1129
Progress of Theoretical Physics, Vol. 106, No. 6, December 2001
Study of Even-Z Nuclei Up to Mg
with the Gogny Force Using AMD
Yoshio Sugawa, Masaaki Kimura and Hisashi Horiuchi
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
(Received May 31, 2001)
Employing the Gogny force as an effective force, we study the ground state properties of
light nuclei using antisymmetrized molecular dynamics (AMD). In this study, we are mainly
concerned with the binding energies and radii of light even-Z isotopes, namely He, Be, C, O,
Ne and Mg. Using a new technique to calculate the density dependent term of the effective
force, we have realized fast and accurate calculations. From a comparison with Skyrme
SIII results within the same AMD framework, we find that the Gogny and SIII forces well
reproduce the experimental binding energies of stable nuclei. The two forces give almost
equal radii, except in the case of 7 Be and 9 Be. For both forces, approximate treatment
of the center-of-mass kinetic energy causes overestimation of the binding energy compared
with the exact treatment. It also causes a decrease of the nuclear deformation compared
with the exact treatment. We also carry out an energy variation after the parity projection.
With regard to the binding energies and radii, parity-projected calculations do not exhibit
a large difference compared to non-projected results, although the density distribution and
clustering features are often significantly changed by the parity projection.
§1.
Introduction
There have been many theoretical studies of nuclear structure using various
effective nuclear forces. However, there are only a few approaches that are free from
model assumptions regarding the wave function. The Cartesian mesh Hartree Fock
method 1) is one such approach within the mean field approximation, and with this
framework many investigations have been conducted. Skyrme forces, whose range is
zero, are used in the studies with the Cartesian mesh Hartree Fock method. Though
many people have attempted to refine the Skyrme force, for some nuclei, Skyrme
forces seem to fail in the description of the deformation properties. We do not know
whether this is due to the insufficiency of the mean field or to the use of a zero range
force. In order to clarify this point, it is important to use finite range effective forces.
Among many finite range effective nuclear forces, the Gogny force 2) is regarded as
one that can be used to account for properties of nuclei over a wide range of mass
like Skyrme forces. However, the finiteness of its range itself has prevented the
detailed investigation of nuclei with this force within the Cartesian mesh Hartree
Fock approach, because the computational time is excessive. For this reason, studies
employing the Gogny force have progressed little, compared to their employing other
simpler interactions such as the Skyrme interaction.
Antisymmetrized molecular dynamics (AMD) is another framework free from
model assumptions. At the expense of limiting the single particle wave function to a
Gaussian or a superposition of Gaussians, the AMD enables realistic and systematic
study of nuclear properties without assumption regarding the wave function. AMD
1130
Y. Sugawa, M. Kimura and H. Horiuchi
has been applied to both nuclear structure study 3), 4) and nuclear reaction simulations 5) and has succeeded in describing nuclear properties of a variety of nuclei 6)
and various kinds of nuclear reactions up to quite heavy systems. 7) Among the many
advantages of AMD over other frameworks, the use of a finite range interaction and
the use of exact parity and angular momentum projection are necessary attributes
of a model for the detailed investigation of nuclear structure.
The study of effective interactions in the AMD framework is yet in a primitive
state of development. In most existing structure studies using the AMD, Volkov
or modified Volkov interactions 8) have been used. Although these are finite range
interactions, their parameters are not global. Rather some of the parameters must be
optimized within each region — from small to large mass — in order to obtain good
agreement with experiment. In the AMD framework, the Gogny force has been used
only in the reaction calculation. There are two reasons that the Gogny interaction
has not been more extensively used in AMD. One reason is that the evaluation of the
density dependent term of the Gogny interaction with sufficient numerical accuracy is
more difficult than with the modified Volkov interaction. Another reason is that the
density dependent term with a fractional power of the density is not uniquely defined
in the case of off-diagonal matrix elements between different Slater determinants.
This latter problem appears when parity and angular momentum projections are
applied exactly.
In order to explore nuclear properties more precisely and more generally, we
developed a new method that allows us to obtain enough accurate expectation value
of the density dependent term of the Gogny interaction in moderate computational
time. This also enabled us to use another effective force. In the present work, we
investigated the properties of isotopes with even proton number up to Mg with the
Gogny and the SIII force. 9) We are mainly concerned with the binding energies and
the radii of these light isotopes. Regarding the nuclear deformation properties, we
will discuss the nuclear deformation properties in a subsequent paper. Comparison
of the Gogny force with the SIII force, whose terms are all of zero range, would
clarify the properties of the Gogny interaction. Because the SIII force has been
used widely in the Hartree Fock framework and there are many calculational results
obtained with this force, we can compare the AMD results with the Hartree Fock
results. This comparison should give important information concerning the AMD
framework, not only the advantages but also disadvantages in comparison with the
Cartesian mesh Hartree Fock that are caused from the above mentioned limitations
of AMD.
In many Hartree Fock calculations, the subtraction of the center-of-mass kinetic
energy is carried out approximately, in general, by omitting the two-body cross
terms of the form pi · pj . In comparison with the exact treatment, this results in an
overestimation of the binding energy that sometimes is about 10 MeV. We find that
accurate prediction of the binding energy with the Gogny and SIII forces is possible
only when we adopt this approximation.
We also carry out an energy variation after the parity projection, which is
thought to be important in the study of nuclear structure. In some specific nuclei, the parity projection enhances the parity-violating intrinsic deformation. In
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1131
this study, by performing the parity-projected calculation for isotopes from He to
Mg, we investigate how the parity projection affects the nuclear structure and which
observables are changed.
The content of this paper is as follows. In §2, we give a brief explanation of
the AMD framework and explain the approximations made in the Hartree Fock
Hamiltonian for the sake of comparison of the AMD and Hartree Fock calculations.
In §3, a newly developed method of the density integral evaluation is explained. In
§4, calculated results and discussions are given. We see that the Gogny and SIII
forces reproduce the experimental binding energy of stable nuclei and give almost
equal values of proton and neutron radii, although both forces yield values that are
smaller than the observed charge radii.
We show that the exact and the approximate treatments of the center-of-mass
kinetic energy give different deformation energy curves. We also see that the effect of
the parity projection is not large for binding energies and radii but is often large for
the density distribution and clustering features. In §5, we summarize the paper. In
the Appendix, detailed description of the numerical technique used in the evaluation
of the density dependent term of the force is given.
§2.
