Semi-spheroidal Quantum Harmonic Oscillator
Semi-spheroidal Quantum Harmonic Oscillator
D. N. Poenaru,1, 2, ∗ R. A. Gherghescu,1, 2 A. V. Solov’yov,1 and W. Greiner1
1Frankfurt Institute for Advanced Studies, J. W. Goethe Universität,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
2 Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH),
P.O. Box MG-6, RO-077125 Bucharest-Magurele, Romania
(Dated: November 15, 2018)
A new single-particle shell model is derived by solving the Schrödinger equation for a semi-
spheroidal potential well. Only the negative parity states of the Z(z) component of the wave function
are allowed, so that new magic numbers are obtained for oblate semi-spheroids, semi-sphere and
prolate semi-spheroids. The semi-spherical magic numbers are identical with those obtained at the
oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... The superdeformed prolate
magic numbers of the semi-spheroidal shape are identical with those obtained at the spherical shape
of the spheroidal harmonic oscillator: 2, 8, 20, 40, 70, 112, 168 ...
PACS numbers: 03.65.Ge, 21.10.Pc, 31.10.+z,
The spheroidal harmonic oscillator have been used in
various branches of Physics. Of particular interest was
the famous single-particle Nilsson model [1] very success-
ful in Nuclear Physics and its variants [2, 3, 4] for atomic
clusters. Major spherical-shells N = 2, 8, 20, 40, 58, 92
have been found [2] in the mass spectra of sodium clus-
ters of N atoms per cluster, and the Clemenger’s shell
model [3] was able to explain this sequence of spherical
magic numbers.
In the present paper we would like to write explicitly
the analytical relationships for the energy levels of the
spheroidal harmonic oscillator and to derive the corre-
sponding solutions for a semi-spheroidal harmonic oscil-
lator which may be useful to study atomic cluster de-
posited on planar surfaces.
For spheroidal equipotential surfaces, generated by a
potential with cylindrical symmetry the states of the va-
lence electrons were found [3] by using an effective single-
particle Hamiltonian with a potential
Mω20R
2 + δ
2 + δ
In order to get analytical solutions we shall neglect an
additional term proportional to (l2 − 〈l2〉n). We plan to
include in the future such a term which needs a numerical
solution. K. L. Clemenger introduced the deformation δ
by expressing the dimensionless two semiaxes (in units
of the radius of a sphere with the same volume, R0 =
1/3, where rs is the Wigner-Seitz radius, 2.117 Å for
Na [5, 6]) as
2 + δ
; c =
2 + δ
The spheroid surface equation in dimensionless cylindri-
cal coordinates ρ and z is given by
= 1 (3)
where a is the minor (major) semiaxis for prolate (oblate)
spheroid and c is the major (minor) semiaxis for prolate
(oblate) spheroid. Volume conservation leads to a2c = 1.
One can separate the variables in the Schrödinger
equation, HΨ = EΨ, written in cylindrical coordinates.
As a result the wave function [7, 8] may be written as
Ψ(η, ξ, ϕ) = ψmnr (η)Φm(ϕ)Znz (ξ) (4)
where each component of the wave function is ortonor-
malized leading to
Φm(ϕ) = e
2π (5)
ψ(η) = Nmnrη
|m|/2e−η/2L
nr (η)
Nmnr =
α⊥(nr+|m|)!
in which η = R20ρ
2/α2⊥ and the quantum numbers m =
(n⊥−2i) with i = 0, 1, ... up to (n⊥−1)/2 for an odd n⊥
or to (n⊥− 2)/2 for an even n⊥. Lmn (x) is the associated
Laguerre polynomial and the constant α⊥ =
h̄/Mω⊥
has the dimension of a length.
Znz(ξ) = Nnze
−ξ2Hnz (ξ)
Nnz =
π2nznz!)
1/2 (7)
where ξ = R0z/αz, αz =
h̄/Mωz, and the main quan-
tum number n = n⊥ + nz = 0, 1, 2, ....
