The volume coefficient K(=incompressibility of the nuclear matter), the Coulomb coefficient K_c, and the volume-symmetry coefficient K_{vs} of the nucleus incompressibility are studied in the framework of the relativistic mean field theory, with aid of the scaling model. It is found that K= 300\pm 50MeV is necessary to account for the empirical values of K_v, K_c, and K_{vs}, simultaneously. The result is independent on the detail descriptions of the potential of the \sigma-meson self-interaction and is almost independent of the strength of the \omega-meson self-interaction

Hasegawa, A

Koide, K

Kouno, H

Mitsumori, T

Nakano, M

Noda, N

CERN Document Server

SAGA-HE-88-95
July 1995
Incompressibility of nuclear matter, and Coulomb and volume-symmetry
coecients of nucleus incompressibility in the relativistic mean eld theory
H. Kouno, T. Mitsumori, N. Noda, K. Koide, A. Hasegawa
Department of Physics, Saga University, Saga 840, Japan
and
M. Nakano
University of Occupational and Environmental Health, Kitakyushu 807, Japan
(PACS numbers: 21.65.+f, 21.30.+y )
ABSTRACT
The volume coecientK(=incompressibility of the nuclear matter), the Coulomb
coecient Kc, and the volume-symmetry coecient Kvs of the nucleus incom-
pressibility are studied in the framework of the relativistic mean eld theory,
with aid of the scaling model. It is found that K = 300  50MeV is neces-
sary to account for the empirical values of Kv, Kc, and Kvs, simultaneously.
The result is independent on the detail descriptions of the potential of the -
meson self-interaction and is almost independent of the strength of the !-meson
self-interaction.
One way to determine the incompressibility K of nuclear matter from the
giant monopole resonance (GMR) data is using the leptodermous expansion[1]
of nucleus incompressibility K(A;Z) as follows.
K(A;Z) = K+KsfA
−1=3 +KvsI
2 +KcZ
2A−4=3 +    ; I = 1−2Z=A; (1)
where the coecients, K, Ksf , Kvs and Kc are volume coecient (incompress-
ibility of nuclear matter), surface term coecient, volume-symmetry coecient
and Coulomb coecient, respectively. We have omitted higher terms in eq. (1).
Although there is uncertainty in the determination of these coecients by us-
ing the present data, Pearson [2] pointed out that there is a strong correlation
between K and Kc. (See table I.) Similar observations are done by Shlomo and
Youngblood [3].
Table I
According to this context, Rudaz et al. [4] studied the relation between
incompressibility and the skewness coecient by using the generalized version
of the relativistic Hartree approximation [5]. The compressional and the surface
properties are studied by Von-Ei et al. [6-8] in the framework of the relativistic
mean eld approximation of the -!- model with the nonlinear  terms. They
found that low incompressibility (K  200MeV) and a large eective nucleon
mass M at the normal density (0:70  M=M  0:75) are favorable for the
nuclear surface properties [8]. On the other hand, using the same model, Bodmer
and Price [9] found that the experimental spin-orbit splitting in light nuclei
supports M  0:60M . The result of the generator coordinate calculations for
breathing-mode GMR by Stoitsov, Ring and Sharma [10] suggestsK  300MeV.
In previous papers[11,12], we have studied the relation between K and Kc
in detail, using the relativistic mean eld theory with the nonlinear  terms
[13] and the one with the nonlinear  and ! terms [14]. We found that, under
the assumption of the scaling model [1], K = 300  50MeV is favorable to
account for K, Kc and Kvs, simultaneously. It seems that this conclusion is
not drastically changed in the use of the relativistic mean eld theory and the
scaling model. In this paper, we examine the conclusion in more general way,
in which the result does not depend on the detail descriptions of the -meson
self-interaction. The reason why we restrict our discussions to K, Kc and Kvs is
that the general discussions, which are independent of the detail of the model (
e.g., types of the interactions, values of the parameters in the Lagrangian, etc.)
are possible to a considerable extent, since, as is shown below, these quantities
are almost analytically estimated by using the result for the nuclear matter, if
we assume the scaling model [1].
We use the relativistic mean eld theory based on the -!- model with the
nonlinear  terms. The Lagrangian density consists of four elds, the nucleon  ,
the scalar -meson , the vector !-meson V, and the vector-isovector  meson
b, i.e.,
LN! =  (iγ@
 −M) 
+
1
2
@@
−
1
2
m2s
2 −
1
4
FF
 +
1
2
m2vVV
 −
1
4
B B
 +
1
2
m2b  b

