Fractional dynamic symmetries and the ground

state properties of nuclei

arXiv:0806.2300v2 [physics.gen-ph] 12 Aug 2008

Richard Herrmann
GigaHedron, Farnweg 71, D-63225 Langen, Germany
E-mail: herrmann@gigahedron.com

Abstract. Based on the Riemann- and Caputo definition of the fractional

derivative we use the fractional extensions of the standard rotation group SO(3) to
construct a higher dimensional representation of a fractional rotation group with
mixed derivative types. An extended symmetric rotor model is derived, which
predicts the sequence of magic proton and neutron numbers accurately. The
ground state properties of nuclei are correctly reproduced within the framework
of this model.

PACS numbers: 21.60.Fw, 21.60.Cs, 05.30.Pr

1. Introduction

The experimental evidence for discontinuities in the sequence of atomic masses, α−

and β- decay systematics and binding energies of nuclei suggests the existence of a set
of magic proton and neutron numbers, which can be described successfully by single
particle shell models with a heuristic spin-orbit term [1], [2]. The most prominent
representative is the phenomenological Nilsson model [3] with an anisotropic oscillator
potential:
3
X 1
V (xi ) = mωi2 x2i − ~ω0 κ(2~l~s + µl2 ) (1)
i=1
2
Although these models are flexible enough to reproduce the experimental results, they
lack a deeper theoretical justification, which becomes obvious, when extrapolating the
parameters κ, µ, which determine the strength of the spin orbit and l2 term to the
region of superheavy elements [5].
Hence it seems tempting to describe the experimental data with alternative
methods. Typical examples are relativistic mean field theories [6],[7], where nucleons
are described by the Dirac-equation and the interaction is mediated by mesons.
Although a spin orbit force is obsolete in these models, different parametrizations
predict different shell closures [8],[9]. Therefore the problem of a theoretical foundation
of magic numbers remains an open question since Elsasser [10] raised the problem 75
years ago.
A fundamental understanding of magic numbers for protons and neutrons may be
achieved if the underlying corresponding symmetry of the nuclear many body system
is determined. Therefore a group theoretical approach seems appropriate.
Fractional dynamic symmetries and the ground state properties of nuclei 2

Group theoretical methods have been successfully applied to problems in nuclear

physics for decades. Elliott [11] has demonstrated, that an average nuclear potential
given by a three dimensional harmonic oscillator corresponds to a SU(3) symmetry.
Low lying collective states have been successfully described within the IBM-model
[12], which contains as one limit the five dimensional harmonic oscillator, which is
directly related to the Bohr-Mottelson Hamiltonian.
In this paper we will determine the symmetry group, which generates a single
particle spectrum similar to (1), but includes the magic numbers right from the
beginning.
Our approach is based on group theoretical methods developed within the
framework of fractional calculus.
The fractional calculus [13]-[16] provides a set of axioms and methods to extend
the coordinate and corresponding derivative definitions in a reasonable way from
integer order n to arbitrary order α:
∂n ∂α
{xn , n } → {xα , α } (2)
∂x ∂x
The definition of the fractional order derivative is not unique, several definitions e.g.
the Feller, Fourier, Riemann, Caputo, Weyl, Riesz, Grünwald fractional derivative
definitions coexist [17]-[25]. A direct consequence of this diversity is the fact, that the
solutions e.g. of a one dimensional wave equation differ significantly depending on the
specific choice of a fractional derivative definition.
Until now it has always been assumed, that the fractional derivative type for an
extension of a fractional differential equation to multi-dimensional space should be
chosen uniquely.
In contrast to this assumption, in this paper we will investigate properties of
higher dimensional rotation groups with mixed Caputo and Riemann type definition
of the fractional derivative. We will demonstrate, that a fundamental dynamic
symmetry is established, which determines the magic numbers for protons and neutron
respectively and furthermore describes the ground state properties of nuclei with
reasonable accuracy.

