LEVEL DENSITY FROM REALISTIC NUCLEAR POTENTIALS
A. CALBOREANU
“Horia Hulubei” Institute of Physics and Nuclear Engineering, Bucharest, Romania
Received October 5, 2006
Nuclear level density of some nuclei is calculated using a realistic set of single
particle states (sps). These states are derived from the parameterization of nuclear
potentials that describe the observed sps over a large number of nuclei. This approach
has the advantage that one can infer level density for nuclei that are inaccessible for a
direct study, but are very important in astrophysical processes such as those close to
the drip lines. Level densities at high excitation energies are very sensitive to the
actual set of sps. The fact that the sps spectrum is finite has extraordinary
consequences upon nuclear reaction yields due to the leveling-off of the level density
at extremely high excitation energies wrongly attributed so far to other nuclear
effects. Single-particle level density parameter a is extracted by fitting the calculated
densities to the standard Bethe formula.
INTRODUCTION
Nuclear state density is one of the earliest concepts in nuclear physics
introduced by Bethe soon after the composition of nuclei was firmly established
[1]. It is a basic nuclear property and an ingredient of the theory of nuclear
reactions. It can be obtained from experimental data on neutron and proton
resonances, from cross section fluctuation analysis and from the slope of the
particle spectra in nuclear reactions. Energy dependence of the cross-sections,
the relative yield of various reaction channels and the gamma emission from
excited nuclei offer additional information about the level density. Though the
most certain way to infer the level density is by direct counting, the method is
restricted to a relatively small excitation energy range above the ground energy.
Even so they are affected by major experimental uncertainties and may reflect
peculiarities with little capacity of generality. However all level density formulae
have to account for the low excitation energy limit.
In spite of its respectable age and notoriety the concept of nuclear level
density is revived from time to time in accord with the new questions to which it
is requested to respond. Bethe’s original formula was adapted to describe
simultaneously the density of neutron resonance spectra and the density of low
energy levels. Other developments where directed to account for isotopic effects,
for odd-even differences, spin distribution, collective effects, or for nuclear
Rom. Journ. Phys., Vol. 51, Nos. 9–10, P. 859–866, Bucharest, 2006
860
A. Calboreanu
2
deformation influence. A host of level density formulae provide an overall
description of the experimental data for a large atomic mass interval, satisfactory
in many applications [2].
At present the advances in nuclear astrophysics and heavy ion reactions
emulate the interest in level density. Both fields, which intersect in some cases,
need a deeper understanding of the behavior of the level density at high energy
of excitation. Hot nuclear matter properties at T ~ 8 MeV/A, characteristic to the
early universe (0.1–1 s) as well as the properties of the level density in
unexplored mass regions far from stability and for Z > 92 elements, are
challenging experimental and theoretical problems.
In the present work we discuss several theoretical concepts and computational approaches of the level density of nuclei. Various models and approximations have merits and limitations as no theoretical model is capable to offer a
satisfactory quantitative description of level density over a large range of atomic
masses and excitation energies. We performed comparative studies with the
classical Fermi gas model, with combinatorial and exact “brute force” calculations, with Monte Carlo simulations, with realistic micro canonic calculations
and with spectral distribution method based on the Gaussian orthogonal ensemble.
