motion of positrons in a crystal. In this case, the positrons collide successively with different chains of atoms in the crystal. Here the collisions with different atomic chains can be treated as random if the motion in the transverse plane is chaotic. The mean value of the spectral density of the emission can be related to the emission spectrum of the positron, B’(b), in the field of a separate atomic chain as follows [13] (&;)=Lndll,jm
dbd=“(b). -CC
(II)
N. F. Shul’ga et al. / Dynamical chaos in parsage through crystals
Here L is the target thickness and b is the impact parameter of the chain. The formula demonstrates that the quantity (8:) is a smooth function of frequency in the case of chaotic motion of an above-barrier particle. The analysis of the Poincart cross sections has shown that along with chaotic motion there can be a regular quasi-periodic motion of an above-barrier positron in the transverse plane even if E, - U, (U, = 4Ze2/d is the characteristic value of the potential energy of interaction between the electron and the chain, Z ] e 1 is the atom charge). The latter takes place, e.g., if the particle moves in the crystal along a crystallographic plane. In this case the emission spectrum is determined by eq. (10). The computer calculations have shown that the periods of the regular motion of positrons along crystallographic planes depend weakly on the initial conditions. Thus, the plane potential in which the positrons move is nearly parabolic, in spite of the fact that the positron collides with a small number (- 5) of atomic chains in the crystal during the period of one oscillation. This is why the positions of the maxima of the emission spectrum for all the particles moving regularly will coincide and, therefore, even a small part of the planar channeling particle can give a noticeable contribution to the emission in the range of the maximum. In particular, regular motion of the particle can occur along the (110) and (112) planes if the positrons move in a silicon crystal near the (111) axis at E, - U,. In this case, in accordance with eq. (lo), there will be maxima in the emission spectrum at the frequencies
179
If the period of the oscillations of particles like this is of the order of T, (T, - 0.3d( E/2E,)‘/‘, E, < UC, UC 1 eV is the value of the potential in the saddle point) then the maximum of the radiation spectral density is in the frequency range w - 4nE2/m2TC. The spectral density of radiation is a smooth function of the frequency w for positrons moving chaotically during the channeling just as in the case of abovebarrier motion. Thus, the presence of particles in a beam that move regularly and quasi-periodically in the transverse plane gives rise to maxima in the emission spectral density. Therefore, analysis of the emission spectrum of positrons in a crystal provides information on the character of the particle motion in the crystal. As an example we analyse results of an experiment on positron passage through a silicon crystal along the (111) axis at E = 1050 MeV [15]. Two maxima, at wi = 1.8 MeV and o2 = 2.8 MeV, were obtained in the emission spectrum. Theoretical estimations based on the analysis given above have shown that the frequency WY’)= 2.2 MeV corresponds to the maximum in the emission spectrum caused by regular motion during axial channeling while w(;~) = 3 MeV corresponds to regular motion along the (110) plane. Thus, the splitting of the maximum in the emission spectrum of positrons that was observed in [15] is caused by regular motion of some particles of the beam in the crystal.
References wi=4
2@‘),lli,
p (
1
where u, is the width of the corresponding planar channel and flyi) is the corresponding critical angle for planar channeling. The calculation has shown that the phase volume corresponding to the planar channeling along the (112) plane is much smaller than the one for the (110) plane. Thus, in the experimental emission spectrum the most distinguished frequency is the one that relates to the electron motion along the (110) plane. We consider now the emission during axial charmeling (proper channeling) of positrons in a crystal. In the field shown in fig. 4 the channeled positrons perform finite motion in the region confined by the dotted line. The potential of the channel in which the particles move is similar to the Henon-Heiles one [4,5]. Motion of a particle in a field like this can be regular or chaotic. The main contribution to the radiation is given by the particles with the maximum amplitude of oscillation. In this case regular motion is performed by the particles moving near the planes in which the channel axis and the nearest atomic chain lie (see figs. lo-13 in ref. [14]).