AMD equation
2.1. Framework of AMD
The wave function of the total system is expressed in terms of a Slater determinant of single particle wave functions. Each single particle wave function has a
part χτ i . The spatial part has a Gausspatial part φi (r), spin part χi and isospin √
sian form centered at Z i and its width is 1/ ν. The spin part is parametrized by a
complex number parameter ξi . We have
|Φ(Z, ξ) = det[ϕi (rj )],
|ϕi (rj ) = φi (r j ) · χi · χτ i ,
|φi (r) =
1
2
1
2
2ν
π
3/4
Zi
exp −ν r − √
ν
1
+ Z 2i ,
2
(2.1)
(2.2)
(2.3)
− ξi χ↓ , and χτ i represents the isospin of a proton or
√
neutron. When we express the complex number vector Z i as Z i = νDi + 2h̄i√ν K i ,
D i represents the spatial position of the wave packet and K i the momentum of
the wave packet. The three complex parameters Z i , and one complex parameter ξi
for individual nucleons and one real parameter ν common to all the nucleons are
determined by the energy variation. The linear superposition of Gaussian packets
for single particle wave functions and/or linear superposition of Slater determinants
have often been used in AMD calculations, but in this paper we adopt the simple
version of AMD described above. The time evolution of the system is calculated
using the variational principle
where χi =
+ ξ i χ↑ +
t2
δ
t1
dt
d
Φ (Z, ξ) | ih̄ dt
− H |Φ (Z, ξ)
Φ (Z, ξ) |Φ (Z, ξ)
= 0,
(2.4)
1132
Y. Sugawa, M. Kimura and H. Horiuchi
where
H =
Φ (Z, ξ) |H|Φ (Z, ξ)
,
Φ (Z, ξ) |Φ (Z, ξ)
A
H =T +V −
which leads to
ih̄
µ
ih̄
µ
i=1 pi
2
(2.6)
,
2mA
∂2I
∂ 2I
Ż
+
ξ̇µ
µ
∂Z ∗ν ∂Z µ
∂Z ∗ν ∂ξµ
(2.5)
∂ 2I
∂ 2I ˙
+
Ż
ξµ
µ
∂ξν∗ ∂Z µ
∂ξν∗ ∂ξµ
=
∂H
,
∂Z ∗ν
=
∂H
,
∂ξν∗
(2.7)
where
I ≡ lnΦ (Z, ξ) |Φ (Z, ξ).
(2.8)
The ground state wave function is obtained with the frictional cooling method, 10)
which is also known as the imaginary time evolution method. The equation of
imaginary time evolution is obtained by multiplying the right-hand side of Eq. (2.7)
by an imaginary factor.
The deformation parameter β and γ are defined as follows:



x2 1/2
= exp
[x2 y 2 z 2 ]1/6
5
2π 
β cos γ +
,
4π
3
y 2 1/2
= exp
[x2 y 2 z 2 ]1/6
5
2π 
β cos γ −
,
4π
3

z 2 1/2
[x2 y 2 z 2 ]1/6

= exp

5
β cos γ .
4π
(2.9)
x2 , y 2 and z 2 are eigenvalues of the tensor quantity ri rj , and satisfy x2 ≤ y 2 ≤ z 2 . The energy surface as a function of the deformation parameter β is calculated using the constrained frictional cooling method developed by Doté et al. 11)
2.2. Effective interactions
In the present study, we employed two effective interactions, namely the Gogny
D1 and the Skyrme III (SIII) forces. The Gogny force has the form
V (r) =
(Wi + Bi Pσ − Hi Pτ − Mi Pτ Pσ )e−r
2 /µ2
i
i=1,2
+ t3 (1 + x3 Pσ )[ρ(R)] δ(r) + iW0σ · [P × δ(r)P ],
(2.10)
which is regarded to be a more realistic interaction than the Skyrme interaction,
since the range of the two-body interaction is finite.
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1133
Table I. Force parameters of the Gogny force.
i=1
i=2
σ
1/3.0
Wi
−402.4
−21.3
x3
1
Bi
−100
−11.77
t3
1350
Hi
−496.2
37.27
W0
115
Mi
−23.56
−68.81
µi
0.7
1.2
Table II. Force parameters of the SIII force.
σ
1
x0
0.45
x1
0
x2
0
x3
1
t0
−1128.75
t1
395
t2
−95
t3
14000.0
W0
120
The Skyrme force has the form
1
V (r) = t0 (1 + x0 Pσ )δ(r) + t1 (1 + x1 Pσ )[P 2 δ(r) + δ(r)P 2 ]
2
+t2 (1 + x2 Pσ )P · δ(r)P
1
(2.11)
+ t3 (1 + x3 Pσ )[ρ(R)]σ δ(r) + iW0σ · [P × δ(r)P ].
6
←−
−→
∂
∂
Here r = r 1 − r 2 , R = (r 1 + r 2 )/2, P =
and P =
. The force parameters
∂r
∂r
are displayed in Tables I, II.
2.3. Parity-projected calculation
In this study, we also perform the parity-projected calculation, which employs
the parity-projected wave function |Φ± , instead of the single Slater determinant |Φ.
We write
|Φ± = |Φ ± Px |Φ.
(2.12)
Here Px is the spatial inversion operator.
There is a problem in calculating the off-diagonal matrix elements of the densitydependent term in the Gogny force with |Φ and Px |Φ, because we do not know how
to define the density ρ(R) for off-diagonal matrix elements. We define the density
ρ(R) to be used for the off-diagonal matrix elements with |Φ and Px |Φ as
ρ(R) ≡
Φ|
− R)Px |Φ
,
Φ|Px |Φ
i δ(r i
(2.13)
and also we define (ρ(R))σ as
(ρ(R))σ ≡ |ρ(R)|σ exp{iσ arg(ρ(R))},
−π < arg(ρ(R)) ≤ π.
(2.14)
When σ = 1, as in the SIII force and the spin is fixed on up or down coupled
to zero, the density dependent two-body force (1 + Pσ )ρ(R)δ(r) is equivalent to the
zero range three-body force of the form δ(ri − r j )δ(rj − r k ). In this case, it is easy to
show that the above definition of ρ(R) is correct. This definition of the off-diagonal
density matrix elements is also suggested in some HFB studies. 12), 13)
1134
Y. Sugawa, M. Kimura and H. Horiuchi
2.4. Approximate calculations of center-of-mass kinetic energy and Coulomb energy
in the Hartree Fock calculation
Although the wave function of AMD is expressed in terms of a single Slater
determinant, as in Hartree Fock method, there are a few differences between Hartree
Fock and AMD method with regard to the calculational treatment of the Hamiltonian. In order to make the comparison clear, here we discuss the differences in the
Hamiltonians.
(1) Treatment of center-of-mass kinetic energy correction
The biggest difference is in the treatment of the center-of-mass kinetic energy
correction. The center-of-mass kinetic energy,
T0 ≡
(
A
i=1 pi )
2mA
2
(2.15)
,
should be subtracted from the expectation value of the total Hamiltonian. In AMD,
Gaussian wave packets whose widths are equal for all nucleons represent the spatial
part of the single particle wave functions. The total wave function of the system is
constructed by taking their Slater determinant. It follows that the wave function
of the center-of-mass exactly separates from that of the internal coordinates. This
enables us to evaluate the center-of-mass kinetic energy T0 in a rigorous way:
H = T + V −
(
A
i=1 pi )
2
2mA
= T + V − T0AMD .
(2.16)
Here T0AMD ≡ 3h̄2 ν/2m is a constant value and depends only on the width parameter
ν. For each combination of the proton number and neutron number, we varied the
width ν from 0.13 to 0.22 and chose ν so that it minimizes the energy H. This
implies that ν and consequently T0AMD depend on proton and neutron number in
the results given below.
