ЯДЕРНАЯ ФИЗИКА, 2007, том 70, № 8, с. 1401-1406
ЯДРА. Теория
NEUTRON SKIN THICKNESS AND NUCLEAR MATTER PROPERTIES
© 2007 S. Yoshida1)*, H. Sagawa2)
Received October 31, 2006
Linear correlations are found among the isovector nuclear matter properties in both the Skyrme—Hartree— Fock (SHF) and the relativistic mean-field models. In addition, we found a kind of correlation between the isovector nuclear matter properties and the incompressibility in the SHF model. The Skyrme parameters are related analytically to nuclear matter properties with the Thomas—Fermi approximation. By using a linear correlation between the neutron skin thickness and the pressure of the neutron matter in the SHF model, we show that the neutron skin thickness of 208Pb gives crucial information about not only the neutron equation of state but also the isovector nuclear matter properties and the parametrization of Skyrme interaction.
PACS:21.30.-x, 21.60.-n, 21.60.Jz, 21.65.+f
1. INTRODUCTION
The Skyrme—Hartree—Fock (SHF) model [1] and the relativistic mean-field (RMF) model [2, 3] have been successful in providing a microscopic description of many properties of the nuclear ground states. Parameter sets in the SHF and the RMF models were determined by using the nuclear binding energy and the saturation density which are experimental quantities known well. Other nuclear matter properties such as incompressibility and symmetry energy which are less well determined from experiments are only used as rough guides in the determination of parameter sets. There are many versions of parameter sets for the SHF and the RMF models. Different parameter sets for the SHF and the RMF give fairly similar results for nuclear ground-state properties such as binding energy and radii of stable nuclei. On the other hands, results for physical quantities under extreme conditions, such as ground-state properties of neutron-rich unstable nuclei and infinite-nuclear-matter properties, depend on parameter sets. In this article we discuss relations among nuclear matter properties of the Hamiltonian density for some different Skyrme and RMF parameter sets. And we show that parameters in the Skyrme interaction except for the spin—orbit interaction can be expressed analytically in terms of isoscalar and isovector nuclear matter properties of Hamiltonian density with the Thomas—Fermi approximation. There exists a clear linear correlation between the pressure of neutron matter and the neutron skin thickness of heavy stable
!)Science Research Center, Hosei University, Tokyo, Japan.
2) Center for Mathematical Science, University of Aizu, Aizu-Wakamatsu, Fukushima, Japan. E-mail: s_yoshi@i.hosei.ac.jp
nuclei in both the SHF [4, 5] and the RMF [5, 6] models, whereas the correlation is weak for the unstable nuclei investigated in [7]. By using this correlation we analytically show relations between isovector nuclear matter properties and the neutron skin thickness of 208Pb. Furthermore we discuss the neutron equation of state (EOS).
2. PHYSICAL PROPERTIES AND SKYRME INTERACTION
The Skyrme interaction VSky is an effective zerorange force with density-dependent and momentum-dependent terms [8],
VSky(ri, r2) = to(1 + XoPa)S(ri - (1)
1
+ ¿2(1+ X2Pa)k • 5(ri - r2)k +
+ 1 ¿3(1+ X3Pa )pa (r)S(ri - r2) + 6
+ iW(ai + &2)k! • S(ri - r2)k,
where k = (V1 - V2)/(2i) acting on the right and
k' = -(V/1 - V2)/(2i) acting on the left are the relative-momentum operator, Pa is the spin exchange operator, a is the Pauli spin matrix, and r = (r1 + + r2)/2. The interaction (1) simulates the G matrix for nuclear Hartree—Fock calculations.