2. Notation

We will investigate the spectrum of multi dimensional fractional rotation groups for
two different definitions of the fractional derivative, namely the Riemann- and Caputo
fractional derivative. Both types are strongly related.
Starting with the definition of the fractional Riemann integral
Z x
1

α
 (R I+ f )(x) =
 dξ (x − ξ)α−1 f (ξ) x≥0
α
 Γ(α) 0
R I f (x) = Z 0 (3)
 α 1 α−1
 (R I−
 f )(x) = dξ (ξ − x) f (ξ) x<0
Γ(α) x
where Γ(z) denotes the Euler Γ-function, the fractional Riemann derivative is defined
as the product of a fractional integration followed by an ordinary differentiation:
α ∂ 1−α
R ∂x = RI (4)
∂x
Fractional dynamic symmetries and the ground state properties of nuclei 3

It is explicitely given by:

Z x
1 ∂

 (
 R +∂ α
f )(x) = dξ (x − ξ)−α f (ξ) x≥0
−

α
Γ(1 α) ∂x 0
R ∂x f (x) = Z 0 (5)
1 ∂
α
dξ (ξ − x)−α f (ξ)

 (R ∂−
 f )(x) = x<0
Γ(1 − α) ∂x x
The Caputo definition of a fractional derivative follows an inverted sequence of
operations (4). An ordinary differentiation is followed by a fractional integration
∂
C ∂xα = R I 1−α (6)
∂x
which results in:
Z x
1 ∂

 (
 C +∂ α
f )(x) = dξ (x − ξ)−α f (ξ) x≥0
−

α
Γ(1 α) 0 ∂ξ
C ∂x f (x) = Z 0 (7)
1 ∂
α
dξ (ξ − x)−α f (ξ)

 (C ∂−
 f )(x) = x<0
Γ(1 − α) x ∂ξ
Applied to a function set f (x) = xnα using the Riemann fractional derivative definition
(5) we obtain:
Γ(1 + nα)
R ∂xα xnα = x(n−1)α (8)
Γ(1 + (n − 1)α)
= R [n] x(n−1)α (9)
where we have introduced the abbreviation R [n].
For the Caputo definition of the fractional derivative it follows for the same
function set:
Γ(1 + nα)

α nα
 x(n−1)α n>0
C ∂x x = Γ(1 + (n − 1)α) (10)

0 n=0
= C [n] x(n−1)α (11)
where we have introduced the abbreviation C [n].
Both derivative definitions only differ in the case n = 0:
C [n] = R [n] − δn0 R [0] (12)
1
= R [n] − δn0 (13)
Γ(1 − α)
where δmn denotes the Kronecker-δ. We will rewrite equations (9) and (11)
simultaneously, introducing the short hand notation
R,C ∂xα xnα = R,C [n] x(n−1)α (14)
We now introduce the fractional angular momentum operators or generators of
infinitesimal rotations in the i, j plane on the N -dimensional Euclidean space:
R,C Lij (α) = R,C i~(xα α α α
i ∂ j − xj ∂ i ) (15)
The commutation relations of the fractional angular momentum operators are
isomorph to the fractional extension of the rotational group SO(N )
R,C [Lij (α), Lkl (α)] = R,C i~fijkl mn Lmn (α) (16)
i, j, k, l, m, n = 1, 2, .., N
Fractional dynamic symmetries and the ground state properties of nuclei 4

Figure 1. Spectrum of the Casimir operator L2 (L, α) from (19) as a function of

the fractional derivative coefficient α. Only the L = 0 state differs for Riemann
and Caputo derivative.

with structure coefficients R,C fijkl mn . Their explicit form depends on the function set
the fractional angular momentum operators act on and on the fractional derivative
type used.
The Casimir-operators of the fractional rotation group R SOα (3) based on the
Riemann fractional derivative definition have been derived in [26] and for C SOα (3)
based on the Caputo fractional derivative definition are given in [27]. We summarize
the major results:
According to the group chain
α α
R,C SO (3) ⊃ R,C SO (2) (17)
there are two Casimir-operators Λi , namely Λ2 = Lz (α) = L12 (α) and Λ3 = L2 (α) =
L212 (α) + L213 (α) + L223 (α). We introduce the two quantum numbers L and M , which
completely determine the eigenfunctions |LM >. It follows
R,C Lz (α)|LM > = R,C ~ sign(M ) [|M |] |LM > (18)
M = −L, −L + 1, ..., ±0, ..., L
2 2
R,C L (α)|LM > = R,C ~ [L][L + 1] |LM > (19)
L = 0, 1, 2, ...
where |M | denotes the absolute value of M . In addition, on the set of eigenfunctions
|LM >, the parity operator Π is diagonal and has the eigenvalues
Π|LM >= (−1)L |LM > (20)
2
In figure 1 the eigenvalues of the Casimir-operator L are shown as a function of α.
Only in the case L = 0 the spectra differ for the Riemann- and Caputo derivative.
While for the Caputo derivative
2
C L (α)|00 >= 0 (21)
because C [0] = 0, using the Riemann derivative for α 6= 1 there is a nonvanishing
contribution
2 2 2 Γ(1 + α)
R L (α)|00 >= R ~ [0][1]|00 >= ~ |00 > (22)
Γ(1 − α)
Fractional dynamic symmetries and the ground state properties of nuclei 5