The eigenvalues are
En = h̄ω⊥(n⊥ + 1) + h̄ωz(nz + 1/2) (8)
The parity of the Hermite polynomials Hnz (ξ) is given
by (−1)nz meaning that the even order Hermite poly-
nomials are even functions H2nz(−ξ) = H2nz (ξ) and
the odd order Hermite polynomials are odd functions
H2nz+1(−ξ) = −H2nz+1(ξ). There is a recurrence rela-
tionship 2zHn = Hn+1+2nHn−1. One hasH0 = 1, H1 =
http://arxiv.org/abs/0704.0847v1
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
(spheroidal deformation)
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8
(spheroidal deformation)
FIG. 1: LEFT: Spheroidal harmonic oscillator energy levels in units of h̄ω0 vs. the deformation parameter δ. Only 6 major
shells (N = 0, 1, 2, ..., 5) have been considered. Each level is labeled by n, n⊥ quantum numbers and is (2n⊥+2)-fold degenerate.
The labels are 0, 0; 1, 0, 1, 1; 2, 0, 2, 1, 2, 2; 3, 0, 3, 1, 3, 2, 3, 3; 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, etc. RIGHT: Semi-spheroidal harmonic
oscillator energy levels in units of h̄ω0 vs. the deformation coordinate δ. Only 9 major shells (N = 0, 1, 2, ..., 8) have been
considered. Each level is labeled by n, n⊥ quantum numbers (with nz = n− n⊥ = 1, 3, 5, ... and is (2n⊥ + 2)-fold degenerate.
The labels are 1, 0; 2, 1; 3, 2, 3, 0; 4, 3, 4, 1; 5, 4, 5, 2, 5, 0; 6, 5, 6, 3, 6, 1, etc. The semi-spherical magic numbers are identical
with those obtained at the oblate spheroidal superdeformed shape (δ = −2/3): 2, 6, 14, 26, 44, 68, 100, 140, ...
2z, H2 = 4z
2−2, H3 = 8z3−12z, H4 = 16z4−48z2+12,
H5 = 32z
5 − 160z3 + 120z, etc.
In units of h̄ω0 the eigenvalues, ǫ = E/(h̄ω0), are given
(2− δ)1/3(2 + δ)2/3
For a prolate spheroid, δ > 0, at n⊥ = 0 the energy level
decreases with deformation except for n = 0, but when
n⊥ = n it increases. For a given prolate deformation and
z0=0.5
z1=1.5
z2=2.5
z3=3.5
z4=4.5
z5=5.5
FIG. 2: LEFT: Harmonic oscillator potential V = V (ξ), the
wave functions Znz = Znz (ξ) for nz = 0, 1, 2, 3, 4, 5 and the
corresponding contributions to the total energy levels ǫz nz =
Enz/h̄ωz = (nz + 1/2) for spherical shapes, δ = 0. ξ =
h̄/Mωz. RIGHT: The similar functions for a semi-
spherical harmonic oscillator potential. Only negative parity
states are retained which are vanishing at ξ = 0 where the
potential wall is infinitely high .
a maximum energy ǫm, there are nmin closed shells and
other levels for high-order shells up to nmax:
nmin =
(2− δ)1/3(2 + δ)2/3ǫm −
2 + δ
nmax =
(2− δ)1/3(2 + δ)2/3ǫm −
2− δ (11)
and similar formulae for oblate deformations, δ < 0. The
low lying energy levels for the six shells (main quantum
number n = 0, 1, 2, 3, 4, 5) can be seen in figure 1. Each
level, labelled by n⊥, n, may accomodate 2n⊥ + 2 parti-
cles. One has 2
(n⊥+1) = (n+1)(n+2) nucleons
in a completely filled shell charcterized by n, and the
total number of states of the low-lying n + 1 shells is
n=0(n+1)(n+2) = (n+1)(n+2)(n+3)/3 leading to
the magic numbers 2, 8, 20, 40, 70, 112, 168... for a spheri-
cal shape. Besides the important degeneracy at a spher-
ical shape (δ = 0), one also have degeneracies at some
superdeformed shapes, e.g. for prolate shapes at the ra-
tio c/a = (2 + δ)/(2 − δ) = 2 i.e. δ = 2/3. More details
may be found in the Table I. The first five shells can
reproduce the experimental magic numbers mentioned
above; in order to describe the other shells Clemenger
introduced the term proportional to (l2 − 〈l2〉n).