In the Hartree Fock approach, no such simple separation of the center-of-mass
wave function is possible, and one conventional way
to determine the center-of-mass
2 kinetic energy correction is to substitute T0aprx ≡ pi /2mA for T0 . This means
that the 2-body term i=j pi · pj /2mA is ignored. Therefore, the Hartree Fock
Hamiltonian H aprx can be rewritten as
H aprx = T + V −
2
p
i
2mA
2
( A
i=j pi · pj
i=1 pi )
=T +V −
+
2mA
2mA
p
·
p
j
i=j i
=H+
.
2mA
(2.17)
The parameters of the effective interactions (the SIII and Gogny forces) have been
searched for and defined within this Hartree Fock framework. Therefore it must
be kept in mind that these force parameters are optimized so that H aprx , not H,
gives the best description of the properties of the nuclei. This can be interpreted
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1135
as meaning that the effective interactions V implicitly include the term i=j pi ·
pj /2mA. In subsequent sections, we describe the differences in the obtained results
caused by the differences of these two Hamiltonians, H and H aprx .
(2) Coulomb energy
In AMD, the functional form of the Coulomb interaction (1/|r i −rj |) is simulated
by the superposition of seven Gaussians. 5) Within this approximation, both direct
and exchange terms are treated exactly. In the Hartree Fock approach, the exchange
term of the Coulomb term has been approximated with several methods, 14) and its
deviations from the exact calculation has been reported to be small. Tajima et al.
reported the expectation values obtained in their SIII calculation. 15) They used the
local potential approximation of the exchange term of the Coulomb force which acts
attractively. Their expectation values of the Coulomb force were found to be about
20% greater than our results.
§3.
Precise calculation of the density dependent part
In previous AMD calculations of the nuclear structure, in which the modified
Volkov force has been used, all matrix elements are obtained analytically. T , V and their Z and ξ derivatives are given as functions of {Z}and {ξ} without any
ambiguity. This is possible because the density-dependent term in the modified
Volkov force can be analytically integrated.
drρ(r)σ+2 ,
σ = 1.
(3.1)
However, since σ = 1/3 in the Gogny force, it is impossible to obtain the analytic expression in this case. In the study of medium energy nuclear collisions, Ono et al. 16)
numerically evaluated this integral using Metropolis sampling. However numerical
errors in the expectation value V and its derivatives in this computation are not
sufficiently small in the study of nuclear structure, although they are small enough
for the study of medium energy nuclear collisions. If one increases the number of the
sampling points, one can obtain more convergent expectation value, but its computational time increases much. We recently developed a new code that evaluates this
integral using simple Cartesian mesh summation. To obtain the same convergence as
in Metropolis sampling, we must use finer mesh. However, the mesh points are now
aligned regularly, and the evaluation of the density at each mesh point can be done
faster by the use of a vector super computer. We give more detailed explanation of
this mesh evaluation method in the Appendix.
Now we explain the rigidity of the calculation regarding the density dependent
part of the interactions. Ono et al. reported that in the evaluation of the binding
energy of 12 C , the numerical error was about 2 MeV when they use 100 A(= 1200)
test particles. In our new method, the number of lattice points is defined by two
parameters, the integration volume, L3 [fm3 ], and size of one cube, a fm. The
calculational time depends on the number of points in the volume (M + 1)3 , where
M ≡ L/a. In Table III, a comparison of our new method with that of Ono et al. 17)
1136
Y. Sugawa, M. Kimura and H. Horiuchi
Table III. Top: Convergence of V in the present code. The integration volume is (12 fm)3 , M is
the number of mesh points in one direction,and dE = |V − VM =24 |. Bottom: Convergence of
V in the code of Ono et al. N is the total number of sampling points, and the dispersion σ is
defined from 100 evaluation of V .
M
(M + 1)3
dE (MeV)
N
σ
12
1728
0.062
1200
2.42
15
3375
0.00077
2400
1.78
18
5832
0.00003
3600
1.43
24
13824
6000
1.19
for the ground state of 12 C is made. The upper table shows the convergence of our
result obtained with various values of M for the L = 12 fm case. dE is the difference
between the expectation value V and its convergent value (M = 24). We find that
even when M = 12, in which case the unit volume of the cell is 1 fm3 , dE is around 60
keV. On the other hand, with the code developed by Ono, the lower table, even when
N (the number of sampling points) is increased, the dispersion of the expectation
value is large. In the present study, we used M = 18, and the discrepancy from
‘exact’ expectation value can be regarded as negligible. When a density dependent
force with σ = 1, such as Skyrme-III or the modified Volkov force, is used, it is
not necessary to evaluate such a discrepancy, because an analytical expression of the
integral (Eq. (3.1)) can be found. However, it is more economical to use this method
than to use analytical integration because the calculational time becomes large in
that case.
§4.
Results and discussion
We made calculation for the ground state properties of He, Be, C, O, Ne and
Mg isotopes. For each isotope, we applied the frictional cooling method to the
Hamiltonian H. From the ground state wave function thus obtained, we calculated
the binding energy E, radii for protons and neutrons and deformation parameters
β.
In order to compare the AMD calculation results with the Hartree Fock results,
we applied two prescriptions. One estimation is to use |Φ (Z, ξ), the ground state
wave function of H obtained above, to evaluate H aprx . We denote the binding
energy so obtained as Ê: Ê−E = − i=j P i ·P j /2mA. Since |Φ (Z, ξ) is optimized
so that H is minimal, it is not necessarily the ground state of H aprx . For this reason,
as the second prescription, we calculated the ground state of H aprx , |Φ(Z, ξ)aprx
only for lighter isotopes with cooling calculation. We attach the suffix aprx to the
quantities calculated with |Φ(Z, ξ)aprx , e.g. E aprx . Comparison between Ê and
E aprx , β and β aprx , and so forth, should indicate whether a different treatment of T0
would give rise to a difference in the observables. We summarize these definitions in
Fig. 1.
First, we compare and analyze the differences that result from the difference in
the treatment of T0 . Next, we examine our results.
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
approximated (Hartree Fock-like)
AMD(exact)
Hamiltonian
H
C.M. kinetic energy
T0
H aprx
hT i = A
Frictional
Cooling
Frictional
Cooling
jà(Z; ò)i
Ground state
wavefunction
jà(Z; ò) iaprx
E
E
Binding Energy
1137
Estimated
Binging Energy
aprx
E = - h à(Z; ò) j H aprxjà(Z; ò)i
Fig. 1. Definitions of the Hamiltonian, ground state wave functions and other observables.
4.1. Effects of the approximate treatment of T0
4.1.1. Magnitude of the difference between E and Ê
In Table IV, E and Ê for the Gogny and the SIII interactions are listed in the
case of N = Z = even nuclei. E is the expectation value of −H for its ground state
wave function |Φ (Z, ξ). Ê is the expectation value of −H aprx for |Φ (Z, ξ). For the
cases of 12 C, 16 O, 20 Ne and 24 Mg nuclei, Ê are more than 5 MeV larger than those
of E in most cases. This difference results from the approximate treatment of the
center-of-mass kinetic energy in the Hartree Fock approach, T0aprx ≡ T /A. In the
approximate treatment, the center-of-mass kinetic energy is overestimated by these
amounts. E and Ê do not differ significantly for 4 He and 8 Be. This is because the
off-diagonal part i=j pi · pj /2mA is nearly zero for 4 He and because 8 Be has a
well-developed α-α structure.
