A SEMICLASSICAL HARTREE-FOCK METHOD

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A SEMICLASSICAL HARTREE-FOCK METHOD

J. Bartel

To cite this version:

J. Bartel. A SEMICLASSICAL HARTREE-FOCK METHOD. Journal de Physique Colloques, 1984,

45 (C6), pp.C6-205-C6-212. �10.1051/jphyscol:1984624�. �jpa-00224225�

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n06, Tome 45, juin 1984 page C6-205

A S E M I C L A S S I C A L HARTREE-FOCK METHOD

I n s t i t u t für Theoretische Physik, Universitat Regensburg, 0-8400 Regensburg, F . R. G.

Résumé - La méthode de la resommation partielle en k est présentée comme une approche semiclassique pour calculer des propriétés du fondamental nucléaire d'une façon selfconsistante

partir d'une interaction nucleon- nucleon effective. Nou montrons que cette méthode peut être facilement généralisée àla description de système excités. Des densités semiclassiques selfconsistantes peuvent par ailleurs être utilisées comme point de

départ idéalpour le calcul d'énergies de déformation nucléaires (par exemple des barrières de fission).

Abstract - We present a semiclassical approach, the partial fi resummation method, to calculate nuclear ground state properties in a selfconsistent way starting from an effective nucleon-nucleon interaction. This method is shown to be easily generalized tu describe excited nuclear systems.

Selfconsistent semiclassical densities can be further used as

optimal starting point for the calculation of nuclear deformation energy surfaces like fission barriers.

There has been a considerable progress over the last ten years to describe static and low energy dyriamical nuclear properties in the Hartree-Fock (HF) framework using density-dependent effective interactions of the Skyrme or Gaussian type /1/. These calculations which are relatively simple and economic for spherïcal shapes become quickly quite involved when deformed shapes or excited systems are considered. A link between this microscopicmethodand nuclear mass formulae of average nuclear properties like the liquid drop or droplet mode1 is given by the Strutinsky theorem, which corresponds to an expansion of the HF energy E(p) around its average (semi- classical) part E(P). The shell correction energy

though being small (G~/~(p)sl%) is responsible for essential nuclear properties like ground state deformations or the existence of a double hump fission barrier. The treatment of shell effects in perturbation, as done in the Strutinsky method, has been shown to reproduce the HF energy within less thanl MeV under the condition that the average density matrix is determined in a selfconsistent way

II - THE RESUMMATION METHOD

In the one-body approximation of the nuclear N-body problem semiclassical quantities can most easily be obtained using the single-particle Bloch density C J

e-% / 3 / . The latter can be used to calculate the density matrix

(r,r4) through an inverse Laplace transform

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984624

C6-206 JOURNAL DE PHYSIQUE

Knowing the exact Bloch density of a problem, the exact density matrix is obtained.

This is however generally not the case and approximations have to be made. The simplest semiclassical approach is the Thomas-Fermi method yielding a density distri- bution p (2) only defined inside the classically allowed region r

rcl, where the classical turning point relis determined through the Fermi energy h by V b _cl

h .

Calculating the Wigner transform of the Bloch density, a systematic expansion in powers of fi is obtained, called the Wigner-Kirkwood expansion, with semiclassical corrections to the Thomas Fermi approximation

~lthoucjh a very rapidly converging expansion for the energy is obtaind / 6 / , densities are confined to the classically allowed region and even diverge at the classical turning point

To cure the turning point problem two methods have been proposed.

One of them, the extended Thomas-Fermi (ETF) approach expresses the kinetic and spin-orbit densities as functionals £0 the local density p (?) which allows for self- consistent calculations with density dependent effective forces in the variational sense. In the other method al1 up to first (or second) derivatives of the central potential are consistently resummed in equ. (2) to al1 orders in fi /3,10/. Resumming

Fig. 1 - Illustration of the local approximation of a potential (Woods-Saxon type) in the resummation method by a linear (dashed line) or a harmonic potential (dash- dotted line).

This linearized form corresponds to a local approximation of V(r) by a constant

d o p e potential. Resumming up to second derivatives /IO/ leads to a local approxi- mation by a harmonic potential as shown in figure 1. Performing the Laplace

inversions (eq. (1)) by the saddle point method, which can be shown to be a

semiclassical expansion /Il/, density distributions are obtained which reproduce the quantum mechanical distributions on the average, washing out al1 quantum oscillations.

