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Shell-model calculation with Hartree-Fock condition
Nuclear Physics A481 (1988) 458-476 North-Holland, Amsterdam
SHELL-MODEL
CALCULATION
Tsutomu
HOSHINO,
WITH
Hiroyuki
HARTREE-FOCK
SAGAWA
CONDITION
and Akito ARIMA
Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan Received 3 November (Revised 7 December
1986 1987)
Abstract: Giant resonances and spectroscopic factors of IhO are studied in the framework of the shell model with (0+2)hw model space. It is found that the Hartree-Fock condition is crucial in order to obtain reasonable excitation energies of giant resonances in IhO. The calculated spectroscopic factors in 15N are quenched only by 10% due to the coupling to lp-2h states. The spectroscopic factor of the s-state in “B is also discussed.
1. Introduction The nuclear shell model has been the most successful and well-established framework for studying nuclear structure problems. The recent development of computers makes it feasible to perform large scale shell-model calculations, for example (0 + 2) hw calculations. In this work we take the nucleus I60 as an example for such shell-model calculations. The Utrecht group I,*) has already studied this problem. They found a large admixture of lp-lh components in the ground state of 160. If one takes a model space generated by the Hartree-Fock single-particle states and uses the interaction which satisfies the stability condition, the HartreeFock condition should be automatically satisfied. If this is the case, there might be very small lp-lh components in the ground state. However, when one uses the harmonic oscillator wave function and an effective interaction which simulates the G-matrix condition condition
as is usually done in the shell-model calculation, the Hartree-Fock is not guaranteed. Thus, one has to take into account the Hartree-Fock in the shell-model calculation in order to obtain realistic wave functions ‘).
At present, the Hartree-Fock calculations have been performed by using two different kinds of effective interactions. The first one uses phenomenological density-dependent forces such as Skyrme-type interactions 4,5) or the one derived by density-matrix expansion method “). These density-dependent forces give extremely successful results for the binding energies, the density distributions and the single-particle energies of doubly-closed shell nuclei. The second one is called the Brueckner-Hartree-Fock method ‘,‘) where a free nucleon-nucleon interaction such as the Reid soft-core potential is used. Many extensive calculations have been done including higher-order diagrams based on the Brueckner-Hartree-Fock method, but still a serious difference between the calculated and experimental saturation curves (so called the Coester line discrepancy) is not yet solved. 0375-9474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
459
T. Hoshino et al. / Shell model calculation
The shell-model been performed
calculations recently
with the density-dependent
‘). In these studies, the configuration
Ohm model space. This approach but the energy calculations
spectra
with
Skyrme-type
are poorly
purely
reproduces reproduced
phenomenological
the binding matrix
space is limited within
energies
in comparison
forces have
of d-s shell nuclei,
with the shell-model
elements
by Wildenthal
and
Chung I”). The shell-model calculations based on the Brueckner G-matrix were also attempted by the Jiilich-Tiibingen group ‘I). In these calculations, they used the Reid soft-core potential as an effective nucleon-nucleon interaction and took into account the many-body effects coming from higher-order diagrams of the G-matrix in the calculations of matrix elements in the Ohw configuration space. In spite of these serious efforts, the results are still poor in comparison with experimental data. In this paper,
we will study the excitation
energies
of giant resonances
and the
spectroscopic factors of I60 by large space shell-model calculations. Since neither the fundamental approach like the Brueckner G-matrix nor the phenomenological Skyrme-type force is far from the realistic situation, we will take the following pragmatic approach for the large scale shell-model calculation taking into account the Hartree-Fock condition. We take as an interaction, the M3Y (Michigan 3-range Yukawa) force which simulates the Reid soft-core potential. In the case of the Skyrme-type force, the rearrangement terms stemming from the variation of the density-dependent force are crucial to give successful results in the Hartree-Fock calculations. In the Brueckner reaction matrix, this density dependence is due to the starting energy and a Pauli rearrangement term 12). In our calculation, we have to simulate this rearrangement effect in order to satisfy the Hartree-Fock condition using the M3Y force. As is discussed in sect. 2.3, the rearrangement term is taken care by the effective potential 6V in eq. (2.9). In this prescription, we will be able to calculate the physical higher-shell components above one major shell in the ground states and also in the excited states eliminating unphysical Ip-lh excitations. In p-shell nuclei, these higher shell components correspond to Od-ls-Of-lp shell ones. Those components are crucial to study the spectroscopic factors in (e, e’p) experiments. The Hartree-Fock condition has been already studied in the case ofthe shell-model calculation
of I60 in ref. ‘), in which he set all matrix
elements
between
the core
of I60 and lp-lh states to be zero. Recent experimental data revealed a difficulty in explaining spectroscopic factors in the proton knock-out reaction and provided a test stone which examines shellmodel calculations very precisely 13). The experimental information of the (e, e’p) reaction is particularly interesting because both the absolute spectroscopic factors and the momentum distributions of single-particle states have been provided with good accuracy. The observed spectroscopic factors 14) are typically one half of the pure single-particle values in p-shell nuclei such as “0, suggesting that the spreading
27 Hosh~no et al. i Shell model ca~culatjon
460
of the spectroscopic
factors
many-particle-many-hole functions account.
obtained
states
find that the theoretical
values.
These
values
is due to the coupling
spectroscopic
in this work in which the coupling
We, however,
the single-particle
of single-particle
states. We calculate
factors
to
using the wave
to 2p-2h states is taken into
spectroscopic
are still much
factors
larger
than
are 93% of the observed
ones 14). The model space (0-t 2)hw is consistently used in the following calculations. The adopted effective interaction and the single-particle energies are shown in sect. 2.2. The effect of the Hartree-Fock condition is examined in sect. 2.3. We will point out that this condition is pa~icularly important in order to obtain reasonable excitation energies of giant resonances. Sect. 3.1 is devoted to the discussion of the results of monopole and quadrupole giant resonances in 160. The calculated spectroscopic factors and the momentum distributions in I60 are compared with the experimental ones in sect. 3.2. A summary is given in sect. 4. We also discuss the case of “C in order to compare with a previous calculation by the Utrecht group ‘).
2. Shell-model 2.1. MODEL
SPACE
AND CENTER-OF-MASS
calculation EFFECT
Shell-model calculations for Op shell nuclei in a large shell-model space are now feasible because of the development of modern computers. We use the OxfordBuenos-Aires-MSU shell-model code 15), which is based on the Lanczos method. In spite of the development of computers, however, shell-model calculations are still basically restricted by the dimensions of the shell-model space. The number of O+ states in I60 is 44 within the present (0+ 2) ho full model space, but turns out to be 4340 within the (0 f 2 + 4) ho model space. We adopt a hamiltonian in a second quantized
form,
where (41 V(Q- Q)/Y% is an antisymmetrized
matrix
element
of the two-body
interaction, and (Y, p, y and 6 are harmonic oscillator single-particle states. If one uses a translationally invariant effective interaction, one can separate the harmiltonian in the usual way into a term which depends only on the center-of-mass (c.m.) coordinate and one which is a function of the relative coordinates (the intrinsic hamiltonian). The hamiltonian of the c.m. motion consists only of a kinetic energy term. In this paper we take harmonic oscillator single-particle wave functions. One usually adds a c.m. harmonic oscillator potential fAfkh2R2 (R = (l/A) Cf=, ri) to (2.1). Then, the calculated wave functions can be expressed as a product of the eigenstate of the cm. motion and that of the intrinsic motion, provided that one
7: Hoshino
adopts
a complete
quanta,
for example
set of states
et al. / Shell
within
two in the present
In this paper, we do maintain
461
model calculation
a certain
number
of harmonic
the form (2.1) of the hamiltonian
i&
&=P
oscillator
calculation. and add the term r6)
(2.2)
+;MAw2R2-;hw
to the hamiltonian (2.1) instead of iAMw2R2, since the HF condition can be only handled in this prescription. We numerically checked that two methods did not make any difference in the final results in the case without the I-IF condition. With an appropriate value for /3 (“IOO), spurious states will be separated well enough from low-energy states which have approximately the OS cm. motion. In the present calculation, we take into account the s-p-sd-pf shells h/2,Of~,2,Of~,2,~PW, lp,,J as the model space. (0s L/2, OP,,,,0~~,2,04/2, Ob, The unperturbed excitation energy is taken up to 2hw. In these shell-model orbits, calculations within the (0 + 2) iiw model space for I60 are possible and the spurious c.m. motion can be approximately separated. 2.2. EFFECTIVE
INTERACTION
AND SINGLE-PARTICLE
ENERGIES
We use the M3Y (Michigan 3-range Yukawa) “) interaction as the two-body interaction Vi,. The parameters of M3Y are determined by a fit to the G-matrix obtained from the Reid soft-core potential in the oscillator basis. This effective NN interaction is composed of a central part in both the spin-singlet even-parity (SE) and the spin-triplet even-parity (TE) channels, a tensor part in the even-parity (TNE) channel, and a spin-orbit part in the odd-parity (LSO) channel. Three Yukawa potentials with different ranges are included for the central part, while two different ranges are adopted for the spin-orbit and also for the tensor part. Each potential is written
in the following
form:
VSEtTEf=
i Vs'T'Y(r,2/Ri)qS(f~PE, i=l
VLso = i VksoL * SY( r12/ Rj) PO, is=, where
Y(X) = exp (-x)/x.