4.1.2. Center-of-mass kinetic energy correction for C isotopes
T0 can be studied in more detail using the ground state calculation of C isotopes
with the SIII force. In Fig. 2, we show the various values of the center-of-mass
Table IV. Comparison of E and Ê for Gogny and SIII.
4
He
8
Be
12
C
16
O
20
Ne
24
Mg
EGogny
28.94
47.41
82.36
124.72
145.52
180.79
ÊGogny
28.94
48.73
87.62
129.70
151.07
187.84
ESIII
25.64
41.83
82.40
120.85
143.71
179.21
ÊSIII
25.64
43.47
88.29
125.82
150.08
186.07
1138
Y. Sugawa, M. Kimura and H. Horiuchi
18
c.m. kinetic energy [MeV]
16
14
12
10
8
E c.m.
E'c.m.
T0
<TAMD>/A
6
4
C isotopes
2
0
6
8
10
12
14
16
mass number
18
20
22
24
Fig. 2. Comparison of the center-of-mass kinetic energy correction for C isotopes. Ec.m. : conven
: prescription of Bultler et al. for
tional kinetic energy correction for the HF calculation. Ec.m.
the same calculation as for Ec.m. . T0 : exact correction in AMD. TAMD /A: value when the
same prescription as for Ec.m. is used in AMD.
kinetic energy as functions of mass number. Ec.m. and Ec.m.
are the center-of-mass
kinetic energies obtained from the Hartree Fock calculation of Sagawa et al. 18) for
1/3
the cases f (A) = 1 and f (A) = 2/(t + 1/3t), respectively, where t = 2A
. Ec.m.
2
2 aprx
corresponds to the conventional prescription, T0
≡
p i /2mA = T /A, and
corresponds to the prescription of Butler et al. 19) T0 is the exact center-of-mass
Ec.m.
kinetic energy of the AMD calculation. We display TAMD /A to make a comparison
,
with Ec.m. . T AMD /A and T0 exhibit qualitative agreement with Ec.m. and Ec.m.
respectively.
Although T0 (the exact value) and Ec.m.
are in the range of 10 − 12 MeV and
aprx
decrease for bigger A, Ec.m. and T0
are in the range of 16 − 17 MeV and increase
with A. From the fact that Ec.m. and T0 differ significantly and their dependences
on A are opposite, it must be noted that the treatment of the center-of-mass kinetic
energy needs caution at least with regard to the argument of binding energy. In the
Hartree Fock approach, the effective interactions, not only the SIII but also all others,
implicitly include the effect of the two-body term of the kinetic energy, i=j pi ·
pj /2mA. Therefore, when we compare the results of the AMD and Hartree Fock
calculations with the same effective interaction, the meaning of the term ‘effective
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1139
interaction’ is different. In the nuclear reaction calculation using the Gogny force, it
has been noted 20) that the binding energies obtained from AMD differ significantly
from the experimental values and that the discrepancy increases as the mass number
increases. In those works, a simple prescription to make the center-of-mass kinetic
energy T̃0 mass-number dependent, 21) T˜0 = T0 + 0.8(A − 1) was used. Now that
we know that this discrepancy results from the effective force itself, we see that it is
more straightforward to change the parameter sets of the effective interactions when
we use them together with the exact subtraction of T0 in the AMD framework. An
alternative treatment for this is to add the i=j pi · pj /2mA term to the effective
interaction, so that the parameter sets have the same meaning as in the Hartree Fock
calculation.
4.1.3. Change of the nuclear structure due to the approximate treatment of T0
Here we make a comparison between |Φ (Z, ξ) and |Φ (Z, ξ)aprx for all He, Be
and C isotopes with the SIII and for several Ne and Mg isotopes with the Gogny
force. The resultant two states are indeed almost the same for many nuclei. However,
differences between |Φ (Z, ξ) and |Φ (Z, ξ)aprx are found for deformed nuclei, as
shown in Table V. For 8 Be and 14 Be, the energy E aprx was found to be more bound
than Ê, which are obtained using re-estimation of the AMD wave function. In the
case of 8 Be with Haprx , we obtained two minima of the energy. One corresponds
to a nearly spherical configuration and the other corresponds to the α-α cluster
configuration, while with H, we obtain only one energy minimum which corresponds
to α-α configuration. The spherical configuration has a binding that is about 1.5 MeV
deeper than the α-α configuration. If we make an angular momentum projection,
the binding energy of the deformed α-α configuration is believed to be lowered by
approximately 5 MeV. So, in the case of Haprx as well as H, the α-α configuration
will be the ground state of 8 Be. The distance between two α clusters in |Φ(Z, ξ)aprx
is smaller than that of |Φ(Z, ξ). In the 14 Be case, β aprx is smaller than β. These
results imply that the minima of the energy surfaces of H and H aprx are different.
Smaller deformation for H aprx was also observed in prolately deformed nuclei of Ne
and Mg with the Gogny force, namely 20 Ne, 22 Ne, 26 Ne, 28 Ne, 22 Mg, 24 Mg and
28 Mg. Therefore we can conclude that there is a tendency for the Hartree Fock-like
treatment of the center-of-mass kinetic energy to yield smaller prolate deformation
than that given by the exact treatment in AMD. This tendency can be understood
as follows. Suppose there are two minima, one with a spherical shape and one with
a prolately deformed shape, which have nearly equal binding energies. In the usual
case, the spherical state has a larger kinetic energy. The energy resulting from a
Table V. Comparison of binding energy and β. Ê and β are obtained by minimization of H.
E aprx and β aprx are obtained by minimization of H aprx .
8
ÊSIII
β
aprx
ESIII
aprx
βSIII
Be
43.47
0.623
43.92
0.505
14
Be
59.98
0.257
60.45
0.156
20
ÊGogny
β
aprx
EGogny
aprx
βGogny
Ne
151.07
0.337
151.99
0.139
22
Ne
169.68
0.218
169.91
0.207
26
Ne
194.14
0.102
194.90
0.083
28
Ne
200.66
0.095
201.10
0.091
22
Mg
161.09
0.239
161.26
0.221
24
Mg
187.84
0.253
188.33
0.249
28
Mg
222.82
0.139
223.07
0.111
1140
Y. Sugawa, M. Kimura and H. Horiuchi
binding energy [MeV]
larger kinetic energy and larger potential energy is close to that resulting from a
smaller kinetic energy and smaller potential energy. When we treat the c.m. energy
exactly, T0 is equal for the two systems if the optimized width ν is the same. However,
in the HF-like treatment, T0aprx is proportional to T , and therefore T0aprx is larger
for the spherical case and we obtain a larger binding energy for the more spherical
state. This effect works so as to diminish the prolate deformation of nucleus.
