# Combinatorial nuclear level-density model Page: 4 of 14

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3

1?

U

0

.660

9

S40 -. .- - .

g 20 I --

7

00 1 2 3 4 5 6 7 8

6 Excitation Energy [McV]

s

4 --

2 _

Excitation Energy [MeV]FIG. 1: (Color online) Rotational enhancement, Krot, for the

nucleus 162Dy as a function of excitation energy. The inset

shows the enhancement compared to the simple enhancement

model of Ref. [26] (red dashed line).

slowly increasing function, of the order of a factor 5. This

prediction is in contrast to the SU(3) model of Ref. [26],

which is often used to model NLD (see eg. Ref. [27]) and

which is shown in the inset of Fig. 1. The SU(3) model

gives almost an order of magnitude larger enhancement

for excitation energies in the region of the neutron separa-

tion energy. Furthermore, there have been experimental

efforts to search for the fade out of collective rotations

at excitation energies far beyond the neutron separation

energy, as given by the SU(3) model. These searches

[28, 29] have given rio experimental support for the fade-

out profile used.

In the present model of combinatorially taking into ac-

count the rotations there is an explicit double counting

of levels. However, this effect is expected to be negligi-

ble for excitation energies below the neutron separation

energy [5].

C. Vibrations

In order to describe vibrational states the Quasi-

particle-Tamm-Dancoff-Approximation (QTDA) is used.

According to the Brink-Axel hypothesis [30, 31] phonons

are built on every intrinsic many-body configuration

Emb. The QTDA equation is solved for each state in or-

der to get phonon excitation energies and wave-functions.

The residual interaction is approximated by the double

stretched Quadrupole-Quadrupole interaction. This in-

teraction is well defined in the case of a harmonic oscil-

lator potential. In the case of a finite-depth potential as

the folded-Yukawa potential the interaction should take

into account additional finite size effects, for example as

is done in Ref. [32]. In the present work the finite-depth

effects are ignored and the double stretched approach is

used as defined in Refs. [33, 34].The QTDA secular equation can be written [35]

12

1 {_ z Q2KI ) 2(U,V.+/U,)2

X2K a ( E, + Ep) - ttw -, (10)

where the effect of Eq. (4) has not been explicitly writ-

ten out. Q2K is the double stretched quadrupole op-

erator, where the components K = 0 and K = 2 are

considered. The roots hw of this equation are the exci-

tation energies of the vibrational phonons, whereas the

poles Epp = (e, - A)2 + .2 are the unperturbed two-

quasiparticle excitations on the many-body configuration

Emb and the e are the single-particle energies.

The self-consistent coupling strength is given by [33]82r Mwo

X2K - ,

A (T2) + g2K 5 AQ20)(11)

where 20 = 1 and g22 = -1. The expectation-values

(T2) and (Q2o) are calculated in double stretched coor-

dinates [33].

Double counting is explicitly avoided by the following

procedure. The phonon wave functions are given byC'J = X,,,aga,

,(12)

where XM, are the wave-function components of all ex-

cited quasi-particle states a a, on top of the many-

particle-many-hole configuration Emb. The level density

is increased by one state at the energy of the phonon duw,

and decreased by the amount given by the wave-function

component X ,~ at the energy of the corresponding pole.

The change in level density due to one phonon is thus

6p(E) = 8(E -Emb -- w)-

X ,,b (E - Emb - (Egp + EqP)) , (13)

where E is the excitation energy relative to the ground-

state.

The vibrational enhancement factor in this method is

in general quite small, of the order of a few percent. This

is in sharp contrast to other methods that describe the

vibrational enhancement in level densities. For example,

the often employed attenuated phonon method gives up

to an order of magnitude enhancement at the neutron

separation energy [10, 12, 36]. Fig. 2 shows the vibra-

tional enhancement as a function of excitation energy for

rs2Dy. The effect is very small, close to 1 % at 7 MeV

excitation energy. For the same nucleus the attenuated

phonon method gives an enhancement factor of about 3

as shown in the inset of Fig. 2.

In calculations for a large number of nuclei it is pos-

sible to extract systematics of the Giant Quadrupole

Resonances (GQR) and test if the double stretched

quadrupole interaction is reasonable. Fig. 3 shows the

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Moller, Peter; Aberg, Sven; Uhrenhoit, Henrik & Ickhikawa, Takatoshi. Combinatorial nuclear level-density model, article, January 1, 2008; [New Mexico]. (digital.library.unt.edu/ark:/67531/metadc933186/m1/4/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.