- Email: [email protected]
Quantum chaos in channeling
Nuclear Instruments and Methods in Physics Research B 1I5 (1996) 328-33
1
NOMB
Beam Interactions with Materials 6 Atoms
EISEVlER
Quantum chaos in channeling V.A. Khodyrev
*
Institute of Nuclear Physics, Moscow State University. Moscow 119899, Russia
Abstract The quantum eigen states of transverse motion in axial channel are studied. The wave functions have been founded using a proposed numerical relaxation procedure. The evolution of particle wave function with depth is calculated and compared with the evolution of classical particle flux. The quantum state localization in momentum space results in a difference of the two results. The correspondence with experiment is discussed.
1. Introduction Quantum systems that are chaotic in their classical limit has been the subject of intense studies in recent years (see book by Gutzwiller [l] for review). The central question here is a quantum-classical correspondence in the description of such systems. The quasiclassical quantization method is applicable only for integrable systems or for those motions of semichaotic systems when the trajectory of motion lies on an invariant surface in phase space. Many aspects of this problem has been discovered during the last years. The following fact, which is considered now as having a general meaning, must be noted in the context of the present paper. Computer analysis of some models [ 1] showed that classical diffusion in chaotic systems resulting from a local instability of trajectories can be strongly suppressed by quantum effects even in the cases where the common criteria of applicability of classical description seem to be satisfied with confidence. The effect of quantum suppression of the classical ‘deterministic’ diffusion has been demonstrated in the experiments with the ionization of Rydberg hydrogen atoms in microwave field [2]. It has been shown [3] that the origin of quantum states localization in phase space responsible for this effect is analogous to the origin of Anderson localization of electron wave functions in the configuration space for amorphous solids [4]. It has been shown [5-71 that the translation symmetry of a transverse potential for axial channeling results in a co-existence of regular and chaotic trajectories [8] of transverse motion what modifies noticeably the statistical description of axial channeling based on the microcanonical distribution [9]. This type of symmetry gives a peculiar
* Fax: + 7-95-93908%; 0168-583X/96/$15.00 SSDI 0168-583X(95)01
e-mail: [email protected].
opportunity to study the quantum chaos, it is a reason why the models with this symmetry are often used in the theoretical analysis. In this case, the set of quantum stationary states, the Bloch waves, is well determined. One can expect that at least some of these waves generate a density distribution different from the classical microcanonical one (with the resonant structure in the classical phase space [8] taken into account>. Some observations of the early channeling studies stimulated the discussion of the manifestation of quantum effects in ions channeling [IO]. The discussion has not resulted in a defined conclusion that time. It is clear now that the fact that a large number of diffracted waves is involved results in qualitatively new phenomena, so the concepts of the theory of electron diffraction used at that time seem nonadequate. In this paper, wave functions for a simple models of particle channeling are calculated. In Section 2 eigen states for two-dimensional transverse motion between rows of atomic strings are discussed. Section 3 presents a comparison of classical and quantum descriptions of particle penetration through thin crystals.
2. Planar channeling Two-dimensional transverse motion with relatively high velocity along the most open planar channel in the potential with simple rectangular symmetry can be roughly modeled using the following Hamiltonian:
where m is the particle mass, x and p are the coordinate and momentum perpendicular to the planar channel with the width d. A constant transverse velocity along the plane is assumed, therefore the collisions with the strings occur with equal time intervals T. The impulse (eikonal) approx-
Copyright 0 1996 Elsevier Science B.V. All rights reserved 452-7
V.A. Khodyreu / Nucl. Insrr. und Meth. in Phys. Res. B I15 (1996) 328-331
K=0.150
329
and u = V,T/h. Divided by 27r, parameter p. gives the number of quantum states corresponding to the phase volume of the Poincar6 surface of section. The classical parameter K = 2 ITY/F. Due to the double periodicity of the Hamiltonian Eq. (1) the basis of the states can be formed by an eigen states of quasi-momentum k and quasi-energy E (k, E = 0 + 1):
q 0.0
-0.5
where X = (k, E), the coefficients U:(T) are periodical. The eigenstates qA can be represented by the values of a?“‘) = a;(~,,) at time moments just before the kicks, T, = J n - 0. Using these variables we can write the propagator on one time period 7’:
K=0.349
(*a( &n+
J
-0.5
0.0
0.5 C
Fig. 1. PoincarC values of K.