The isoscalar part h(p) and the isovector part es (p) of the Hamiltonian density Hnm are defined by
(2)
+ ^ii(1 + xP){k,2£(ri - Г2) + 5(ri - r2)k2} +
h(p) = lim Hnm,
I ^0
^ 1 d2 (Нш
^ = 2 Й M—
(3)
OSHF, »RMF
L, MeV 140 г
100
60
20
-20
16
о 10
15
14*8 « m9
17
Kym MeV
100
11
5
о
13
о
4
о
° 0 0 20 .19
nO °6 oj
о 12
о
о
26
KSym, MeV
30
34
38
/, MeV
-100
-300
-500
11
о
10
20 • «19
16 14
» V
15
18 •
17
О 13
9 3 О 7 о°0° 8
О
2 О
41
° .
о
12
о
100
-100
-300
-500
20 11 8 19 О 5
13 О 0„ °сРб°9°3
14
1615. ОФ»
10/18
17
26
/, MeV 38
34
30
34 38
/, MeV
7
о
30
12
о
4-о1
о
J_I_I_I_I_I_I_I
" A 15* d ф17 18..14
_16 • 010
" 19 • О 8*5 - О 20 5 6 О - О 7 °11 3 О 12 91 О 2 о
О13 | 1 о4 1 1
-20 20
L, MeV 140
100
60
20
60
100 140
L, MeV
-20
15 " 16 • 17 • 10 • e 18 14 • •
- о
- 5
11 ° " 6 19 О Q ^ • 3 О °9
"О 20 13 12°
| I 2 О 1 О 4° | I
Kym MeV
100
-100
-300
-500
200 250 300 350 400
K, MeV
200 250 300 350 400
K, MeV
b
a
3
5
2
c
Fig. 1. Correlations among nuclear matter properties calculated by using 13 parameter sets of the SHF (closed circles) and 7 parameter sets of the RMF (closed circles) models.
ЯДЕРНАЯ ФИЗИКА том 70 № 8 2007
Table 1. Power a of the density-dependent force in the Skyrme interaction (1)
Parameter set a Parameter set a
SI 1.0 Skya 1/3
SIII 1.0 SkI3 0.25
SIV 1.0 SkI4 0.25
SVI 1.0 SkM 1/6
SkX 0.8 SkM* 1/6
MSkA 0.7179 SG2 1/6
SLy4 1/6
where I is the asymmetry parameter I = (pn — — PP)/(pn + PP) and pn(pP) is the density of neutrons (protons). The physical properties of infinite symmetric nuclear matter with pn = pp can be obtained from the following six equations:
o - — (h
dP\P
P=Pnm
—Eo —
h(Pnm) pnm
(in SHF),
(4)
(5)
—Eo — — M (in RMF),
pnm
K =9P^TP2 (-
d2 dp2
P=Pnm
J = £è (Pnm ),
d
L — 3pdp£& (p)
P=Pnm
Ksym — 9p
d2
Lsym — 9p dp2£S (p)
(6)
(7)
(8)
P=Pnm
Table 2. Optimal values of the parameters in the Skyrme interaction, the nuclear matter properties and the effective mass at a = 0.1 and 0.2 for pnm = 0.16 fm~3, E0 = = 16 MeV, and K = 220 MeV, where Q1 = 3i1 + t2(5 + + 4x2) and Q2 = 3tx — t2(4 + 5x2)
a to t3 Qi m*/m
0.1 —3619.2 20343 905.85 0.6961
0.2 —2419.5 14835 395.24 0.8400
Snp a xo X3 Q2
0.14 0.1 0.58330 0.70839 361.93
0.14 0.2 0.53574 0.73269 —86.772
0.16 0.1 0.53573 0.69173 — 167.05
0.16 0.2 0.47366 0.71742 —612.22
Snp a J L Ksym
0.14 0.1 28.390 23.371 —202.54
0.14 0.2 29.053 26.221 — 196.84
0.16 0.1 29.650 40.248 — 138.01
0.16 0.2 30.317 43.088 — 132.33
where pnm, E0, K, and J are the nuclear saturation density, the binding energy per nucleon, the incom-pressibility of symmetric nuclear matter, and the symmetry energy, respectively. The quantities pnm, E0, and K are determined by the isoscalar part of the Hamiltonian density, whereas J, L, and Ksym are obtained from the isovector part of the Hamiltonian density. At first, we discuss relations among physical properties of infinite nuclear matter in the SHF model for 13 different parameter sets (SI, SIII, SIV, SVI, Skya, SkM, SkM*, SLy4, MSkA, SkI3, SkI4, SkX, SGII) taken from [1, 9-17] and the RMF model for 7 different parameter sets (NL3, NLSH, NLC, TM1, TM2, DD-ME1, DD-ME2) taken from [18— 23]. The nonlinear potential of u meson is added in
the parameter sets of TM1 and TM2. The meson— nucleon couplings depend on the total density in DD-ME1 and DD-ME2 parameter sets, while the couplings are constants in the other five parameter sets. In addition, parameters in the nonlinear potential of a mesons are fixed to zero in DD-ME1 and DD-ME2 parameter sets.