[l] Yu. L. Bolotin, V. Yu. Gonchar, V.I. Truten’ and N.F. Shul’ga, Ukr. Fiz. Zh. 31 (1986) 14 (in Russian); Dokl. Akad. Nauk SSSR 296 (1987) 1104 (in Russian); Phys. Lett. 123 (1987) 357. [2] N.F. Shul’ga, in: Problemy Tbeoreticheskoy Fiziki. Sb. Nauchn. Trudov (Naukova Dumka, Kiev, 1986) p. 298 (in Russian). [3] AI. Akhiezer and N.F. Shul’ga, Sex. Phys. Usp. 30 (1987) p. 197 [4] G.M. Zaslavsky, in: Stokhasticbnost’ Dinamicheskikh System (Nauka, Moscow, 1984) p. 217 (in Russian). [5] A.J. Lichtenberg and M.A. Lieberman, in: Regular and Stochastic Motion (Springer, Berlin, 1983). [6] I. Lmdhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (1965) no. 14. [7] M.A. Kumakhov and G. Shirmer, in: Atomnye StoIknovenia v Kristallakh (Atomizdat, 1980) p. 192 (in Russian). [8] M.L. Ter-Mikhaelyan, in: High-energy Electromagnetic Processes in Condensed Media (Wiley, New York, 1972). [9] AI. Akhiezer and N.F. Shul’ga, Sov. Phys. Usp. 25 (1982) 541. [lo] N.V. La&in, A.S. Mazmanishvih and N.F. Shul’ga, Dokl. Akad. Nauk SSSR 277 (1984) 850; Zh. Eksp. Teor. Fiz. 88 (1985) p. 763 (in Russian); Phys. Lett. 112A (1985) 240. III. CHANNELING.
DECHANNELING
180
N.l? Shurga et al. / Dynamical chaos in passage through crystals
[ll] L.D. Landau and I.Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92 (1953) 735 (in Russian). [12] J.F. Bak et al., Preprint CERN-EP/87-87 (May 1987); Nucl. Phys. 302B (1988) 525. [13] N.F. Shui’ga, V.I. Truten’ and S.P. Fomin, Sov. Phys. JETP 60 (1984) 145.
[14] T.P. Grozdanov et al., Phys. Rev. A33 (1986) 55. [15] D.I. Adejshvili et al., Ukr. Fiz. Zh. 31 (1986) 1460 (in Russian).
174
DYNAMICAL N.F. SHUL’GA,
CHAOS IN Tm N.V. LASKIN
and Methods
MOTION OF FAST CHARGED
in Physics Research
PARTICLES
B48 (1990) 174-180 North-Holland
IN CRYSTALS
and V.I. TRUTEN
Institute of Physics and Technology of the Ukrainian Acad. Sciences, 310108, Kharkov.
USSR
Rest&s of an investigation of the dynamical chaos phenomenon in the case of passage of fast charged particles through a crystat along one of the crystal axes are presented. Here we discuss the conditions for dynamical chaos arising in axial channeling and above-barrier motion of relativistic electrons and positrons in a crystal and some consequences of this in some of the physical processes that accompany the passage of the particle through the crystal, such as coherent radiation, multiple scattering by atomic chains, the fine structure of the radiation spectrum, the effect of thermal oscillations of atoms on the character of particle motion, and the effect of multiple scattering on the coherent radiation by fast particles. The results of this theory are compared with results of an experiment carried out at CERN on the effect of multiple scattering on the coherent radiation of ultrarelativistic electrons and positrons. The experiment is shown to confirm the main predictions of this theory.
1. Possibility of regular and chaotic motion of a fast charged particle in the field of a crystal atom chain Since a crystal has a periodic structure, it is natural to assume that any partic!e motion in this structure would be regular, quasi-periodic. In practice, however, this assumption is not valid and along with the regular motion there can occur a chaotic motion of the particle [l-3]. The essence of the matter is that even if the particle moves in the most simple fields dependent on two coordinates, an essentially irregular, chaotic motion of the particle is possible along with the regular one [4,5]. The character of the motion depends on the number of the motion integrals. Thus, for example, any finite motion of the particle will be quasi-periodic in a system with two degrees of freedom which has two motion integrals. If, however, the system has only one integral, then the motion may be stochastic. This is the case for a fast charged particle that moves in a crystal at a small angle to one of the crystal axes. Indeed, in this case the motion of the particle is mainly determined by the continuous potential U(x, y) of the crystal atom chains, i.e., by the lattice potential averaged over the coordinate z parallel to the crystal axis along which the particle moves [6]. In a field like this the particle momentum component parallel to the chain axis is conserved. The motion in the orthogonal plane is given by
pt=
-g/J(p),
(1)
where p = (x. y) and E is the particle energy. Note that a system of units with h = c = 1 is applied. Thus, we come to the problem of the two-dimensional motion of a particle in the periodic field U(x, y).