What is espec~ally gratifying is to notice that the surface of the systern is very

well reproduced, as shown in figure 2. This feature is of crucial importance if

we aim at using these semiclassical densities in an iterative HF-like cycle with

density-dependent effective interactions like those of the Skyrme type /12-14/. Such

a selfconsistent (semiclassical) program is indeed possible and yields nuclear

ground state energies and density distributions in good agreement with the results

of the corresponding HF-cycle /15/. As an example we show in figure 3 the self-

consistent proton and neutron density- and potential distributions calculated with

the. Skyrme SI11 force /13/ for the nucleus 208~b. The ground state energies and rms

Fig. 2 - Comparison of the exact, quantum-mechanical density distribution (full line) of 92 particles in a spherical Woods-Saxon potential (Vo

- ^{44 MeV,}

Ro

1.27 ~ l / 3 fm, a

0.67 fm) with the semiclassical densities/ll/ obtained with the saddle-point method from the harmonised approximation /IO/ (dashdotted line) and the Bhaduri approach /3/ (dashed line) to the resummed Bloch densities.

Fig. 3 - Comparison of the selfconsistent semiclassical density- and potential- distributions (full line) for protons and neutrons with the corresponding HF-results as obtained for the nucleus 2 0 8 ~ b with the Skyrme SI11 force.

radii for a number of closed shell nuclei are shown in table 1 as obtained with the modified SkM force of ref. /14/. For the cornparison of binding energies, a

Strutinsky shell correction energy 6E has of course to be taken into account. The agreement with the corresponding HF results seems quite satisfactory. Starting £rom the selfconsistent semiclassical calculations, quantum oscillations can be taken into account in a perturbative way as done for the energy by the Strutinsky method.

This presents the semiclassical method like an interesting alternative for full HF

calculations.

JOURNAL DE PHYSIQUE

Table 1 - Comparison of selfconsistent semiclassical (sk) energies (in MeV) and neutron and proton rms radii with the corresponding HF-result obtained with modified SkM force-/14/. The energies obtained in the ETF approach have also been reported.

An application where shell oscillations are of no interest and even hinder the pro- cedure, is the determination or test of mass formula paranetrisations like the liquid drop or the droplet mode1 /15,16/. In that way one can determine unambiguously surface /14/ or curvature properties associated with a given nucleon-nucleon inter- action, which is for instance not possible by an analysis of the results of corres- ponding HF calculations.

Up to now we have limited ourselves to spherical shapes and to non excited systems.

Before coming back to "cold" nuclei in section

we will briefly explain how the semiclassical fi-resummation method can be generalized to the description of excited nuclear systems.

III - SELFCONSISTENT SEMICLASSICAL E,BTHODS FOR EXCITED NUCLEI

Considering excited systens we will always suppose that the system is in a thermal equilibrium which can be characterized by a (nuclear) temperature T. The question how this equilibrium situation has been reached is closely connected to the equi-

libration processes of intrinsic excitation,a question which is beyond the scope of the present study and cannot be answered here. Still, highly excited systems have become subject of intensive studies in the last few years mainly for applications in high energy heavy-ion reactions /17/ anü in nuclear astrophysics /18/. Indeed, hot dense nuclear matter is expected in the ultimate evolution stage of a massive star, namely in its gravitational collapse to a neutron star or a black hole. Self- coiisistent microscopic calculations in the framework of the HF-method at finite temperature are quite standard today

and can be used in the perspective to calculate the equation of state of natter at densities near nuclear natter saturation density

Considering the complexity of these calculations and keeping in mind that shell effects disappear at excitations correspondinq to a tenperature of about 2.5 to 3 ) l e V /19/, the question arises, whether it is not possible to treat these systems semiclassically.

The variational quantity of a system which can be characterized by a temperature T is the free energy F

E - TS, where the intrinsic energy E(T) of the system can be calculated for a Skyrme-force - as in the case T

O - as the integral over an

+ -+

energy density which is a functional of the local densities

(g) ,

( 2 ) , ^{J (r)} . ^These

will of course be different for different temperatures. In the resummation method

this tenperature dependence can be incorporated very easily by writin~ /.16,21/

where

The entropy S is then simply given by /16/

where

To include a temperature dependence in the ETF formalism is less trivial. One possibility consists in adding the "cold" semiclassical corrections r 2 ^and

to the "hot" TF functional T T ~ [ ~ ] /21-23/. The often used "low temperature expan- sion" which is simply an approximation of the former method has been shown to be inadequate in the nuclear surface and yield quantitatively wrong results /21/. As an example, we show in figure 4 the neutron and proton rms radii of 2 0 8 ~ b as a function of the temperature T. Both'the resummation method and the ETF approach, are in good agreement with the HF data (note the scale) whereas the low temperature expansion drastically fails for T

3 MeV. Finally it has been possible to generalize the ETF method to finite temperatures in a rigorous way /24/, which leads to a still better agreement with the HF results and those of the resummation method /24/.