We calculated the energy surfaces using both the Gogny and SIII forces in the
case of 20 Ne and 24 Mg. Both calculations were done by constraining the deformation
parameter β using the constrained frictional cooling method. 11) In Fig. 3, energy
surfaces for 20 Ne are shown. The upper and the lower panels correspond to the
Hamiltonian H and to the HF-like Hamiltonian H aprx , respectively. The slope of
H is found to be more moderate than that of H aprx for both the Gogny and SIII
forces. This changes the position of the minimum energy in the case of 20 Ne with
the Gogny force. There is no such large shift in the case of the SIII force, since the
surface is steeper. In addition to the difference in the treatment of the center-ofmass kinetic energy, the angular momentum projection also changes the shape of the
energy surface, because, in general, as the deformation becomes larger, the rotational
energy also becomes larger. For example, in the case of 20 Ne, the angular momentum
projection lowers the binding energy by about 5 MeV at β = 0.4. For this reason,
we must take the effect of the angular momentum projection into account in this
-13 0
exact c.m. kinetic energy prescription; H
20
Ne
SIII
Gogny
-13 5
-14 0
-14 5
-15 0
0
0.1
0.2
0.3
0.4
0.5
0.6
binding energy [MeV]
deformation parameter
-13 4
approximate c.m. kinetic energy prescription; H aprx
20
Ne
-13 9
SIII
Gogny
-14 4
-14 9
-15 4
0
0.1
0.2
0.3
0.4
0.5
0.6
deformation parameter
Fig. 3. Energy surfaces of 20 Ne for the Gogny and SIII forces. The top panel is for the AMD
Hamiltonian H, and bottom panel is for the HF-like Hamiltonian H aprx .
binding energy [MeV]
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
-15 5
24
exact c.m. kinetic
energy prescription; H
-16 0
1141
Mg
-16 5
-17 0
-17 5
SIII
Gogny
-18 0
-18 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
binding energy [MeV]
deformation parameter
-16 0
24
approximate c.m. kinetic
energy prescription; H aprx
-16 5
Mg
-17 0
-17 5
-18 0
SIII
Gogny
-18 5
-19 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
deformation parameter
Fig. 4. Energy surfaces of 24 Mg for the Gogny and SIII forces. The top panel is for AMD Hamiltonian H, and bottom panel is for the HF-like Hamiltonian H aprx .
nucleus.
Figure 4 presents a similar picture for 24 Mg. In this nucleus, the energy increases
faster than 20 Ne, and the position of the minimum energy is not so different for H
and H aprx . The angular momentum projection of the minimum energy does not
change the position of the minimum for the same reason.
4.2. Comparison of the Gogny and SIII
In this section, we attempt to extract the characteristic differences between the
Gogny force and the SIII force within the same AMD framework. In Figs. 5 (a)-(l),
calculated binding energies are shown. As explained above, we cannot compare E,
the binding energy of the Hamiltonian H, with the experimental value. Therefore
we compare Ê, the expectation value of H aprx for the ground state wave function
of H, with experiments. When we compare ÊGogny and ÊSIII with the experimental
values, we find that the results produced by the Gogny force are in better agreement
with experiments than the results produced by the SIII force for almost all isotopes
in our calculation. The biggest difference between the two interactions was found for
nuclei around 8 Be [Figs. 5 (c) and (d)]. The Gogny force reproduced the sharp rise
of the binding energy at 8 Be, while the binding energy calculated from the SIII force
exhibits monotonous increase. However the two interactions yield similar tendency
in general.
Isotope radii obtained using these two interactions do not differ greatly for heavy
isotopes, as shown in Figs. 6 (a)-(l). However, the largest difference is in Be isotopes
1142
Y. Sugawa, M. Kimura and H. Horiuchi
35
(a)
30
25
20
15
Experiment
10
E Gogny
E Gogny
Blumel
5
He isotopes with Gogny
binding energy (MeV)
binding energy (MeV)
35
0
4
10
50
40
Experiment
E Gogny
E Gogny
Blumel
30
20
Be isotopes with Gogny
0
6
7
8
9 10 11 12 13
mass number (Be isotopes)
14
Experiment
E SIII
E SIII
Sugahara
15
10
Kruppa
5
Tajima
4
80
60
Experiment
E Gogny
40
E Gogny
Blumel
20
70
8
10
12
14
16
18
20
60
50
Experiment
E SIII
E SIII
Sugahara
Tajima
Li
Takami
40
30
20
10 Be isotopes with SIII
0
6
7
8
9 10 11 12 13
mass number (Be isotopes)
120
100
80
Experiment
ESIII
E SIII
Tajima
Sagawa
60
40
20
8
10
12
14
16
18
20
22
mass number (C isotopes)
180
(g)
O isotopes with Gogny
160
binding energy (MeV)
binding energy (MeV)
14
(f) C isotopes with SIII
0
22
180
140
120
100
80
Experiment
E Gogny
60
40
12
14 16 18 20 22 24
mass number (O isotopes)
(h)
O isotopes with SIII
140
120
100
80
Experiment
E SIII
60
E SIII
Tajima
40
E Gogny
Blumel
20
0
10
(d)
mass number (C isotopes)
160
8
5
6
7
9
mass number (He isotopes)
140
100
0
20
(e) C isotopes with Gogny
binding energy (MeV)
binding energy (MeV)
140
120
25
80
(c)
60
10
(b)
0
binding energy (MeV)
binding energy (MeV)
80
70
8
5
6
7
9
mass number (He isotopes)
30
20
26
0
12
Fig. 5. (continued)
14 16 18 20 22 24
mass number (O isotopes)
26
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
220
220
200
160
160
140
120
100
Experiment
80
60
40
0
280
260
240
220
E Gogny
Ne isotopes with Gogny
E Gogny
Blumel
16 18 20 22 24 26 28 30
mass number (Ne isotopes)
(k)
140
120
Experiment
E Gogny
E Gogny
Blumel
40
20
0
20
100
80
22 24 26 28 30 32 34
mass number (Mg isotopes)
40
Ne isotopes with SIII
16 18
20 22 24 26 28 30
mass number (Ne isotopes)
32
(l) Mg isotopes with SIII
200
180
160
140
120
100
80
Experiment
E SIII
E SIII
Tajima
60
40
36
Experiment
E SIII
E SIII
Tajima
60
280
260
240
220
Mg isotopes with Gogny
160
60
140
0
32
(j)
120
20
200
180
100
80
binding energy (MeV)
180
180
20
binding energy (MeV)
(i)
binding energy (MeV)
binding energy (MeV)
200
1143
20
0
20
22 24 26 28 30 32 34
mass number (Mg isotopes)
36
Fig. 5. Comparison of binding energies for the Gogny and SIII forces. (a),(c),(e),(g),(i) and (k)
display the binding energies of He, Be, C, O, Ne and Mg isotopes obtained from AMD and HF
calculations with the Gogny force, and (b),(d),(f),(h),(j),(l) are those calculated with the SIII
force. For the sake of comparison, results of the Hartree-Fock calculations carried out by Tajima
et al., 26) Sugahara et al., 29) Kruppa et al., 30) Li et al., 31) Takami et al., 22) Sagawa et al. 18)
and Blumel et al. 27) are shown.
(neutron)
again. One difference is that RGogny
(neutron)
is always larger than RSIII
. Another is
(neutron)
that RSIII
has a clear peak at 8 Be, but it is small at the neighboring isotopes
7 Be and 9 Be while R(neutron) is large at these isotopes, too. Since these isotopes
Gogny
have 3 He + α and 5 He + α cluster structure, the distance between the two clusters
is smaller for the SIII force than for the Gogny force.