The potential energy U(x, y) has deep minima (for negatively charged particles, for example electrons) or considerable maxima (for positrons) at the coordinate values of x and y that determine the atomic-chain positions in the orthogonal plane, and it has saddle points lying on the straight lines that connect the neighbouring chains. The character of the particle motion in a field like this is determined by the number of motion integrals of eq. (1). One of the motion integrals is well known. It is the energy integral of the transverse motion [6], E,
= ;Erj2 + U(p),
lEll
KE.
(2)
Depending on the value of E, , the particle motion in the crystal can be both finite (channeling or, for positive particles, proper channeling) and infinite (above-barrier motion) with respect to the atomic chains. As regards the second motion integral, its existence is not obligatory in the problem under consideration, this conclusion holding true both for channeled and above-barrier particles [l]. In this report some results of an investigation of dynamical chaos phenomena during the passage of relativistic electrons and positrons through a crystal along one of the crystal axes are presented. Primary attention is paid to the condition of the dynamical chaos phenomenon arising when the motion of the particles in the crystal is above-barrier and to some consequences of the phenomenon for physical processes that accompany the passage of the particles through the crystal - such as multiple scattering by atomic chains, coherent radiation, influence of multiple scattering on the coherent radiation, and the fine structure of the radiation spectrum.
175
N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
2. Scattering of fast particles by crystal atom chains A fast particle moving at a small angle J/ to the axis, but with transverse energy above the barrier between channels, successively collides with different crystal atom chains parallel to the z-axis. The collisions may or may not be correlated. The correlations take place, for example, in the case when J, >> J/, ($, is the critical angle for axial channeling) [6] and when in addition the angle a (a * $) between the incident beam and one of the close-packed lattice planes is small. In this case the relative variation of the impact parameter between collisions of the particle with nei~bou~ng atomic chains is small so that the particle trajectory will be changed smoothly with penetration depth. Besides, the trajectory will mainly be determined by the continuous potential U,(x) of the plane, to which the motion is nearly parallel,
.t=
-$$_r,(x),
@=2@
af w 4 3Z
z)-f(&
z)],
(4)
where Q, is the azimuthal angle of particle scattering by atomic chains in the orthogonal plane, f ( I&,z) is the particle-dist~bution function in the angle cp at the depth z, #(b) is a function of the particle deflection in the field of an atomic chain, n is the density of atoms and d is the atomic spacing along the axis z. The total angle of particle scattering 8, is related to + as 13= 211, sin ($1/2).
( 1-exp
[ - 2ndr$/_I
db sin2( ?)I).
(5) If, in addition, \I/ * J/,, we have
iP=
where x is the coordinate orthogonal to the plane. Thus, we have come across a one-dimensional problem of particle motion in the field U,(x). It is clear the motion in a field like this will be a regular quasi-periodic one (in the case under consideration the transverse-motion energy relative to the axis and to the plane will both be conserved). In the chaotic particle motion in the (x, v) plane the impact parameter varies strongly between successive collisions of the particle, thus collisions with different atomic chains can be treated as a random process. In this case one can consider multiple scattering of the particle by atomic chains. But in contrast to an amorphous medium, in the case under consideration multiple scattering occurs not on separate atoms but on clusters of atoms, i.e. crystal atom chains parallel to the crystal axis. To treat this process one usually uses the random chain model in which the process of the particle scattering in the periodic field of the atomic chains is assumed to be replaced by a process of scattering by the atomic chains located irregularly but parallel to each other [3,6,7]. In this case the multiple scattering can be considered using the kinetic equation
=nd$f” db[f(+++(b), --co
Note, however, that the statistics of random events (that is collisions with atomic chains) during the particle passage through the crystal can be different. Thus, it is important to know to what degree the above mentioned simplest model of multiple scattering is valid. With that end in view the values of the mean squares of the multiple-scattering angle of above-barrier electrons and positrons, obtained by computer simulation of particle passage through a crystal, were compared with results which were obtained using a theory based on the random chain model. As is generally known [3], in the random chain model we have
z/_“,
dh[-&/_m_
dyU(b,
Y)]2,
(6)
The computer simulations of the mean square value of the multiple-sitters angle in the periodic field of atomic chains was based on the formula p(z)
= $
,E 44’ I=1
sin2( 2),
(7)
