Transcript Slide 1

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
REU 2007 Cyclotron Institute
Texas A&M University

Slater Determinant

Collective Modes

General Description

There are several collective modes in nuclei. A few are illustrated below.
Monopole
Dipole
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
Quadrupole
 0 (1,..., A ) 
Isoscalar
(T=0)
1 (1)
1 ( 2 )
...
1 ( A )
 2 (1)
2 (2)
...
2 ( A)
1
.
A!
.

G
G
RPA
(0)
(1  V ph G
(0)
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.
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.
.
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.
 A (1)  A ( 2 ) ...  A ( A )
Isovector
(T=1)
L=0

Results
The RPA Green’s function is obtained from
L=1
The arguments 1, 2…, A above denote the coordinates of the particles
(position and spin) while λ are the single-particle wave functions.
L=2

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.

In 80Zr, due to the large single-particle separation energies, the
corresponding threshold enhancements in the excitation strength are
negligible.
The formalism must account for the possibility of the excitation of a
particle beyond the binding energy threshold (i.e. into the continuum).
By interchanging any two rows the sign of the wave function will change
(i.e. antisymmetry).
Figure 1.The IVGDR free and RPA response functions for 28O
(in arbitrary units).
Isovector Giant Dipole Resonance


Brief History



Variational Principle

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.
Continuum Effects

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
*
 
*


( 2 )V (12 )  ( 2 ) d ( 2 )   (1)      (1),

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).


Collective Motion Models

Nucleon-nucleon interaction
Liquid Drop Model


 n (r , t )   0 n (r ) 
and the proton density is given by
 p (r , t )   0 p (r ) 
Z   on
A r
N   op
A r

In our calculation we used a simplified Skyrme type effective nucleonnucleon interaction given by
 ri  r j
V ij  t 0 ( ri  r j )  t 3  
6
 2
1
 cos(  t ),



Hamiltonian Operator


 ( ri  r j ),





i j
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.

f ( ri ) .
i

The nuclear response is given by the Strength Function S(E):
S (E ) 

2
0 F n
 ( E  E n ),
0
and
n
denote the ground and excited states, respectively.

In the Green’s function formalism
1
S ( E )  Im Tr ( fGf ) .

Another useful value is the transition density. In our case it can be
obtained using the relation

28
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).
The bare particle-hole Green’s function is defined as
G
(0)

1
1
*
(r , r ' ,  )     h (r ) 

h
H0 h  H0 h 
O rb its
0 s1 /2
0 p 3 /2
0 p 1 /2
0 d 5 /2
0 d 3 /2
1 s1 /2
Linear Response Theory

E
S ( E )E

1

f ( r ' )  Im G ( r ' , r , E )  d r '.



 h ( r ' ).

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.
0 f7 /2
0 f5 /2
1 p 3 /2
1 p 1 /2
0 g9 /2
0 g7 /2
1 d 5 /2
1 d 3 /2
2 s1 /2
60
O
Ca
80
Figure 3. Similar to Figure 1 for the symmetric nucleus 80Zr.
Discussion
H F S in g le P a rticle e n e rg ie s (M e V )
Microscopic Treatment

i
Mean-Field Approximation
Figure 2. Similar to Figure 1 for 60Ca.
Work supported by:
F 
Random Phase Approximation
 T   V ( ij ).
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).
du
We define a scattering operator F such as
 t (r , E ) 
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
i 1

t0=-1600 MeV fm3
t3=12500 MeV fm4
α=1/3

H 
v
In our calculations we have used the values of:
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).
Hartree-Fock Mean-Field Approximation
dv
 2 mE 
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.
where
Microscopic Description

Work done at:
General Remarks


u lj ( r ) v lj ( r ) / W .
n
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.

2
.
dr
dr
The irregular solution is found from the boundary conditions as r:
where the parameters t0, t3 and α are obtained by fitting the HF results to
experimental data.
 cos(  t ).

Nuclear Response

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
2m

H0  E
W u
  1, 2 ,... A.

1
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
we obtained after some calculations the HF equations


*
T1  (1)       ( 2 )V (12 )  ( 2 ) d ( 2 )    (1)  

 

Acknowledgments
Because of the possibility of particle excitation into the continuum the
single-particle Green’s function is obtained from
g lj ( r , r ' , E ) 
In common RPA calculations with discretized continuum these strength
enhancements incorrectly appear as low-lying excited states, also
termed soft dipole modes.
Zr

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.
P ro to n N e u tro n P ro to n N e u tro n P ro to n N e u tro n
-2 6 .8 2 4
-1 7 .8 8 2
-1 7 .8 8 2
----------7 .6 1 4
-7 .6 1 4
-5 .2 2 5
-2 6 .8 2 4
-1 7 .8 8 2
-1 7 .8 8 2
-2 9 .9 1 9
-2 3 .7 7 6
-2 3 .7 7 6
-2 9 .9 1 9
-2 3 .7 7 6
-2 3 .7 7 6
-3 0 .7 9 2
-2 5 .5 6 9
-2 5 .5 6 9
-3 0 .7 9 2
-2 5 .5 6 9
-2 5 .5 6 9

-7 .6 1 4
-7 .6 1 4
-5 .2 2 5
----------
-1 6 .3 5 0
-1 6 .3 5 0
-1 3 .5 9 1
----------7 .9 5 2
-7 .9 5 2
-4 .6 7 4
-4 .6 7 4
-1 6 .3 5 0
-1 6 .3 5 0
-1 3 .5 9 1
-1 9 .1 3 9
-1 9 .1 3 9
-1 6 .4 3 8
-1 9 .1 3 9
-1 9 .1 3 9
-1 6 .4 3 8
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:
-7 .9 5 2
-7 .9 5 2
-4 .6 7 4
-4 .6 7 4
----------
-1 1 .7 1 4
-1 1 .7 1 4
-7 .9 9 8
-7 .9 9 8
----------3 .5 3 6
-3 .5 3 6
-0 .2 5 5
-0 .2 5 5
-0 .2 5 5
-1 1 .7 1 4
-1 1 .7 1 4
-7 .9 9 8
-7 .9 9 8
----------3 .5 3 6
-3 .5 3 6
-0 .2 5 5
-0 .2 5 5
-0 .2 5 5
(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
excitation energies.