The tensor
sr2=3(o,
3 hd(n.
operator
is defined
CZ)/&-(~,
(2.3)
as
. ~2),
(2.4)
and PSC7), PE and PO denote the projection operators for the spin-singlet (triplet), even-parity and odd-parity channels, respectively. The parameters are tabulated in table I. The longest range R, for the central part is chosen to simulate the OPEP tail at large distance. The longest range R, used for the tensor force is taken to fit to that of the OPEP tail closely in the form as r*Y. The force with the range R2 for
462
T. Hoshino et al. / Shell model calculation TABLE 1 Parameters
Channel
R, = 0.25 fm
of the M3Y
R, = 0.4 fm
SE TE TNE
12 454 21 227
-3835 -6622 -1259.6
LSO
-3 733
-427.3
SE TE TNE LSO
potential 1.414 fm for central R,=
0.700 fm for tensor -10.463 -10.463 -28.41
(MeV) (MeV) (MeV/fm*) (MeV)
singlet even. triplet even. - tensor even. - spin-orbit odd.
every channel simulates roughly the “cT-exchange” process. The short-range force with R, = 0.25 fm is also included to improve the fit. The two-body LS force is included in the M3Y force, but the calculated splitting of Op shell orbits is not large enough to explain the experimental splitting (6.32 MeV) between the 4P and $- states found in “N. We shift up the kinetic energy of the OP,,~ shell by 3.0 MeV to produce a proper energy spacing. The calculated single particle energy of the Of,,> shell is higher than that of the 1p3,2 shell when one uses the M3Y force. We lower the Of,,, and Of,,, levels by 5.0 MeV. The single particle energies for both I60 and ‘*C are shown in table 2. The size parameter is computed according to the following formula 18): hw =45A-‘/‘-25A-*“[MeV], b=m. The size parameter account
reproduces
the finite size effect of nucleon pcharge(r) =
J
charge r.m.s. radius
of I60 taking
as follows:
pproton( r’){ m2}-3’2
With a2 = 0.4 fm*, we obtain 2.3. HARTREE-FOCK
(2.5)
the observed
exp ( - (r - r’)*/a’)
d3r’
.
r, = 2.68 fm for the charge r.m.s. radius
into
(2.6) of 160.
CONDITION
It is generally impossible to reproduce the saturation condition of the HartreeFock theory only with a density-independent two-body interaction. One has to take into account a many-body force or density-dependent one. For example, a densitydependent interaction is often used in the Hartree-Fock (HF) calculation 4,6). As mentioned above, we adopt the interaction M3Y in the following calculations. However, since this interaction does not depend the nuclear density, the saturation condition is not fulfilled in the HF calculation by including only the contributions shown in fig. la. This might be improved if one takes into account higher-order
T. Hoshino
463
et al. / Shell model calculation TABLE 2
Kinetic energies and single-particle
energies in I60 and ‘*C
?I
I60
Shell KE
SPE
-54.34
11.16
-46.37
OP,,,
17.40
-27.70
18.61
-18.06
OP,,,
20.40
-20.58
21.61
-13.38
Od S/2
24.35
-6.68
26.05
KE
SPE
OSI/.?
10.44
(MeV)
2.19
IS I/Z
24.35
-5.50
26.05
Od3/2
24.35
-0.07
26.05
-0.25 6.3 1
Of,,,
26.32
6.29
28.49
12.59
31.32
8.09
33.49
13.94
31.32
9.73
33.49
15.05
26.32
11.73
28.49
17.16
diagrams of the Brueckner G-matrix shown in fig. Id [refs. ‘,12)], which are called as the Pauli blocking term and the starting energy term. Since our aim is the shell-model calculation in the large configuration space, we do not calculate directly the higher-order diagrams of the Brueckner-Hartree-Fock theory. Instead, we will take into account these effects adding an extra potential to the M3Y force in order to satisfy the Hartree-Fock condition. The HF condition
is written (Q,l#>+
for the two-body
interaction
as follows:
c (QYI VIPY), = &,pe, 7 y occupied
(2.7)
where E, is the single-particle energy of the ath orbit. We have to take into account effectively this condition for the following reasons: (i) Violation of this condition pushes down the ground-state energy unrealistically. At the same time, excitation energies of giant resonances became too high (the 0: state of I60 is predicted at E, = 45 MeV, while the observed giant monopole resonance has E, = 25 MeV). (ii) Transition strengths to 0’ states are not realistic. We calculated the overlaps between the harmonic oscillator single-particle wave functions with the oscillator parameter b = 1.73 fm and the HF wave functions calculated with a Skyrme-type density-dependent SC11 interaction 19) in the OS and Op shell-model orbits for 160. The overlaps are more than 99.9% for OS,,, and OP,,~, and 99.8% for the Op,,, state, respectively. These values confirm that harmonic oscillator wave functions are quite good approximations to single-particle wave functions. Comparing the contributions of the kinetic energy and the M3Y interaction
464
7: Hoshino
V J.j
__--+
t
.v
v __-
n+l.l.j
Ai
n+l,l,j
Jj
n+l,l,j
et al. / Shell model calculation
0 sz0
L-I--
(4 fib)
(c)
_-.-
~
_----
0
Fig. 1. Diagrammatic representations of the Hartree-Fock condition. The diagram (a) represents the Hartree-Fock condition written in eq. (2.8). The contribution of the diagram (b) in the calculation of neighbouring nuclei is excluded, but physical lp-2h states corresponding to the diagram (c) are taken into account. The diagrams (dt are examples of higher order terms.
in table 3, we can see that the sum of the two-body matrix elements are much larger than the off-diagonal kinetic energies in absolute value. Thus, the HF condition is badly violated in the case of the M3Y force. Similar two-body matrix elements are obtained in the case of the SGII interaction when one switches off the rearrangement term as is shown in the sixth column of table 3. However, when the rearrangement effects is included, the HF condition is fulfilled almost perfectly particle states l~~,,~ and lp,,, are above the threshold. Therefore, that the rearrangement
effect has a crucial
even though the we can conclude
role in the HF calculation
with the
density-dependent Skyrme-type force. We have to take into account the rearrangement effect also in the shell-model calculation with the M3Y interaction in order to fulfill the HF condition. In order to simulate in the particle-hole
the rearrangement
matrix (pltlh)+
elements,
effect, we add a repulsive
so that the HF condition
c (PYl(V,,,+6V)Ihy),=O, y occupied
force 6V only
is strictly
fulfilled: (2.81
where (2.91 The parameter V, is fixed to satisfy the condition (2.8) for each orbit. Thus, the diagrams shown in fig. la are cancelled completely. The matrix element corresponding to the diagram lb for the neighboring nucleus is also cancelled by that of the
Comparison
of the contributions
of the kinetic
energy
and the M,Y interactions SGII (HF)
M3Y (H.O.)
(Pl$~)
P/f’
Xi (PjlVlW
It/ VI tplflh)
C, tpjl Vlhj)
C, tpjlvlhj)
+rearrange
It/
VI
lSl,,/OS,,2
8.56
-16.0
0.55
8.30
-13.0
-8.1
0.62
1PW/OP3,,
11.0
-15.9
0.69
5.66
-7.6
-5.4
0.71
lP,/,/OP,,Z
11.0
-16.9
0.65
5.13
-7.0
-4.9
0.70
The symbol “i-rearrange” means the addition of a rearrangement term to Ii (~$1 V!hj). Abbreviations of H.O. and HF indicate that calculations are done with harmonic oscillator wave functions and Hartree-Fock wave functions, respectively. Because lp states of the SGII interaction are in the continuum, the values of t and V are smaller than those of the M3Y inleraction. The particle wave functions 1p3,,a, lp,,, are calculated in the spherical box with the radius R = 10 fm. kinetic
energy
plus SV, while the physical
contribution. The same rearrangement
lp-2h
effect is important
state shown
in fig. lc has a finite
for the diagonal
part of the single
particle potential C, occupied(c~rf( Vhl(3y+ GV)jary),, though the effect is smaller than that for the off-diagonal case discussed above. This effect, however, is already taken into account by choosing appropriate single-panicle wave functions. It is furthermore confirmed that the M3Y interaction produces reasonable two-body matrix elements provided that the single-particle wave functions are properly chosen. Namely hw of the harmonic oscillator wave function is adjusted to the observed nuclear radius (see eq. (2.5)). Thus we use the M3Y interaction in the following as the residual interaction except for the HF condition (2.8). As was mentioned before, this condition excludes ground
state and gives reasonable
excitation
energies
the lp-lh
component
in the
of the 0: and 2: states of 160.
