Transcript Isovector Giant Dipole Resonance 60Ca, 28O and 80Zr

Continuum Hartree-Fock Random Phase Approximation
Description of Isovector Giant Dipole Resonance in
28O, 60Ca and 80Zr.
Emilian Nica
Texas A&M University
Advisor: Dr.Shalom Shlomo
1
Plan
Introduction
I.
a.
b.
c.
d.
Theory
II.
a.
b.
c.
d.
e.
III.
Overview
Collective Modes
Isovector Giant Dipole Resonance
Collective Motion Models
Overview
Hartree-Fock Mean-Field Approximation
Nucleon-nucleon Interaction
Random Phase Approximation
Nuclear Response
Results and Discussion
2
Part I
Introduction
3
Overview

General Remarks

The present discussion deals with the study of the Isovector Giant
Dipole Resonance (IVGDR) in symmetric (80Zr) and neutron-rich (28O,
60Ca) nuclei.

The goal of our calculations was to determine whether the small peaks
in the RPA response function were indeed due to low-energy resonance
effects or to particle emission threshold energies.

We used a Hartree-Fock (HF) Continuum Random Phase
Approximation (CRPA) to determine the nuclear response of the three
nuclei under study.

The highly accurate HF-CRPA was applied to account for particle
excitation into the continuum and to exclude the most common sources
of error.
4
Collective Modes

General Description

There are several collective modes in nuclei. A few are illustrated below.
Monopole
Dipole
Quadrupole
L=0
L=1
L=2
Isoscalar
(T=0)
Isovector
(T=1)

In the isoscalar case, the neutron and proton ensembles oscillate in
phase (i.e. dashed line). For the isovector, the neutrons and protons
oscillate out of phase.
5
Isovector Giant Dipole Resonance

Brief History

The Isovector Giant Dipole Resonance (IVGDR) was the first collective
motion seen in nuclei. It was discovered in 1947 by G.C. Baldwin and
G.S. Klaiber.
 This was done by studying the photon absorption cross section for
different nuclei. The giant resonance corresponds to the prominent peak
in the graph below.
6
Collective Motion Models

Liquid Drop Model

In the Liquid Drop model the IVGDR is described as the oscillation of
the neutron liquid against the proton liquid with a restoring force related
to the neutron-proton interaction.
 The neutron density is given by
Z  on
 n (r, t )   0 n (r ) 
 cos( t ),
A r
and the proton density is given by
N op
 p (r, t )  0 p (r) 
 cos(t ).
A r
In the expressions above ρ0n and ρ0p are the neutron and proton
saturation densities respectively , 2ε is the small amplitude of the
oscillation and  is the oscillating frequency.

Microscopic Description

In the microscopic treatment of collective motion the nuclear wave
function is described as a linear combination of particle-hole excitations.
 We used the Hartree-Fock (HF) based Continuum Random Phase
Approximation (CRPA) to calculate the response function of the IVGDR
in symmetric (80Zr) and neutron-rich nuclei (28O and 60Ca).
7
Part II
Theory
8
Overview

Microscopic Model

The microscopic model involves the use of HF-CRPA calculations to
describe the collective motion in nuclei. In essence, its use involves the
collective effect of single-particle excitations on the structure of the
nucleus.

The HF-CRPA calculations model the collective motion through changes
in the ground-state nuclear wave function induced by particle-hole
excitations.

The collective aspect of the motion is accounted for by considering all
possible single-particle excitations and the overall effect these have on
the nuclear properties.

Our calculations were based on a Green’s function HF-CRPA. This
allows for solutions in coordinate space as opposed to configuration
space and for a simplification of the procedure.
9
Hartree-Fock Mean-Field Approximation

General Remarks

The Hartree-Fock Mean-Field Approximation is a method used to solve
the nuclear many-body problem.

The Hartree-Fock calculations allow us to determine the nuclear
ground-state wave function and energy.

