Transcript Slide 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
Introduction
Mean-Field Approximation
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 of 17F8 and the recently measured spectra of 23F14.
Using UWS, we obtained good agreement with experimental data.
•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
Mean-Field Approximation
0
2
4
6
Ui(r)
where:
Mean-Field
Approximation
•Single-particle wave-functions Φi are determined by the independent
single-particle
potentials
R
0
8
10
12
•Due to spherical symmetry, the solution is separable into radial component
component (spherical harmonics)
; and the isospin function
:
; angularR
0
0
-10
Single-Particle Schrödinger Equation:
-20
2
4
Ui(r)
6
8
10
12
•The ground state wave-function should give the lowest expectation value for
the Hamiltonian
We obtain the Hartree-Fock Equations:
-20
-30
-30
-40
A-Nucleon Wave-Function:
•We want to obtain minimum of E with the constraint that the sum of the singleparticle wave-function integrals over all space is A, to conserve the number of
nucleons:
HΨ=EΨ
•In Mean-Field Approximation each nucleon interacts independently with a
potential formed by other nucleons
Single-Particle Schrödinger Equation:
Hartree-Fock Method (cont.)
•The Hamiltonian operator is sum of
kinetic and potential energy operators:
•The anti-symmetric
groundfor
state
wave-function
of a nucleus generally
can be written
as a Slater
determinant
•Many-body
problem
nuclear
wave-function
cannot
be solved
of a matrix whose elements are single-particle wave-functions
analytically
(HΨ=EΨ)
-10
Hartree-Fock Method
Mean-Field
Approximation
Mean Field
(cont.)
-40
A-Nucleon Wave-Function:
-50
-50
Vo
Vo
-60
-60
A=Anti-Symmetrization operator for fermions
A=Anti-Symmetrization operator for fermions
Determining the Skyrme Parameters
Skyrme Interaction (cont.)
•After all substitutions and making the coefficients of all variations equal to zero,
we have the Hartree-Fock Equations:
Values of 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
•mτ*(r), Uτ(r), and W τ(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 W τ (r) and keep reiterating
the HF Method until the wave-functions converge
Parameter
kde0
sgII
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
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
x1
-0.3087 (0.0165)
-0.05880
S-O
2p orbits in 56Ni
x2
-0.9495 (0.0179)
1.4250
Eo
90Zr, 116Sn, 144Sm, 208Pb
x3
1.1445 (0.0882)
0.06044
ρcr
Nuclear Matter
Wo (MeV fm5)
128.96 (3.33)
105.00

0.1676 (0.0163)
0.16667
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.
22O
Nucleon Density from Hartree-Fock kde0
Interaction
Woods-Saxon Potential (cont.)
Woods-Saxon Potential
where:
Standard Parameterization:
kde0 r2UHF Fit to r2UWS
208Pb
fm
100
fm
0
0
0
We adopt the parameterization:
0
2
4
6
8
10
kde0 r2UHF Fit to r2UWS
2
4
6
8
10
12
-500
12
-1000
-200
Protons
-1500
-300
Vo=-58.298
-2000
a=0.621
-500
fm
-2500
0
0
-50
τz)
0
2
4
6
8
12
fm
2
4
6
8
10
12
-4 00
-6 00
-1 00
-1 50
Neutrons
-2 00
Vo=-52.798
-2 50
R=3.420
-3 00
The parameters were determined from the UHF calculated for a wide range of nuclei.
10
-2 00 0
a=0.534
-3 50
MeV fm2
τz)
MeV fm2
a=ao(1- αa
R=7.355
a=0.520
50
USO=-VSO(1- αv
Protons
-Vo=68.256
R=3.800
-400
τz)
MeV fm2
τz]
R = ro[(A-1)1/3+d][1-αR
Uo=-Vo(1- αv
MeV fm2
-100
-8 00
-1 0 00
-1 2 00
Neutrons
-1 4 00
-Vo=60.875
-1 6 00
R=7.055
-1 8 00
a=0.636
-2 0 00
Spin-Orbit Splittings for 17F and 23F
Single Particle Energies (in MeV) for 16O
protons
neutrons
1s1/2
kde0
sgII
Woods-Saxon
Particle
State
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
408
31.58
31.37
1p3/2
18.4
16.19
1p1/2
12.1
1d5/2
2s1/2
1d3/2
neutrons
Experimental
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
30.03
1s1/2
41.94
40.60
38.24
17.11
16.64
1p3/2
27.67
26.53
25.88
10.17
11.57
13.11
1p1/2
23.24
21.19
21.66
22.82
0.60
2.37
3.75
2.96
1d5/2
13.24
14.03
12.72
12.97
0.10
0.12
0.98
1.50
2s1/2
10.97
9.06
8.22
10.06
-3.65
-2.69
-2.02
1d3/2
9.18
4.89
5.38
7.46
protons
Particle
State
Single Particle Energies (in MeV) for
Conclusions
22O
•
We find that the single-particle energies obtained from Skyrme HartreeFock 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.
•
Grant numbers:
PHY-0355200
PHY-463291-00001
Grant number:
DOE-FG03-93ER40773
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.
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.