surface
of section
for ‘kicked
rotator’
at two
imation for the scattering is used, whose intensity is determined by the value of V,,. This periodically driven system, named ‘kicked rotator’, is often considered in studies of classical and quantum chaos and represents the general properties of chaotic dynamics [8]. This model was considered also in the discussion of classical chaos manifestation in channeling [7]. In classical description it is convenient, using reduced variables q = (T/md)p, 5 = x/d and T = r/T, to consider the map of the values of these variables after one period T:
q,, , =
4,
sin(2n5,).
-K
PI
5n+, =L+cl”+I~ which is determined K=27r-
Pa)
by the only parameter
V,T* md*
’
For two values of K, Fig. I shows Poincart? surface of section [8] received by successive iteration of the map Eq. (2) for several initial conditions. The cross sections are periodical over q. At K = 0.150, the chaos is beginning to develop near the separatrix of the strongest resonance. At K = 0.349 we have the developed chaotic component resulting particularly in unlimited diffusion in the q direction. When written using the variables 5 and T the Schrijdinger equation for this system contains two parameters inversely proportional to the Plank constant: p, = md2/hT
1)
j-
k))”
=e ~2irr~e-lZpCi~e~lVJ,(~)a~I;),
I
(5)
where J, are the Bessel functions. A following relaxation method was used for the numerical finding of eigen vectors A, = {a;) and corresponding values of E from Eq. (5). We can write Eq. (5) as A?+‘) = e~“UA~), where U is the unitary matrix. Supposing at first that the corresponding value of E is known we can filter an arbitrary self vector A, from some initial vector A’“” successively acting on it by a matrix (I + eC”U)/2. This procedure does not change the weight of the finding vector in the iterated one and diminishes the weight of other eigen vector by a factor cos(Ae), where AC is the difference between quasi-energy values. The effectiveness of suppression of eigen vectors with close values of E can be gained if we will use the propagator on several periods of kicks. The actual value of E can be determined simultaneously if we will adjust it so that the module of iterated vectors will decrease minimally. After the subtraction of the projection of the initial vector Ainit on the found one the filtering is repeated in order to find the vector of the next quasi-stationary state. Using this procedure, the wave functions of eigen states of Hamiltonian Eq. (1) were determined for two sets of parameters: p. = 900, u = 21.5 and k = 590, v = 32.5. The corresponding K values are 0.15 and 0.349 as in Fig. 1. The value of quasi-momentum k was taken equal to 0.7. The order of parameters of the second set represents the conditions of axial-planar channeling transition of hundreds of keV protons in a crystal like Si. The specific representation of the set of momentum distributions 1a:I’, the positions in coordinates; meanvalue-width of these distributions, is shown in Fig. 2. For K = 0.15 the series of states with equal mean momentum are clearly seen which correspond to the confinement of trajectory in one of the most open channels (the resonant regions in Fig. I). The sequence of states with changing values of (j) corresponds to underbarrier motion. It is clear that the states corresponding to the chaotic separatrix
IV. CHANNELING
VA. Khodyrev /Nucl.