The Hartree-Fock method and results provide a basis (i.e. wave function
, Hamiltonian and single-particle energies) for the introduction of
particle-hole excitations in the RPA.
10
Hartree-Fock Mean-Field Approximation

Hamiltonian Operator

The total Hamiltonian of the many-body nuclear system can be written
as a sum of the single-particle kinetic energies and two-body
interactions (potentials)
A
A
H   Ti  V (ij ).
i 1


i j
V is the short-range, nucleon-nucleon interaction and it is written here
in general form. The potential V can be chosen a specific form (i.e.
Skyrme interaction).
Mean-Field Approximation

The many-body Schrodinger equation H  E is difficult to solve. We
will use a mean-field approximation to considerably simplify our models.
In a mean-field approximation each particle moves independently of
other nucleons in a central potential U representing the interaction of a
nucleon with all the other nucleons.
11

Slater Determinant

Because of the antisymmetrization of the overall nuclear wave function
Φ0 in the mean-field approximation, it can be written as a determinant
known as the Slater determinant
1 (1) 1 (2) ... 1 ( A)
2 (1) 2 (2) ... 2 ( A)
 0 (1,..., A) 
1
A!
.
.
.
 A (1)  A (2) ...  A ( A)
The arguments 1, 2…, A above denote the coordinates of the particles
(position and spin) while λ are the single-particle wave functions.

By interchanging any two rows the sign of the wave function will change
(i.e. antisymmetry).
12

Variational Principle

By applying the variational principle to the expectation value of the
Hamiltonian and normalizing through the use of Lagrange multipliers

  *0 (1,..., A) H 0 0 (1,..., A)      *    0
we obtained after some calculations the HF equations




T1 (1)    * (2)V (12) (2)d (2) (1)    * (2)V (12) (2)d (2)  (1)    (1),



  1,2,...A.

The HF equations can be solved in an iterative way by “guessing” an
initial set of wave functions λ, solving the equations above for new
single-particle wave functions and repeating the process until
convergence (i.e. ελ converge).
13
Nucleon-nucleon interaction

In our calculation we used a simplified Skyrme type effective nucleonnucleon interaction given by
1   ri  r j 
 (ri  r j ),
Vij  t0 (ri  r j )  t3  
6
 2 
where the parameters t0, t3 and α are obtained by fitting the HF results to
experimental data.

In our calculations we have used the values of:
t0=-1600 MeV fm3
t3=12500 MeV fm4
α=1/3
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Random Phase Approximation

Microscopic Treatment

In the microscopic treatment of collective vibrations within the RPA
method the new wave function is a linear combination of all possible
single-particle excitations.
 In our treatment of the IVGDR we will use a Green’s function formalism
instead of a “standard” RPA procedure (i.e. configuration space).

Linear Response Theory

The bare particle-hole Green’s function is defined as


1
1
G (0) (r, r' ,  )  h* (r) 

h (r' ).
H




H




h
0
h
 0 h

In the preceding equation, h is the occupied single-particle wave
function with the corresponding single-particle energy εh, H0 is the HF
Hamiltonian and  is the excitation energy of the nucleus.
15
Random Phase Approximation

The RPA Green’s function is obtained from
GRPA  G(0) (1  VphG(0) )1.
The term Vph in the expression above stands for the particle-hole
interaction and it is directly linked to the RPA procedure since it
represents the additional interaction introduced by single-particle
excitations.

The formalism must account for the possibility of the excitation of a
particle beyond the binding energy threshold (i.e. into the continuum).
16
Random Phase Approximation

Continuum Effects

Because of the possibility of particle excitation into the continuum the
single-particle Green’s function is obtained from
glj (r, r' , E ) 
1
2m
  2 ulj (r )vlj (r ) / W .
H0  E

ulj is the regular solution to the HF Hamiltonian and vlj is the irregular
solution (i.e. numerical solution starting from a low r to r and vice
versa). The r and r stand for lesser and greater of r and r’,
respectively, while W is the Wronskian
W u

dv
du
v
.
dr
dr
The boundary condition for the irregular solution at r is:
 2m E 
v(r ) ~ expi
r .
2




For positive energies E, this describes an outgoing wave
asymptotically.
 We note that in our calculations we have used a very small value for
the smearing parameter : /2=0.01 MeV.
17
Nuclear Response

General Remarks

We define a scattering operator F such as
F   f (ri ).
i

The nuclear response is given by the Strength Function S(E):
S ( E )   0 F n  ( E  En ),
2
n
where 0 and n denote the ground and excited states, respectively.

In the Green’s function formalism
S (E) 

1

Im Tr ( fGf ).
Another useful value is the transition density. In our case it can be
obtained using the relation
E
1

t (r, E ) 
f
(
r
'
)
Im
G
(
r
'
,
r
,
E
)
dr'.