FERMI GAS DISTRIBUTIONS
In most of the nuclear level density theories the nucleus is viewed as an
assemble of nucleons moving independently in a potential well that generate a
set of single particle states (sps) To determine the density of states means to
calculate the number of ways by which nucleons can realize the total energy of
the system by adding up their individual energies. Bethe, assimilating the
nucleons to a Fermi gas filling an equally spaced set of single particle states and
applying the standard statistical methods, produced his celebrated formula for
the level density
ρ(E ) =
π1/ 2
e2
1/
4
5
/
4
12 a E
aE
(1)
where E is the excitation energy of the nucleus and a is the level density
parameter related to the mean densities of single particle states around the Fermi
energy gn ( ε F ) and g p ( ε F )
a = 1 π2 ⎡⎣ gn ( ε ) + g p ( ε ) ⎤⎦
6
(2)
Though conceptually correct Bethe’s formula fails to predict the behavior
of level densities at both low and high excitation energies regions. These limits
are due to the physical assumptions made for its derivation. At low energy, i.e.,
3
Level density from realistic nuclear potentials
861
few MeV above the ground state, discrepancies are due to the residual interactions
of all sorts and to the nuclear deformation in the middle of shells. The progress
in the theoretical description of the structure of nuclei made possible a more
realistic consideration of nucleon sps and the effect of residual interactions. As
no theoretical calculations can account for all levels in a given energy interval,
for practical reasons semi empirical formulae have been devised to deal with the
problem [2]. These formulae realize a fit of experimental data with a modified
Bethe formula introducing as many extra parameters as necessary for the aimed
precision. While these expressions have a great advantage for many applications in
reactor physics, nuclear analysis methods, or radiation protection and safeguard,
they contain little fundamental substance. Also these formulae are restricted to a
narrow range of validity of the basic physical quantities of interest (mass, atomic
number, energy, angular momenta, etc.).
REALISTIC SHELL MODEL CALCULATIONS
New theoretical methods have been devised to deal with the density of states
of complex systems. Nuclear spectroscopy has found means to deal conveniently
with states very far from the ground states. It is possible to find average properties such as distribution of states, transition strengths, or any other observable in
a quite straightforward way. The computing facilities made possible to solve
some problems completely, for instance to find all eigenvalues of a given
configuration with a limited number of active nucleons. When averaged over an
energy interval they feature a remarkable Gaussian distribution similar to the
Gaussian Orthogonal Ensemble (GOE) [3]. To calculate the distribution of states
for a given configuration it is necessary to calculate the low order moments of
the many particle Hamiltonian H over the respective subset of many particle
states configuration. The first moment will give the centroid, the second moment
the width, while the higher order moments the skewness, the excess, the kurtozis
and so on. The main assumptions of the spectral distribution method are:
– the central limit theorem operates for the shell model Hamiltonian
– only the lowest order moments are significant
– the moments can be calculated using propagation techiques [3], connecting
the original single particle configuration (base) to the many particle interactive
system.
The density of states is given by te expression:
ρ ( E ) = f ( D, σ ) e
( Ec − E )2
σ2
[1 + higher terms]
(3)
where f contains the dimensionality D of the configuration, Ec is the centroid
enegy and σ is the width
862
A. Calboreanu
Ec = H = 1 Tr ( H ) = 1
D
D
∑ φHφ
4
σ2 = H 2 − H
2
(4)
To illustrate the merits of this approach we consider a one body Hamiltonian
acting on P particles distributed over N states, equally spaced by energy δ. The
density of states in this case is
ρ( E , P ) =
C NP β2 ( E − Ec )2
e
πδ
(5)
where the Fermi energy, the centroid energy and the width are
EF =
P −1
∑ εi
0
σ2 =
⎡ N
⎤
E0 = 1 ⎢
εi − E F ⎥
2⎢
⎥⎦
⎣ N − P +1
P(N − P) ⎡
N ( N − 1) ⎢⎣
∑
∑ εi2 − N1 ( ∑ εi )
2⎤
⎥⎦
(6)
(7)
For a system of equidistant sps {0, 1, 2, …, N} the density distribution is
Gaussian and we have
σ2 = δ P ( N − P )( N + 1)
12
2
E F = 1 P ( P − 1)δ
2
(8)
where δ is the distance between two neighbor states.
Fig. 1 shows a three dimensional representation of the level density for P
particles distributed over 30 sps uniformly spaced (δ = 1). One can remark the
peculiar bell shape of the distribution due to the limitation of the sps space. After
reaching a maximum at the energy E0, where the occupation of the states is
evenly distributed, the density decreases as the particles occupy preferentially
the upper side of the sps spectrum. There is also a maximum attainable excitation
energy realized if the P uppermost sps are occupied. In real cases the probability
of such situation is however reduced by the existence of unbound sps that absorb
gradually a part of nucleons. These nucleons become free and the initial nucleus
looses its original identity. This approach has however serious drawbacks related
to the difficulties to deal with multi configurations of sps and to account for both
neutrons and protons simultaneously.