Instr. and Meth. in Phys. Res. B 11s 11996) 328-331
son with the classical description. This representation can be obtained by the transformation of wave function: Xk(X, P) =1(x, PlW12,
(6)
where 1x, p > is a coherent state centered on x and p in coordinate and momentum: (X’I x, P> 1
=
[-I [ ml*
0
Mean
dk~entum j
100
Fig. 2. Distribution of pairs of mean-value-width of momentum distribution for the eigen states of Hamiltonian Eq. (1) for two sets of parameters.
layer (Fig. 1) are positioned on this diagram at the top of the series with ( j) = 0. One can also see a gap in ( j) between these states and the first underbarrier state. The gap origin is a partition of momentum distribution when the separatrix layer is crossed (Fig. 1). The falling to pieces momentum distribution for separatrix layer may be associated with the instability of classical trajectory. At K = 0.349, there are only a few states of channeling, the distribution of points in Fig. 2 is disordered significantly. For several states we present here in Fig. 3 the Husimi representation in phase space allowing the direct compari-
Fig. 3. Husimi transformation pointed in Fig. 2.
of wave functions
for the states
l/4
exp -
(XI-x)*
202
.
+ip(x’-x)
(7)
I
The dispersion u determines the relation between coordinate and momentum resolutions and may be adjusted for obtaining the clear phase portrait. Fig. 3 presents this portrait for the states noted in Fig. 2. States 1A, 3A and 1B resemble the classical state of regular motion. Thus, there is the classical-quantum correspondence in the existence of channeling states which are not destroyed by the ‘discreteness’ of the channel walls. The state 2A corresponds to the separatrix layer (Fig. l), its chaotic nature is noticeably exhibited in the distribution of 1V(x, t)l* (not presented here). The state 2B is displayed in Fig. 2 as one of randomly distributed points and may be associated with the classical chaotic motion, however the Husimi distribution is not uniform, it is unique for each such state. The phenomenon of scarring of the wave function localized on the isolated unstable classical trajectory [ 1 I] is presented by the state 3B. This trajectory goes through the potential maximums in the direction (12) and its stability in quantum description is one of the remarkable features of quantum chaos. A general violation of the classical-quantum correspondence consists in the localization of quantum states in momentum space. Fortunately, this circumstance ensures success of the numerical method, because, in any case, the matrix U needs to be restricted.
3. Axial channeling The superposition of large number of Bloch waves with strongly complicated structure is effective for classical motion in the random direction in a three-dimensional periodic potential. Therefore, the effects of specific structure of quantum states, if they exist, can be detected more probably in axial channeling. In this connection, it is symptomatic that the discussion of quantum effects [lo] has been stimulated by the observation of diffraction-like effects in the case of ion beam direction near the axial channels. The alternative to the Kikuchi diffraction may be the following explanation of these effects. If the momentum for some Bloch states is widely distributed the filling of these states after the penetration of plane wave through crystal surface can result in the appearance of side waves in the direction far from the initial direction. The wide
V.A. Khdyrev/Nucl.
Instr. and Meth. in Phys. Res. B 11.5 (1996) 328-331
331
momentum distribution (although localized) is a specific detail of the chaotic states. It seems, however, that such type of effects is of second order, the first-order one being the quantum suppression of classical diffusion due to the localization of quantum states [12]. In order to receive a representation about the scale of possible effects, we have calculated the successive diffraction of initial plane wave on the crystal atomic layers. The same model with the S-periodic in time but with two-dimensional potential was used. The form of the potential in transverse plane was taken as thermally smoothed Lindhard potential [9]. The calculation was carried out for a simple cubic structure of a crystal with Z, = 14 and particles with unit charge and a mass equal to half of the proton mass. The last choice is forced by the circumstance that even in this case about 500 X 500 reciprocal lattice vectors are needed to be accounted. The determination of eigen states wave functions as in the previous section demands a very large volume of computer calculations. However, we can come to some conclusions comparing classical and quantum description of particle penetration through a crystal in different directions. Fig. 4 shows one of such results for the velocity of a particle, equal to 3.5 times the Bohr velocity, its initial direction of motion
alignment is 30,, where 0, is the Lindhard critical angle [9]>. It can be seen that the classical picture of the development of a ring-like angular distribution is not reproduced in the quantum description for this thick layer: the quantum momentum distribution is localized. It is unlikely that this ideal picture can claim to be a reliable prediction. The effect of noise, although not well studied, may severely destroy quantum interference [ 131. However, it is well known from the experiments that there is a strong delay in the ring equilibration relative to the classical estimation [9]: the intensity in momentum distribution remains concentrated in the original direction even at large depth [ 141. In conclusion, it seems interesting to analyze the transmission channeling experiments in order to look for the effects of quantum chaos, the suppression of classical diffusion. If this will be the case, channeling, due to the possibility of experimental conditions variation, can be a nice opportunity to study these phenomena.