where N is the number of particles in the beam and (pi is the azimuthal scattering angle of particle i at the depth z which is obtained bynumerical solution of eq. (1) with different initial conditions (that is, impact parameters). As an example we consider scattering of electrons and positrons which move with an energy of 30 GeV in a silicon crystal of 50 Pm thickness at a small angle to the (111) axis. The results of the calculations are given in fig. 1. The solid curve relates to the theory based on the random chain model. The results of computer simulation are given by the symbols A and 0 (the symbols relate to different orientations of the n- and y-axes against the beam). c is the mean square of the multiple-scattering angle in an amorphous medium. The results obtained demonstrate that for (c,5 \I/,the results of the theory based on the random chain model and computer calculation are the same. For $J > +, the results of the simulation are smaller than the corresponding results of the theory. The difference increases with 4, being greater for positrons than for electrons. As an analysis of Poincare cross sections demonstrates [l], this result can be explained by the fact that the regions of phase space that correspond to the regular motion expand as 4 increases. They expand considerably more for positrons than for electrons and affect the particle multiple scattering in the crystal. Indeed, if there are regions in phase space that correspond to III. CHANNELING,
DECHANNELING
N. F. Shui’ga et al. / Dynamical chaos in passage through crystals
176 e-
5, (444)
Fig. 1. Dependence of the mean scattering angles of electrons (a) and positrons (b) on orientation in a silicon crystal.
regular motion, then the particle moving chaotically cannot get into these regions. Thus, the phase space regions of regular motion have an effect on the multiple scattering of particles in a crystal, the influence increasing as the mentioned regions expand and the time during which the particle is near the regions becomes longer. The influence of thermal oscillations of atoms and the electron subsystem of the lattice on the scattering was not included in the results deduced above. Note that these factors have not only to give rise to multiple scattering on fluctuations of the lattice potential but also to decrease the phase space regions for regular motion, which in turn will increase the multiple scattering on atomic chains.
an even multiple of the radiation formation length I, = 2E( E - o)/m2u (the coherence length) multiplied by 27. If the particle motion in the transverse plane is chaotic, then its collisions with atomic chains can be considered as random. In this case the spectrum and polarization of the radiation are found to be smooth functions of the frequency (see fig. 2), the radiation being determined only by peculiarities of the particle interaction with the field of a separate atomic chain in the frequency region for which the length 1, is small compared to the mean free path between successive collisions with chains. In this case the peculiar behaviour of the particle motion in the field of a chain affects its radiation. Thus, correlations between electron collisions with atoms in a chain increase the mean value of the angle of its scattering in the crystal as compared to the scattering angles in an amorphous medium, and this in turn intensifies the electron emission in the range of low frequencies as compared to emission in an amorphous medium. But the asymmetry of the electron scattering in the field of an atomic chain (the scattering takes place along the azimuthal angle cp) gives rise to linear polarization of the radiation. In particular, it is easy to verify that the polarization degree is 50% in the range of frequencies satisfying the condition I, > 2 R/J, (R is the screening radius of the atom potential). It is worth noting an analogous result can be obtained if one uses the exact formulas of the theory of coherent radiation in which all simplifications connected with dynamical chaos are avoided. However, it requires tiresome computer calculations.
I
3. Coherent radiation in the case of regular and chaotic motions of fast particles in a crystal In the theory of coherent radiation by ultrarelativistic electrons one usually considers the case of the electron motion at a small angle 4 (but 4 S- JI,) to one of the crystal axes and, besides, the angle a (a =Z 4) between the particle momentum and a crystallographic plane is small too [8,9]. In this case the electron successively collides with atomic chains located in the plane, the collisions being regular. The regularity of the electron interactions with the atomic chains gives rise to sharp maxima in the spectral density and the radiation polarization (fig. 2). The positions of the maxima correspond to the frequencies w, for which the particle free path length I = a/# between collisions with the neighbouring chains (a is the chain spacing) is equal to
Fig. 2. The emission spectrum of electrons with an energy of 1 GeV which move at the angle J, = 2 mrad to the (100) axis [9]. The solid curve corresponds to motion in the vicinity of the (100) plane; the dash-dotted line correspond to chaotic motion in the plane orthogonal to the (100) axis; the dotted line is the emission spectrum in an amorphous medium.