MICRO CANONIC LEVEL DENSITY
From the above results it appears that a realistic treatment of the sps
spectrum is necessary in order to describe the behavior of the level density at
high excitation energies. Even the crude model employed has revealed that the
level density does not increase indefinitely with energy. This conclusion has
5
Level density from realistic nuclear potentials
863
anticipated in fact experimental data on the spectra of particles emitted from
highly excited nuclei (100 MeV and above). One of the first experiments
revealing a flattening of the level density at higher energies was performed by
Nebbia and collab. [4]. They associated this flattening (with a variation of the
level density parameter with energy from a = A/8 MeV–1 at E = 20 MeV, to
a = A/13 MeV–1 at E = 400 MeV) to a change of the effective mass of nucleons
in excited nuclei.
In this paper we follow the standard statistical methods, considering a
nucleus as a micro canonical assemble composed of N neutrons and P protons
occupying a set of sps generated by a realistic potential. Such potentials based on
systematics of the experimental single particle states throughout the atomic mass
table have been reported in the literature [5, 6].
N is the
If, for a nucleus A with N neutrons and Z protons, {ε1N , ε2N , …, ε m
}
set of bound single particle neutron states and {ε1Z , ε2Z , …, ε nZ } is the set of
bound single particle proton states, the following conditions for the occupancies
niN and niZ must be fulfilled [7]
∑ niN εiN + ∑ niZ εiZ = E
∑ niN = N ∑ niZ = Z
niN
⎡
⎛ ε N − λ N ⎞⎤
⎟⎥
= ⎢1 + exp ⎜ i
T
⎝
⎠⎦
⎣
−1
niZ
⎡
⎛ εZ − λ Z ⎞⎤
⎟⎥
= ⎢1 + exp ⎜ i
⎝ T
⎠⎦
⎣
(9)
(10)
−1
(11)
The entropy of the system S is a sum of partial entropies of neutrons and
protons SN and SZ
S = S N + SZ
(12)
where
SN = −
∑ ( niN ln ( niN ) + (1 − niN ln (1 − niN ) ))
(13)
and similarly for SZ.
The density of states reads
ρ ( E, N , Z ) =
1
1 exp ( S )
3/ 2
Δ
( 2π )
(14)
The d’Alabertian Δ is
( ∑ PiN )( ∑ PiZ )( ∑ εiN PiN + ∑ εiZ PiZ ) −
2
2
− ( ∑ εiN Pi N ) ( ∑ Pi Z ) − ( ∑ εiZ Pi Z ) ( ∑ Pi N )
Δ=
where Pi N = 1 − niN and Pi Z = 1 − niZ .
(15)
864
A. Calboreanu
6
After solving numerically the system of equations (9–11), all the
subsequent quantities are straightforwardly computed. We have performed a
simple test for the 20Ne nucleus. In Fig. 2 the results of these preliminary
calculations are displayed, along with those of two alternative methods: a
Monte Carlo simulation (separate dots) and an exact “brute force” calculation
based a program written by Zimmer [8] Besides the excellent agreement one
can notice the bell shaped distribution encountered in the previous spectral
distribution method calculation with one type of particles distributed over an
equidistant set of sps. This is an indication of the validity of the central limit
theorem even in more complicated cases. Theoretical implications of this fact
will be discussed elsewhere.