The influence of the HF condition on the wave functions of both I60 and iSN is shown in table 4. We see that the lp-lh component is only 0.8% in ‘60g,S. with the HF condition, while it is 27.7% without the condition. It should be noted that the lp-lh
component
does not vanish
because
of the coupling
between
the lp-lh
and
the 2p-2h states. We also calculate Of states in 12C in order to see how the HF condition affects the lp-lh components and energies of those states. We change the size b-parameter according to the formula (2.5). The nucleus “C is a typical intermediate coupling one. Namely, a OAw model space calculation shows that the ground state of “C contains 45% of (Op3,2)6(Op,,2)2 and 47% of (Op3,2)8(Op,,2)o configuration. The components of the wave function of the ground state which is calculated in the (0+2)hw model space are shown in table 5. The component of the (OS)~(O~)’ is 80.6%. This should be compared with the result of the Utrecht group calculation lS2). They obtained results that the mixing probability of the lp-lh (0~)’ (Of, 1~)’ states
T. Hoshino ef al. / Shelf model calculation
466
TABLE 4
Configurations
of the ground
state of I60
Configuration
With HF
(Os)‘YOpY (OS)-‘(Odls)’
86.6% 0.0% 0.8% 12.6%
(oP)-‘(OflP)’ (Op)-z(Odls)z Binding
energy
( MeV)
Without 67.0% 7.0% 20.7% 5.3%
145.8
Configurations
HF
172.8
of “N
Configuration (with HF) W4(OPY’
85.6% 1.5% 3.7% 9.2%
(Os)-‘(Op)-‘(Odls)’ (OP)-2(OflP)’ (Op)-3(0dls)*
TAKE
Configurations
5
of the ground
state of ‘*C
Configuration
With HF
(os)4(OP)x (OS)-‘(Odls)’
80.6% 6.0% 8.9% 4.1% 0.4%
m-‘(Oflp)’ (0p)-2(0dls)2 tOs)-‘(OP)2 Binding
energy
( MeV)
97.4
was about 20%, and the Op-Oh component in the case of 160, the mixing of the lp-lh
83.7% 1.8% 3.7% 10.8%
Without
HF
64.3% 10.8% 17.5% 6.4% 1.0% 108.2
(OS)~(O~)~ was only 64%. As mentioned components gives also in “C unrealistic
values to physical quantities such as the excitation the r.m.s. radius, although the excitation energies with those calculated in 160.
energies of giant resonances and are not very high in comparison
3. Results 3.1. GIANT
RESONANCES
The calculated rupole transitions
response functions SF(&) for 160 are drawn for isoscalar quadin fig. 2 and for isoscalar monopole transitions in fig. 3, respectively.
7: Hoshino e/ al. / Shell model calculation I
I
I
“0
I 20
0 0
467
E2 T=O
I 60
40
60
% [Levi Fig. 2. The strength
distribution
for isoscalar quadrupole transitions space is taken in the calculation.
I_
60
in IhO. The (0+2)hw
full model
I "0 EO T=O
#
40 -
20 -
I 20
0 0
1 40
I 60
60
Ex [MeVl Fig. 3. The strength
They are defined
distribution
for the isoscalar
monopole
transitions
in IhO
as s FE*=F I(ftF~hli)l’S(Ex-El), = 4 WE&
Taking the long-wavelength follows:
F ~2~
limit,
i+f)s(E,
-
4) ,
(A =0,2).
we can write the transition
(3.1) operators
FEh as
(3.2)
468
T. Hoshino et al. / Shell model calculation
(3.3) for the quadrupole energy
and monopole
of the isoscalar
case, respectively.
giant quadrupole
resonance
Experimentally (GQR)
is found
the centroid to be 21 MeV
for I60 and that of the isoscalar monopole resonance (GMR) is estimated at about 3 MeV above the GQR from various experiments *OX2i).We can see from fig. 2 that the calculated levels appear slightly higher than the experimental ones. The lowest states with the largest transition strengths have the excitation energies E, = 23.3 MeV for O+ and E, = 27.5 MeV for 2+. The centroid energies of the monopole and quadrupole modes are 23.9 MeV and 29.9 MeV, respectively. The relative position of the GQR and the GMR is thus different from that shown in the experiment. This is related to the fact that the single-particle energies of the Of shells are not low enough to increase the component (0~))’ (Of) ’ in the 2+ state which might push down the energies of the 2+ states. In our calculation with the kinetic energies shown in table 2, this component (Op))‘(Of)’ in the 2+ state amounts to 22%, while the Ot states do not involve the Of shells to construct their wave functions. Therefore, when we change the single-particle energies of the Of shells, the relative position of the giant resonances is expected to become correct. The full width at the half-maximum (FWHM) is experimentally 7.5 MeV for the GQR of I60 [ref. *I)]. The second moment u which is related to the FWHM is calculated by using the following relation, l- FWHM=2&izu,
(3.4)
where
mk =
5 C5E
EkSF(E)
dE.
(3.5)
A gaussian strength distribution is assumed in deriving (3.4), although the shape of the GQR is known experimentally to be very different from the gaussian type distribution. The calculated strengths of the quadrupole transition are distributed over a wide energy range compared with those of the monopole. This fact confirms the results of Hoshino and Arima 22). The calculated FWHM is 6.86 MeV for the GQR, while it is 4.62 MeV for the GMR in an energy range between 23 and 45 MeV. We find that the calculated strength distribution is different from the gaussian type as expected from experiment. The width is originated from two kinds of coupling; one is the coupling to the continuum state and the other is the coupling to manyparticle-many-hole states, which gives rise to the so-called spreading width. In the present calculation the escaping width is not taken into account. The escaping width, however, is not negligible for the monopole transition. On the other hand, the escaping width of the quadrupole transition is reported to be small and
T. Hoshino et al. / Shell model calculation does
not contribute
much to the absolute
for the GQR is large enough The energy-weighted two-body
interaction
to explain
width. The calculated most of the observed
sum rule is theoretically is independent
in this way for the quadrupole
determined
of the momentum.
transition
469
width of 6.86 MeV one of 7.5 MeV. by assuming
that the
The upper limit determined
of I60 is 2176 e* fm4 MeV, ref. 23). The
calculated value up to the tenth state exhausts 81.0% of the total limit. Even if we go up to the 40th state (E, = 45.4 MeV), this increases only to 85.8%. The value for the monopole is 2194 e2 fm4 MeV. The calculated value from 23 to 45 MeV exhausts only 67.4% of the limit. The strength function of isoscalar monopole transitions in ‘*C is shown in fig. 4 together with the one calculated without the HF condition. If the HF condition is imposed, the calculated centroid energy is 31.3 MeV and the FWHM is 12.0 MeV, while without and 9.91 MeV, up to 50 MeV value without
the condition the centroid energy and the FWHM become 47.4 MeV respectively. The energy-weighted sum rule for monopole transitions amounts to 850 e2 fm4 MeV, which is 60.9% of the total limit 23). The the condition is 297 e* fm4 MeV, which is considerably smaller even
if one takes into account
the fact that the r.m.s. radius
is decreased.
10 5 -0
Fig. 4. The strength
distribution
3.2. SPECTROSCOPIC
Information by the nucleon of instruments
10
20
30 E, WV1
40
for the isoscalar monopole transitions condition, (b) without the condition.
50
60
in “C: (a) with the Hartree-Fock
FACTOR
on the validity of the concept of the single-particle motion is obtained knock-out reaction as well as the stripping reaction. The development for electron scattering makes it feasible to perform the knock-out
T. Hoshino et al. / Shell model calculation
470
reaction
(e, e’p) in the quasi-elastic
probes *“). The spectroscopic defined
region
strength
more precisely
than that using hadronic
for the single-particle
state (Y(= n, Z,j, m,) is
as
where
(3.7) where [J] = m. The quantity SL” is called the shell-model spectroscopic factor. The triple-bar matrix element is a reduced matrix element both for spin and isospin space. Surprisingly, the spectroscopic factors observed in the (e, e’p) experiment with satisfactory statistics are reported to be smaller than the values obtained by many-body theories. The spectroscopic strength is about 0.5, [ref. r4)], while the theoretical value is 1.0 in the pure shell-model limit. The spectroscopic factors calculated by using the present wave functions are shown in fig. 5 for the 160(e, e’p)15N reaction. As seen from fig. 5, the strength is concentrated in the lowest excited i- and sP states. The experimental spectroscopic strength derived from the distorted wave impulse approximation analysis 14) is 0.45-0.59 for fP and 0.53-0.58 for G- depending on the optical potential parameters, while the present calculated
0.10 0.08 0.06 0.04 0.02 0.00 0.00 0.06 0.04 0.02 0.00
0
10
20
30
40
Ex [MeVl Fig. 5. Spectroscopic
strength distributions in the 160(e , ~‘P)‘~N reactions. $- states in the lower case, and $- states in the upper
The final states of “N case.
are
471
T. Hoshino ef al. / Shell model calcularion
values
are 0.935 for $l and 0.936 for $r summing
lp. The momentum
distribution
defined
up the contributions
from Op and
as
where 4,,,,(p) is the Fourier transform of the single-particle wave function, is shown in fig. 6 for the case of the i; and $; states. In the calculation, we introduced a normalization factor A,; defined as
A/,
CN’,:= 7eNeUnp
(3.9)
9
n =,,
in order to obtain the same value of the whole volume integral one. The value r], is an absorption factor defined in ref. 14).
