Observables predicted by HF theory
•  Total binding energy of the nucleus in its ground state
⇒ separation energies for p / n (= BE differences)
•  Ground state density distribution of protons and neutrons
⇒ mean square radii
⇒ EM multipole moments (for deformed g.s. density)
•  single particle levels (energies, angular momenta, parity)
⇒ shell gaps (explains magic numbers)
⇒ Fermi levels for p / n
NOTE: HF is a ground state theory.
For excited states we need Random Phase Approximation = RPA
1/5/16
VolkerOberacker,VanderbiltUniversity
1
Static Hartree-Fock: numerical implementation
Represent wave functions for all A nucleons on 3-D Cartesian lattice
Grid points:
Wave function on the lattice becomes a complex-numbered array
of dimension
1/5/16
VolkerOberacker,VanderbiltUniversity
2
Basis-Splines (7th order), periodic boundary condition
Expand wave function in terms of products of Basis-Spline functions
1/5/16
VolkerOberacker,VanderbiltUniversity
3
Stable nucleus 16O
HFB-2D calculation, SLy4 N-N interaction
16
16
O protons
20
20
continuum
0
1p1/2
1p3/2
V(r)
-20
-40
-40
-60
-60
1/5/16
2
4
r (fm)
1p1/2
1p3/2
-20
1s1/2
-80
0
continuum
V(r)
0
O neutrons
6
8
-80
0
VolkerOberacker,Vanderbilt
1s1/2
2
4
r (fm)
6
8
4
Neutron dripline nucleus 24O (T1/2 = 65 ms)
HFB-2D calculation, SLy4 N-N interaction
24
24
O protons
20
20
continuum
-20
0
V(r)
1/5/16
1s1/2
-40
1s1/2
-60
-60
-80
0
2s1/2
1d5/2
1p1/2
1p3/2
-20
1p1/2
1p3/2
-40
continuum
V(r)
0
O neutrons
2
4
r (fm)
6
8
-80
0
VolkerOberacker,Vanderbilt
2
4
r (fm)
6
8
5
48Ca: HF predicts spherical density distribution,
in agreement with exp. data
620 static iterations, CPU time = 2m 21s (16 INTEL Xeon CPUs)
Convergence of total binding
energy ΔE / E = -5.7 × 10-9
HF binding energy
= -431.10 MeV
exp. binding energy
= -416.00 MeV
accuracy = 3.4%
1/5/16
V.Oberacker,Vanderbilt
6
Charge densities for various nuclei (electron scattering data vs. theory)
Ref: J.W. Negele, Rev. Mod. Phys. 54, 913 (1982)
1/5/16
VolkerOberacker,VanderbiltUniversity
7
rms charge radii for several isotope chains.
HF calculation with different Skyrme forces, compared
to exp. data (filled and open diamonds)
M. Bender, P. Heenen, and P.G. Reinhard,
Rev. Mod. Phys. 75, 122 (2003)
1/5/16
VolkerOberacker,VanderbiltUniversity
8
HF neutron single-particle
energies, using several
Skyrme forces, compared
to experiment
M. Bender, P. Heenen,
and P.G. Reinhard,
Rev. Mod. Phys. 75,
122 (2003)
1/5/16
VolkerOberacker,VanderbiltUniversity
9
HF binding energies, calculated with different Skyrme forces.
Plotted is energy difference between experiment and theory.
M.Bender,P.Heenen,andP.G.Reinhard,
Rev.Mod.Phys.75,122(2003)
1/5/16
VolkerOberacker,VanderbiltUniversity
10
249Bk:
HF predicts positive quadrupole + hexadecapole
deformation, in agreement with exp. data
6,300 static iterations, CPU time = 2h 50m (16 INTEL Xeon CPUs)
Convergence of total binding
energy ΔE / E = 4.2 × 10-10
HF binding energy
= -1,875.11 MeV
exp. binding energy
= -1,864.02 MeV
accuracy = 0.6%
1/5/16
V.Oberacker,Vanderbilt
11
relative energy change, abs (dE / E)
10
10
-4
10
10
10
static Hartree-Fock iteration
-5
249
Bk
97
-6
-7
10
10
10
-3
-8
-9
-10
-11
10 0
1/5/16
2000
4000
number of iterations
V.Oberacker,Vanderbilt
6000
12
Constrained Hartree-Fock calculations: collective energy surface
Ref: Ring & Schuck, chapter 7.6
Add electric multipole operators Qi to variational principle
'
*
HF
HF
HF
HF
δ ) < Φ0 | H | Φ0 > −∑ λi < Φ0 | Qi | Φ0 > , = 0
(
+
i
Lagrange multipliers
€
The values of Lagrange multipliers are determined from the constraint
that the electric multipole operators have a given expectation value
< Φ0HF | Qi | Φ0HF >= qi
given
As a result of constraints, one obtains the “collective energy surface”
E(qi )
€
1/5/16
or more explicitly
E(q20 ,q22 ,q30 ,q40 ,...)
VolkerOberacker,VanderbiltUniversity
13
HF calculation with quadrupole
constraint, for chain of gadolinium
isotopes, using several Skyrme
forces.
upper figure: collective potential
energy surface
lower figure: ground state
deformation, compared to exp.
data
M. Bender, P. Heenen,
and P.G. Reinhard,
Rev. Mod. Phys. 75,
122 (2003)
1/5/16
VolkerOberacker,VanderbiltUniversity
14
3-D Skyrme HF calculation, with quadrupole constraint:
double-humped fission barrier for 240Pu
1/5/16
VolkerOberacker,VanderbiltUniversity
15
“Cranked” Hartree-Fock calculations: rotational bands
Ref: Ring & Schuck, chapter 7.7
Add total angular momentum operator to variational principle
δ [ < Φ0HF | H | Φ0HF > −ω ( I ) < Φ0HF | J x | Φ0HF >] = 0
Lagrange multiplier = cranking frequency
The cranking frequency ω(I) depends on ang. mom. quantum number I;
its value is determined from the constraint condition
< Φ0HF (ω ( I )) | J x | Φ0HF (ω ( I )) >= I(I +1)
As a result of this constraint, one obtains “rotational bands”
€
E(I)
which agree much better with experiment than simple collective models
(microscopic moments of inertia change as function of ang. momentum!)
1/5/16
VolkerOberacker,VanderbiltUniversity
€
16
Cranked HF calculation in 3-D, for two Skyrme forces
Ref: M. Yamagami and K. Matsuyanagi, Nucl. Phys. A672 (2000) 123
1/5/16
VolkerOberacker,VanderbiltUniversity
17
Cranked HF calculation in 3-D, for two Skyrme forces
Ref: M. Yamagami and K. Matsuyanagi, Nucl. Phys. A672 (2000) 123
1/5/16
VolkerOberacker,VanderbiltUniversity
18