Single Particle Energies

in Skyrme Hartree-Fock and Woods-Saxon Potentials

Brian D. Newman

Cyclotron Institute

Texas A&M University

Mentor: Dr. Shalom Shlomo

Introduction

Atomic nuclei exhibit the interesting phenomenon of single-particle motion that can be described within the mean field approximation for the many-body system. We have carried out Hartree-Fock calculations for a wide range of nuclei, using the Skyrme-type interactions. We have examined the resulting mean field potentials UHF by fitting r2UHF to r2UWS, where UWS is the commonly used Woods-Saxon potential. We consider, in particular, the asymmetry (x=(N-Z)/A) dependence in UWS and the spin-orbit splitting in the spectra of 17F8 and the recently measured spectra of 23F14. Using UWS, we obtained good agreement with experimental data.

Mean-Field Approximation•Many-body problem for nuclear wave-function generally cannot be solved analytically

•In Mean-Field Approximation each nucleon interacts independently with a potential formed by other nucleons

Single-Particle Schrödinger Equation:

A-Nucleon Wave-Function:

A=Anti-Symmetrization operator for fermions

Mean-Field Approximation

HΨ=EΨ

-60

-50

-40

-30

-20

-10

00 2 4 6 8 10 12

R

Vo

Ui(r)

•Single-particle wave-functions Φi are determined by the independent single-particle potentials

•Due to spherical symmetry, the solution is separable into radial component ; angular component (spherical harmonics) ; and the isospin function :

Mean Field (cont.)•The anti-symmetric ground state wave-function of a nucleus can be written as a Slater determinant of a matrix whose elements are single-particle wave-functions

Hartree-Fock Method

•The Hamiltonian operator is sum of kinetic and potential energy operators:

where:

•The ground state wave-function should give the lowest expectation value for the Hamiltonian

Hartree-Fock Method (cont.)

•We want to obtain minimum of E with the constraint that the sum of the single-particle wave-function integrals over all space is A, to conserve the number of nucleons:

We obtain the Hartree-Fock Equations:

Hartree-Fock Method with Skyrme Interaction

•The Skyrme two-body NN interaction potential is given by:

operates on the right side

operates on the left side

ijP is the spin exchange operator

to, t1, t2, t3, xo, x1, x2, x3, , and Wo are the ten Skyrme parameters

Skyrme Interaction (cont.)

•After all substitutions and making the coefficients of all variations equal to zero, we have the Hartree-Fock Equations:

•mτ*(r), Uτ(r), and Wτ(r) are given in terms of Skyrme parameters, nucleon densities, and their derivatives

•If we have a reasonable first guess for the single-particle wave-functions, i.e. harmonic oscillator, we can determine mτ*, Uτ (r), and Wτ (r) and keep reiterating the HF Method until the wave-functions converge

Determining the Skyrme Parameters•Skyrme Parameters were determined by a fit of Hartree-Fock results to experimental data

•Example: kde0 interaction was obtained with the following dataProperties Nuclei

B 16,24O, 34Si, 40,48Ca, 48,56,68,78Ni, 88Sr, 90Zr, 100,132Sn, 208Pb

rch16O, 40,48Ca, 56Ni, 88Sr, 90Zr, 208Pb

rv(υ1d5/2) 17O

rv(υ1f7/2) 41Ca

S-O 2p orbits in 56Ni

Eo90Zr, 116Sn, 144Sm, 208Pb

ρcr Nuclear MatterTable: Selected experimental data for the binding energy B,

charge rms radius rch , rms radii of valence neutron orbits rv, spin-orbit splitting S-O, breathing mode constrained energy Eo, and critical density ρcr used in the fit to determine the parameters of

the Skyrme interaction.