-300
-250
-200
-150
Momentum
-100
-50
as the experimental
0
pe [MeV/c]
Fig. 6. The momentum distributions of the single-particle states in IhO. The solid curve (circles) represents the calculated (experimental) distribution of P,,~, while the dashed curve (open circles) shows the calculated (experimental) one of pi,?. The experimental data are taken from ref. 14).
Recently, the momentum distributions of 4’ states in “B have been experimentally measured in NIKHEF-K “). Their analysis takes into account the (0+2)8w model space for the “C ground state, but only the 1h w model space for the $’ states in “B. This analysis is doubtful because of the adopted model space. Namely, if the (0+ 2) hw model space for the “C nucleus is taken, the model space for the t’ states in “B should include (1+3)hw states. As a schematic example, we calculate the spectroscopic factor of the i- state in 15N taking either a O&J or a (0+2)hw model space for “N and the (0+2)hw model space for IhO without the HF condition. The
472
spectroscopic
T. ~oshino
factor
et al. / Shell model c~~cul~t~~n
of the state turns
out to be 0.685 in the Oirw model
15N, while it is 0.972 when the fuli (0+2)&w and 160. One can see that the inconsistent and final states decreases
model space is assumed
treatment
the spectroscopic
space for
both for “N
of the model space in the initial
strength
significantly.
In our calculation, the ground state of “C is calculated within the Ohw model space and the i” states in “B within the lfio model space. Our results show that
0.0 0
1
L
I
10
20
30
.
1.1
,
40 %
50
60
I
I
70
60
MN
Fig. 7. Spectroscopic strength distributions in the “C(e, e’p)“B reaction. The final states are 4” states of “B. The ground state of “C is calculated within the Ohw model space.
101
I 0
100 Momentum
150
I
I
I
I
I 50
200
250
pn [MeV/c]
Fig. 8. The momentum distribution of the s,,, state of “C. The final state of “B is located experimentally at 6.79 MeV. The ground state of “C is computed with the Ohw model space.
T. Hoshino et al. / Shell model calculation
the centroid moment
energy
is 48.0 MeV in the missing
is 4.92 MeV (r,,,, distribution
distribution
= 11.59 MeV). The calculated
of the first +’ state in “B is 0.0190. The strength the momentum
energy
without
473
spectroscopic
distribution
the normalization
and the second
is shown
constant
strength
in fig. 7 and
A in fig. 8, together
with the experimental distribution ‘) of the 5’ state at 6. 79 MeV. Since the in “Cg_ is deeply bound, the spectroscopic strengths are distributed in a wide energy range in contrast to the case of 160(er e’p)r’N. The absolute the calculated momentum distribution is larger than the experimental one, the shapes resemble each other.
s,,~ state relatively value of although
4. Summary and discussion We have reported the effect of the HF condition on the shell-model calculations of I60 and ‘*C. The model space consists of configurations within the (0+ 2)hw excitation energy. We used the M3Y interaction as the effective two-body interaction in the calculations of the single-particle energies and the two-body matrix elements. The violation effect in the this effect by effect in the
of the HF condition is caused mainly by the neglect of the Brueckner G-matrix such as shown in fig. Id. We take introducing the force 6V in eq. (2.9) which simulates the Brueckner G-matrix (or equivalently the rearrangement
higher-order into account higher-order effect of the
density-dependent force). There is no unique way to select the force 6V in the effective interaction. In the present study, the effective potential (2.9) is good enough to avoid the mixture of large lp-lh components in the ground state due to the violation of the HF condition while small physical lp-lh components still exist through the coupling of 2p-2h states. The lp-lh states are contained only less than 1% in the wave function of the ground
state in 160. The importance
of the HF condition
was already
in refs. 3Z25). We performed the shell-model calculations with the b-parameter (2.5). This prescription seems reasonable because one reproduces observed
ground-state
binding
energies
of “C and I60 and the density
suggested
fixed by eq. very well the distribution
of I60 together with reasonable excitation energies and strength distributions of the giant resonances. The charge distribution shown in fig. 9 matches well the experimental one and produces the calculated charge r.m.s. radius r = 2.68 fm which is in reasonable agreement with the experimental one r,(exp) = 2.72 fm [ref. ““)I. The difference can be improved by changing the value of the b-parameter. The calculated excitation energy of the quadrupole giant resonance is 29.9 MeV in average, while that of the monopole resonance is 23.9 MeV. These values are quite reasonable in comparison with the experimental excitation energies. The widths of the quadrupole and the monopole giant resonances are calculated to be 6.86 MeV and 4.62 MeV, respectively. In the quadrupole case, the experimental one is reported to be 7.5 MeV. This result suggests that the main origin of the width of the quadrupole
T. Hoshino et al. / Shell model calculation
474
0.06
0.04
0.02
0.00
0
1
2
3 =
Fig. 9. The charge with the finite-size
4
5
6
7
r.fml
distribution of the ground state of 160. The solid curve is the charge density calculated effect by using in eq. (2.6). The dashed curve is the proton density. The experimental distribution is shown by the dotted line and shadowed area *‘).
giant resonances can be explained as a damping width due to the coupling to 2p-2h states. One of the advantages of this calculation with the HF condition is the inclusion of the physical diagram shown in fig. lc excluding the unphysical diagram fig. lb where the hole line is a spectator. The second advantage is that the adopted oscillator b-parameter is close to the value which is commonly used in the literature. In ref. ‘), the b-parameter
used is 15% larger than our value in order to reproduce
the empirical
r.m.s. radius because of the large mixing of lp-lh amplitudes in the ground state. It is expected that a larger shell-model space brings in a reduction of the spectroscopic strength and momentum distribution. We, however, stress that one should enlarge consistently the model space of the initial and final states. As was shown in the case of “N, the inconsistent treatment of the model space gives a superficial decrease of the spectroscopic factors. The spectroscopic strengths obtained in the present calculations are 0.935 for the ground state in ‘*N(i-) and 0.936 for the first excited state in ‘5N($). We should notice that these spectroscopic factors do not correspond to the occupation probabilities of OP,,~ and Op,,, orbits in “0, which are 0.969 and 0.983, respectively. Inclusion of the ground-state correlation doesn’t considerably affect the spectroscopic factor, which suggests that single-particle or single-hole structure still remains even if one takes into account the ground state correlation. A similar result has been obtained in a different truncated space by Zuker, Buck and McGrory “), who of neighboring nuclei around took OP,P, Od,,, and 1~~ orbits for the calculation I60 allowing up to 4p-4h excitations as a model space.
475
T. Hoshino et al. / Shell model calculation
These spectroscopic the origin different scopic
factors
of this discrepancy, set of optical
strength
are much larger than the experimental it is very desirable
model parameters
from the experimental
to reanalyze
the data by using a
which are used for extracting cross section.
ones. To see the spectro-
The peak of the momentum
distribution is different from the experimental one as shown also an interesting open problem for future calculations.
in fig. 6. This point is
We wish to thank K. Yazaki and K. Shimizu for valuable discussions and W. Bentz for his critical reading of the manuscript. We are very grateful to B.A. Brown who offered us the shell model code and useful advice on the computation. We also thank the members of the Theoretical Nuclear Physics Group in the University of Tokyo for their discussions. The calculations have been performed Laboratory, University of Tokyo.