Let us consider a particular shape (half of an oblate or
prolate spheroid) of a semi-spheroidal cluster deposited
on a surface with the z axis perpendicular on the sur-
face and the ρ axis in the surface plane. Then the semi-
20 40 60 80 100 120 140
20 40 60 80 100 120 140
20 40 60 80 100 120 140 160
20 40 60 80 100 120 140
= - 1
= - 2/3
= - 0.4
= - 0.8/3
= 2/3
= 0.4
= 0.8/3
FIG. 3: Variation of shell corrections with N for Na clusters. TOP: δ = 0. The semi-spherical magic numbers are identical
with those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, ... For a prolate superdeformed
(δ = 2/3) shape the magic numbers are identical with those obtained at the spherical shape: 2, 8, 20, 40, 70, 112, 168, ...
Other magic numbers are given in Table I. Oblate and prolate shapes are considered on the left-hand side and right-hand side,
respectively.
spheroidal surface equation is given by
(a/c)2(c2 − z2) z ≥ 0
0 z < 0
The radius of the semi-sphere obtained for the defor-
mation δ = 0 is Rs, given by the volume conservation,
(1/2)(4πR3s/3) = 4πR
0/3, leading to Rs = 2
1/3R0. We
shall give ρ, z, a, c in units of Rs instead of R0. Accord-
ing to the volume conservation, a2cR3s/2 = R
0 so that
a2c = 1. Other kind of shapes obtained from a spheroid
by removing less or more than its half (as in the liquid
drop calculations [9]) will be considered in the future; in
this case it is not possible to obtain analytical solutions.
The new potential well we have to consider in order to
solve the quantum mechanical problem is shown in the
right-hand side of the figure 2. The potential along the
symmetry axis, Vz(z), has a wall of an infinitely large
height at z = 0, and concerns only positive values of z
∞ z = 0
MR2sω
2/2 z ≥ 0 (13)
In this case the wave functions should vanish in the
origin, where the potential wall is infinitely high, so
that only negative parity Hermite polynomials (nz odd)
should be taken into consideration.
From the energy levels given in figure 1 we have to
select only those corresponding to this condition. In this
way the former lowest level with n = 0, n⊥ = 0 should
be excluded. From the two leveles with n = 1 we can
retain the level with n⊥ = 0 i.e. nz = 1. This will be
the lowest level for the semi-spherical harmonic oscillator
and will accomodate 2n⊥+2 = 2 atoms. From the three
levels with n = 2 only the one with nz = n⊥ = 1 with
2n⊥ + 2 = 4 degeneracy is retained so that the first two
magic numbers at spherical shape (δ = 0) are now 2
followed by 6, etc. Some deformed magic numbers may
be found in the Table I and as position of minima in
Fig. 3.
Each level, labelled by n⊥, n, may accomodate 2n⊥+2
particles. When n is an odd number, one should only
have even n⊥ in order to select the odd nz = n − n⊥.
The contribution of the shells with odd n to the semi-
spherical magic numbers will be
neven
(2n⊥ + 2) =
(n+ 1)2
leading to the sequence 2, 8, 18 for n = 1, 3, 5. The con-
tribution of the shells with even n to the semi-spherical
TABLE I: TOP: Deformed magic numbers of the spheroidal harmonic oscillator.
BOTTOM: Deformed magic numbers of the semi-spheroidal harmonic oscillator.
