In this article, we have applied the methods of chaos theory to channeling phenomena of positive charged particles in crystal lattices. In particular, we studied the transition between two ordered types of motion; i.e., motion parallel to a crystal axis (axial channeling) and to a crystal plane (planar channeling), respectively. The transition between these two regimes turns out to occur through an angular range in which the particle motion is highly disordered and the region of phase space spanned by the particle is much larger than the one swept in the two ordered motions. We have evaluated the maximum Lyapunov exponent with the method put forward by Rosenstein et al [Physica D65, 117 (1993)] and by Kantz [Phys. Lett. A185, 77 (1994)]. Moreover, we estimated the correlation dimension by using the Grassberger-Procaccia method. We found that at the transition the system exhibits a very complex behavior showing an exponential divergence of the trajectories corresponding to a positive Lyapunov exponent and a noninteger value of the correlation dimension. These results turn out to be linked to a physical interpretation. The Lyapunov exponents are in agreement with the model by Akhiezer et al [Phys. Rep.203, 289 (1991)], based on the equivalence between the ion motion along the crystal plane described as a “string of strings” and the “kicked” rotator. The nonintegral value of the correlation dimension can be explained by the nonconservation of transverse energy at the transition.

1.
D. S.
Gemmell
,
Rev. Mod. Phys.
46
,
129
(
1974
).
2.

Strictly speaking, the confined motion of a positive particle along an axial channel bounded by nearest-neighbor strings is called “proper” channeling or “hyperchanneling.” The term “axial” channeling includes motions in which the positive particle wanders from one axial channel to another without coming too close to the atomic strings.

3.
G.
DellaMea
,
A. V.
Drigo
,
S. L.
Russo
,
P.
Mazzoldi
,
G. G.
Bentini
,
A.
Desalvo
, and
R.
Rosa
,
Phys. Rev. B
7
,
4029
(
1973
).
4.
A.
Desalvo
and
R.
Rosa
,
Phys. Rev. B
9
,
4605
(
1974
).
5.
M. A.
Kumakhov
and
R.
Wedell
,
Phys. Status Solidi B
76
,
119
(
1976
).
6.
J. C.
Kimball
,
G.
Petschel
, and
N.
Cue
,
Nucl. Instrum. Methods Phys. Res. B
33
,
53
(
1988
).
7.
A. I.
Akhiezer
,
V. I.
Truten’
, and
N. F.
Shul’ga
,
Phys. Rep.
203
,
289
(
1991
).
8.
A. I.
Akhiezer
,
V. I.
Truten’
, and
N. F.
Shul’ga
,
Nucl. Instrum. Methods Phys. Res. B
67
,
207
(
1992
).
9.
H. S.
Dumas
,
J. A.
Ellison
, and
F.
Golse
,
Physica D
146
,
341
(
2000
).
10.
J.
Lindhard
,
Kgl. Danske Videnskab Selskab Mat. Fys. Medd.
34
(
1965
), No. 14.
11.

For a comprehensive and recent account of chaos in Hamiltonian systems see the book by Zaslavsky (Ref. 39).

12.
S.
Giannerini
and
R.
Rosa
,
Stud. Nonlinear Dyn. Econom.
(
2004
), Vol.
8
, Iss.
2
, No. 11.
13.
C. D.
Cutler
, in
Dimension Estimation and Models
, edited by
H.
Tong
(
World Scientific
,
Singapore
,
1993
), pp.
1
107
.
14.
H. S.
Dumas
,
Nucl. Instrum. Methods Phys. Res. B
234
,
3
(
2005
).
15.

The author divides the phase space of initial conditions in four zones: (i) far from any channeling direction, (ii) at or near an axial channeling direction, (iii) at or near a planar channeling direction, and (iv) a small complementary set forming thin boundaries between the sets (i), (ii), and (iii).

16.

According to the foregoing introduction, the expression “as-yet-undescribed” should be read as “as yet not thoroughly described” or something equivalent. We mention that an older paper by another of these authors (Ref. 40) quoted our previous work.