by using VAX-l l/780 at the Meson Science
References 1) P.W.M. Glaudemans, in Proc. Int. Symp. on nuclear shell models, Drexel University, Philadelphia, 1984, p. 2 2) G. van der Steenhoven, Phys. Lett. Bl56 (1985) 151 3) S.S.M. Wong, Phys. Lett. 20 (1966) 188 4) D. Vautherin and D.M. Brink, Phys. Rev. CS (1972) 626 5) M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 6) J. Negele, Phys. Rev. Cl (1970) 1260 7) K. Ando, H. Bando and E. M. Krenciglowa, Prog. Theor. Phys. Suppl. No. 65 (1979) 38 8) V.R. Pandharipanda and R.B. Wiringa, Rev. Mod. Phys. 51 (1979) 821; H. Kiimmel, K.H. Liihrmann and J.G. Zabolitzky, Phys. Reports 36 (1978) 1; B.D. Day, Phys. Rev. C24 (1981) 1203 9) H. Sagawa, B.A. Brown and 0. Scholten, Phys. Lett. B159 (1985) 228; A. Arima, N. Onishi, T. lnoue, Y. Akiyama and T. Suzuki, Nucl. Phys. A459 (1986) 286 10) B.H. Wildenthal and W. Chung, Phys. Rev. C22 (1980) 2260 11) H. Miither, A. Polls and T.T.S. Kuo, Nucl. Phys. A435 (1985) 548; H. Miither, A. Polls, P.K. Rath and A. Faessler, Nucl. Phys. A442 (1985) 68 12) J. Sharpin, H. Miither, T.T.S. Kuo and A. Faessler, Nucl. Phys. A293 (1977) 61 13) P.K.A. De Witt Huberts, Nucl. Phys. A446 (1985) 301~ 14) M. Bernheim, A. Bussiere, J. Mougey, D. Royer, D. Tarnowski, S. Turck-Chieze, S. Frullani, S. Both, C. Giusti, F.D. Pacati, G.P. Capitani, E. De Sanctis and G.J. Wagner, Nucl. Phys. A375 (1982) 381 15) B.A. Brown, A. Etchegoyen and W.D.M. Rae, A shell-mode1 program “OXBACH” (Aug. 1985) 16) F. Palumbo and D. Prosperi, Nucl. Phys. All5 (1968) 296; D.H. Gloeckner and R.D. Lawson, Phys. Lett. B53 (1974) 313 17) G.F. Bertsch, J. Borysowicz, H. McManus, and W.G. Love, Nucl. A284 (1977) 399 18) G.F. Bertsch, The practitioner’s shell model (North-Holland, Amsterdam, 1972); J. Blomqvist and A. Molinari, Nucl. Phys. Al06 (1968) 545 19) Nguyen van Giai and H. Sagawa, Phys. Lett. B106 (1981) 379 20) S. Drozdz, V. Klemt, J. Speth and J. Wambach, Nucl. Phys. A451 (1986) 11 21) K.T. KnGpBe, G.J. Wagner, H. Breuer, M. Rogge and C. Mayer-Boricke, Phys. Rev. Lett. 35 (1975) 779 22) T. Hoshino and A. Arima, Phys. Rev. Lett. 37 (1976) 266
476
T. Hoshino
et al. / Shell model calculation
23) A. Bohr and B. Mottelson, Nuclear structure (Benjamin, Reading, 1975) vol. 2, p. 404 24) S. Frullani and J. Mougey, Adv. in Nucl. Phys. 14 (1984) 25) A. Arima, in Proc. Int. Symp. on nuclear shell models, Drexel University, Philadelphia, 1984, p. 579 26) H. Miska, B. Norum, M.V. Hynes, W. Bertozzi, S. Kowalski, F.N. Rad, C.P. Sargent, T. Sasanuma and B.L. Berman, Phys. Lett. B83 (1979) 165 27) A.P. Zuker, B. Buck and J.B. McGrory, Phys. Rev. Lett. 21 (1968) 39
SHELL-MODEL
CALCULATION
Tsutomu
HOSHINO,
WITH
Hiroyuki
HARTREE-FOCK
SAGAWA
CONDITION
and Akito ARIMA
Department of Physics, Faculty of Science, University of Tokyo, Tokyo, Japan Received 3 November (Revised 7 December
1986 1987)
Abstract: Giant resonances and spectroscopic factors of IhO are studied in the framework of the shell model with (0+2)hw model space. It is found that the Hartree-Fock condition is crucial in order to obtain reasonable excitation energies of giant resonances in IhO. The calculated spectroscopic factors in 15N are quenched only by 10% due to the coupling to lp-2h states. The spectroscopic factor of the s-state in “B is also discussed.
1. Introduction The nuclear shell model has been the most successful and well-established framework for studying nuclear structure problems. The recent development of computers makes it feasible to perform large scale shell-model calculations, for example (0 + 2) hw calculations. In this work we take the nucleus I60 as an example for such shell-model calculations. The Utrecht group I,*) has already studied this problem. They found a large admixture of lp-lh components in the ground state of 160. If one takes a model space generated by the Hartree-Fock single-particle states and uses the interaction which satisfies the stability condition, the HartreeFock condition should be automatically satisfied. If this is the case, there might be very small lp-lh components in the ground state. However, when one uses the harmonic oscillator wave function and an effective interaction which simulates the G-matrix condition condition
as is usually done in the shell-model calculation, the Hartree-Fock is not guaranteed. Thus, one has to take into account the Hartree-Fock in the shell-model calculation in order to obtain realistic wave functions ‘).
At present, the Hartree-Fock calculations have been performed by using two different kinds of effective interactions. The first one uses phenomenological density-dependent forces such as Skyrme-type interactions 4,5) or the one derived by density-matrix expansion method “). These density-dependent forces give extremely successful results for the binding energies, the density distributions and the single-particle energies of doubly-closed shell nuclei. The second one is called the Brueckner-Hartree-Fock method ‘,‘) where a free nucleon-nucleon interaction such as the Reid soft-core potential is used. Many extensive calculations have been done including higher-order diagrams based on the Brueckner-Hartree-Fock method, but still a serious difference between the calculated and experimental saturation curves (so called the Coester line discrepancy) is not yet solved. 0375-9474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
459
T. Hoshino et al. / Shell model calculation
The shell-model been performed
calculations recently
with the density-dependent
‘). In these studies, the configuration
Ohm model space. This approach but the energy calculations
spectra
with
Skyrme-type
are poorly
purely
reproduces reproduced
phenomenological
the binding matrix
space is limited within
energies
in comparison
forces have
of d-s shell nuclei,
with the shell-model
elements
by Wildenthal
and
Chung I”). The shell-model calculations based on the Brueckner G-matrix were also attempted by the Jiilich-Tiibingen group ‘I). In these calculations, they used the Reid soft-core potential as an effective nucleon-nucleon interaction and took into account the many-body effects coming from higher-order diagrams of the G-matrix in the calculations of matrix elements in the Ohw configuration space. In spite of these serious efforts, the results are still poor in comparison with experimental data. In this paper,
we will study the excitation
energies
of giant resonances
and the
spectroscopic factors of I60 by large space shell-model calculations. Since neither the fundamental approach like the Brueckner G-matrix nor the phenomenological Skyrme-type force is far from the realistic situation, we will take the following pragmatic approach for the large scale shell-model calculation taking into account the Hartree-Fock condition. We take as an interaction, the M3Y (Michigan 3-range Yukawa) force which simulates the Reid soft-core potential. In the case of the Skyrme-type force, the rearrangement terms stemming from the variation of the density-dependent force are crucial to give successful results in the Hartree-Fock calculations. In the Brueckner reaction matrix, this density dependence is due to the starting energy and a Pauli rearrangement term 12). In our calculation, we have to simulate this rearrangement effect in order to satisfy the Hartree-Fock condition using the M3Y force. As is discussed in sect. 2.3, the rearrangement term is taken care by the effective potential 6V in eq. (2.9). In this prescription, we will be able to calculate the physical higher-shell components above one major shell in the ground states and also in the excited states eliminating unphysical Ip-lh excitations. In p-shell nuclei, these higher shell components correspond to Od-ls-Of-lp shell ones. Those components are crucial to study the spectroscopic factors in (e, e’p) experiments. The Hartree-Fock condition has been already studied in the case ofthe shell-model calculation
of I60 in ref. ‘), in which he set all matrix
elements
between
the core
of I60 and lp-lh states to be zero. Recent experimental data revealed a difficulty in explaining spectroscopic factors in the proton knock-out reaction and provided a test stone which examines shellmodel calculations very precisely 13). The experimental information of the (e, e’p) reaction is particularly interesting because both the absolute spectroscopic factors and the momentum distributions of single-particle states have been provided with good accuracy. The observed spectroscopic factors 14) are typically one half of the pure single-particle values in p-shell nuclei such as “0, suggesting that the spreading
27 Hosh~no et al. i Shell model ca~culatjon
460
of the spectroscopic
factors
many-particle-many-hole functions account.
obtained
states
find that the theoretical
values.
These
values
is due to the coupling
spectroscopic
in this work in which the coupling
We, however,
the single-particle
of single-particle
states. We calculate
factors
to
using the wave
to 2p-2h states is taken into
spectroscopic
are still much
factors
larger
than
are 93% of the observed
ones 14). The model space (0-t 2)hw is consistently used in the following calculations. The adopted effective interaction and the single-particle energies are shown in sect. 2.2. The effect of the Hartree-Fock condition is examined in sect. 2.3. We will point out that this condition is pa~icularly important in order to obtain reasonable excitation energies of giant resonances. Sect. 3.1 is devoted to the discussion of the results of monopole and quadrupole giant resonances in 160. The calculated spectroscopic factors and the momentum distributions in I60 are compared with the experimental ones in sect. 3.2. A summary is given in sect. 4. We also discuss the case of “C in order to compare with a previous calculation by the Utrecht group ‘).