+gs   − gv  γ V
 − g  γ

2
 b − U() ;
F = @V − @V; B = @b − @b − gb  b; (2)
where ms, mv , m, gs, gv and g are -meson mass, !-meson mass, -meson
mass, -nucleon coupling, !-nucleon coupling, and -nucleon coupling, respec-
tively. The U() is a nonlinear self-interaction potential of  meson eld . For
example, in ref. [11], we have used the quartic-cubic terms of  as in ref. [13],
i.e.,
U() =
1
3
b3 +
1
4
c4; (3)
where b and c are the constant parameters which are determined phenomeno-
logically. However, in this paper, we do not give an explicit expression of U()
and discuss the problem in more general way, without any assumption of U().
In the scaling model [1], K, Kc and Kvs in eq. (1), are given by
K = 920
@2Eb
@2
j=0 ; (4)
Kc = −
3q2el
5R0

9K0
K
+ 8

; (5)
Kvs = Ksym − L

9
K0
K
+ 6

; (6)
where , 0, Eb and qel are the baryon density, the normal baryon density, the
binding energy per nucleon, and the electric charge of proton, respectively, and
R0 = [3=(40)]
1=3,
K0 = 330
d3Eb
d3
j=0 ; (7)
L = 30
dJ
d
j=0 ; Ksym = 9
2
0
d2J
d2
j=0 : J =
1
2
2
@2Eb
@23
j3=0: (8)
The quantity such as K0 is sometimes called "skewness".
In the mean eld theory with the Lagrangian (2), L and Ksym are given by
[7]
L =
3
8M2
C2 +

2

2kFk
0
F
EF
−
k2FE
0
F
E2F

( = 0); (9)
and
Ksym =
3
2
2

2k02F
EF
+
2kF k
00
F
EF
−
4kF k
0
FE
0
F
E2F
+
2k2FE
02
F
E3F
−
k2FE
00
F
E2F

( = 0); (10)
where kF (= [3
2=2]1=3) is the Fermi momentum, and C = gM=m,
k0F =
dkF
d
=
kF
3
; k00F =
d2kF
d2
=
−2kF
92
; (11)
EF =
q
k2F +M
2; E0F =
dEF
d
; E00F =
d2EF
d2
: (12)
M in eq. (12) is the eective nucleon mass. Furthermore, at  = 0, E
0
F and
E00F are related to K and K
0 in the following relations, respectively [7,11].
E0F =
K
90
−
C2v
M2
; Cv =
gvM
mv
: (13)
E00F =
K +K0
320
: (14)
At  = 0, Cv and C are also related to M
 as follows.
C2v =
M2
0

