Transcript Document

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 spinorbit 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
HΨ=EΨ
•In Mean-Field Approximation each nucleon interacts
independently with a potential formed by other nucleons
Mean-Field Approximation
R
0
0
-10
Single-Particle Schrödinger Equation:
-20
-30
A-Nucleon Wave-Function:
-40
Vo
-50
-60
A=Anti-Symmetrization operator for fermions
2
Ui(r)
4
6
8
10
12
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
:
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
Pij 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 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, spinorbit 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:
(1- αv
ro
τz )
with
a
ro=1.27 fm
Woods-Saxon Potential (cont.)
We adopt the parameterization:
τz]
R = ro[(A-1)1/3+d][1-αR
Uo=-Vo(1- αv
USO=-VSO(1- αv
a=ao(1+ αa|
τz)
τz)
|)
The parameters were determined from the UHF calculated for a
wide range of nuclei.
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
22O
kde0 r2UHF Fit to r2UWS
fm
100
0
0
2
4
6
8
10
12
MeV fm2
-100
-200
Protons
-300
-Vo=58.298
-400
R=3.800
a=0.520
-500
fm
50
0
MeV fm2
-50
0
2
4
6
8
10
-100
-150
Neutrons
-200
-250
-Vo=52.798
R=3.420
-300
-350
a=0.534
12
208Pb
kde0 r2UHF Fit to r2UWS
fm
0
0
2
4
6
8
10
12
-500
MeV fm2
-1000
-1500
Protons
-Vo=68.256
-2000
R=7.355
a=0.621
-2500
fm
0
-200 0
2
4
6
8
10
-400
MeV fm2
-600
-800
-1000
-1200
Neutrons
-1400
-Vo=60.875
-1600
R=7.055
-1800
a=0.636
-2000
12
Single Particle Energies (in MeV) for 16O
Particle
State
Experimental
protons
neutrons
1s1/2
kde0
sgII
WoodsSaxon
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
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
Single Particle Energies (in MeV) for 22O
protons
neutrons
Particle
State
kde0
sgII
WoodsSaxon
1s1/2
37.97
36.92
28.87
1p3/2
20.32
21.69
17.04
1p1/2
17.37
16.85
14.43
Experimental
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
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.
Acknowledgments
Grant numbers:
PHY-0355200
PHY-463291-00001
Grant number:
DOE-FG03-93ER40773