3. The Caputo-Riemann-Riemann symmetric rotor

We now use group theoretical methods to construct higher dimensional representations

of the fractional rotation groups R,C SOα (3).
As an example of physical relevance we introduce the group CRR G with the
following chain of subalgebras:
CRR G ⊃ C SOα (3) ⊃ R SOα (3) ⊃ R SOα (3) (23)
The Hamiltonian H can now be written in terms of the Casimir operators of
the algebras appearing in the chain and can be analytically diagonalized in the
corresponding basis. The Hamiltonian is:
ω1 2 ω2 2 ω3 2
H= C L1 (α) + R L2 (α) + R L3 (α) (24)
~ ~ ~
with the free parameters ω1 , ω2 , ω3 and the basis is |L1 M1 L2 M2 L3 M3 >. Furthermore,
we impose the following symmetries:
First, the wave functions should be invariant under parity transformations, which
according to (20) leads to the conditions
L1 = 2n1 L2 = 2n2 L3 = 2n3 , n1 , n2 , n3 = 0, 1, 2, 3, ... (25)
second, we require
C Lz1 (α)|L1 M1 L2 M2 L3 M3 > = C + ~[L1 ]|L1 M1 L2 M2 L3 M3 > (26)
R Lz2 (α)|L1 M1 L2 M2 L3 M3 > = R + ~[L2 ]|L1 M1 L2 M2 L3 M3 > (27)
R Lz3 (α)|L1 M1 L2 M2 L3 M3 > = R + ~[L3 ]|L1 M1 L2 M2 L3 M3 > (28)
which reduces the multiplicity of a given |2n1 M1 2n2 M2 2n3 M3 > set to 1.
With these conditions, the eigenvalues of the Hamiltonian (24) are given as
E(α) = ~ω1 C [2n1 ][2n1 + 1] +
~ω2 R [2n2 ][2n2 + 1] + ~ω3 R [2n3 ][2n3 + 1] (29)
3
X Γ(1 + (2ni + 1)α) Γ(1 + α)
= ~ωi − δn1 0 ~ω1 (30)
i=1
Γ(1 + (2ni − 1)α) Γ(1 − α)
n1 , n2 , n3 = 0, 1, 2, ..
on a basis |2n1 2n1 2n2 2n2 2n3 2n3 >.
This is the major result of our derivation. We call this model the Caputo-
Riemann-Riemann symmetric rotor. What makes this model remarkable is its
behaviour near α = 1/2.
On the left of figure 2 we have plotted the energy levels in the vicinity of α ≈ 1/2
for the case
ω1 = ω2 = ω3 = ω0 (31)
which we denote as the spherical case.
For the idealized case α = 1/2, using the relation Γ(1 + z) = zΓ(z) the level
spectrum (30) is simply given by:
3 1
E(α = 1/2) = ~ω0 (n1 + n2 + n3 + − δn1 0 ) (32)
2 2
For n1 6= 0 this is the well known spectrum of the 3-dimensional harmonic oscillator.
Assuming a twofold spin degeneracy of the energy levels, we introduce the quantum
number N as
N = n1 + n2 + n3 (33)
Fractional dynamic symmetries and the ground state properties of nuclei 6

Figure 2. On the left the energy spectrum E(α) from (30) for the spherical
case (31) in units of ~ω0 for the Caputo-Riemann-Riemann symmetric rotor is
presented. The right diagram shows the neutron energy levels for the spherical
nucleus 298
114 X calculated within the framework of the asymmetric two center shell
model (ATCSM) [28] near the ground state as a function of increasing strength
of the spin-orbit term (κ0 κ~l~s) increasing from 50% to 100% of the recommended
κ value, while the µl2 value is kept constant. The transition from magic numbers
of the standard 3-dimensional harmonic oscillator levels (34) to the shifted set of
magic numbers (37) is pointed out with thick lines. Left and right figure therefore
show a similar behaviour for the energy levels.