being in plane (012) near the (100)
(Springer-Verlag. Heidelberg, 1990). J.E. Baytield and P.M. Koch, Phys. Rev. Lett. 33 (1974) 258. D.R. Gempel and R.E. Prange, Phys. Rev. A 29 (1984) 1639. P.W. Anderson, Rev. Mod. Phys. 50 (1978) 191. V.A. Khodyrev, Phys. Lett. A 111 (1987) 67. J.C. Kimball. G. Petschel and N. Cue, Nucl. Instr. and Meth. B 33 (1988) 53. [7] A.I. Akhiezer, V.I. Truten and N.F. Shulga, Phys. Rep. 203 (1991) 289. [8] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, Heidelberg, 1983). [9] I. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (14) (1965). [lo] L.T. Chadderton. Philos. Mag. 18 (1968) 1017. [I l] E.J. Heller, Phys. Rev. Lett. 53 (1984) 1515. [12] G. Casati, B.V. Chirikov, F.M. lzrailev and J. Ford, in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. G. Casati and J. Ford, Lecture Notes in Physics Vol. 90 (Springer, Berlin, 1979) p. 334. [ 131 L. Benet, T.H. Seligman and H.A. Weidenmuller, Phys. Rev. Lett. 71 (1993) 529. [14] D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129.
Classical
axial direction
(mis-
Quantum
Fig. 4. Transverse momentum distribution after particle transition through the indicated number of crystal atomic layers. The results of classical and quantum description are shown.
References [I] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics [2] [3] [4] [5] [6]
IV. CHANNELING
1
NOMB
Beam Interactions with Materials 6 Atoms
EISEVlER
Quantum chaos in channeling V.A. Khodyrev
*
Institute of Nuclear Physics, Moscow State University. Moscow 119899, Russia
Abstract The quantum eigen states of transverse motion in axial channel are studied. The wave functions have been founded using a proposed numerical relaxation procedure. The evolution of particle wave function with depth is calculated and compared with the evolution of classical particle flux. The quantum state localization in momentum space results in a difference of the two results. The correspondence with experiment is discussed.
1. Introduction Quantum systems that are chaotic in their classical limit has been the subject of intense studies in recent years (see book by Gutzwiller [l] for review). The central question here is a quantum-classical correspondence in the description of such systems. The quasiclassical quantization method is applicable only for integrable systems or for those motions of semichaotic systems when the trajectory of motion lies on an invariant surface in phase space. Many aspects of this problem has been discovered during the last years. The following fact, which is considered now as having a general meaning, must be noted in the context of the present paper. Computer analysis of some models [ 1] showed that classical diffusion in chaotic systems resulting from a local instability of trajectories can be strongly suppressed by quantum effects even in the cases where the common criteria of applicability of classical description seem to be satisfied with confidence. The effect of quantum suppression of the classical ‘deterministic’ diffusion has been demonstrated in the experiments with the ionization of Rydberg hydrogen atoms in microwave field [2]. It has been shown [3] that the origin of quantum states localization in phase space responsible for this effect is analogous to the origin of Anderson localization of electron wave functions in the configuration space for amorphous solids [4]. It has been shown [5-71 that the translation symmetry of a transverse potential for axial channeling results in a co-existence of regular and chaotic trajectories [8] of transverse motion what modifies noticeably the statistical description of axial channeling based on the microcanonical distribution [9]. This type of symmetry gives a peculiar
* Fax: + 7-95-93908%; 0168-583X/96/$15.00 SSDI 0168-583X(95)01
e-mail: [email protected].