M − a1 −
q
k2F +M
2

; (15)
[13] and
C2 =
8M2
0

a4 −
k2F
6EF

; (16)
[15] where a1 and a4 are the binding energy and the symmetry energy at  = 0,
respectively. We remark that eqs. (5), (6), and (9)(16) have no explicit
dependence on U().
From eq. (5), K0 are determined, if 0, K and Kc are given. Therefore, from
the eqs. (5), (6), and (9)(16), it is seen thatKvs is determined, if 0, M , a1, a4,
K, Kc, and M
 are given, without giving the detail descriptions for U(). Using
these equations, we calculate Kvs. In the calculations, we put 0 = 0:16fm
−3,
M = 939MeV, a1 = 15:75MeV and a4 = 30:0MeV. For K and Kc, we use the
values in table I. We assume thatM = 0:5M  0:94M , the phenomenologically
acceptable values. (We remark the upper bound for M is gotten, if we put
Cv = 0 in eq. (15). ) In g. 1 (solid lines), we show Kvs as a function of M
 for
two sets of K and Kc in table I. In the gures, Kvs decreases as M
 increases.
This is understood as follows. As M increases, L and Ksym decreases, because
of large EF in the denominators in eqs. (9) and (10). Small Ksym makes Kvs
smaller (more negative) and, on the contrary, small L makes Kvs larger (less
negative), since (9K0=K + 6) > 0 in eq. (6). In the cases of g. 1, the eect of
the small Ksym overcomes that of the small L, therefore, Kvs decreases. In the
case of K = 300MeV and Kc = −3:990MeV, Kvs = −289  −98MeV. These
values are in good agreement with the corresponding empirical values in table
I. We also remark that the uncertainty of Kvs in changing M
 is comparable
to the empirical error bar of Kvs in table I. In the case of K = 250MeV and
Kc = −0:7065MeV, Kvs = 50  410MeV. These values are somewhat larger
than the empirical ones. In table II(a), we summarize the range of the calculated
Kvs for each set of K and Kc in table I. Comparing the table I and table II(a),
we see that K = 300  50MeV is necessary to account for K, Kc and Kvs,
simultaneously.
Fig. 1(a),(b), Table II(a),(b)
We remark the following three points.
(1) The results are independent of the form of U(), since eqs. (5), (6), and
(9)(16) are required for any type of U().
(2) The question, whether there are coupling parameters, which reproduce
the set of K and Kc in table I, or not, is still open, and the answer for the
question depends on the detail descriptions of U(). For example, if we use the
quartic-cubic potential (3), we could not nd the coupling parameters (i.e., gs,
gv, b, and c ), which reproduce K = 200MeV and Kc = 2:577MeV [11]. (We
remark that K and Kc are independent on g in the mean eld theory. ) Also,
using eq. (3), we get the parameter set for K = 300MeV and Kc = −3:990MeV,
only in the case of M = 0:83M [11]. In those cases, parameter set is uniquely
determined or could not be found, since the number of the parameters is not
larger than the number of the inputs, i.e., K, Kc, and two conditions for the
saturation. If the higher terms of  are added to (3), the wider range of M
may be available. However, K = 300  50MeV is necessary to reproduce the
empirical values of K, Kc, and Kvs simultaneously, for any type of U().
(3) The results do not have a strong-dependence on 0, a1 and a4, since
the calculated Kvs is much more sensitive to the ratio K
0=K than to those
quantities.
Next we add the following attractive term of vector self-interaction (VSI)
[14,12] to the Lagrangian (2).
LV SI =
g2vY
2
4
m2v(VV
)2; (17)
, where Y is a positive constant which determine the strength of VSI. Although,
according to this modication, the formalism for calculating Kvs, which is de-
scribed above, is slightly modied, it is still possible to study M-Kvs relation
without giving the detail descriptions of U(), as in the case of no VSI, i.e.,
Y = 0. In g. 1, we also show Kvs as a function of M
, in the cases of y = 1
and y = 5, where y = g2v0Y=m
2
v [12]. In those gures, it is seen that the
VSI makes Kvs smaller in the small M
 region. Kvs is almost independent
on y in the large M region, where the value of M dominates the features
of the equations of state. As a result, the uncertainty of Kvs in changing
M becomes smaller. In the case of K = 300MeV and Kc = −3:990MeV,
Kvs = −298  −282(−303  −289)MeV with y = 1:0(5:0). In the case of
K = 250MeV and Kc = −0:7065MeV, Kvs = 50  147(49  86)MeV with
y = 1:0(5:0). In each case, Kvs with y = 5:0 is not much dierent from that