17.

We must emphasize that for our incident energies (1MeV) and small incidence angles at the transition (some tenths of a degree; i.e., some milliradians) the transverse energy is of the order of some tens of eV. As a consequence the variations of transverse energy, which are of the order of some eV (see below, Fig. 16), are appreciable, while the corresponding variations in the parallel component of the energy are negligible compared to its value.

18.
19.
A.
Desalvo
,
R.
Rosa
, and
F.
Zignani
,
Radiat. Eff.
27
,
89
(
1975
).
20.
A.
Carnera
,
G.
DellaMea
,
A. V.
Drigo
,
S. L.
Russo
,
P.
Mazzoldi
,
G. G.
Bentini
,
A.
Desalvo
, and
R.
Rosa
Phys. Rev. B
18
,
995
(
1978
).
21.
G.
Lulli
,
E.
Albertazzi
,
M.
Bianconi
,
R.
Nipoti
,
M.
Cervera
,
A.
Carnera
, and
C.
Cellini
,
J. Appl. Phys.
82
,
5958
(
1997
).
22.
L. D.
Landau
and
E. M.
Lifschitz
,
Mechanics
(
Pergamon
,
London
,
1960
).
23.
R. S.
Nelson
and
M. W.
Thompson
,
Philos. Mag.
8
,
1677
(
1963
).
24.
M. T.
Rosenstein
,
J.
Collins
, and
C. D.
Luca
,
Physica D
65
,
117
(
1993
).
26.
H.
Kantz
and
T.
Schreiber
,
Non Linear Time Series Analysis
(
Cambridge University Press
, Cambridge,
1997
).
27.

Implementations of the algorithm can be found in the TISEAN package (Ref. 41) and in the R package tseriesChaos (Ref. 42). Both of them are freely available.

28.
S.
Giannerini
and
R.
Rosa
,
Physica D
155
,
101
(
2001
).
29.
S.
Giannerini
,
D. L.
Gonzalez
, and
R.
Rosa
, Int. J. Bifurcation Chaos (in press).
30.
P.
Grassberger
and
I.
Procaccia
,
Phys. Rev. Lett.
50
,
346
(
1983
).
31.
P.
Grassberger
and
I.
Procaccia
,
Physica D
9
,
189
(
1983
).
32.
K.
Lenkeit
and
R.
Wedell
,
Phys. Status Solidi B
98
,
235
(
1980
).
33.
Y. V.
Bulgakov
and
V. I.
Shulga
,
Radiat. Eff.
28
,
15
(
1976
).
34.

The energy E or, more exactly, Ec2, which appears in the corresponding equation by Akhiezer et al. (Refs. 7 and 8) valid for the relativistic case, reduces to the mass of the particle M in the nonrelativistic limit. In this connection, we notice that by passing to the nonrelativistic case we must make explicit the velocity of the particle v in some of their formulas, since they put vc=1.

35.
U.
Bill
,
R.
Sizmann
,
C.
Varelas
, and
K. E.
Rehm
,
Radiat. Eff.
27
,
59
(
1975
).
36.
J. A.
Davies
,
J.
Denhartog
, and
J. L.
Whitton
,
Phys. Rev.
165
,
345
(
1968
).
37.
T.-Y.
Li
and
J. A.
Yorke
,
Am. Math. Monthly
82
,
985
(
1975
).
38.
R. C.
Hilborn
and
N. B.
Tufillaro
,
Am. J. Phys.
65
,
822
(
1997
).
39.
G. M.
Zaslavsky
,
Hamiltonian Chaos and Fractional Dynamics
(
Oxford University Press
,
Oxford
,
2005
).
40.
T. J.
Burns
and
J. A.
Ellison
,
Phys. Rev. B
29
,
2790
(
1984
).
42.
R Development Core Team,
R: A Language and Environment for Statistical Computing
(
R Foundation for Statistical Computing
,
Vienna
,
2005
), ISBN 3-900051-07-0; URL: http://www.R-project.org
You do not currently have access to this content.