opportunity to study the quantum chaos, it is a reason why the models with this symmetry are often used in the theoretical analysis. In this case, the set of quantum stationary states, the Bloch waves, is well determined. One can expect that at least some of these waves generate a density distribution different from the classical microcanonical one (with the resonant structure in the classical phase space [8] taken into account>. Some observations of the early channeling studies stimulated the discussion of the manifestation of quantum effects in ions channeling [IO]. The discussion has not resulted in a defined conclusion that time. It is clear now that the fact that a large number of diffracted waves is involved results in qualitatively new phenomena, so the concepts of the theory of electron diffraction used at that time seem nonadequate. In this paper, wave functions for a simple models of particle channeling are calculated. In Section 2 eigen states for two-dimensional transverse motion between rows of atomic strings are discussed. Section 3 presents a comparison of classical and quantum descriptions of particle penetration through thin crystals.
2. Planar channeling Two-dimensional transverse motion with relatively high velocity along the most open planar channel in the potential with simple rectangular symmetry can be roughly modeled using the following Hamiltonian:
where m is the particle mass, x and p are the coordinate and momentum perpendicular to the planar channel with the width d. A constant transverse velocity along the plane is assumed, therefore the collisions with the strings occur with equal time intervals T. The impulse (eikonal) approx-
Copyright 0 1996 Elsevier Science B.V. All rights reserved 452-7
V.A. Khodyreu / Nucl. Insrr. und Meth. in Phys. Res. B I15 (1996) 328-331
K=0.150
329
and u = V,T/h. Divided by 27r, parameter p. gives the number of quantum states corresponding to the phase volume of the Poincar6 surface of section. The classical parameter K = 2 ITY/F. Due to the double periodicity of the Hamiltonian Eq. (1) the basis of the states can be formed by an eigen states of quasi-momentum k and quasi-energy E (k, E = 0 + 1):
q 0.0
-0.5
where X = (k, E), the coefficients U:(T) are periodical. The eigenstates qA can be represented by the values of a?“‘) = a;(~,,) at time moments just before the kicks, T, = J n - 0. Using these variables we can write the propagator on one time period 7’:
K=0.349
(*a( &n+
J
-0.5
0.0
0.5 C
Fig. 1. PoincarC values of K.
surface
of section
for ‘kicked
rotator’
at two
imation for the scattering is used, whose intensity is determined by the value of V,,. This periodically driven system, named ‘kicked rotator’, is often considered in studies of classical and quantum chaos and represents the general properties of chaotic dynamics [8]. This model was considered also in the discussion of classical chaos manifestation in channeling [7]. In classical description it is convenient, using reduced variables q = (T/md)p, 5 = x/d and T = r/T, to consider the map of the values of these variables after one period T:
q,, , =
4,
sin(2n5,).