N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
Thus, under dynamical chaos conditions the formulas that determine the electron radiation in a crystal are greatly simplified. In this case, one can find a relation between the quantities that determine the scattering of and the radiation from a fast particle in a crystal.
4. Effect of multiple scattering on the coherent radiation The coherence length increases sharply as the particle energy grows and the frequency of the emitted photon gets lower. If the length exceeds the free path length of the particle between successive collisions with chains in a crystal and if the motion in the transverse plane is chaotic, then one has to take into account the effect of multiple scattering on the radiation. This problem has been solved in ref. [lo] where a path integration method is used. In particular, using the random chain model, the effect of multiple scattering on the coherent radiation spectrum is shown to be determined by the formula
where J’Ln is the electron radiation spectrum in an amorphous medium in the range of low frequencies, N is the ratio of the mean square angle values of particle scattering in the crystal and the amorphous medium and a(x) is a function that determines the multiple scattering effect on the coherent radiation (see eq. (6.4) in ref. [3]). The quantity w, is determined by the expression oF = 2 No,,, where wLP is the frequency at which the multiple scattering affects the bremsstrahlung in an amorphous medium [8,11]. For a wide range of angles E+=ZOGd,
St,
111
+(R/d
B # 2 #,) the quantity N is large compared to unity (N - R/4d#, see also fig. I), which is why multiple scattering should affect the radiation in a crystal more than in an amorphous medium. Lately the effect under consideration has been investigated at CERN for the passage of electrons and positrons with 10 and 20 GeV energy through a silicon crystal of 100 urn thickness in the vicinity of a (110) axis. Fig. 3 shows the results of the experiment relating to the angular range # 2 #, where comparison of theoretical and experimental data is possible. The results obtained in ref. [12] confirm the main predictions of the coherent radiation theory based on the random chain mode: for a wide range of frequencies o, the spectral density of radiation in the crystal greatly exceeds the spectral density of radiation in an amorphous medium and depends weakly on w; but at low frequencies, w 5 200 MeV, the intensity of coherent radiation decreases. However, in order to make a quantitative comparison between theory and experiment some additional data are needed. Indeed, according to the coherent radiation theory, in the case considered and for an ideal lattice, N = 121/-‘. Such a value of N, however, leads to values which exceed the experimental data by a factor of 2. The difference can be caused by a decrease of the mean scattering angles in the crystal as compared to the values given by the random chain model (see fig. 1) and by crystal imperfections, i.e. lattice defects or impurities. The mean scattering angles were not determined in the experiment considered, therefore the quantity N in eq. (8) can be treated as a phenomenological normalizing parameter. If one has found the value of N comparing the theoretical and experimental data obtained at some definite angle in the frequency range, where the
+ - 10Gd,
si,<1/O)
o (MeV) Fig. 3. The emission spectral density of positrons small angles to the (110) axis in a silicon crystal
(left) and electrons (right) with 20 and 10 GeV energy, respectively, which move at the thickness of which is 100 pm. The dots correspond to experiment [12], the solid lines to theory [lo]. III. CHANNELING,
DECHANNELING
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N. F. Shul’ga et al. / Dynamical chaos in passage through crystals
multiple scattering does not affect the radiation, then the radiation spectrum can in accordance with eq. (8) be determined for the total frequency range both at this angle value and other values of q, and at other energy values as well. As a result we obtain NeXp= 5.3V’. The radiation spectra corresponding to this value of N are shown with solid lines in fig. 3. The results obtained demonstrate attenuation of the coherent radiation intensity in the low-frequency range, which has been observed experimentally, to be due to the effect on the radiation of multiple scattering of particles by atomic chains. The quantity N in eq. (8) is the ratio of the mean squares of the particle scattering angles in the crystal and in an amorphous medium. Therefore the value of N can be independently determined under the concrete conditions of the experiment from analysis of the angular distribution of particles scattered by the crystal. Realization of the “complete” experiment, in which the radiation spectra as well as the angular distributions of the scattered particles are measured, provides an opportunity not only to determine the experimental value of N but to verify independently the dynamical chaos hypothesis that is the basis of the coherent radiation suppression effect.