2. Shell-model 2.1. MODEL
SPACE
AND CENTER-OF-MASS
calculation EFFECT
Shell-model calculations for Op shell nuclei in a large shell-model space are now feasible because of the development of modern computers. We use the OxfordBuenos-Aires-MSU shell-model code 15), which is based on the Lanczos method. In spite of the development of computers, however, shell-model calculations are still basically restricted by the dimensions of the shell-model space. The number of O+ states in I60 is 44 within the present (0+ 2) ho full model space, but turns out to be 4340 within the (0 f 2 + 4) ho model space. We adopt a hamiltonian in a second quantized
form,
where (41 V(Q- Q)/Y% is an antisymmetrized
matrix
element
of the two-body
interaction, and (Y, p, y and 6 are harmonic oscillator single-particle states. If one uses a translationally invariant effective interaction, one can separate the harmiltonian in the usual way into a term which depends only on the center-of-mass (c.m.) coordinate and one which is a function of the relative coordinates (the intrinsic hamiltonian). The hamiltonian of the c.m. motion consists only of a kinetic energy term. In this paper we take harmonic oscillator single-particle wave functions. One usually adds a c.m. harmonic oscillator potential fAfkh2R2 (R = (l/A) Cf=, ri) to (2.1). Then, the calculated wave functions can be expressed as a product of the eigenstate of the cm. motion and that of the intrinsic motion, provided that one
7: Hoshino
adopts
a complete
quanta,
for example
set of states
et al. / Shell
within
two in the present
In this paper, we do maintain
461
model calculation
a certain
number
of harmonic
the form (2.1) of the hamiltonian
i&
&=P
oscillator
calculation. and add the term r6)
(2.2)
+;MAw2R2-;hw
to the hamiltonian (2.1) instead of iAMw2R2, since the HF condition can be only handled in this prescription. We numerically checked that two methods did not make any difference in the final results in the case without the I-IF condition. With an appropriate value for /3 (“IOO), spurious states will be separated well enough from low-energy states which have approximately the OS cm. motion. In the present calculation, we take into account the s-p-sd-pf shells h/2,Of~,2,Of~,2,~PW, lp,,J as the model space. (0s L/2, OP,,,,0~~,2,04/2, Ob, The unperturbed excitation energy is taken up to 2hw. In these shell-model orbits, calculations within the (0 + 2) iiw model space for I60 are possible and the spurious c.m. motion can be approximately separated. 2.2. EFFECTIVE
INTERACTION
AND SINGLE-PARTICLE
ENERGIES
We use the M3Y (Michigan 3-range Yukawa) “) interaction as the two-body interaction Vi,. The parameters of M3Y are determined by a fit to the G-matrix obtained from the Reid soft-core potential in the oscillator basis. This effective NN interaction is composed of a central part in both the spin-singlet even-parity (SE) and the spin-triplet even-parity (TE) channels, a tensor part in the even-parity (TNE) channel, and a spin-orbit part in the odd-parity (LSO) channel. Three Yukawa potentials with different ranges are included for the central part, while two different ranges are adopted for the spin-orbit and also for the tensor part. Each potential is written
in the following
form:
VSEtTEf=
i Vs'T'Y(r,2/Ri)qS(f~PE, i=l
VLso = i VksoL * SY( r12/ Rj) PO, is=, where
Y(X) = exp (-x)/x.
The tensor
sr2=3(o,
3 hd(n.
operator
is defined
CZ)/&-(~,
(2.3)
as
. ~2),
(2.4)
and PSC7), PE and PO denote the projection operators for the spin-singlet (triplet), even-parity and odd-parity channels, respectively. The parameters are tabulated in table I. The longest range R, for the central part is chosen to simulate the OPEP tail at large distance. The longest range R, used for the tensor force is taken to fit to that of the OPEP tail closely in the form as r*Y. The force with the range R2 for
462
T. Hoshino et al. / Shell model calculation TABLE 1 Parameters
Channel
R, = 0.25 fm
of the M3Y
R, = 0.4 fm
SE TE TNE
12 454 21 227
-3835 -6622 -1259.6
LSO
-3 733
-427.3
SE TE TNE LSO
potential 1.414 fm for central R,=
0.700 fm for tensor -10.463 -10.463 -28.41
(MeV) (MeV) (MeV/fm*) (MeV)
singlet even. triplet even. - tensor even. - spin-orbit odd.
every channel simulates roughly the “cT-exchange” process. The short-range force with R, = 0.25 fm is also included to improve the fit. The two-body LS force is included in the M3Y force, but the calculated splitting of Op shell orbits is not large enough to explain the experimental splitting (6.32 MeV) between the 4P and $- states found in “N. We shift up the kinetic energy of the OP,,~ shell by 3.0 MeV to produce a proper energy spacing. The calculated single particle energy of the Of,,> shell is higher than that of the 1p3,2 shell when one uses the M3Y force. We lower the Of,,, and Of,,, levels by 5.0 MeV. The single particle energies for both I60 and ‘*C are shown in table 2. The size parameter is computed according to the following formula 18): hw =45A-‘/‘-25A-*“[MeV], b=m. The size parameter account
reproduces
the finite size effect of nucleon pcharge(r) =
J
charge r.m.s. radius
of I60 taking
as follows:
pproton( r’){ m2}-3’2
With a2 = 0.4 fm*, we obtain 2.3. HARTREE-FOCK
(2.5)
the observed
exp ( - (r - r’)*/a’)
d3r’
.
r, = 2.68 fm for the charge r.m.s. radius
into
(2.6) of 160.
CONDITION
It is generally impossible to reproduce the saturation condition of the HartreeFock theory only with a density-independent two-body interaction. One has to take into account a many-body force or density-dependent one. For example, a densitydependent interaction is often used in the Hartree-Fock (HF) calculation 4,6). As mentioned above, we adopt the interaction M3Y in the following calculations. However, since this interaction does not depend the nuclear density, the saturation condition is not fulfilled in the HF calculation by including only the contributions shown in fig. la. This might be improved if one takes into account higher-order
T. Hoshino
463
et al. / Shell model calculation TABLE 2
Kinetic energies and single-particle
energies in I60 and ‘*C
?I
I60
Shell KE
SPE
-54.34
11.16
-46.37
OP,,,
17.40
-27.70
18.61
-18.06
OP,,,
20.40
-20.58
21.61
-13.38
Od S/2
24.35
-6.68
26.05
KE
SPE
OSI/.?
10.44
(MeV)
2.19
IS I/Z
24.35
-5.50
26.05
Od3/2
24.35
-0.07
26.05
-0.25 6.3 1
Of,,,
26.32
6.29
28.49
12.59
31.32
8.09
33.49
13.94
31.32
9.73
33.49
15.05
26.32
11.73
28.49
17.16
diagrams of the Brueckner G-matrix shown in fig. Id [refs. ‘,12)], which are called as the Pauli blocking term and the starting energy term. Since our aim is the shell-model calculation in the large configuration space, we do not calculate directly the higher-order diagrams of the Brueckner-Hartree-Fock theory. Instead, we will take into account these effects adding an extra potential to the M3Y force in order to satisfy the Hartree-Fock condition. The HF condition
is written (Q,l#>+
for the two-body
interaction
as follows:
c (QYI VIPY), = &,pe, 7 y occupied
(2.7)
where E, is the single-particle energy of the ath orbit. We have to take into account effectively this condition for the following reasons: (i) Violation of this condition pushes down the ground-state energy unrealistically. At the same time, excitation energies of giant resonances became too high (the 0: state of I60 is predicted at E, = 45 MeV, while the observed giant monopole resonance has E, = 25 MeV). (ii) Transition strengths to 0’ states are not realistic. We calculated the overlaps between the harmonic oscillator single-particle wave functions with the oscillator parameter b = 1.73 fm and the HF wave functions calculated with a Skyrme-type density-dependent SC11 interaction 19) in the OS and Op shell-model orbits for 160. The overlaps are more than 99.9% for OS,,, and OP,,~, and 99.8% for the Op,,, state, respectively. These values confirm that harmonic oscillator wave functions are quite good approximations to single-particle wave functions. Comparing the contributions of the kinetic energy and the M3Y interaction
464
7: Hoshino
V J.j
__--+
t
.v
v __-
n+l.l.j
Ai
n+l,l,j
Jj
n+l,l,j
et al. / Shell model calculation
0 sz0
L-I--
(4 fib)
(c)
_-.-
~
_----
0
Fig. 1. Diagrammatic representations of the Hartree-Fock condition. The diagram (a) represents the Hartree-Fock condition written in eq. (2.8). The contribution of the diagram (b) in the calculation of neighbouring nuclei is excluded, but physical lp-2h states corresponding to the diagram (c) are taken into account. The diagrams (dt are examples of higher order terms.
in table 3, we can see that the sum of the two-body matrix elements are much larger than the off-diagonal kinetic energies in absolute value. Thus, the HF condition is badly violated in the case of the M3Y force. Similar two-body matrix elements are obtained in the case of the SGII interaction when one switches off the rearrangement term as is shown in the sixth column of table 3. However, when the rearrangement effects is included, the HF condition is fulfilled almost perfectly particle states l~~,,~ and lp,,, are above the threshold. Therefore, that the rearrangement
effect has a crucial
even though the we can conclude
role in the HF calculation
with the
density-dependent Skyrme-type force. We have to take into account the rearrangement effect also in the shell-model calculation with the M3Y interaction in order to fulfill the HF condition. In order to simulate in the particle-hole
the rearrangement
matrix (pltlh)+
elements,
effect, we add a repulsive
so that the HF condition
c (PYl(V,,,+6V)Ihy),=O, y occupied
force 6V only
is strictly
fulfilled: (2.81
where (2.91 The parameter V, is fixed to satisfy the condition (2.8) for each orbit. Thus, the diagrams shown in fig. la are cancelled completely. The matrix element corresponding to the diagram lb for the neighboring nucleus is also cancelled by that of the
Comparison
of the contributions
of the kinetic
energy
and the M,Y interactions SGII (HF)
M3Y (H.O.)