-K
PI
5n+, =L+cl”+I~ which is determined K=27r-
Pa)
by the only parameter
V,T* md*
’
For two values of K, Fig. I shows Poincart? surface of section [8] received by successive iteration of the map Eq. (2) for several initial conditions. The cross sections are periodical over q. At K = 0.150, the chaos is beginning to develop near the separatrix of the strongest resonance. At K = 0.349 we have the developed chaotic component resulting particularly in unlimited diffusion in the q direction. When written using the variables 5 and T the Schrijdinger equation for this system contains two parameters inversely proportional to the Plank constant: p, = md2/hT
1)
j-
k))”
=e ~2irr~e-lZpCi~e~lVJ,(~)a~I;),
I
(5)
where J, are the Bessel functions. A following relaxation method was used for the numerical finding of eigen vectors A, = {a;) and corresponding values of E from Eq. (5). We can write Eq. (5) as A?+‘) = e~“UA~), where U is the unitary matrix. Supposing at first that the corresponding value of E is known we can filter an arbitrary self vector A, from some initial vector A’“” successively acting on it by a matrix (I + eC”U)/2. This procedure does not change the weight of the finding vector in the iterated one and diminishes the weight of other eigen vector by a factor cos(Ae), where AC is the difference between quasi-energy values. The effectiveness of suppression of eigen vectors with close values of E can be gained if we will use the propagator on several periods of kicks. The actual value of E can be determined simultaneously if we will adjust it so that the module of iterated vectors will decrease minimally. After the subtraction of the projection of the initial vector Ainit on the found one the filtering is repeated in order to find the vector of the next quasi-stationary state. Using this procedure, the wave functions of eigen states of Hamiltonian Eq. (1) were determined for two sets of parameters: p. = 900, u = 21.5 and k = 590, v = 32.5. The corresponding K values are 0.15 and 0.349 as in Fig. 1. The value of quasi-momentum k was taken equal to 0.7. The order of parameters of the second set represents the conditions of axial-planar channeling transition of hundreds of keV protons in a crystal like Si. The specific representation of the set of momentum distributions 1a:I’, the positions in coordinates; meanvalue-width of these distributions, is shown in Fig. 2. For K = 0.15 the series of states with equal mean momentum are clearly seen which correspond to the confinement of trajectory in one of the most open channels (the resonant regions in Fig. I). The sequence of states with changing values of (j) corresponds to underbarrier motion. It is clear that the states corresponding to the chaotic separatrix
IV. CHANNELING
VA. Khodyrev /Nucl.
Instr. and Meth. in Phys. Res. B 11s 11996) 328-331
son with the classical description. This representation can be obtained by the transformation of wave function: Xk(X, P) =1(x, PlW12,
(6)
where 1x, p > is a coherent state centered on x and p in coordinate and momentum: (X’I x, P> 1
=
[-I [ ml*
0
Mean
dk~entum j
100
Fig. 2. Distribution of pairs of mean-value-width of momentum distribution for the eigen states of Hamiltonian Eq. (1) for two sets of parameters.
layer (Fig. 1) are positioned on this diagram at the top of the series with ( j) = 0. One can also see a gap in ( j) between these states and the first underbarrier state. The gap origin is a partition of momentum distribution when the separatrix layer is crossed (Fig. 1). The falling to pieces momentum distribution for separatrix layer may be associated with the instability of classical trajectory. At K = 0.349, there are only a few states of channeling, the distribution of points in Fig. 2 is disordered significantly. For several states we present here in Fig. 3 the Husimi representation in phase space allowing the direct compari-
Fig. 3. Husimi transformation pointed in Fig. 2.
of wave functions
for the states
l/4
exp -
(XI-x)*
202
.
+ip(x’-x)
(7)
I
The dispersion u determines the relation between coordinate and momentum resolutions and may be adjusted for obtaining the clear phase portrait. Fig. 3 presents this portrait for the states noted in Fig. 2. States 1A, 3A and 1B resemble the classical state of regular motion. Thus, there is the classical-quantum correspondence in the existence of channeling states which are not destroyed by the ‘discreteness’ of the channel walls. The state 2A corresponds to the separatrix layer (Fig. l), its chaotic nature is noticeably exhibited in the distribution of 1V(x, t)l* (not presented here). The state 2B is displayed in Fig. 2 as one of randomly distributed points and may be associated with the classical chaotic motion, however the Husimi distribution is not uniform, it is unique for each such state. The phenomenon of scarring of the wave function localized on the isolated unstable classical trajectory [ 1 I] is presented by the state 3B. This trajectory goes through the potential maximums in the direction (12) and its stability in quantum description is one of the remarkable features of quantum chaos. A general violation of the classical-quantum correspondence consists in the localization of quantum states in momentum space. Fortunately, this circumstance ensures success of the numerical method, because, in any case, the matrix U needs to be restricted.
