Transcript Microscopic Description of the Breathing Mode and Nuclear Compressibility By: David Carson Fuls, Stephen F.

Microscopic Description of the Breathing Mode and Nuclear Compressibility
By: David Carson Fuls, Stephen F. Austin State University
REU Cyclotron 2005, Mentor: Dr. Shalom Shlomo
Giant Resonance
Introduction
We have,
0.04
where  (r ) is equal to,
0
d 

 (r )    3o  r o .
dr 

d 2 ( E / A)
K k
dk 2f
2
f
4.5
6
7.5
9
10.5
12
13.5
15
  0 .1
0.00
0
1.5
3
4.5
6
7.5
9
10.5
12
13.5
15
-0.04
Ground State
The ground state of the nucleus with A nucleons is given by
an antisymmetric wave function which is, in the mean-field
approximation, given by a Slater determinant.
 1 (r1 ,  1 , 1 ) 2 (r1 ,  1 , 1 ) ...  A (r1 ,  1 , 1 ) 


1
 1 (r2 ,  2 , 2 ) 2 (r2 ,  2 , 2 ) ...  A (r2 ,  2 , 2 ) 

det





A! 

  (r ,  , )  (r ,  , ) ...  (r ,  , ) 
2
A
A
A
A A
A
A 
 1 A A A
In the spherical case, the single-particle wave function i (r, , ) is
given in terms of the radial R (r ), the spherical spin harmonic
Y jlm (r,  ), and the isospin  m ( ) functions:
i (r,  , ) 
The Skyrme interaction parameters (ti, xi, α, and Wo) are
obtained by fitting the HF results to the experimental data.
This interaction is written in terms of delta functions  (ri  rj )
which make the integrals in the HF equations easier to carry
out.
r[ fm]
-0.08
2
Classical Picture of the
Breathing Mode
o
We consider the isoscalar
breathing mode in which the
neutrons and protons move in
phase (∆T=0, ∆S=0).
where Vph is the particle-hole interaction and the free particle-hole
Green’s function is defined as,


1
1
Go (r, r' , E )  i * (r) 

i (r' ),
h



E
h



E
i
o
i
 o i

where i is the single-particle wave function, єi is the single-particle
energy, and ho is the single-particle Hamiltonian.
Sm
208
Pb
15.85+/-0.20
15.40+/-0.40
13.96+/-0.20

 mi ,
i
F   f (ri )


3

j ( j  1)  l (l  1)  
2



 
1 d 
h
4



 U (r ) 
W (r ) R (r )    R (r ),
 8 2 m* (r )  
r
dr
r







i 1
1 2
where f (r )  r for monopole excitation, to obtain the strength
2
function
S ( E )   0 F n  ( E  En ) 
2
n
where the effective mass m* (r) , the central potential
U (r ), and the spin-orbit potential W (r ) are written in
terms of the Skyrme parameters, matter density,
charge density, and current density.
With an initial guess of the single-particle wave functions
1
V   V (ri , r j ),
2
V (ri , r j ' )  V
NN
ij
V
(usually the harmonic oscillator wave functions because
they are known analytically) we can find the matter
density, kinetic density, current density, and charge
density. Once we know these values, we can use them
to find the effective mass, central potential, and the spinorbit potential. We then use these functions in the HF
equations to find the new radial wave functions. We
repeat the whole procedure with these new wave
functions until convergence is reached.
Co ul
ij
.
h2 A
ˆ
E   H total    2   *i (r ) i (r )dr
8 m i 1
A
   *i (r )* j (r ' )V (r, r ') i (r ) j (r ' )drdr '
i j
A
   *i (r )* j (r ' )V (r, r ') i (r ' ) j (r )drdr '.
a) TAMU Data: D. H. Youngblood et al, Phys. Rev. C 69, 034315 (2004); C 69,
054312(2004).
b) Nguyen Van Giai and H. Sagawa, Phys. Lett. B106, 379 (1981).
c) TAMU Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310
(2005).
1

Im[Tr ( f  G  f )]
After doing the fully self-consistent HF+RPA
calculations for the centroid energy of the
breathing mode in the four nuclei, using the
two Skyrme interactions SG2 and KDE0, we
have deduced a value of
and the transition density.
 RPA  t (r, E ) 
E
1
  f (r' )  [ ImG(r, r' , E )]d 3r'.

S ( E )  E
K = 230 +/- 20 MeV.
A Note on Self-Consistency
In numerical implementation of HF based RPA theory, it is the job of the
theorist to limit the numerical errors so that these are lower than the
experimental errors. Some available HF+RPA calculations omit parts of
Acknowledgments
Work done at:
the particle-hole interaction that are numerically difficult to implement,
such as the spin-orbit or Coulomb parts. Omission of these terms leads to
self-consistency violation, and the shift in the centroid energy can be on
the order of 1 MeV or 5 times the experimental error. The calculations we
have carried out are fully self-consistent.
Note:
E
K,
K
E
2
K
E
For example: E = 14 MeV (in 208Pb), and K = 230 MeV,
then a ΔE = 1 MeV leads to ΔK = 35 MeV.
i j
In the scaling model, we have the matter density oscillates as
Now we apply the variation principle to derive the HartreeFock equations. We minimize
E   Hˆ total  ,
 (r )   3 (t )  ( (t )r ),
2E