S ( E )E


18
Part III
Results and Discussion
19
HF Single Particle energies (MeV)
28
60
O
Ca
80
Zr
Orbits Proton Neutron Proton Neutron Proton Neutron
0s1/2
0p3/2
0p1/2
0d5/2
0d3/2
1s1/2
0f7/2
0f5/2
1p3/2
1p1/2
0g9/2
0g7/2
1d5/2
1d3/2
2s1/2
-26.824
-17.882
-17.882
----------7.614
-7.614
-5.225
-26.824 -29.919
-17.882 -23.776
-17.882 -23.776
-29.919 -30.792
-23.776 -25.569
-23.776 -25.569
-30.792
-25.569
-25.569
-7.614
-7.614
-5.225
----------
-16.350 -19.139
-16.350 -19.139
-13.591 -16.438
-19.139
-19.139
-16.438
-7.952 -11.714
-7.952 -11.714
-4.674 -7.998
-4.674 -7.998
---------- ----------3.536
-3.536
-0.255
-0.255
-0.255
-11.714
-11.714
-7.998
-7.998
----------3.536
-3.536
-0.255
-0.255
-0.255
-16.350
-16.350
-13.591
----------7.952
-7.952
-4.674
-4.674
Results
Figure 1.The IVGDR free and RPA response functions for 60Ca
(in arbitrary units).
21
Results
Figure 2. Similar to Figure 1 for 28O.
22
Results
Figure 3. Similar to Figure 1 for the symmetric nucleus 80Zr.
23
Discussion

The figures show Im fGf (solid line) and Re fGf (dashed line)
of the free (top) and RPA (bottom) of the response functions
for 28O, 60Ca and 80Zr.

For a resonance Im fGf has a peak at a resonance energy E=ER,
and Re fGf decreases with E, going through zero at ER.

Example 60Ca:
(i)
(ii)
(iii)
(iv)
Threshold energies: Sn=4.674 MeV, Sp=13.591 MeV.
The sharp peaks at 8.398, 8.917 and 11.676 MeV are due to proton
bound to bound particle-hole excitations π0d -> π0f, π1s -> π1p
and π0d -> π1p, respectively.
The neutron particle-hole excitation are all to the continuum. Note
the threshold effect of enhancement in Im fGf above 4.674 MeV
and at around 6 MeV.
In the RPA excitation strength ( Im fGf ) we observe the
collective IVGDR above 11 MeV with threshold enhancement at low
24
excitation energies.
Summary
We have carried out the HF-based-Continuum RPA calculation of the
IVGDR response function for the symmetric nucleus 80Zr and the
neutron-rich nuclei 28O and 60Ca.
We have demonstrated the important threshold effect of enhancement
in the IVGDR excitation strength at low excitation energies in neutronrich nuclei associated with loosely bound orbits.
In 80Zr, due to the large single-particle separation energies, the
corresponding threshold enhancements in the excitation strength are
negligible.
In common RPA calculations with discretized continuum these strength
enhancements incorrectly appear as low-lying excited states, also
termed soft dipole modes.
25
Acknowledgments
Work done at:
26
Work supported by:
27
Hartree-Fock Mean-Field Approximation

Antisymmetry

The Pauli exclusion principle states that no two identical fermions can
occupy the same state. A consequence of this is that interchanging the
positions of 2 particles will change the sign of the overall wave function.

Let Ψab(r12) be a wavefunction of two particles with r12 the distance
between the particles and a , b the occupied orbits. The Pauli exclusion
principle implies that Ψab(r12) → 0 as r12 → 0.

If Ψab(r12)= Ψa(r1) Ψb (r2) it is not necessary that the above condition is
respected. A better expression would be
ab (r12 )  a (r1 )b (r2 )  a (r2 )b (r1 )
, which vanishes when r1=r2.
28
Random Phase Approximation

General Remarks:

The Random Phase Approximation (RPA) is a method to describe
excitations of ground state particles. For a single particle excitation the
RPA shifts a particle from its ground state into a higher energy state. A
schematic illustration of this is shown below.

In a Slater Determinant formalism this corresponds to the annihilation of
one state (i.e. a row) and the creation of a new state in place of the
previous one. The energy of the system is the eigenvalue corresponding
to the new nuclear wave function.
29