Values of the Skyrme Parameters

 

Parameter kde0 (2005) sgII (1985)

to (MeV fm3) -2526.51 (140.63) -2645.00

t1 (MeV fm5) 430.94 (16.67) 340.00

t2 (MeV fm5) -398.38 (27.31) -41.90

t3 (MeV fm3(1+)) 14235.5 (680.73) 1559.00

xo 0.7583 (0.0655) 0.09000

x1 -0.3087 (0.0165) -0.05880

x2 -0.9495 (0.0179) 1.4250

x3 1.1445 (0.0882) 0.06044

Wo (MeV fm5) 128.96 (3.33) 105.00

0.1676 (0.0163) 0.16667

Woods-Saxon Potential

Standard Parameterization: a

with

ro

(1- αv τz ) ro=1.27 fm

We adopt the parameterization:

R = ro[(A-1)1/3+d][1-αR τz]

Uo=-Vo(1- αv τz)

USO=-VSO(1- αv τz)

a=ao(1+ αa| |)

The parameters were determined from the UHF calculated for a wide range of nuclei.

Woods-Saxon Potential (cont.)

Woods-Saxon Potential (cont.)

Schrödinger's Equation:

Separable Solution:

where:

Numerical Solution:

Starting from uo and u1, we find u2 and continue to get u3, u4, …

Nucleon Density from Hartree-Fock kde0 Interaction

-350

-300

-250

-200

-150

-100

-50

0

50

0 2 4 6 8 10 12

-500

-400

-300

-200

-100

0

100

0 2 4 6 8 10 12

22O kde0 r2UHF Fit to r2UWS

Protons

-Vo=58.298

R=3.800

a=0.520

Neutrons

-Vo=52.798

R=3.420

a=0.534

MeV

fm

2

MeV

fm

2

fm

fm

-2000

-1800

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

0 2 4 6 8 10 12

-2500

-2000

-1500

-1000

-500

0

0 2 4 6 8 10 12

208Pb kde0 r2UHF Fit to r2UWS

Protons

-Vo=68.256

R=7.355

a=0.621

Neutrons

-Vo=60.875

R=7.055

a=0.636

MeV

fm

2

MeV

fm

2

fm

fm

Single Particle Energies (in MeV) for 16O

Particle State

Experimental kde0 sgIIWoods-

Saxon

1s1/2 35.74 35.09 33.84

1p3/2 21.8 20.05 20.63 20.10

1p1/2 15.7 13.88 14.98 16.56

1d5/2 4.14 5.89 7.03 6.44

2s1/2 3.27 3.20 3.99 4.68

1d3/2 -0.94 -1.02 0.11 1.13

1s1/2 408 31.58 31.37 30.03

1p3/2 18.4 16.19 17.11 16.64

1p1/2 12.1 10.17 11.57 13.11

1d5/2 0.60 2.37 3.75 2.96

2s1/2 0.10 0.12 0.98 1.50

1d3/2 -4.40 -3.65 -2.69 -2.02

neutrons

protons

Single Particle Energies (in MeV) for 22O

Particle State

Experimental kde0 sgIIWoods-

Saxon

1s1/2 37.97 36.92 28.87

1p3/2 20.32 21.69 17.04

1p1/2 17.37 16.85 14.43

1d5/2 6.85 5.42 8.36 5.48

2s1/2 2.74 3.99 5.93 4.52

1d3/2 0.34 1.03 1.65

1s1/2 41.94 40.60 38.24

1p3/2 27.67 26.53 25.88

1p1/2 23.24 21.19 21.66 22.82

1d5/2 13.24 14.03 12.72 12.97

2s1/2 10.97 9.06 8.22 10.06

1d3/2 9.18 4.89 5.38 7.46

neutrons

protons

Spin-Orbit Splittings for 17F and 23F

Experimental values of single-particle energy levels (in MeV) for 17F and 23F, along with predicted values from Skyrme Hartree-Fock and Woods-Saxon calculations.

Conclusions

• We find that the single-particle energies obtained from Skyrme Hartree-Fock calculations strongly depend on the Skyrme interaction.

• By examining the Hartree-Fock single-particle potential UHF, calculated for a wide range of nuclei, we have determined the asymmetry dependence in the Woods-Saxon potential well.

• We obtained good agreement between the experimental data for the single-particle energies for the protons in 17F and 23F, with those obtained using the Woods-Saxon potential.

Grant numbers: PHY-0355200PHY-463291-00001

Grant number: DOE-FG03-93ER40773

Acknowledgments