Consequently we obtain a first set nmagic 1 of magic numbers nmagic

1
nmagic 1 = (N + 1)(N + 2)(N + 3) N = 1, 2, 3, ... (34)
3
= 8, 20, 40, 70, 112, 168, 240, ... (35)
which correspond to the standard 3-dimensional harmonic oscillator at energies
E = ~ω0 (N + 3/2) (36)
In addition, for n1 = 0, which corresponds to the |00 2n2 2n2 2n3 2n3 > states, we
obtain a second set nmagic 2 of magic numbers
1
nmagic 2 = (N + 1)(5 + (N + 1)2 ) N = 0, 1, 2, 3, ... (37)
3
= 2, 6, 14, 28, 50, 82, 126, 184, 258, ... (38)
at energies
E = ~ω0 (N + 1) (39)
which is shifted by the amount − 21 ~ω0 compared to the standard 3-dimensional
harmonic oscillator values.
Fractional dynamic symmetries and the ground state properties of nuclei 7

From figure 2 it follows, that for α < 1/2 the second set nmagic 2 of energy levels
falls off more rapidly than the levels of set nmagic 1 . As a consequence for decreasing
α the magic numbers nmagic 1 die out successively. On the other hand, for α > 1/2
the same effect causes the magic numbers nmagic 1 to survive.
We want to emphasize, that the described behaviour for the energy levels in
the region α < 1/2 may be directly compared to the influence of a ls-term in
phenomenological shell models. As an example, on the right hand side of figure 2
a sequence of neutron levels for the superheavy element 298 114 X calculated with the
asymmetric two center shell model (ATCSM) [28] with increasing strength of the ls-
term from 50% to 100% is plotted. It shows, that the n = 168 gap breaks down at
about 70% and the n = 112 gap at about 90% of the recommended κ-value for the ls-
term. This corresponds to an α ≈ 0.46 value, since in the Caputo-Riemann-Riemann
symmetric rotor the n = 168 gap breaks down at α = 0.466, the n = 112 gap at
α = 0.460 and the n = 70 gap vanishes at α = 0.453.
We conclude, that the Caputo-Riemann-Riemann symmetric rotor predicts a well
defined set of magic numbers. This set is a direct consequence of the underlying
dynamic symmetries of the three fractional rotation groups involved. It is indeed
remarkable, that the same set of magic numbers is realized in nature as magic proton
and neutron numbers.
In the next section we will demonstrate, that the proposed analytical model is an
appropriate tool to describe the ground state properties of nuclei.

4. Ground state properties of nuclei

We will use the Caputo-Riemann-Riemann symmetric rotor (30) as a dynamic shell

model for a description of the microscopic part of the total energy Etot of the nucleus.
Etot = Emacroscopic + Emicroscopic (40)
= Emacroscopic + δU + δP (41)
where δU and δP denote the shell- and pairing energy contributions.
For the macroscopic contribution we use the finite range liquid drop model
(FRLDM) proposed by Möller [31] using the original parameters, except the value
for the constant energy contribution a0 .
As the primary deformation parameter we use the ellipsoidal deformation Q:
b ω3 ω3
Q= = = (42)
a ω1 ω2
where a, b are the semi-axes of a rotational symmetric ellipsoid. Consequently a value
Q < 1 describes prolate and a value of Q > 1 descibes oblate shapes. In order to
relate the ellipsiodal deformation Q to the quadrupole deformation 2 used by Möller,
we define:
Q = 1 − 1.430852 + 0.70766922 (43)
Furthermore we extend the original FRLDM-model introducing an additional
curvature energy term VR (Q), which describes the interaction of the nucleus with
the collective curved coordinate space [32]:
VR (Q) = −aR BR A−5/3 (44)
Fractional dynamic symmetries and the ground state properties of nuclei 8

Figure 3. As a test of the plateau condition ∂U/∂γ = 0 for the Strutinsky shell
correction method, the total shell correction energy δU = δUP + δUN is plotted
as a function of γ for different nuclei.