We can say that as the general trend the Gogny and SIII forces give similar
binding energies and radii. But in Be isotopes around 8 Be, there are differences
between these forces. One of the important subjects to be pursued in the study of
unstable nuclei is the problem of the cluster structure. Kanada-En’yo et al. 4) carried
1144
Y. Sugawa, M. Kimura and H. Horiuchi
neutron r.m.s. radii [fm]
proton r.m.s. radii [fm]
2.5
(a) proton radii of He isotopes
2
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
1.5
3
(b) neutron radii of He isotopes
2.5
2
Gogny
SIII
SIII (Tajima)
SIII (1body)
1.5
1
1
2
4
6
8
10
2
12
4
3
(c) proton radii of Be isotopes
2.5
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
2
1.5
3.5
6
7
8
9
10
11
12
13
14
(d) neutron radii of Be isotopes
Gogny
SIII
SIII (Tajima)
SIII (1body)
2
1.5
5
6
2.6
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
2.2
neutron r.m.s. radii [fm]
proton r.m.s. radii [fm]
2.8
2.4
7
8
9
10
11
12
13
14
15
mass number (Be isotopes)
(e) proton radii of C isotopes
2
3.5
(f) neutron radii of C isotopes
3
2.5
Gogny
SIII
SIII (Tajima)
SIII (1body)
2
1.5
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
mass number (C isotopes)
mass number (C isotopes)
(g) proton radii of O isotopes
2.8
2.6
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
2
neutron r.m.s. radii [fm]
proton r.m.s. radii [fm]
12
2.5
15
3
2.2
10
3
mass number (Be isotopes)
2.4
8
1
5
3
6
mass number (He isotopes)
neutron r.m.s. radii [fm]
proton r.m.s. radii [fm]
mass number (He isotopes)
3.5
(h) neutron radii of O isotopes
3
Gogny
SIII
SIII (Tajima)
SIII (1body)
2.5
2
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
mass number (O isotopes)
mass number (O isotopes)
Fig. 6. (continued)
proton r.m.s. radii [fm]
3.5
(i) proton radii of Ne isotopes
3
2.5
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
neutron r.m.s. radii [fm]
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
2
4
3
2.5
Gogny
SIII
SIII (Tajima)
SIII (1body)
2
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
mass number (Ne isotopes)
3.5
(k) proton radii of Mg isotopes
3
Gogny
SIII
SIII (Tajima)
SIII (1body)
exp
2
mass number (Ne isotopes)
neutron r.m.s. radii [fm]
proton r.m.s. radii [fm]
(j) neutron radii of Ne isotopes
3.5
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
2.5
1145
4
(l) neutron radii of Mg isotopes
3.5
3
2.5
Gogny
SIII
SIII (Tajima)
SIII (1body)
2
1920 212223 2425 262728 293031 3233343536 3738
mass number (Mg isotopes)
1920 212223 2425 262728 293031 3233343536 3738
mass number (Mg isotopes)
Fig. 6. Comparison of isotope radii for the Gogny and SIII forces. (a)-(l) are the proton and
neutron radii of He, Be, C, O, Ne and Mg isotopes obtained from the AMD and HF(HF+BCS)
calculations with the Gogny and SIII forces. Squares and triangles represent the results of
the AMD calculation. The unfilled circles are the results of HF+BCS calculation performed
by Tajima et al. 26) The crosses represent the results of the AMD calculation employing an
approximate treatment of the center-of-mass (see text). The filled circles are the experimental
values. 32), 33)
out the AMD study using the MV1 force and found that, although the clustering seen
in 8 Be diminishes in Be isotopes toward the neutron magic isotope 12 Be, it becomes
stronger again in the neutron drip-line isotope 14 Be. Takami et al. 22) reported
the density distributions of Be isotopes obtained from their Cartesian mesh HF
calculation with the SIII force. Their density distribution supports the result of
Kanada-En’yo et al. Namely, the prominent α-α clustering in 8 Be, a decrease of the
clustering toward the N = 8 isotope 12 Be, and an increase of clustering in the neutron
drip-line isotope 14 Be. In Fig. 8, we display the density distribution of Be isotopes
obtained in the present AMD calculation with the Gogny force. We see behavior that
is similar to that reported in Refs. 4) and 22) in the density distribution as a function
of neutron number. This tendency is also seen with the SIII force. But neutrons
of 12 Be are always spherical, due to the magicity of neutron magic number N =
8. We could not find the deformed neutrons states of 12 Be with 2p-2h character,
which experiments and many other theoretical studies point out. 23) - 25) To study
this nucleus, we need more careful treatment of the AMD wave functions, such as an
angular momentum projection. More detailed discussion of clustering will be given
in our forthcoming paper.
1146
Y. Sugawa, M. Kimura and H. Horiuchi
4
r.m.s. radii [fm]
3
2
r2
r2
r2
1
r2
0
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
mass number
1/2
Fig. 7. Comparison of r2 1/2 and rˆ2 for ν = 0.16 calculated using AMD with the Gogny force
for N = Z nuclei. r2 1/2 can be approximated as 1.2A1/3 . Dotted line represent the difference
1/2
(overestimation) between rˆ2 and r2 1/2 .
8
Be
10
Be
12
Be
14
Be
Fig. 8. Proton density distributions of even-even Be isotopes. Clear α-α cluster structure appears
in 8 Be. This cluster structure becomes obscure toward neutron magic number nuclei 12 Be and
becomes clear again for the neutron drip-line nucleus 14 Be.
4.3. Comparison between AMD and HF(HF+BCS) results
In Fig. 5, we also show the results of the HF(HF+BCS) calculations. The
HF+BCS results with the SIII force are taken from the calculation of Tajima et al. 26)
Using the Cartesian mesh method. The HF results with the Gogny force are taken
from the paper by Blumel et al. 27) Since there are only a few isotopes for which there
is a large difference in the potential energy surface that is caused by approximate
treatments of T0 , we directly compare the result of the AMD optimization and
HF(HF+BCS) calculations.
The binding energy ÊGogny agrees quite well with the Gogny HF binding energy
for oxygen isotopes. The agreement of the binding energy with Gogny-Hartree Fock
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1147
is also seen in Ne and Mg isotopes except for 20 Ne and 24 Mg [Figs. 5 (i) and (k)].
However, the binding energy of ÊSIII is smaller than Cartesian mesh HF+BCS results
especially for neutron rich region of deformed C and Mg isotopes (Figs. 5 (f) and (l)).
From the fact that we use the effective nuclear force, the Skyrme SIII force used in
the Hartree Fock Bogoliubov study, it is clear that this disagreement comes from the
limitations of the present simple version of the AMD functional space. The proton
(neutron) distributions are determined by Z, ξ and ν. Though Z and ξ are independently varied for each nucleon, ν is the same for all nucleons. Additionally, each
Gaussian wave packet is limited to a spherical shape. There is no such limitation
in the Cartesian mesh HF method. There are possible ways to improve this disagreement without using the superposition of Gaussian wave packets and/or Slater
determinants. One possible way is to use deformed Gaussian wave packets instead
of spherical ones, and another is to use different values of the Gaussian width ν for
protons and neutrons. In a subsequent paper, we will discuss the former prescription
briefly and show that this modification gives considerable improvement.