Further it is a challenge to see the effect of the limitation of the number of
sps for a system close to a real nucleus encountered in experiments. A clear
evidence for the effect of limitation of sps contributing to the level density was
reported by Nebbia et al. [4]. They found that in order to obtain the experimental
slopes of alpha particle spectra emitted in 14N induced reaction on 154Sm the
level density parameter a entering the Bethe formula must be decreased steadily
from A/8 MeV–1 to A/13 MeV–1 in the excitation energy range from 50 MeV to
400 MeV. We have chosen the nucleus 160Tb as a possible residual nucleus after
the alpha emission. The conclusions will not differ significantly if another
residual nucleus close to it is selected. In Table 1 the sps considered in the level
density calculation for this nucleus are listed. They have been calculated from
the global optical potentials determined from a systematics of sps of spherical
nuclei [5, 6]. In Fig. 3 the density of states of 160Tb is plotted. We note that
lowest quasi-bound proton state (2f) has been included amongst the genuinely
bound states. Also the 3 s neutron state was included in calculation due to its
proximity (0.1 MeV) to the surface of the potential well. The obvious
observation is the quality of the agreement between these calculations and the
experiment. The values of the parameter a of the Bethe formula that would
render the same values of the level density are very close to those extracted from
experimental alpha spectra by Nebbia et al. No other parameters have been
introduced in the theoretical calculation besides the temperature T and the
chemical potential λn and λp that were calculated according to the procedure
described above. This is in spite that an unambiguous determination of the alpha
emitter is lacking. A question may arise as whether a number of nucleons have
disappeared following their placement in continuum. This certainly happens and
a full consideration of scattering states is desirable. Preliminary calculations [9]
have shown that, on average, up to eight protons and six neutrons are probably
placed in continuum at 450 MeV excitation energy of the primary compound
system. However this fact does not affect in any way our conclusions, or the
quality of agreement.
7
Level density from realistic nuclear potentials
865
Table 1
95
The single particle states of 160
generated by the spherical potential
65 Tb
of Ref. [5]. The last 4s neutron state is unbound by 0.10 and 0.65 MeV
respectively. The last 2f proton state is unbound by 1.61 MeV. The
energies are scaled from the deepest (1s) states: 42.11 MeV for
neutrons and 35.08 MeV for protons
Neutrons
Protons
sp
π
ε0
sp
π
ε0
s
p
p
2
4
2
0.0
4.53
5.07
s
p
p
2
4
2
0.0
3.90
4.52
d
d
s
f
6
4
2
8
9.83
11.12
12.32
15.78
d
d
s
f
6
4
2
8
8.74
10.14
11.97
14.37
f
p
p
g
6
4
2
10
18.10
19.41
20.31
22.30
f
p
p
g
6
4
2
10
16.82
18.87
19.75
20.68
g
d
s
d
h
8
6
2
4
12
25.89
26.79
28.64
28.44
29.32
g
d
s
d
h
8
6
2
4
12
24.38
26.15
27.77
28.36
27.60
f
h
p
f
i
p
s
8
10
4
6
14
2
2
34.28
34.32
36.30
36.63
36.78
37.12
42.21
8
10
6
32.69
33.60
36.69
CONCLUSIONS
Nuclear level density remains one of the most interesting concepts of
nuclear physics. Its evolution during more than six decades has revealed new
facets. It was a permanent challenge for theoreticians and a constant target for
experimentalists. Its application spreads from reactor physics, activation methods,
radiation protection and shielding, safeguard, new materials, to the new horizons
of fundamental research of nuclear structure, astrophysics and cosmology.
866
A. Calboreanu
8
Present calculations revealed how important is to use the entire spectrum of
the single particle states to get reliable level densities of highly excited nuclei.
The method described in this work has a high power of prediction, though it
relies on a very few parameters that have themselves a clear physical
significance. The basic ingredient is the set of single particle states that are
determined independently of their application, from analysis nuclear reactions at
low and medium energies. A problem may arise as to what extent sps spectrum
determined from at relatively low energies remains valid in highly excited
nuclear systems. The problem was the subject of many investigations and
temperature dependent mean field calculation have shown that this sps spectrum
is little affected even at temperatures as high as 6 MeV [10]. A second problem
refers to the role of continuum states. As stated before, preliminary theoretical
investigations have shown that a number of constitutive nucleons are placed in
continuum at high excitation energies. Models to deal with the problem have
been described elsewhere [9]. However a thorough investigation is necessary for
a large sample of nuclei. The problem of phase transition in hot nuclei is another
challenge for theoretical investigation related to the problem of nuclear level
density.
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