where A is the nucleon number, aR is the curvature parameter given in [MeV] and the
relative curvature energy BR (Q) given as:
 2
16/3 199 − 288 log(2)
BR (Q) = 9 Q (45)
(2 + Q2 )(266 − 67Q2 + 96(Q2 − 4) log(2))
which is normalized relative to a sphere BR (Q = 1) = 1.
Therefore the total energy may be splitted into
Etot = Emac + Emic (46)
where
Emac (a0 , aR ) = FRLDM(a0 , Q = 1) + VR (Q = 1, aR ) (47)
Emic (a0 , aR , Q) = + δU + δP + FRLDM(a0 , Q) + VR (Q, aR )
− (FRLDM(a0 , Q = 1) + VR (Q = 1, aR )) (48)
with two free parameters a0 , aR , which will be used for a least square fit with the
experimental data.
For calculation of the shell corrections we use the Strutinsky method [29],[30].
Since we expect that the shell corrections are the dominant contribution to the
microscopic energy, for a first comparison with experimental data we will neglect
the pairing energy term.
In order to calculate the shell corrections, we introduce the following parameters:
1
~ω0 = 38A− 3 [M eV ] (49)
− 13
ω1 = ω0 Q (50)
ω2 = ω1 (51)
2
ω3 = ω0 Q 3 (52)

 0.46 + 0.000220 Z
 Z > 50
αZ = 0.2469 + 0.00448 Z 28 < Z ≤ 50 (53)

0.2793 + 0.00332 Z Z ≤ 28


 0.41 + 0.000200 N
 N > 50
αN = 0.3118 + 0.00216 N 28 < N ≤ 50 (54)

0.2793 + 0.00332 N N ≤ 28

Fractional dynamic symmetries and the ground state properties of nuclei 9

Figure 4. Comparison of calculated shell corrections δU from the Caputo-

Riemann-Riemann symmetric rotor (30) with the parameter set (49)-(56) (thick
line) with the tabulated Emic from Möller [31] (dashed line). Upper two rows
show values for a given Z as a function of N , lower two rows for a given N as a
function of Z. Bars indicate the experimentally known region. The original 2
values from [31] are used, which is the main source of error.

γ = 1.1 ~ω0 (55)

m =4 (56)
a0 = 2.409[MeV] (57)
aR = 15.0[MeV] (58)
Input parameters are the number of protons Z, number of neutrons N , the nucleon
number A = N + Z, and the ground state quadrupole deformation 2 .
The values obtained include the frequencies (50),(51),(52), which result from a
least square fit and quadratic approximation of equipotential surfaces, the fractional
derivative coefficients for protons (53) and neutrons (54) which determine the level
spectrum for protons and neutrons for the proton and neutron part of the shell
correction energy respectively from a fit of the set of nuclids 56 100 132 208
28 Ni, 50 Sn, 50 Sn, 82 Pb
100
and from the requirement, that the neutron shell correction for 50 Sn should amount
about −5.1[M eV ], (55) from the plateau condition ∂U/∂γ = 0 (see figure 3) and (56)
the order of included Hermite polynomials for the Strutinski shell correction method.
Finally ~ω0 , a0 , aR from a fit of the experimental mass excess given in [38].
We compare our results with for the microscopic energy contribution Emic with
data from Möller et. al. [31] and use their tabulated 2 values. They have not only
listed data for experimental masses but also predictions for regions, not yet confirmed
by experiment.
Fractional dynamic symmetries and the ground state properties of nuclei 10

Figure 5. Comparison of calculated Emic from the Caputo-Riemann-Riemann

symmetric rotor (30) with the parameter set (49)-(56), minimized with respect
to 2 with the experimental masses from Audi [38] as a function of N . From top
to bottom the minimized 2 values, theoretical Emic , experimental microscopic
contribution from the difference of experimental mass excess and and macroscopic
FRLDM energy and error in [M eV ] are plotted.

In figure 4 we compare the calculated δU values with the tabulated Emic , which
is justified for almost spherical shapes (2 ≈ 0). The results agree very well within the
expected errors (which are estimated ≈ 2 [M eV ] for the pairing energy and 0.5 [M eV ]
for Emic ), especially in the region of experimentally known nuclei.
A remarkable difference between the calculated shell correction and tabulated
Emic from Möller occurs for superheavy elements (N = 184, last picture in figure
4). While phenomenological shell models predict a pronounced minimum in the shell
correction energy for Z = 114 [33]-[37] the situation is quite different for the rotor
model, where two magic shell closures at Z = 112 and Z = 126 are given, but the
Z = 112 shell closure is not strong enough to produce a local minimum in the shell
correction energy plot as a function of Z. Instead, between Z = 112 and Z = 126,
Fractional dynamic symmetries and the ground state properties of nuclei 11