Now, we compare the radii (Fig. 6) obtained in the AMD and Hartree Fock calculations. For He isotopes, the Skyrme Hartree Fock results give radii of He isotopes
(proton)
(neutron)
that are larger than RAMD and RAMD . This is because in AMD calculation we
2
define the radius as, r2 = A1 Φ| A
i=1 (r − r G ) |Φ/Φ|Φ with r G representing the
center-of-mass coordinate, while in the Skyrme Hartree Fock calculation it is defined
2
as rˆ2 = A1 Φ| A
i=1 r |Φ/Φ|Φ, omitting the effect of the center-of-mass. When
we calculate both r2 and rˆ2 using AMD, there is non-negligible overestimation of
rˆ2 . This difference is significant for small mass number, as shown in Fig. 7. Indeed,
in Figs. 6(a) and (b), the HF+BCS results overestimate the proton and neutron radii
of 4 He.
The triangles in Fig. 6 represent the radii obtained with the latter definition from
AMD-SIII ground states. Except for the most neutron-rich and proton-rich nuclei,
(neutron)
AMD-SIII agrees with the Skyrme Hartree Fock calculation. Values of RAMD
for
heavier nuclei are almost equal to the HF+BCS results. In contrast to the binding
energy, the obtained radii seem to be only slightly affected by the restriction of the
functional space of the present simple version of AMD.
4.4. Effects of the parity projection
In this subsection we present the results of parity-projected calculations. Parity
projection is regarded as an important factor in the nuclear study of structure.
Indeed, in some specific nuclei such as 20 Ne, it is known that the parity projection
enhances the parity-violating intrinsic deformation 28) and changes the shape of the
energy surface as a function of the deformation parameter β. Here we again consider
the binding energies and radii. In Fig. 9, the binding energies and matter radii of
He, C and Ne isotopes are plotted. Compared to the calculation without parity
projection, these isotopes are found to be bound more deeply by about 1–2 MeV.
These values are not very large. Though the binding energies have changed slightly,
the radii of these isotopes are almost unchanged. In contrast to the fact that the
parity projection does not change the nuclear radii, it affects the nuclear density
1148
Y. Sugawa, M. Kimura and H. Horiuchi
3
30
25
20
non-parity-projected
parity-projected
exp
15
10
5
(a) binding energies of He isotopes
matter r.m.s. radii [fm]
binding energy [MeV]
35
0
(b) matter radii of He isotopes
2.6
2.2
1.8
non-parity-projected
parity-projected
1.4
1
2
4
6
8
10
12
2
4
mass number (He isotopes)
matter r.m.s. radii [fm]
binding energy [MeV]
110
non-parity-projected
parity-projected
exp
90
70
(c) binding energies of C isotopes
8
10
12
(d) matter radii of C isotopes
2.8
2.6
2.4
non-parity-projected
parity-projected
2.2
2
50
10
12
14
16
18
20
mass number (C isotopes)
22
10
24
12
14
16
18
20
22
24
mass number (C isotopes)
220
3.5
matter r.m.s. radii [fm]
binding energy [MeV]
6
mass number (He isotopes)
3
130
200
180
non-parity-projected
parity-projected
exp
160
140
(e) binding energies of Ne isotopes
120
(f) matter radii of Ne isotopes
3.3
3.1
2.9
non-parity-projected
parity-projected
2.7
2.5
18
20
22
24
26
28
30
32
34
18
mass number (Ne isotopes)
20
22
24
26
28
30
32
34
mass number (Ne isotopes)
Fig. 9. Binding energies and matter radii of He, C and Ne isotopes. The triangles (squares) represent the results of parity-projected (non-projected) AMD calculations, while the filled circles
represent the experimental values.
distributions drastically in some nuclei. For example, when the parity is projected,
O+α cluster structure appears as the ground state of 20 Ne, while the ground state of
this nucleus is, without the parity projection, parity symmetric and mean field-like.
We illustrate this point in Fig. 10. We will show and explain the details of this effect
in a subsequent paper.
§5.
Summary
Having developed a new code that enables the calculation of the density dependent term precisely enough within reasonable computation time, we can now
progress toward a more accurate investigation of nuclear structures and nuclear reactions using density dependent forces like the Gogny interaction.
With this code, we calculated ground states of He, Be, C, O, Ne and Mg isotopes with the Gogny and SIII forces. The exact treatment of the center-of-mass
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
non projected
Fig. 10. Total density distributions of
without (with) parity projection.
1149
parity projected
20
Ne. The left (right) panel is the result of the calculation
kinetic energy resulted in non-negligible underestimation of the binding energy. The
difference between the center-of-mass energy correction for this and approximated
one used in the Hartree Fock approach affects not only the binding energy but also
the deformation of several isotopes. This implies that the treatment of the centerof-mass kinetic energy term changes the energy surface and the deformation of the
ground state in some cases. Within the common framework of AMD, the Gogny and
SIII forces reproduced the experimental binding of stable nuclei. These two forces
give almost equal radii, except in the case of 7 Be and 9 Be. In neutron-rich nuclei,
both forces (and especially the SIII force) yield some deviations from the experimental binding energy. Comparison of the Cartesian mesh Hartree Fock calculations
with the SIII force revealed some limitations of the present simplest version of the
AMD wave function. In particular, although the AMD calculation with the SIII force
similar to those results with Hartree Fock calculation for nuclei close to the N = Z
region, it yields an underestimation of the binding energy in deformed neutron-rich
nuclei. However, for the nuclear radius, AMD and HF give similar results. Results
for the Gogny force agree well with those calculated within the HF by Blumel et
al. for both binding energies and radii. Parity projection made the nuclear bindings
slightly deeper by about 1–2 MeV. However, parity projection did not change the
nuclear radii. The density distribution and the clustering features, by contrast, are
often changed significantly by the parity projection, as we saw in 20 Ne.
Our results show several important things. One is the importance of the treatment of the center-of-mass kinetic energy, since this changes the meaning of effective
interaction significantly. This is also important in the application of the nuclear
reaction calculation, where this effect has been neglected and phenomenological parameters have been introduced to obtain agreement with experimental binding energies, in order to use the same parameter set as in HF(HF+BCS). The second thing
is the fact that the Gogny force gives values of the binding energies and radii that
agree with experimental values equally well as or slightly better than those given by
the SIII force. The modified Volkov I force has been used in the AMD analysis of
properties of light nuclei because it has a finite range and also because its matrix
elements can be evaluated analytically. Now the Gogny force is another possibility
1150
Y. Sugawa, M. Kimura and H. Horiuchi
as the effective interaction to be used in AMD, since it is free from the problem that
is inherent in the Volkov force: the force parameter set that can be used to describe
the properties of some isotopes cannot be used to study other isotopes whose mass
number differ by too much. The third thing is the confirmation of the existence of the
cluster structure in unstable nuclei. Especially in Be isotopes, we have found with
any of three effective forces (the MV1, Gogny and SIII) that 8 Be has a prominent
α-α clustering, this clustering diminishes toward the neutron magic isotope 12 Be,
and it becomes stronger again at the neutron drip-line isotope 14 Be. In this study,
neutrons of 12 Be are always spherical, and we could not find the deformed neutron
state that has the 2p-2h character. To study this nucleus, we need a more careful
treatment of the AMD wave functions, such as an angular momentum projection.