(Pl$~)
P/f’
Xi (PjlVlW
It/ VI tplflh)
C, tpjl Vlhj)
C, tpjlvlhj)
+rearrange
It/
VI
lSl,,/OS,,2
8.56
-16.0
0.55
8.30
-13.0
-8.1
0.62
1PW/OP3,,
11.0
-15.9
0.69
5.66
-7.6
-5.4
0.71
lP,/,/OP,,Z
11.0
-16.9
0.65
5.13
-7.0
-4.9
0.70
The symbol “i-rearrange” means the addition of a rearrangement term to Ii (~$1 V!hj). Abbreviations of H.O. and HF indicate that calculations are done with harmonic oscillator wave functions and Hartree-Fock wave functions, respectively. Because lp states of the SGII interaction are in the continuum, the values of t and V are smaller than those of the M3Y inleraction. The particle wave functions 1p3,,a, lp,,, are calculated in the spherical box with the radius R = 10 fm. kinetic
energy
plus SV, while the physical
contribution. The same rearrangement
lp-2h
effect is important
state shown
in fig. lc has a finite
for the diagonal
part of the single
particle potential C, occupied(c~rf( Vhl(3y+ GV)jary),, though the effect is smaller than that for the off-diagonal case discussed above. This effect, however, is already taken into account by choosing appropriate single-panicle wave functions. It is furthermore confirmed that the M3Y interaction produces reasonable two-body matrix elements provided that the single-particle wave functions are properly chosen. Namely hw of the harmonic oscillator wave function is adjusted to the observed nuclear radius (see eq. (2.5)). Thus we use the M3Y interaction in the following as the residual interaction except for the HF condition (2.8). As was mentioned before, this condition excludes ground
state and gives reasonable
excitation
energies
the lp-lh
component
in the
of the 0: and 2: states of 160.
The influence of the HF condition on the wave functions of both I60 and iSN is shown in table 4. We see that the lp-lh component is only 0.8% in ‘60g,S. with the HF condition, while it is 27.7% without the condition. It should be noted that the lp-lh
component
does not vanish
because
of the coupling
between
the lp-lh
and
the 2p-2h states. We also calculate Of states in 12C in order to see how the HF condition affects the lp-lh components and energies of those states. We change the size b-parameter according to the formula (2.5). The nucleus “C is a typical intermediate coupling one. Namely, a OAw model space calculation shows that the ground state of “C contains 45% of (Op3,2)6(Op,,2)2 and 47% of (Op3,2)8(Op,,2)o configuration. The components of the wave function of the ground state which is calculated in the (0+2)hw model space are shown in table 5. The component of the (OS)~(O~)’ is 80.6%. This should be compared with the result of the Utrecht group calculation lS2). They obtained results that the mixing probability of the lp-lh (0~)’ (Of, 1~)’ states
T. Hoshino ef al. / Shelf model calculation
466
TABLE 4
Configurations
of the ground
state of I60
Configuration
With HF
(Os)‘YOpY (OS)-‘(Odls)’
86.6% 0.0% 0.8% 12.6%
(oP)-‘(OflP)’ (Op)-z(Odls)z Binding
energy
( MeV)
Without 67.0% 7.0% 20.7% 5.3%
145.8
Configurations
HF
172.8
of “N
Configuration (with HF) W4(OPY’
85.6% 1.5% 3.7% 9.2%
(Os)-‘(Op)-‘(Odls)’ (OP)-2(OflP)’ (Op)-3(0dls)*
TAKE
Configurations
5
of the ground
state of ‘*C
Configuration
With HF
(os)4(OP)x (OS)-‘(Odls)’
80.6% 6.0% 8.9% 4.1% 0.4%
m-‘(Oflp)’ (0p)-2(0dls)2 tOs)-‘(OP)2 Binding
energy
( MeV)
97.4
was about 20%, and the Op-Oh component in the case of 160, the mixing of the lp-lh
83.7% 1.8% 3.7% 10.8%
Without
HF
64.3% 10.8% 17.5% 6.4% 1.0% 108.2
(OS)~(O~)~ was only 64%. As mentioned components gives also in “C unrealistic
values to physical quantities such as the excitation the r.m.s. radius, although the excitation energies with those calculated in 160.
energies of giant resonances and are not very high in comparison
3. Results 3.1. GIANT
RESONANCES
The calculated rupole transitions
response functions SF(&) for 160 are drawn for isoscalar quadin fig. 2 and for isoscalar monopole transitions in fig. 3, respectively.
7: Hoshino e/ al. / Shell model calculation I
I
I
“0
I 20
0 0
467
E2 T=O
I 60
40
60
% [Levi Fig. 2. The strength
distribution
for isoscalar quadrupole transitions space is taken in the calculation.
I_
60
in IhO. The (0+2)hw
full model
I "0 EO T=O
#
40 -
20 -
I 20
0 0
1 40
I 60
60
Ex [MeVl Fig. 3. The strength
They are defined
distribution
for the isoscalar
monopole
transitions
in IhO
as s FE*=F I(ftF~hli)l’S(Ex-El), = 4 WE&
Taking the long-wavelength follows:
F ~2~
limit,
i+f)s(E,
-
4) ,
(A =0,2).
we can write the transition
(3.1) operators
FEh as
(3.2)
468
T. Hoshino et al. / Shell model calculation
(3.3) for the quadrupole energy
and monopole
of the isoscalar
case, respectively.
giant quadrupole
resonance
Experimentally (GQR)
is found
the centroid to be 21 MeV
for I60 and that of the isoscalar monopole resonance (GMR) is estimated at about 3 MeV above the GQR from various experiments *OX2i).We can see from fig. 2 that the calculated levels appear slightly higher than the experimental ones. The lowest states with the largest transition strengths have the excitation energies E, = 23.3 MeV for O+ and E, = 27.5 MeV for 2+. The centroid energies of the monopole and quadrupole modes are 23.9 MeV and 29.9 MeV, respectively. The relative position of the GQR and the GMR is thus different from that shown in the experiment. This is related to the fact that the single-particle energies of the Of shells are not low enough to increase the component (0~))’ (Of) ’ in the 2+ state which might push down the energies of the 2+ states. In our calculation with the kinetic energies shown in table 2, this component (Op))‘(Of)’ in the 2+ state amounts to 22%, while the Ot states do not involve the Of shells to construct their wave functions. Therefore, when we change the single-particle energies of the Of shells, the relative position of the giant resonances is expected to become correct. The full width at the half-maximum (FWHM) is experimentally 7.5 MeV for the GQR of I60 [ref. *I)]. The second moment u which is related to the FWHM is calculated by using the following relation, l- FWHM=2&izu,
(3.4)
where
mk =
5 C5E
EkSF(E)
dE.
(3.5)
A gaussian strength distribution is assumed in deriving (3.4), although the shape of the GQR is known experimentally to be very different from the gaussian type distribution. The calculated strengths of the quadrupole transition are distributed over a wide energy range compared with those of the monopole. This fact confirms the results of Hoshino and Arima 22). The calculated FWHM is 6.86 MeV for the GQR, while it is 4.62 MeV for the GMR in an energy range between 23 and 45 MeV. We find that the calculated strength distribution is different from the gaussian type as expected from experiment. The width is originated from two kinds of coupling; one is the coupling to the continuum state and the other is the coupling to manyparticle-many-hole states, which gives rise to the so-called spreading width. In the present calculation the escaping width is not taken into account. The escaping width, however, is not negligible for the monopole transition. On the other hand, the escaping width of the quadrupole transition is reported to be small and
T. Hoshino et al. / Shell model calculation does
not contribute
much to the absolute
for the GQR is large enough The energy-weighted two-body
interaction
to explain
width. The calculated most of the observed
sum rule is theoretically is independent
in this way for the quadrupole
determined
of the momentum.
transition
469
width of 6.86 MeV one of 7.5 MeV. by assuming
that the
The upper limit determined
of I60 is 2176 e* fm4 MeV, ref. 23). The
calculated value up to the tenth state exhausts 81.0% of the total limit. Even if we go up to the 40th state (E, = 45.4 MeV), this increases only to 85.8%. The value for the monopole is 2194 e2 fm4 MeV. The calculated value from 23 to 45 MeV exhausts only 67.4% of the limit. The strength function of isoscalar monopole transitions in ‘*C is shown in fig. 4 together with the one calculated without the HF condition. If the HF condition is imposed, the calculated centroid energy is 31.3 MeV and the FWHM is 12.0 MeV, while without and 9.91 MeV, up to 50 MeV value without
the condition the centroid energy and the FWHM become 47.4 MeV respectively. The energy-weighted sum rule for monopole transitions amounts to 850 e2 fm4 MeV, which is 60.9% of the total limit 23). The the condition is 297 e* fm4 MeV, which is considerably smaller even
if one takes into account
the fact that the r.m.s. radius
is decreased.
10 5 -0
Fig. 4. The strength
distribution
3.2. SPECTROSCOPIC
Information by the nucleon of instruments
10
20
30 E, WV1
40
for the isoscalar monopole transitions condition, (b) without the condition.