5. Emission by relativistic positrons and electrons in a crystalline medium during their regular and chaotic motion Now we consider the radiation by an ultrarelativistic positron beam passing through a crystal along one of the crystallographic axes and show the possibility of obtaining information on the character of the particle motion in the crystal from the spectrum of the radiation emission. To specify the conditions we consider the beam to fall on a silicon crystal along the (111) axis. Fig. 4 shows the equipotential surfaces of the continuous potential energy U(x, y) in the case considered. Firstly let us note that in the case of positron beam incidence on a crystal along its axis, the particles are separated into two groups corresponding to finite and infinite motion in the orthogonal plane. The particles will have finite trajectories in the field shown in fig. 4 if the points of their penetration into the crystal are inside the region confined by the dotted line. The analysis based on the Poincare cross-section technique [l] shows that the positron trajectory in the field under consideration, both in the case of channeling (that is finite motion) and for above-barrier motion, can be regular as well as chaotic. We want to find out how the behaviour of particle motion affects the emission of radiation. To simplify the problem we consider the case in which the emission can be treated in the dipole approximation of classic electrodynamics.
Fig. 4. Equipotential contours of the continuous potential energy of interaction of positrons with a silicon crystal, for particles moving along the (111) axis. Energy values are in eV.
The spectral energy density emitted by a separate particle in a crystal in the case of present interest is given by the expression [9]
dr@(t)
W(q)=/_mw
ei@,
where S = m2w/2 E2 and p(t) is the positron trajectory in the transverse plane. If the motion in the transverse plane is regular and periodic with period T, then eq. (9) can be written as ~?;=Nr;;*+[l-2%(1-i)] 2
X
=dtji(t)
eig’
,
where Nr is the number of oscillations and g = 2an/T, n=l,2, . . . . The formula shows that the emission spectrum of a separate particle which moves regularly has a sharp maximum the position of which is given by the condition 6 = 2a/T. At first we consider emission of radiation during above-barrier motion of positrons in a crystal. In this case, the positrons collide successively with different chains of atoms in the crystal. Here the collisions with different atomic chains can be treated as random if the motion in the transverse plane is chaotic. The mean value of the spectral density of the emission can be related to the emission spectrum of the positron, B’(b), in the field of a separate atomic chain as follows [13] (&;)=Lndll,jm
dbd=“(b). -CC
(II)
N. F. Shul’ga et al. / Dynamical chaos in parsage through crystals
Here L is the target thickness and b is the impact parameter of the chain. The formula demonstrates that the quantity (8:) is a smooth function of frequency in the case of chaotic motion of an above-barrier particle. The analysis of the Poincart cross sections has shown that along with chaotic motion there can be a regular quasi-periodic motion of an above-barrier positron in the transverse plane even if E, - U, (U, = 4Ze2/d is the characteristic value of the potential energy of interaction between the electron and the chain, Z ] e 1 is the atom charge). The latter takes place, e.g., if the particle moves in the crystal along a crystallographic plane. In this case the emission spectrum is determined by eq. (10). The computer calculations have shown that the periods of the regular motion of positrons along crystallographic planes depend weakly on the initial conditions. Thus, the plane potential in which the positrons move is nearly parabolic, in spite of the fact that the positron collides with a small number (- 5) of atomic chains in the crystal during the period of one oscillation. This is why the positions of the maxima of the emission spectrum for all the particles moving regularly will coincide and, therefore, even a small part of the planar channeling particle can give a noticeable contribution to the emission in the range of the maximum. In particular, regular motion of the particle can occur along the (110) and (112) planes if the positrons move in a silicon crystal near the (111) axis at E, - U,. In this case, in accordance with eq. (lo), there will be maxima in the emission spectrum at the frequencies
179
If the period of the oscillations of particles like this is of the order of T, (T, - 0.3d( E/2E,)‘/‘, E, < UC, UC 1 eV is the value of the potential in the saddle point) then the maximum of the radiation spectral density is in the frequency range w - 4nE2/m2TC. The spectral density of radiation is a smooth function of the frequency w for positrons moving chaotically during the channeling just as in the case of abovebarrier motion. Thus, the presence of particles in a beam that move regularly and quasi-periodically in the transverse plane gives rise to maxima in the emission spectral density. Therefore, analysis of the emission spectrum of positrons in a crystal provides information on the character of the particle motion in the crystal. As an example we analyse results of an experiment on positron passage through a silicon crystal along the (111) axis at E = 1050 MeV [15]. Two maxima, at wi = 1.8 MeV and o2 = 2.8 MeV, were obtained in the emission spectrum. Theoretical estimations based on the analysis given above have shown that the frequency WY’)= 2.2 MeV corresponds to the maximum in the emission spectrum caused by regular motion during axial channeling while w(;~) = 3 MeV corresponds to regular motion along the (110) plane. Thus, the splitting of the maximum in the emission spectrum of positrons that was observed in [15] is caused by regular motion of some particles of the beam in the crystal.
