1. Single Particle Energies
in Skyrme Hartree-Fock and Woods-Saxon Potentials
Brian D. Newman
Cyclotron Institute
Texas A&M University
Mentor: Dr. Shalom Shlomo

2. 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.

3. 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
HΨ=EΨ
Mean-Field Approximation R
Ui(r)
Single-Particle Schrödinger Equation:
A-Nucleon Wave-Function: Vo
A=Anti-Symmetrization operator for fermions

4. 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
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 :

5. 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

6. 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:

7. 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
is the spin exchange operator
to, t1, t2, t3, xo, x1, x2, x3, , and Wo are the ten Skyrme parameters

8. 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

9. 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 data
Properties
Nuclei B
16,24O, 34Si, 40,48Ca, 48,56,68,78Ni, 88Sr, 90Zr, 100,132Sn, 208Pb
rch
16O, 40,48Ca, 56Ni, 88Sr, 90Zr, 208Pb
rv(υ1d5/2)
17O
rv(υ1f7/2)
41Ca
S-O
2p orbits in 56Ni Eo
90Zr, 116Sn, 144Sm, 208Pb
ρcr
Nuclear Matter
Table: 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.

10. Values of the Skyrme Parameters
kde0 (2005)
sgII (1985)
to (MeV fm3)
(140.63)
t1 (MeV fm5)
(16.67)
340.00
t2 (MeV fm5)
(27.31)
-41.90
t3 (MeV fm3(1+))
(680.73) xo
(0.0655) x1
(0.0165) x2
(0.0179)
1.4250 x3
(0.0882)
Wo (MeV fm5)
(3.33)
105.00 
(0.0163)

11. Woods-Saxon Potential
Standard Parameterization: ro a
(1- αv τz )
ro=1.27 fm
with

12. Woods-Saxon Potential (cont.)
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.

13. 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, …

14. Nucleon Density from Hartree-Fock kde0 Interaction

15. 22O kde0 r2UHF Fit to r2UWS fm MeV fm2 fm MeV fm2 Protons -Vo=58.298
a=0.520 fm
MeV fm2
Neutrons
-Vo=52.798
R=3.420
a=0.534

16. 208Pb kde0 r2UHF Fit to r2UWS fm MeV fm2 fm MeV fm2 Protons -Vo=68.256
a=0.621 fm
MeV fm2
Neutrons
-Vo=60.875
R=7.055
a=0.636

17. Single Particle Energies (in MeV) for 16O
Particle State
Experimental
kde0
sgII
Woods-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
408
31.58
31.37
30.03
18.4
16.19
17.11
16.64
12.1
10.17
11.57
13.11
0.60
2.37
3.75
2.96
0.10
0.12
0.98
1.50
-4.40
-3.65
-2.69
-2.02
neutrons
protons

18. Single Particle Energies (in MeV) for 22O
Particle State
Experimental
kde0
sgII
Woods-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
41.94
40.60
38.24
27.67
26.53
25.88
23.24
21.19
21.66
22.82
13.24
14.03
12.72
12.97
10.97
9.06
8.22
10.06
9.18
4.89
5.38
7.46
neutrons
protons

19. 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.

20. 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.

21. Grant number: DOE-FG03-93ER40773
Acknowledgments
Grant numbers: PHY
PHY
Grant number: DOE-FG03-93ER40773