50
60
in “C: (a) with the Hartree-Fock
FACTOR
on the validity of the concept of the single-particle motion is obtained knock-out reaction as well as the stripping reaction. The development for electron scattering makes it feasible to perform the knock-out
T. Hoshino et al. / Shell model calculation
470
reaction
(e, e’p) in the quasi-elastic
probes *“). The spectroscopic defined
region
strength
more precisely
than that using hadronic
for the single-particle
state (Y(= n, Z,j, m,) is
as
where
(3.7) where [J] = m. The quantity SL” is called the shell-model spectroscopic factor. The triple-bar matrix element is a reduced matrix element both for spin and isospin space. Surprisingly, the spectroscopic factors observed in the (e, e’p) experiment with satisfactory statistics are reported to be smaller than the values obtained by many-body theories. The spectroscopic strength is about 0.5, [ref. r4)], while the theoretical value is 1.0 in the pure shell-model limit. The spectroscopic factors calculated by using the present wave functions are shown in fig. 5 for the 160(e, e’p)15N reaction. As seen from fig. 5, the strength is concentrated in the lowest excited i- and sP states. The experimental spectroscopic strength derived from the distorted wave impulse approximation analysis 14) is 0.45-0.59 for fP and 0.53-0.58 for G- depending on the optical potential parameters, while the present calculated
0.10 0.08 0.06 0.04 0.02 0.00 0.00 0.06 0.04 0.02 0.00
0
10
20
30
40
Ex [MeVl Fig. 5. Spectroscopic
strength distributions in the 160(e , ~‘P)‘~N reactions. $- states in the lower case, and $- states in the upper
The final states of “N case.
are
471
T. Hoshino ef al. / Shell model calcularion
values
are 0.935 for $l and 0.936 for $r summing
lp. The momentum
distribution
defined
up the contributions
from Op and
as
where 4,,,,(p) is the Fourier transform of the single-particle wave function, is shown in fig. 6 for the case of the i; and $; states. In the calculation, we introduced a normalization factor A,; defined as
A/,
CN’,:= 7eNeUnp
(3.9)
9
n =,,
in order to obtain the same value of the whole volume integral one. The value r], is an absorption factor defined in ref. 14).
-300
-250
-200
-150
Momentum
-100
-50
as the experimental
0
pe [MeV/c]
Fig. 6. The momentum distributions of the single-particle states in IhO. The solid curve (circles) represents the calculated (experimental) distribution of P,,~, while the dashed curve (open circles) shows the calculated (experimental) one of pi,?. The experimental data are taken from ref. 14).
Recently, the momentum distributions of 4’ states in “B have been experimentally measured in NIKHEF-K “). Their analysis takes into account the (0+2)8w model space for the “C ground state, but only the 1h w model space for the $’ states in “B. This analysis is doubtful because of the adopted model space. Namely, if the (0+ 2) hw model space for the “C nucleus is taken, the model space for the t’ states in “B should include (1+3)hw states. As a schematic example, we calculate the spectroscopic factor of the i- state in 15N taking either a O&J or a (0+2)hw model space for “N and the (0+2)hw model space for IhO without the HF condition. The
472
spectroscopic
T. ~oshino
factor
et al. / Shell model c~~cul~t~~n
of the state turns
out to be 0.685 in the Oirw model
15N, while it is 0.972 when the fuli (0+2)&w and 160. One can see that the inconsistent and final states decreases
model space is assumed
treatment
the spectroscopic
space for
both for “N
of the model space in the initial
strength
significantly.
In our calculation, the ground state of “C is calculated within the Ohw model space and the i” states in “B within the lfio model space. Our results show that
0.0 0
1
L
I
10
20
30
.
1.1
,
40 %
50
60
I
I
70
60
MN
Fig. 7. Spectroscopic strength distributions in the “C(e, e’p)“B reaction. The final states are 4” states of “B. The ground state of “C is calculated within the Ohw model space.
101
I 0
100 Momentum
150
I
I
I
I
I 50
200
250
pn [MeV/c]
Fig. 8. The momentum distribution of the s,,, state of “C. The final state of “B is located experimentally at 6.79 MeV. The ground state of “C is computed with the Ohw model space.
T. Hoshino et al. / Shell model calculation
the centroid moment
energy
is 48.0 MeV in the missing
is 4.92 MeV (r,,,, distribution
distribution
= 11.59 MeV). The calculated
of the first +’ state in “B is 0.0190. The strength the momentum
energy
without
473
spectroscopic
distribution
the normalization
and the second
is shown
constant
strength
in fig. 7 and
A in fig. 8, together
with the experimental distribution ‘) of the 5’ state at 6. 79 MeV. Since the in “Cg_ is deeply bound, the spectroscopic strengths are distributed in a wide energy range in contrast to the case of 160(er e’p)r’N. The absolute the calculated momentum distribution is larger than the experimental one, the shapes resemble each other.
s,,~ state relatively value of although
4. Summary and discussion We have reported the effect of the HF condition on the shell-model calculations of I60 and ‘*C. The model space consists of configurations within the (0+ 2)hw excitation energy. We used the M3Y interaction as the effective two-body interaction in the calculations of the single-particle energies and the two-body matrix elements. The violation effect in the this effect by effect in the
of the HF condition is caused mainly by the neglect of the Brueckner G-matrix such as shown in fig. Id. We take introducing the force 6V in eq. (2.9) which simulates the Brueckner G-matrix (or equivalently the rearrangement
higher-order into account higher-order effect of the
density-dependent force). There is no unique way to select the force 6V in the effective interaction. In the present study, the effective potential (2.9) is good enough to avoid the mixture of large lp-lh components in the ground state due to the violation of the HF condition while small physical lp-lh components still exist through the coupling of 2p-2h states. The lp-lh states are contained only less than 1% in the wave function of the ground
state in 160. The importance
of the HF condition
was already
in refs. 3Z25). We performed the shell-model calculations with the b-parameter (2.5). This prescription seems reasonable because one reproduces observed
ground-state
binding
energies
of “C and I60 and the density
suggested
fixed by eq. very well the distribution
of I60 together with reasonable excitation energies and strength distributions of the giant resonances. The charge distribution shown in fig. 9 matches well the experimental one and produces the calculated charge r.m.s. radius r = 2.68 fm which is in reasonable agreement with the experimental one r,(exp) = 2.72 fm [ref. ““)I. The difference can be improved by changing the value of the b-parameter. The calculated excitation energy of the quadrupole giant resonance is 29.9 MeV in average, while that of the monopole resonance is 23.9 MeV. These values are quite reasonable in comparison with the experimental excitation energies. The widths of the quadrupole and the monopole giant resonances are calculated to be 6.86 MeV and 4.62 MeV, respectively. In the quadrupole case, the experimental one is reported to be 7.5 MeV. This result suggests that the main origin of the width of the quadrupole
T. Hoshino et al. / Shell model calculation
474
0.06
0.04
0.02
0.00
0
1
2
3 =
Fig. 9. The charge with the finite-size
4
5
6
7
r.fml
distribution of the ground state of 160. The solid curve is the charge density calculated effect by using in eq. (2.6). The dashed curve is the proton density. The experimental distribution is shown by the dotted line and shadowed area *‘).
giant resonances can be explained as a damping width due to the coupling to 2p-2h states. One of the advantages of this calculation with the HF condition is the inclusion of the physical diagram shown in fig. lc excluding the unphysical diagram fig. lb where the hole line is a spectator. The second advantage is that the adopted oscillator b-parameter is close to the value which is commonly used in the literature. In ref. ‘), the b-parameter
used is 15% larger than our value in order to reproduce
the empirical
r.m.s. radius because of the large mixing of lp-lh amplitudes in the ground state. It is expected that a larger shell-model space brings in a reduction of the spectroscopic strength and momentum distribution. We, however, stress that one should enlarge consistently the model space of the initial and final states. As was shown in the case of “N, the inconsistent treatment of the model space gives a superficial decrease of the spectroscopic factors. The spectroscopic strengths obtained in the present calculations are 0.935 for the ground state in ‘*N(i-) and 0.936 for the first excited state in ‘5N($). We should notice that these spectroscopic factors do not correspond to the occupation probabilities of OP,,~ and Op,,, orbits in “0, which are 0.969 and 0.983, respectively. Inclusion of the ground-state correlation doesn’t considerably affect the spectroscopic factor, which suggests that single-particle or single-hole structure still remains even if one takes into account the ground state correlation. A similar result has been obtained in a different truncated space by Zuker, Buck and McGrory “), who of neighboring nuclei around took OP,P, Od,,, and 1~~ orbits for the calculation I60 allowing up to 4p-4h excitations as a model space.
475
T. Hoshino et al. / Shell model calculation
These spectroscopic the origin different scopic
factors
of this discrepancy, set of optical
strength
are much larger than the experimental it is very desirable
model parameters
from the experimental
to reanalyze
the data by using a
which are used for extracting cross section.
ones. To see the spectro-
The peak of the momentum
distribution is different from the experimental one as shown also an interesting open problem for future calculations.
in fig. 6. This point is
We wish to thank K. Yazaki and K. Shimizu for valuable discussions and W. Bentz for his critical reading of the manuscript. We are very grateful to B.A. Brown who offered us the shell model code and useful advice on the computation. We also thank the members of the Theoretical Nuclear Physics Group in the University of Tokyo for their discussions. The calculations have been performed Laboratory, University of Tokyo.
by using VAX-l l/780 at the Meson Science
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