there emerges a slightly falling energy plateau, which makes the full region promising
candidates for stable, long-lived superheavy elements.
While this result contradicts predictions made with phenomenological shell
models, it supports recent results obtained with relativistic mean field models [7],
which predict a similar behaviour in the region of super heavy elements as the proposed
rotor model.
In figure 5 we have covered the complete region of available experimental data
for nuclids and compare the calculated theoretical microscopic energy contribution
minimized with respect to the deformation with the experimental values. The influence
of shell closures is very clear. The rms-error is about 2.4[MeV]. The maximum
deviation occures between closed magic shells. Therefore in the next section we will
introduce a generalization of the proposed fractional rotor model, which not only
determines the magic numbers accurately but in addition determines the fine structure
of the single particle spectrum correctly.

5. Fine structure of the single particle spectrum - the extended

Caputo-Riemann-Riemann symmetric rotor

In the previous section we have demonstrated, that the Caputo-Riemann-Riemann

symmetric rotor correctly determines the magic numbers in the single particle spectra
for neutrons and protons. However, there remains a significant difference between
calculated and experimental ground state masses for nuclei with nucleon numbers
far from magic shell closures. This is a strong indication for the fact, that the fine
structure of the single particle levels is not yet correctly reproduced.
We therefore propose the following generalization of the Caputo-Riemann-
Riemann symmetric rotor group:
C3C2R3R3 G ⊃ C SOα (3) ⊃ C SOα (2) ⊃ R SOα (3) ⊃ R SOα (3) (59)
with the Casimir operators (18) and (19) it follows for the Hamiltonian H:
ω1 2 ω2 2 ω3 2
H= C L1 (α) + Bω0 C Lz1 (α) + R L2 (α) + R L3 (α) (60)
~ ~ ~
with the free parameters ω1 , ω2 , ω3 , B, where B may be called fractional magnetic field
strength in units [~ω0 ].
Imposing the same symmetries (25),(26) as in the case of the symmetric Caputo-
Riemann-Riemann rotor, the eigenvalues of the Hamiltonian (60) are given as
E(α) = ~ω1 C [2n1 ][2n1 + 1] + B~ω0 C [2n1 ] +
~ω2 R [2n2 ][2n2 + 1] + ~ω3 R [2n3 ][2n3 + 1] (61)
3
X Γ(1 + (2ni + 1)α) Γ(1 + α)
= ~ωi − δn1 0 ~ω1
i=1
Γ(1 + (2ni − 1)α) Γ(1 − α)
Γ(1 + (2n1 )α) 1
+ B~ω0 − δn1 0 B~ω0
Γ(1 + (2n1 − 1)α) Γ(1 − α)
n1 , n2 , n3 = 0, 1, 2, .. (62)
on a basis |2n1 2n1 2n2 2n2 2n3 2n3 >.
We call this model the extended Caputo-Riemann-Riemann symmetric rotor.
The additional C Lz1 (α) term yields a level splitting of the harmonic oscillator set of
magic numbers nmagic 1 (34), while the multiplicity of the nmagic 2 set (37) remains
Fractional dynamic symmetries and the ground state properties of nuclei 12

Figure 6. For α = 1/2, on the left side the level spectrum for the extended
Caputo-Riemann-Riemann symmetric rotor (62) is plotted as a function of the
fractional magnetic field strength B. The levels are labeled according to the
corresponding [N ljz ] Nilsson scheme and the multiplicity is given. For α = 0.48
the resulting level sequence near N ≈ 126 is plotted on the right. At B ≈ 0.25
the resulting spectrum coincides with the corresponding spherical Nilsson level
spectrum. Brackets indicate the proposed appropriate combinations of rotor
levels.

