21
Microscopic Description of the Breathing Mode and Nuclear Compressibility Presented By: David Carson Fuls Cyclotron REU Program 2005 Mentor: Dr. Shalom Shlomo
Microscopic Description of the Breathing Mode and Nuclear
Compressibility
Presented By: David Carson Fuls
Cyclotron REU Program 2005
Mentor: Dr. Shalom Shlomo
IntroductionWe use the microscopic Hartree-Fock (HF) based Random-Phase-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 [fm-3]
ρ = 0.16 fm-3
E/A
[M
eV]
Nuclear Matter Incompressibility
2
18
1][][
o
oo KEE
The value of K is directly related to the second derivative of the equation of state (EOS) of symmetric nuclear matter.
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.
E/A = -16 MeV
ofo
d
AEd
dk
AEdkK
kff
2
22
2
22 )/(
9)/(
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.
We consider the isoscalar
breathing mode in which the
neutrons and protons move in
phase (∆T=0, ∆S=0).),(),( trtr o
o
In the scaling model, we have the matter density oscillates as
We consider small oscillations, so є is very close to zero (≤ 0.1). Performing a Taylor expansion of the density
we obtain,
.3)cos()(),(
dr
drtrtr o
oo
),)(()()( 3 rttr
),cos(1)( tt
,...)(
)))(()((]1)([
))(()())(()(),(
1)(
3
1)(
33
t
t
t
rttt
rttrtttr
.2
h
E
We have,
Where is equal to
This nicely agrees
with the transition density obtained from RPA calculations.
dr
drr o
o
3)(
)cos()()(),( trrtr o
)(r-0.04
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0 1.5 3 4.5 6 7.5 9 10.5 12 13.5 15
][ fmr
)(r
)(r
)(ro
)()( rro
)()( rro
-0.08
-0.04
0.00
0.04
0.08
0 1.5 3 4.5 6 7.5 9 10.5 12 13.5 15
1.0
)(r
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.
In the spherical case, the single-particle wave function is given in terms of the radial , the spherical spin harmonic , and the isospin functions:
),,( ri
),,(...),,(),,(
),,(...),,(),,(
),,(...),,(),,(
det!
1
21
22222222221
11111121111
AAAAAAAAAA
A
A
A
rrr
rrr
rrr
)(),()(
),,(
mjlmi rY
r
rRr
)(rR
),( rY jlm)(
m
The total Hamiltonian of the nucleus is written as
a sum of the kinetic T and potential V energies
Where
The total energy E
,ˆ VTH total
')()'()',()'()(
')'()()',()'()(
)()(8
ˆ
**
**
1
*2
2
rrrrrrrr
rrrrrrrr
rrr
ddV
ddV
dm
hHE
jiji
jiji
ii
A
ji
A
ji
A
itotal
,82 2
22
i
i
i
i
m
h
m
pT
.)',( Coulij
NNijji VVV rr
,),(2
1 jiVV rr
Now we apply the variation principle to derive the Hartree-Fock equations. We minimize
by varying with the constraint of particle number conservation,
and obtain the Hartree-Fock equations
,ˆ totalHE
,)()(1
3
,,
2
A
ii Add rrrr
).(')()'()',()'(
')'()()',()'()(8
1
*
1
*2
2
rrrrrrr
rrrrrrr
iijij
jiji
A
j
A
j
dV
dVm
h
Ai ,..2,1
i
For the two-body nuclear potential Vij, we take a Skyrme type effective NN interaction given by,
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 which make the integrals in the HF equations easier to carry out.
.'))((
)(2
)1(6
1')()1(
]')()()[1(2
1)()1(
0
3322
221100
ijjijiij
jiji
ijijjiijij
ijjijiijijjiijij
iW
PxtPxt
PxtPxtV
kσσrrk
rrrr
krrk
krrrrkrr
)( ji rr
For a spherical case the HF equations can be reduced to,
where the effective mass , the central potential , and the spin-orbit potential are written in terms of the Skyrme parameters, matter density, charge density, and current density.
),()()(43
)1()1(
)(8
1)(
)()(8
)()1(
)()(8
*2
2
'*2
2
2"
*2
2
rRrRrWr
lljj
rm
h
dr
d
rrU
rRrm
h
dr
drR
r
llrR
rm
h
)(* rm)(rU )(rW
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 spin- orbit 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 Expt. KDE0* Orbits Expt. KDE0*
1s1/2 -50+11 -38.21 1s1/2 - -47.77
1p3/2 - -26.42 1p3/2 - -34.90
1p1/2 -34+6 -22.34 1p1/2 - -30.78
1d5/2 - -14.51 1d5/2 - -22.08
2s1/2 -10.9 -9.66 2s1/2 -18.1 -17.00
1d3/2 -8.3 -7.53 1d3/2 -15.6 -14.97
1f7/2 -1.4 -2.76 1f7/2 -8.3 -9.60
2p3/2 -6.2 -4.98
Protons Neutrons
*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 RPA-Green’s function which is given by
where Vph is the particle-hole interaction and the free particle-hole Green’s function is defined as,
where is the single-particle wave function, єi is the single-particle energy, and ho is the single-particle Hamiltonian.
,)1( 1 opho GVGG
),'(11
)(*),',( rrrr iioioi
io EhEhEG
i
We use the scattering operator F
where for monopole excitation, to obtain the strength function
and the transition density.
)(1
A
iifF r
)](Im[1
)(0)(2
fGfTrEEnFESn
n
'.)],',(Im1
[)'()(
),( 3rrrrr dEGfEES
EEt
RPA
2
2
1)( rf r
A Note on Self-ConsistencyIn 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:
For example: E = 14 MeV (in 208Pb), and K = 230 MeV,
then a ΔE = 1 MeV leads to ΔK = 35 MeV.
,KE .2E
E
K
K
90Zr
116Sn
144Sm
208Pb
E [MeV]
S(E
) [fm
4 /M
eV]
Isoscalar Monopole Strength Functions
Fully Self-Consistent HF Based RPA Results For Breathing Mode Energy (in MeV)
Intergral Width Experimenta) SG2b) KDE0c)
90Zr 0--60 17.9 18.110--35 17.81+/-0.30 17.9 18.0
116Sn 0--60 16.2 16.610--35 15.85+/-0.20 16.2 16.6
144Sm 0--60 15.3 15.510--35 15.40+/-0.40 15.3 15.4
208Pb 0--60 13.6 13.810--35 13.96+/-0.20 13.6 13.8
K =215 K =229J =29 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