The fourth thing is that we have made clear the applicability of the simplest version
of the AMD wave function. Our calculation underestimates the binding energy of
deformed neutron-rich nuclei. There will be some prescriptions to improve this disagreement without introducing the superposition of Gaussian wave packets and/or
Slater determinants. One possible method is to use deformed Gaussian wave packets
instead of spherical ones and another prescription is to use different Gaussian widths
for protons and neutrons instead of a Gaussian width ν common to all nucleons. In
subsequent paper, we will explain the former method and give some calculation results. Finally, parity projection makes the nuclear bindings deeper without changing
the nuclear radii. This feature is common to all calculated isotopes. Parity projection also affects the nuclear deformation and the intrinsic density distribution. In
particular, a drastic change of intrinsic density distributions results in some nuclei,
such as 20 Ne. We will report the results concerning density distributions, nuclear
deformation and clustering features in a subsequent paper.
Appendix
Here we treat the density-dependent two-body part of the interactions, which is
of the form of δ(r)[ρ (R)]σ . To obtain the matrix elements of this term, we need to
evaluate
dr[ρ (r)]σ+2 ,
(A.1)
where
ρ (r) =
2ν
π
3/2 e
−2ν
i,k∈α
2
∗
r− Z i2√+Zν k
(BS)ik (BS)−1
ki ,
(A.2)
(BS)ik ≡ eZ i ·Z k χ†i χk .
∗
If σ is not a natural number, numerical evaluation is necessary for this integral. We
simply changed Eq. (A.1) into
∆
[ρ (r n )]σ+2 ,
n
(A.3)
Study of Even-Z Nuclei Up to Mg with the Gogny Force Using AMD
1151
where ∆ is the volume of a cell. If we rewrite
ρ (rn ) =
2ν
π
3/2
e−2ν rn
2
√
2 ν(Z ∗i +Z k )·r n
×e
−(Z i +Z k )
(BS)ik (BS)−1
ki e
∗
2
/2
ik
(A.4)
,
the advantage of the mesh summation becomes clearer. The term in the summation,
2
∗
(BS) (BS)−1 e−(Z i +Z k ) /2 , is common to all n. When we take r as a Cartesian
ik
ki
√
n
mesh point, the last term Aik,n ≡ e2√ ν(Z i +Z k )·rn can be calculated using the recur∗
sion relation, since Aik,n = Aik,n+1 e ν(Z i +Z k )·∆r , where ∆r is a constant vector for
neighboring rn and rn+1 .
This relation reduces the calculational time significantly and thus enables precise
determination of expectation values.
∗
References
1) P. Bonche, H. Flocard, P-H. Heenen, S. J. Krieger and M. S. Weiss, Nucl. Phys. A 443
(1985), 39.
2) J. Decharge and D. Gongy, Phys. Rev. C 21 (1980), 1568.
3) Y. Kanada-En’yo, H. Horiuchi and A. Ono, Phys. Rev. C 52 (1995), 638.
4) Y. Kanada-En’yo and H. Horiuchi Phys. Rev. C 52 (1995), 647.
5) A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi, Prog. Theor. Phys. 87 (1992), 1185.
6) H. Horiuchi, Y. Kanada-En’yo, A. Doté and M. Kimura, Proceedings of the 1st AsianPacific Conference, Tokyo, 1999, ed. S. Oryu, M. Kamimura and S. Ishikawa, Few-Body
System Supplement 12 (2000), 125.
7) A. Ono, Proceedings of the 7th International Conference on Clustering Aspects of Nuclear
Structure and Dynamics Rab. 1999, ed. M. Karolija, Z. Basrak and R. Caplar (World
Scientific, Singapore, 2000), p. 294.
8) T. Ando, K. Ikeda and A. Tohsaki, Prog. Theor. Phys. 64 (1980), 1608.
9) M. Beiner, H. Flocard, Nguyen van giai and P. Quentin, Nucl. Phys. A 238 (1975), 29.
10) H. Horiuchi, Y. Kanada-Enyo and A. Ono, Z. Phys. A 349 (1994), 279.
11) A. Doté, Y. Kanada-En’yo and H. Horiuchi, Phys. Rev. C 56 (1997), 1884.
12) A. Valor, P.-H. Heenen and P. Bonche, Nucl. Phys. A 671 (2000), 145.
13) R. Rodriguez-Guzman, J. L. Egido and L. M. Robledo, Phys. Lett. B 474 (2000), 15.
14) C. T-Schnaider and P. Quentin, Phys. Lett. B 49 (1974), 397 and references therein.
15) N. Tajima, S. Takahara and N. Onishi, Nucl. Phys. A 603 (1996), 23.
16) A. Ono and H. Horiuchi, Phys. Rev. C 51 (1995), 299.
17) A. Ono, H. Horiuchi and T. Maruyama, Phys. Rev. C 48 (1993), 2946.
18) H. Sagawa and H. Toki, J. of Phys. G 13 (1983), 453.
19) M. N. Butler, D. W. L. Sprung and J. Martorell, Nucl. Phys. A 422 (1984), 157.
20) A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi, Phys. Rev. Lett. B 68 (1992), 2898.
21) A. Ono, H. Horiuchi, T. Maruyama and A. Ohnishi, Prog. Theor. Phys. 87 (1992), 1185.
22) S. Takami, K. Yabana and K. Ikeda, Prog. Theor. Phys. 96 (1996), 407.
23) A. Navin, D. W. Anthony, T. Aumann, T. Baumann, D. Bazin, Y. Blumenfeld, B. A.
Brown, T. Glasmacher, P. G. Hansen, R. W. Ibbotson, P. A. Lofy, V. Maddalena, K.
Miller, T. Nakamura, B. V. Pritychenko, B. M. Sherrill, E. Spears, M. Steiner, J. A.
Tostevin, J. Yurkon and A. Wagner1. Phys. Rev. Lett. 85 (2000), 266.
24) T. Suzuki and T. Otsuka. Phys. Rev. C 56 (1997), 847.
25) Y. Kanada-En’yo, private communications.
26) N. Tajima, S. Takahara and N. Onishi, Nucl. Phys. A 603 (1996), 23.
27) R. Blumel and K. Dietrich, Nucl. Phys. A 471 (1987), 453.
28) Y. Kanada-En’yo and H. Horiuchi, Prog. Theor. Phys. 93 (1995), 115.
29) Y. Sugahara, K. Sumiyoshi, H. Toki and A. Ozawa, Prog. Theor. Phys. 96 (1996), 1165.
30) A. T. Kruppa, P.-H. Heenen, H. Flocard and R. J. Liotta, Phys. Rev. Lett. 79 (1997),
2220.
1152
Y. Sugawa, M. Kimura and H. Horiuchi
31) X. Li and P.-H. Heenen, Phys. Rev. C 54 (1996), 1617.
32) H. de Vries, C. W. de Jager and C. de Vries, Atomic Data and Nucl. Data Tabl. 36 (1987),
495.
33) G. Fricke, C. Bernhardt, K. Heilig, L. A. Schaller, L. Schellenberg, E. B. Shera and C. W.
Dejager, Atomic Data and Nucl. Data Tabl. 60 (1995), 177.