unchanged, since this set is characterized by n1 = 0. This is exactly the behaviour

needed to describe the experimentally observed fine structure, as can be deduced from
the right hand side of figure 2.
In order to clearly demonstrate the influence of the additional term, we first
investigate the level spectrum for the spherical (31) and idealized case α = 1/2.
The level spectrum (62) simply results as:
3 1
E(α = 1/2) = ~ω0 (n1 + n2 + n3 + − δn1 0 )
2 2
n1 ! 1
+ B~ω0 ( − δn 0 ) (63)
Γ(1/2 + n1 ) Γ(1/2) 1
3 1
= ~ω0 (n1 + n2 + n3 + − δn1 0 )
2 2
B~ω0 2n1 n1 !
+ √ ( − δn1 0 ) (64)
π (2n1 − 1)!!
where !! denotes the double factorial.
On the left side of figure 6 this spectrum is plotted in units [~ω0 ]. Single levels are
labeled according to the Nilsson-scheme and multiplicities are given in brackets. For
small fractional field strength B the resulting spectrum exactly follows the schematic
level diagram of a phenomenological shell model with spin-orbit term, as demonstrated
e.g. by Goeppert-Mayer [1].
A small deviation from the ideal α = 1/2 value reproduces the experimental
spectra accurately: For α = 0.48 the resulting level spectrum is given on the right hand
side of figure 6. Obviously there is an interference of two effects: First, for α 6= 1/2 now
the degenerated levels of both magic sets split up and second the fractional magnetic
Fractional dynamic symmetries and the ground state properties of nuclei 13

Figure 7. Comparison of calculated Emic from the extended Caputo-Riemann-

Riemann symmetric rotor (62) with the parameter set (65)-(68), minimized with
respect to 2 with the experimental masses from Audi [38] as a function of N .
From top to bottom the minimized 2 values, theoretical and experimental masses
and error in [M eV ] are plotted.

field B acts on the subset nmagic 1 . For B ≈ 0.25 the spectrum may be directly
compared with the spherical Nilsson level scheme, which is given for neutrons between
82 ≤ N ≤ 126 as 2f 27 , 1h 92 , 1i 13 3 5 1
2 , 3p 2 , 2f 2 , 3p 2 , see e.g. results of [39], which
corresponds to a sequence of sub-shells at 90, 100, 114, 118, 124, 126. This sequence is
correctly reproduced with the extended Caputo-Riemann-Riemann symmetric rotor.
With the parameter set, which is obtained by a fit with the experimental masses
of Ca-, Sn- and Pb-isotopes
1
~ω0 = 28A− 3 [M eV ] (65)
(
0.480 + 0.00022 Z Z > 50
αZ = (66)
0.324 + 0.00332 Z Z ≤ 50
Fractional dynamic symmetries and the ground state properties of nuclei 14
(
0.446 + 0.00022 N N > 29
αN = (67)
0.356 + 0.00332 N N ≤ 29
B = 0.27 (68)
the experimental masses are reproduced with an rms-error of 1.7[MeV]. Results are
given in figure 7.
The deformation parameters, obtained by minimization of the total energy, are to
a large extend consistent with values given in [31] e.g. for 264 Hs108 we obtain 2 = 0.22,
which conforms with Möller‘s (2 = 0.2) and Rutz‘s results [8]. However, there occur
discrepancies mostly for exotic nuclei. For example our calculations determine the
nucleus 42 Si to be almost spherical, while Möller predicts a definitely oblate shape.
Finally, defining a nucleus with 2 > 0.05 as prolate and with 2 < −0.05 as oblate
the amount of prolate shapes is about 74% of all deformed nuclei. This is close to the
value of 82% [40], obtained with the Nilsson model using the standard parameters.
Summarizing the results presented, the proposed extended Caputo-Riemann-
Riemann symmetric rotor describes the ground state properties of nuclei with
reasonable accuracy. We have demonstrated, that the nuclear shell structure may
indeed be successfully described on the basis of a dynamical symmetry model.
The advantages of this model, compared to phenomenological shell and relativistic
mean field models respectively are obvious:
Magic numbers are predicted, they are not the result of a fit with a
phenomenological ls-term. There are no potential-terms or parametrized Skyrme-
forces involved and finally, all results may be calculated analytically.
The results obtained encourage further investigations in this field. The next
steps should include the pairing energy term and parameters should be determined
by a more sophisticated fit procedure. With these additional contributions the model
will most probably describe nuclear properties with at least similar accuracy as the
models currently used.

6. Conclusion

Based on the Riemann- and Caputo definition of the fractional derivative we used
the fractional extensions of the standard rotation group SO(3) to construct a higher
dimensional representation of a fractional rotation group with mixed derivative types.
We obtained an extended symmetric rotor model, which predicts the sequence of
magic proton and neutron numbers accurately. Furthermore we have shown, that the
ground state properties of nuclei can be reproduced correctly within the framework of
this model.
Hence we have demonstrated, that a dynamic symmetry, generated by mixed
fractional type rotation groups is indeed realized in nature.