- '08pb Skyrme ïïi ^-

- ^HF -.-.-

- ETF* f -

-- -

- resum.

- r~

I I 1 1

Fig. 4 - The neutron and proton rms radius of '08pb obtained in different models with the Skyrme force SI11 as a function of the temperature T.

Another interesting application of the semiclassical method is given by the calcula- tion of nuclear deformation energy surfaces.

IV - AN APPROXIMATION TO FISSION BARRIER CALCULATIONS

To calculate deformation energies, i.e. the energy of

system as a function of its deformation one generally has to perform a cofistrained HF (CHF) calculation imposing a certain deformation to the system by imposing a aonstraint to one or several multipole moments (see ref. /1/ for a review). A good approximation to the CHF approach is given by the expectation value method (EVM), which estimates the deform- ation energy Eidef) by the expectation value of the total Skyrme Hamiltonian taken between Slater determinants

depending on one or several deformation parameters R i 1 ' 2 5 , 1 4 1 '

The Q are constructed with the single-particle wave functions which are eigenstates

of the one-body Hamiltonian

JOURNAL DE PHYSIQUE

+ * + ^-+

In previous calculations /25,14/, Vq (r) , m/mq (r) and Sq (r) were f itted to Fermi-type functions reproducing the distribut~ons obtained in a spherical HF calculation and the Coulomb potential Vc (9) was neglected in Hq. Deforming these functions with the {c,h} parametrization used in Strutinsky calculations /26/ leads to a correct deformation behaviour, although the absolute value of the deformation energy is shifted up by some 20 MeV, mainly due to the neglection of Ve in eq. (7). Instead of performing a spherical HF calculation it seems quite appealing to use selfcon- sistent semiclassical densities to calculate the quantities entering in Hq (eq. (7)) like Vq (l') ^{, m / G} (r) ^{, .Sq} (2) as well as the Coulomb-potential Vc (l') . Similar approaches have been discussed /8,27/. As a first result of such calculations we show in fig.

Fig. 5 - Contourplot of the deformation energy surface of

calculated in the expectation value method with the modified Sul force. The energy is shown as a function of the deformation parameters {c,h} of ref. /26/.

the fission barrier of 2 4 0 ~ ~ as obtained with the modified SkM force. This parametri- zation was obtained /14/ from the Skyrme force SkM /27/ by adjusting the parameters tl and t2, determining the surface properties of the force, such that the semi- classical fission barrier calculated within the ETF method reproduced the empirically known liquid drop barrier. The ground state (c

1.25, h

- 0.2), the first barrier

(C

1.25, h

0.1) and the fission isomeric state (c

1.35, h

0.05) are clearly seen. These results are preliminary in the sens that the second barrier (c = 1.85 h = - 0.25) and the descent to the scission point are still missing. Comparing this deformation energy with experinental data, various corrections have to be taken into account /29/, including a truncation error due to an expansion of the eigen- function in an oscillator basis, to spurious translational and rotational energies and to the undue imposition of axial and left-right reflection symmetries.

2 3 0 ~ h

Table 2 - Comparison of

z * O ~ u

HF EVM

experimental charge radii,

-P exp HF EVM

rc

qudrupole and hexadecapole

5.79 5.81

5'90 5'91

moments for ground state

~ 2 . ~ .

Q : .

"

~ 4 . ~ * ~ . ( ~ ~ )

9.00I0.05 9.10 9 - 7 1 1 .09*0.15 1.23 i.25

24.5 25.4 1.25 1.28

11.58t0.08 11 - 6 2 1 1 - 2 9

(g-'.) and isOmeric state (f . ^i. s) with the

1.1510.28 7 - 2 2 1.23

results obtained in a full HF calculation and the

36 4 32.6 33.4

expectation value method

explained in the text.

3.89 3.87

Applying these corrections the result of Our calculation is consistent with experi- mental data. To illustrate this fact, we compare in table 2 the ground state rms charge radius as well as the quadrupole and hexadecapole moments for ground state and fission isomeric state,so obtained,with the results of a full HF calculation and, as far as available, to the corresponding experinental data. The agreement seems very satisfactory, which gives good confidence to the reliability of the expectation value method used.

V - CONCLUSION

The partial fi resummation method has been shown to be a very reliable and practical tool to estimate nuclear properties £rom a given effective nucleon-nucleon inter- action. The method has been shown to be easily extended to the description of excited nuclei. Using the result of these semiclassical calculations as an input we have further found an interesting way to estimate nuclear deformation energy surfaces in general, especially used here to calculate fission barriers of actinide nuclei. Finally, the semiclassical method being a very economical and flexible approach it certainly has to play an important role in the long standing project of the improvement of an effective force.

The author is grateful to M. Brack for fruitful discussions.

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