OBLATE PROLATE
δ a/c Magic numbers δ a/c Magic numbers
−0.8/3 17/13 2, 8, 18, 20, 34, 38, 58, 64, 92,
100, 136, 148, ...
0.8/3 13/17 2, 8, 20, 22, 42, 46, 76, 82, 124,
134 ...
−0.4 1.5 2, 6, 8, 14, 18, 28, 34, 48, 58, 76,
90, 114, 132, ...
0.4 2/3 2, 8, 10, 22, 26, 46, 54, 66, 84,
96, 114, 138, 156, ...
−2/3 2 2, 6, 14, 26, 44, 68, 100, 140, ... 2/3 0.5 2, 4, 10, 16, 28, 40, 60, 80, 110,
140, ...
−1 3 2, 6, 12, 22, 36, 54, 78, 108, 144,
1 1/3 4, 12, 18, 24, 36, 48, 60, 80, 100,
120, 150, ...
−0.8/3 17/13 2, 6, 12, 22, 26, 36, 42, 56, 64,
82, 92, 114, 126, 154, ...
0.8/3 13/17 2, 6, 8, 14, 18, 28, 34, 48, 58, 76,
90, 114, 132, ...
−0.4 1.5 2, 6, 12, 22, 36, 54, 78, 108, 144,
0.4 2/3 2, 8, 18, 20, 34, 38, 50, 58, 64,
80, 92, 100, ...
−2/3 2 2, 6, 12, 20, 32, 48, 68, 92, 122,
158, ...
2/3 0.5 2, 8, 20, 40, 70, 112, 168, ...
−1 3 2, 6, 20, 30, 42, 58, 78, 102, 130,
1 1/3 2, 8, 10, 14, 22, 26, 46, 54, 66,
84, 96, 114, 138, 156, ...
magic numbers will be
(2n⊥ + 2) =
n(n+ 2)
which gives the sequence 4, 12, 24 for n = 2, 4, 6. This
should be interlaced with the preceding one so that the
magic numbers will be 2, 2+4 = 6, 6+8 = 14, 14+12 =
26, 26+18 = 44, 44+24 = 68, as shown at the right-hand
side of the Fig. 1.
The equation (9) from the harmonic oscillator, in units
of h̄ω0 is still valid, but one should only allow the values
of n and n⊥ for which nz = n−n⊥ ≥ 1 are odd numbers.
The ortonormalization condition of the Znz component
of the wave function became
Zn′z(z)Znz(z)dz = δn′znz (16)
with nz = 1, 3, 5, ..., n for odd n and nz = 1, 3, 5, ..., n− 1
for even n. Consequently the normalization factor is
times the preceding one
Znz(ξ) =
2Nnze
−ξ2Hnz(ξ)
Nnz =
π2nznz!)
1/2 (17)
For a nucleus with mass number A the shell gap is
given by h̄ω00 = 41A
1/3 MeV. For an atomic cluster [10]
the single-particle shell gap is given by
h̄ω0(N) =
13.72 eV Å
rsN1/3
which is 3.0613N−1/3 eV in case of Na clusters. Since we
consider solely monovalent elements, N in this eq. is the
number of atoms and t denotes the electronic spillout for
the neutral cluster according to [10].
The shell correction energy, δU [11], in figure 3 shows
minima at the oblate and prolate magic numbers given
in the lower part of the table I. The striking result is
that the superdeformed prolate magic numbers of the
semi-spheroidal shape are identical with those obtained
at the spherical shape of the spheroidal harmonic oscil-
lator. We expect that this kind of symmetry will not
be present anylonger for the Hamiltonian including the
term proportional to (l2−〈l2〉n) and/or the more complex
equipotential surface we shall study in the future.
∗ poenaru@fias.uni-frankfurt.de
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[11] V. M. Strutinsky, Nuclear Physics, A 95, 420 (1967).
mailto:poenaru@fias.uni-frankfurt.de