References wi=4
2@‘),lli,
p (
1
where u, is the width of the corresponding planar channel and flyi) is the corresponding critical angle for planar channeling. The calculation has shown that the phase volume corresponding to the planar channeling along the (112) plane is much smaller than the one for the (110) plane. Thus, in the experimental emission spectrum the most distinguished frequency is the one that relates to the electron motion along the (110) plane. We consider now the emission during axial charmeling (proper channeling) of positrons in a crystal. In the field shown in fig. 4 the channeled positrons perform finite motion in the region confined by the dotted line. The potential of the channel in which the particles move is similar to the Henon-Heiles one [4,5]. Motion of a particle in a field like this can be regular or chaotic. The main contribution to the radiation is given by the particles with the maximum amplitude of oscillation. In this case regular motion is performed by the particles moving near the planes in which the channel axis and the nearest atomic chain lie (see figs. lo-13 in ref. [14]).
[l] Yu. L. Bolotin, V. Yu. Gonchar, V.I. Truten’ and N.F. Shul’ga, Ukr. Fiz. Zh. 31 (1986) 14 (in Russian); Dokl. Akad. Nauk SSSR 296 (1987) 1104 (in Russian); Phys. Lett. 123 (1987) 357. [2] N.F. Shul’ga, in: Problemy Tbeoreticheskoy Fiziki. Sb. Nauchn. Trudov (Naukova Dumka, Kiev, 1986) p. 298 (in Russian). [3] AI. Akhiezer and N.F. Shul’ga, Sex. Phys. Usp. 30 (1987) p. 197 [4] G.M. Zaslavsky, in: Stokhasticbnost’ Dinamicheskikh System (Nauka, Moscow, 1984) p. 217 (in Russian). [5] A.J. Lichtenberg and M.A. Lieberman, in: Regular and Stochastic Motion (Springer, Berlin, 1983). [6] I. Lmdhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (1965) no. 14. [7] M.A. Kumakhov and G. Shirmer, in: Atomnye StoIknovenia v Kristallakh (Atomizdat, 1980) p. 192 (in Russian). [8] M.L. Ter-Mikhaelyan, in: High-energy Electromagnetic Processes in Condensed Media (Wiley, New York, 1972). [9] AI. Akhiezer and N.F. Shul’ga, Sov. Phys. Usp. 25 (1982) 541. [lo] N.V. La&in, A.S. Mazmanishvih and N.F. Shul’ga, Dokl. Akad. Nauk SSSR 277 (1984) 850; Zh. Eksp. Teor. Fiz. 88 (1985) p. 763 (in Russian); Phys. Lett. 112A (1985) 240. III. CHANNELING.
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N.l? Shurga et al. / Dynamical chaos in passage through crystals
[ll] L.D. Landau and I.Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92 (1953) 735 (in Russian). [12] J.F. Bak et al., Preprint CERN-EP/87-87 (May 1987); Nucl. Phys. 302B (1988) 525. [13] N.F. Shui’ga, V.I. Truten’ and S.P. Fomin, Sov. Phys. JETP 60 (1984) 145.
[14] T.P. Grozdanov et al., Phys. Rev. A33 (1986) 55. [15] D.I. Adejshvili et al., Ukr. Fiz. Zh. 31 (1986) 1460 (in Russian).