XXXVI international conference on plasma physics and CF, February 9 – 13, 2009, Zvenigorod.
SHELL EFFECTS AND PERIODIC ORBITS IN ATOMIC-ION AND NUCLEUS-CLUSTER
SYSTEMS
G.V. Shpatakovskaya
IMM RAS, Moscow, RF, shpagalya@yandex.ru
The role of the periodic orbits in quantum physics has been discussed in the literature since Bohr
(see, for example, references in the review [1]). In following this question is considered for
spherically symmetric potentials.
Research of the bound interacting fermion system (free atoms and ions, plasma, metal clusters,
nuclei) using the density functional method reduces to the single-particle states in а self-consistent
potential. In spherical approximation a self-consistent attractive potential in such a system can be
assigned to one of the two possible groups [2]: 1) finite at zero point (potentials in spherical clusters
and nuclei) and 2) having Coulomb singularity at zero point (intra-atomic potentials). The group
specificity appears in the gradient, oscillating and shell effects.
To calculate the semi-classical particle density distribution and thermo-dynamical properties of a
system it is enough to know the particle classical motion characteristics only for energy equal to the
chemical potential  . In particular, radial action S  (l ) for the particle with the energy  and
orbital momentum l contains all the necessary information for the shell effects description. The
shell effects are connected with the periodic traces and result, for example, in the magic numbers in
metal cluster mass spectra and in the irregular dependence of atomic properties on the atomic
number.
In small clusters there are two important periodic orbits: the particle having a zero orbital
momentum l  0 traverses the centre along the linear orbit, and the particle with the highest
possible orbital momentum l  l moves along the circular orbit. In the clusters having a hard
potential the k  angle ( k  3, 4,5,... ) closed orbits appear when cluster size increases [3]. For
example oscillations with beats in the experimental sodium cluster mass spectra are the result of
triangular and square orbits interference. In the large soft potential clusters the five-point-star orbit
makes the main contribution [4] that appears in the different, similar to sine, oscillations in the
aluminum cluster mass spectra.
It is shown, that cluster situation can’t come true in the atomic and ion systems under any atomic
number. There are always only two periodic orbits which appear in the measurable quantities: a
linear motion in the s  state of the particle which reflects from the centre and a circular movement
with the maximum orbital momentum l  l , the latter contribution being dominant [5].
This work was supported in part by the Russian Foundation for Basic Research (project nos. 08-0100291 and 06-01-00097).
References
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[2].
[3].
[4].
[5].
M.B. Brack, R.K. Bhaduri, Semi-classical Physics, AWPC, 1997
G.V. Shpatakovskaya, JETP Letters, 2001, Vol. 73, P. 268
G.V. Shpatakovskaya, JETP, 2000, Vol. 91, P. 76
G.V. Shpatakovskaya, JETP Letters, 2000, Vol. 72, P. 269
B.G. Englert and J. Schwinger, Phys. Rev. 1985, V. A32, P.26
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