7. Acknowledgment

We thank A. Friedrich for useful discussions.

8. References
[1] Goeppert-Mayer M Phys. Rev. 75 (1949) 1669.
[2] Haxel F P , Jensen J H D and Suess H D Phys. Rev. 75 (1949) 1769.
Fractional dynamic symmetries and the ground state properties of nuclei 15

[3] Nilsson S G Kgl. Danske Videnskab. Selsk. Mat.-Fys. Medd. 29, (1955) 431.
[4] Nilsson S G et al. Nucl. Phys. A 131, (1969) 1.
[5] Hofmann S and Münzenberg G Reviews of Modern Physics, 72, (2000) 733.
[6] Rufa M et al. Phys. Rev. C 38 (1988) 390.
[7] Bender M, Nazarewicz W, Reinhard P-G Phys. Lett. B515 (2001) 42.
[8] Rutz K et al. Phys. Rev. C 56 (1997) 238.
[9] Kruppa A T et al. Phys. Rev. C 61 (2000) 034313.
[10] Elsasser W M J. Phys. Radium 5 (1934) 635.
[11] Elliott J P Proc. Roy. Soc. London A245, (1958) 128.
[12] Iachello F and Arima A, 1987 The Interacting Boson Model Cambridge University Press,
Cambridge.
[13] Miller K and Ross B 1993 An Introduction to Fractional Calculus and Fractional Differential
Equations Wiley, New York.
[14] Oldham K B and Spanier J 2006 The Fractional Calculus, Dover Publications, Mineola, New
York.
[15] Podlubny I 1999 Fractional Differential equations, Academic Press, New York.
[16] Herrmann R 2008 Fraktionale Infinitesimalrechnung - Eine Einführung für Physiker, BoD,
Norderstedt, Germany
[17] Leibniz G F Sep 30, 1695 Correspondence with l‘Hospital, manuscript.
[18] Euler L Commentarii academiae scientiarum Petropolitanae 5, (1738) pp. 36-57.
[19] Liouville J 1832 J. École Polytech., 13, 1-162.
[20] Riemann B Jan 14, 1847 Versuch einer allgemeinen Auffassung der Integration und
Differentiation in: Weber H (Ed.), Bernhard Riemann’s gesammelte mathematische Werke
und wissenschaftlicher Nachlass, Dover Publications (1953), 353.
[21] Caputo M Geophys. J. R. Astr. Soc. 13, (1967) 529.
[22] Weyl H Vierteljahresschrift der Naturforschenden Gesellschaft in Zürich 62, (1917) 296.
[23] Feller W Comm. Sem. Mathem. Universite de Lund, (1952) 73-81.
[24] Riesz M Acta Math. 81, (1949) 1.
[25] Grünwald A K Z. angew. Math. und Physik 12, (1867) 441.
[26] Herrmann R J. Phys. G: Nucl. Part. Phys. 34, (2007), 607.
[27] Herrmann R arxiv:math-ph/0510099.
[28] Maruhn J and Greiner W Z. Physik 251, (1972) 431.
[29] Strutinsky V M Nucl. Phys. A95, (1967) 420.
[30] Strutinsky V M Nucl. Phys. A122, (1968) 1.
[31] Möller et. al. Atomic Data Nucl. Data Tables 59, (1995) 185.
[32] Herrmann R arxiv:gen-ph/0801.0298.
[33] Myers W D and Swiatecki W J Nucl. Phys. 81, (1966) 1.
[34] Sobiczewski A, Gareev F A and Kalinkin B N Phys. Lett. 22 (1966), 500.
[35] Meldner H Ark. Fys. 36, (1967) 593.
[36] Mosel U, Fink B and Greiner W Contribution to ”Memorandum Hessischer Kernphysiker”
Darmstadt, Frankfurt, Marburg, (1966).
[37] Mosel U and Greiner Z. f. Physik 217 (1968) 256, 222 (1968) 261.
[38] Audi G, Wapstra A H and Thibault C Nucl. Phys. A729 (2003) 337.
[39] Scharnweber D, Mosel U and Greiner W Phys. Rev. Lett. 24 (1970) 601.
[40] Tajima N and Suzuki N Phys. Rev. C 64 (2001) 037301.