with y = 1:0. Kvs is hardly changed, if y increases much more. In table II(b),
we summarize the range of the calculated Kvs in the cases of y = 5:0. Com-
paring this table with table I, we see that K = 300  50MeV is necessary to
account for the empirical values of K, Kc, and Kvs at the same time, as is in
the case of no VSI. Although the VSI makes Kvs smaller (more negative), the
change of Kvs is comparable to the magnitude of the empirical error bars at
the empirical available value of K( 300MeV), and, therefore, the conclusion is
hardly changed. The result is also independent on the detail of U() as in the
case of no VSI.
In summary, we have studied K, Kc, and Kvs by using the relativistic mean
eld theories with the nonlinear  term and with the nonlinear  and ! terms,
with aid of the scaling model. It is found that, in both cases, K = 30050MeV
is necessary to account for the empirical values of K, Kc, and Kvs at the
same time. The result is independent on the detail descriptions of U() and is
almost independent on the strength of VSI. It seems that this conclusion is not
drastically changed, if we use any type of the relativistic mean-eld theory and
the scaling model, since the calculated Kvs is most sensitive to the ratio K
0=K,
which is adjusted to the empirical values.
Acknowledgment: The authors gratefully acknowledge the computing time
granted by the Research Center for Nuclear Physics (RCNP).
References
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183(1992).
[5] E.K. Heide and S. Rudaz, Phys. Lett. B262, 375(1991).
[6] D. Von-Ei, J.M. Pearson, W. Stocker and M.K. Weigel,
Phys Lett. B324, 279(1994).
[7] D. Von-Ei, J.M. Pearson, W. Stocker and M.K. Weigel,
Phys. Rev. C50, 831(1994).
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[9] A.R. Bodmer and C.E. Price, Nucl. Phys. A505, 123(1989).
[10] M.V. Stoitsov, P. Ring and M.M. Sharma, Phys. Rev. C50, 1445(1994).
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M. Nakano, Phys. Rev. C51, 1754(1995).
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to be published in Phys. Rev. C.
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[14] A.R. Bodmer, Nucl. Phys. A526, 703(1991).
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Relativistic Nuclear Many-Body Problem in: Advances in nuclear physics,
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Table and Figure Captions
Table I
The sets of the empirical values of K, Kc and Kvs in the table 3 in ref. [2].
(According to the conclusion in ref. [2], we only show the data in the cases of
K = 150  350MeV.) All quantities in the table are shown in MeV.
Table II
Range of the calculated Kvs using the sets of K and Kc in table I as inputs. (a)
The result in the the mean-eld theory only with  meson self-interaction. (b)
The result in the mean-eld theory with  and ! mesons self-interactions with
y = 5:0. : In each table, "upper bound", "mean value", and "lower bound"
mean that the results are obtained by using the upper bound, the mean value,
and the lower bound of Kc in table I, respectively. All quantities in the table
are shown in MeV.
Fig. 1 Kvs as a function of M
. (a) The cases of K = 300MeV and
Kc = −3:990MeV. (b) The cases of K = 250MeV and Kc = −0:7065MeV: In
each gure, the solid line is the result in the the mean-eld theory only with 
meson self-interaction, and the dotted and the dashed lines are the results in
the mean-eld theory with  and ! mesons self-interactions with y = 1:0 and
y = 5:0, respectively.
Set 1 Set 2 Set 3 Set 4 Set 5
K 150.0 200.0 250.0 300.0 350.0
Kc 5:861 2:06 2:577 2:06 −0:7065 2:06 −3:990 2:06 −7:274 2:06
Kvs 66:83  101 −46:94 101 −160:7 101 −274:5 101 −388:3 101
Table I
upper bound mean value lower bound
K = 150 9401738 7281426 5171113
K = 200 6011230 389 918 178 605
K = 250 262 722 50 410 -161 97
K = 300 -77 214 -289 -98 -506-411
K = 350 -418-294 -651-606 -921-840
Table II(a)
upper bound mean value lower bound
K = 150 9391103 728 863 516 623
K = 200 600 714 389 474 177 235
K = 250 261 325 49 86 -165-154
K = 300 -80 -63 -303-289 -542-500
K = 350 -451-416 -691-628 -931-839
Table II(b)


Incompressibility of nuclear matter, and Coulomb and volume-symmetry coefficients of nucleus incompressibility in the relativistic mean field theory

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