3. Axial channeling The superposition of large number of Bloch waves with strongly complicated structure is effective for classical motion in the random direction in a three-dimensional periodic potential. Therefore, the effects of specific structure of quantum states, if they exist, can be detected more probably in axial channeling. In this connection, it is symptomatic that the discussion of quantum effects [lo] has been stimulated by the observation of diffraction-like effects in the case of ion beam direction near the axial channels. The alternative to the Kikuchi diffraction may be the following explanation of these effects. If the momentum for some Bloch states is widely distributed the filling of these states after the penetration of plane wave through crystal surface can result in the appearance of side waves in the direction far from the initial direction. The wide
V.A. Khdyrev/Nucl.
Instr. and Meth. in Phys. Res. B 11.5 (1996) 328-331
331
momentum distribution (although localized) is a specific detail of the chaotic states. It seems, however, that such type of effects is of second order, the first-order one being the quantum suppression of classical diffusion due to the localization of quantum states [12]. In order to receive a representation about the scale of possible effects, we have calculated the successive diffraction of initial plane wave on the crystal atomic layers. The same model with the S-periodic in time but with two-dimensional potential was used. The form of the potential in transverse plane was taken as thermally smoothed Lindhard potential [9]. The calculation was carried out for a simple cubic structure of a crystal with Z, = 14 and particles with unit charge and a mass equal to half of the proton mass. The last choice is forced by the circumstance that even in this case about 500 X 500 reciprocal lattice vectors are needed to be accounted. The determination of eigen states wave functions as in the previous section demands a very large volume of computer calculations. However, we can come to some conclusions comparing classical and quantum description of particle penetration through a crystal in different directions. Fig. 4 shows one of such results for the velocity of a particle, equal to 3.5 times the Bohr velocity, its initial direction of motion
alignment is 30,, where 0, is the Lindhard critical angle [9]>. It can be seen that the classical picture of the development of a ring-like angular distribution is not reproduced in the quantum description for this thick layer: the quantum momentum distribution is localized. It is unlikely that this ideal picture can claim to be a reliable prediction. The effect of noise, although not well studied, may severely destroy quantum interference [ 131. However, it is well known from the experiments that there is a strong delay in the ring equilibration relative to the classical estimation [9]: the intensity in momentum distribution remains concentrated in the original direction even at large depth [ 141. In conclusion, it seems interesting to analyze the transmission channeling experiments in order to look for the effects of quantum chaos, the suppression of classical diffusion. If this will be the case, channeling, due to the possibility of experimental conditions variation, can be a nice opportunity to study these phenomena.
being in plane (012) near the (100)
(Springer-Verlag. Heidelberg, 1990). J.E. Baytield and P.M. Koch, Phys. Rev. Lett. 33 (1974) 258. D.R. Gempel and R.E. Prange, Phys. Rev. A 29 (1984) 1639. P.W. Anderson, Rev. Mod. Phys. 50 (1978) 191. V.A. Khodyrev, Phys. Lett. A 111 (1987) 67. J.C. Kimball. G. Petschel and N. Cue, Nucl. Instr. and Meth. B 33 (1988) 53. [7] A.I. Akhiezer, V.I. Truten and N.F. Shulga, Phys. Rep. 203 (1991) 289. [8] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, Heidelberg, 1983). [9] I. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 34 (14) (1965). [lo] L.T. Chadderton. Philos. Mag. 18 (1968) 1017. [I l] E.J. Heller, Phys. Rev. Lett. 53 (1984) 1515. [12] G. Casati, B.V. Chirikov, F.M. lzrailev and J. Ford, in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. G. Casati and J. Ford, Lecture Notes in Physics Vol. 90 (Springer, Berlin, 1979) p. 334. [ 131 L. Benet, T.H. Seligman and H.A. Weidenmuller, Phys. Rev. Lett. 71 (1993) 529. [14] D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129.
Classical
axial direction
(mis-
Quantum
Fig. 4. Transverse momentum distribution after particle transition through the indicated number of crystal atomic layers. The results of classical and quantum description are shown.
References [I] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics [2] [3] [4] [5] [6]
IV. CHANNELING