Transcript Microscopic Description of the Breathing Mode and Nuclear

Microscopic Description of the
Breathing Mode and Nuclear
Compressibility
Presented By: David Carson Fuls
Cyclotron REU Program 2005
Mentor: Dr. Shalom Shlomo
Introduction
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.
Nuclear Matter Incompressibility
E/A [MeV]
The value of K is directly related to
the second derivative of the equation
of state (EOS) of symmetric nuclear
matter.
ρ = 0.16 fm-3
E/A = -16 MeV
ρ [fm-3]
1    o 

E[  ]  E[  o ]  K 
18   o 
2
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.
d 2 ( E / A)
K k
dk 2f
2
f
d 2 ( E / A)
 9
d 2 
2
k fo
o
Classical Picture of the
Breathing Mode
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.
o
We consider the isoscalar
breathing mode in which the
neutrons and protons move in
phase (∆T=0, ∆S=0).
 (r, t )  o   (r, t )
In the scaling model, we have the matter density oscillates as
 (r )   3 (t )  ( (t )r ),
 (t )  1   cos(t ),
2E

.
h
We consider small oscillations, so є is very close to zero (≤ 0.1).
Performing a Taylor expansion of the density
 (r , t )   3 (t )  ( (t )r )   3 (t )  ( (t )r )  (t )1
( (t )3  ( (t )r ))
 [ (t )  1]
 ... ,
 (t )
 ( t ) 1
we obtain,
do 

 (r , t )  o (r )   cos(t ) 3o  r
.

dr 

0.24
We have,
o (r )   (r )
0.20
 (r, t )  o (r )   (r ) cos(t )
o (r )
0.16
 (r )
o (r )   (r )
0.12
0.08
0.04
Where  (r ) is equal to
0.00
0
1.5
3
4.5
6
7.5
9
10.5
12
13.5
15
-0.04
do 

 (r )    3o  r

dr 

0.08
  0 .1
0.04
 (r )
This  (r ) nicely agrees
with the transition density
obtained from RPA
calculations.
0.00
0
1.5
3
4.5
6
7.5
9
-0.04
-0.08
r[ fm]
10.5
12
13.5
15
Microscopic Description of the
Breathing Mode
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
  (r ,  , ) 2 (r2 ,  2 , 2 ) ...  A (r2 ,  2 , 2 ) 

det 1 2 2 2





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,  , ) 
R (r )
Y jlm (r ,  )  m ( )
r
The total Hamiltonian of the nucleus is written as
a sum of the kinetic T and potential V energies
Hˆ total  T  V ,
pi2
h2
T 
 2
2mi
8
i
m ,
i
V
1
V (ri , r j ),

2
V (ri , r' j )  VijNN  VijCoul .
Where
The total energy E
E   Hˆ total   
h2
8 2 m
A
*

   (r) (r)dr
i 1
i
i
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 '
i j
Now we apply the variation principle to derive the HartreeFock equations. We minimize
E   Hˆ total  ,
by varying  i with the constraint of particle number
conservation,
A
3

(
r
)
d
r


(
r
)
d
 i
   , r  A,
i 1
2
 ,
and obtain the Hartree-Fock equations
A
h2
 2  i (r )    * j (r ' )V (r, r ') i (r ) j (r ' )dr '
8 m
j 1
A
   * j (r ' )V (r, r ') i (r ' ) j (r )dr '   i  i (r ).
j 1
i  1,2,..A
For the two-body nuclear potential Vij, we take a Skyrme type
effective NN interaction given by,
1


Vij  t0 (1  x0 Pij ) (ri  r j )  t1 (1  x1 Pij )[k ij2  (ri  r j )   (ri  r j )k 'ij2 ]
2
1


  ri  r j 
 (ri  r j ) 
 t 2 (1  x2 Pij )k ij (ri  r j )k 'ij  t3 (1  x3 Pij )  
6
 2 
iW0k ij (ri  r j )(σ i  σ j )  k 'ij .
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.
For a spherical case the HF equations can be
reduced to,
 '
l (l  1)
h2
h2
 "
 d 

 R (r )

R
(
r
)

R
(
r
)



2 *
2
2 *



8 m (r ) 
r
 dr  8 m (r ) 


3

j
(
j

1
)

l
(
l

1
)

 


   
1 d 
h2
4 
 2 *  
 U (r ) 
W (r ) R (r )    R (r ),
r
dr
8

m
(
r
)
r







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.
Method of Solving the HF Equations
With an initial guess of the single-particle wave functions
(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.
Single-Particle Energies (in MeV) for 40Ca
Orbits
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
1f7/2
Expt.
Protons
-50+11
KDE0*
Orbits
-10.9
-8.3
-38.21
-26.42
-22.34
-14.51
-9.66
-7.53
-1.4
-2.76
-34+6
1s1/2
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
Expt.
Neutrons
-18.1
-15.6
KDE0*
-47.77
-34.90
-30.78
-22.08
-17.00
-14.97
1f7/2
2p3/2
-8.3
-6.2
-9.60
-4.98
*TAMU Skyrme Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au,
Phys. Rev. C 72, 014310 (2005).
Giant Resonance
In HF based RPA theory, giant resonances are described as coherent
superpositions of particle hole excitations of the ground state.
In the Green’s Function formulation of RPA, one starts with the RPAGreen’s function which is given by
1
G  Go (1  VphGo ) ,
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' ),
i
 ho   i  E ho   i  E 
where i is the single-particle wave function, єi is the single-particle
energy, and ho is the single-particle Hamiltonian.
We use the scattering operator F
A
F   f (ri )
i 1
1
where f (r )  r 2 for monopole excitation, to obtain the strength
2
function
S ( E )   0 F n  ( E  En ) 
2
n
1

Im[Tr ( f  G  f )]
and the transition density.

RPA
E
1
 t (r, E ) 
  f (r' )  [ ImG(r, r' , E )]d 3r'.

S ( E )  E
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
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.
Isoscalar Monopole Strength Functions
S(E) [fm4/MeV]
90Zr
116Sn
144Sm
208Pb
E [MeV]
Fully Self-Consistent HF Based RPA Results For
Breathing Mode Energy (in MeV)
90
116
144
Zr
Sn
Sm
208
Pb
Intergral Width
0--60
10--35
0--60
10--35
0--60
10--35
0--60
10--35
Experimenta)
17.81+/-0.30
15.85+/-0.20
15.40+/-0.40
13.96+/-0.20
SG2b)
17.9
17.9
16.2
16.2
15.3
15.3
13.6
13.6
K =215
J =29
KDE0c)
18.1
18.0
16.6
16.6
15.5
15.4
13.8
13.8
K =229
J =33
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).
Conclusion
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
K = 230 +/- 20 MeV.
Acknowledgments
Work done at:
Work supported by:
Grant numbers: PHY-0355200
PHY-463291-00001
Grant number: DOE-FG03-93ER40773