.
h
 (t )  1   cos(t ),
We consider small oscillations, so є is very close to zero (≤ 0.1).
Performing a Taylor expansion of the density,
by varying i with the constraint of particle number
conservation,
A
   (r )
i 1
 (r , t )   (t )  ( (t )r )   (t )  ( (t )r )  (t )1
3
i
2
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
dr     , (r )d r  A,
3
 ,
3
( (t )3  ( (t )r ))
 [ (t )  1]
 ... ,
 (t )
 ( t ) 1
we obtain,
and obtain the Hartree-Fock equations,

h
2
8 m
2
 i (r )    * j (r ' )V (r, r ') i (r ) j (r ' )dr '
j 1
    j (r ' )V (r, r ') i (r ' ) j (r )dr '   i  i (r ).
*
j 1
Expt.
Protons
-50+11
-34+6
-10.9
-8.3
KDE0*
Orbits
-38.21
-26.42
-22.34
-14.51
-9.66
-7.53
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
Expt.
Neutrons
-18.1
-15.6
i  1,2,..A
1f7/2
-1.4
1f7/2
2p3/2
-8.3
-6.2
-2.76
Isoscalar Monopole Strength Functions
KDE0*
-47.77
-34.90
-30.78
-22.08
-17.00
-14.97
A
A
d 

 (r , t )  o (r )   cos(t ) 3o  r o .
dr 

Single-Particle Energies (in MeV) for 40Ca
Orbits
KDE0c)
18.1
18.0
16.6
16.6
15.5
15.4
13.8
13.8
K =229
J =33
A
 '
l (l  1)
h2
h2
 "
 d 


R
(
r
)

R
(
r
)



 dr  8 2 m* (r )  R (r )
8 2 m* (r ) 
r2




Method of Solving the HF Equations
The total energy E is,
 (r, t )  o   (r, t )
144
Sn
17.81+/-0.30
SG2b)
17.9
17.9
16.2
16.2
15.3
15.3
13.6
13.6
K =215
J =29
We use the scattering operator F
a sum of the kinetic T and potential V energies,
where,
G  Go (1  VphGo )1,
116
Experimenta)
R (r )
Y jlm (r ,  )  m ( )
r
The total Hamiltonian of the nucleus is written as
pi2
h2
T 
 2
2 mi
8
In the Green’s Function formulation of RPA, one starts with the RPAGreen’s function which is given by
Zr
Intergral Width
0--60
10--35
0--60
10--35
0--60
10--35
0--60
10--35
Conclusion
For a spherical case the HF equations can be
reduced to,
Hˆ total  T  V ,
In the classical description of
the breathing mode, the
nucleus is modeled after a
drop of liquid that oscillates by
expanding and contracting
about its spherical shape.
90
iW0k ij (ri  r j )(σ i  σ j )  k 'ij .
Microscopic Description of the
Breathing Mode
o
Fully Self-Consistent HF Based RPA Results For
Breathing Mode Energy (in MeV)
0.08
This  (r ) nicely agrees
with the transition density
obtained from RPA
calculations.
d 2 ( E / A)
 9
d 2 
1


Vij  t0 (1  x0 Pij ) (ri  r j )  t1 (1  x1 Pij )[k ij2  (ri  r j )   (ri  r j )k 'ij2 ]
2
 ri  r j 
1


 (ri  r j ) 
 t 2 (1  x2 Pij )k ij (ri  r j )k 'ij  t3 (1  x3 Pij )   
6
2


In HF based RPA theory, giant resonances are described as coherent
superpositions of particle hole excitations of the ground state.
-9.60
-4.98
Work supported by:
90Zr
S(E) [fm4/MeV]
E/A [MeV]
3
 (r )
2
k fo
1.5
0.04
Once we know that a two-body
interaction is successful in
determining the centroid energy of
the monopole resonance, we can
use that interaction to find the EOS
and from that we can find the value
of K.
ρ [fm-3]
0.00
-0.04
The value of K is directly related to
the second derivative of the equation
of state (EOS) of symmetric nuclear
matter.
E/A = -16 MeV
o (r )   (r )
0.12
0.08
Nuclear Matter Incompressibility
1    o 

E[  ]  E[  o ]  K 
18   o 
o (r )
0.16
 (r )
For the two-body nuclear potential Vij, we take a Skyrme type
effective NN interaction given by,
o (r )   (r )
0.20
 (r, t )  o (r )   (r ) cos(t ),
We use the microscopic Hartree-Fock (HF) based RandomPhase-Approximation (RPA) theory to describe the
breathing mode in the 90Zr, 116Sn, 144Sm, and 208Pb nuclei,
which are very sensitive to the nuclear matter
incompressibility coefficient K. The value of K is directly
related to the curvature of the equation of state, which is a
very important quantity in the study of properties of nuclear
matter, heavy ion collisions, neutron stars, and supernova.
We present results of fully self-consistent HF+RPA
calculations for the centroid energies of the breathing
modes in the four nuclei using several Skyrme type
nucleon-nucleon (NN) interactions and compare the results
with available experimental data to deduce a value for K.
ρ = 0.16 fm-3
0.24
116Sn
Grant numbers: PHY-0355200
PHY-463291-00001
144Sm
208Pb
*TAMU Skyrme Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au,
Phys. Rev. C 72, 014310 (2005).
E [MeV]
Grant number: DOE-FG03-93ER40773