Figures 1a— 1f show ranges of values of K, J, L, and Ksym in the SHF and the RMF models. The numbers in Fig. 1 denote the different parameter sets: 1 for SI, 2 for SIII, 3 for SIV, 4 for SVI, 5 for Skya, 6 for SkM, 7 for SkM*, 8 for SLy4, 9 for MSkA, 10 for SkI3, 11 for SkI4, 12 for SkX, 13 for SGII, 14 for NLSH, 15 for NL3, 16 for NLC, 17 for TM1, 18 for TM2, 19 for DD-ME1, and 20 for DD-ME2. Clear linear correlations in both the Skyrme and the RMF parameter sets are found among the isovector nuclear matter properties as shown in Figs. 1a—1c. At the first glance it seems that there is not any clear correlations between the incompressibility K and the isovector nuclear matter properties in Figs. 1 d— 1 f. However, we can see a kind of correlation when we arrange the results according to the value of a. The figures indicate the presence of some type of singularity at a = 2/3, which reverses the correlations in Figs. 1 d— 1 f. The values of a in the SHF model are listed in Table 1. We can classify the Skyrme parameter sets into two groups: a > 2/3 and a < 2/3.
J, MeV 50
Eq. (10):
45 40 35 30 25 45 40 35 30 25
K, MeV — 200
bnp = 0.14 fm a
S„p, fm
K = 220 MeV m*/m x 1.0 A 0.9 Д 0.8 О 0.7
-J
b
к
•'S /га
0 0.1 0.2
0.4 0.5 0.6 a
Fig. 2. Symmetry energy J as a function of a for pnm = = 0.16 fm~3 and Eo = 16 MeV: (a) for neutron skin thickness Snp of 208Pb fixed to be 0.14 fm, (b) for incompressibility K fixed to be 220 MeV Points indicate the effective mass.
K <
9h2 10m
cp2/ + 15EQ.
(10)
Because of E0 = 16 MeV and pnm = 0.16 fm"3
in general, ^ЦшcPnin + 15E0j is almost equal to
306 MeV. This is the reason of the singularity shown by dotted lines in Fig. 1d— 1/. Recently, the value of the incompressibility K determined to be 230 ± ± 10 MeV from experimental data on giant monopole resonances [24, 25]. Therefore this value of K suggests a < 2/3.
3. NEUTRON SKIN THICKNESS
AND NUCLEAR MATTER PROPERTIES
The neutron skin thickness 5np is defined as the difference between the rms neutron and proton radii. The pressure Pshf(x) of neutron matter in the SHF model is expressed by a and six nuclear matter properties defined by Eqs. (4)—(9) as a function of the ratio x = pn/pnm, where pn is a neutron density. We checked that there was the clear linear correlation between the neutron skin thickness 5np of 208Pb and the calculated pressure P(x) of neutron matter for the Skyrme interaction used in Fig. 1 at the different ratio x from 0.90 to 1.10 by the same way in [7]. These correlations are characterized by functions A(x) and B(x) as P(x) = A(x)5np + B(x). Moreover, by the method of least square we found that A(x) and B(x) are parametrized as quadratic functions of x a
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