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A simple microscopic approach for the study of giant resonances
Nuclear Physics ONorth-Holland
A441 (1985) 291-315 Publishing Company
A SIMPLE MICROSCOPIC APPROACH FOR THE STUDY OF GIANT RESONANCES D.E. MEDJADI L.aboratoire de Physique Thkorique*, Bordeaux, France and Ecole Not-male Supt+ieure***, Algiers, Algeria and P. QUENTIN Laboratoire de Physique Theorique**, Bordeaux, France and Theoretical Division, Los Alamos**‘,
USA
Received 9 July 1984 (Revised 5 February 1985) Abstract: A simple approach for the description of monopole and quadrupole giant resonances of light nuclei in the framework of the generator coordinate method is proposed. Various effective forces, including recent Skyrme force parametrizations, have been used. Excitation energies, transition matrix elements and densities, sum rules and collective wave functions are presented and discussed.
1. Introduction A fully microscopic treatment of giant resonances is currently available in terms of self-consistent RPA calculations using effective nucleon-nucleon interactions ‘). Another possible approach consists in performing calculations within the framework of the generator coordinate method*) whose connection with RPA was established long ago 3), in particular in the gaussian-overlap approximation case4). Calculations along these lines using the most successful effective interactions to date are not easy to perform. They are indeed almost restricted, so far, to light nuclei within a rather limited variational space for single-particle wave functions 5-7). Alternative yet equivalent approaches (for the giant monopole mode only) make use of the hyperspherical harmonic approximation8-‘0). * Equipe de Recherche associ& au CNRS. Postal address: Laboratoire de Physique Theorique, Universitt de Bordeaux-I, Rue du Solarium, Gradignan, France. ** Postal address: Ecole Normale Superieure, Vieux Kouba (Algiers), Algeria. *** Postal address: Group T9, MS B279, LANL, Los Alamos, NM 87545, USA. * Permanent address. 291
33170
D.E. Medjadi, P. Quentin / Microscopic approach
292
It is the aim of this paper to present a nonexpansive yet accurate enough to investigate the giant-resonance properties of current effective-interaction
approach parame-
trizations, especially those corresponding to recently proposed Skyrme-type forces 11-13). 0 ur approach, which corresponds to a simple version of generator coordinate calculations, is presented in sect. 2, whereas calculational details are discussed in sect. 3. Sect. 4 is devoted to a survey of our results, and some conclusions are drawn in sect. 5.
2. Description of the approach A discretized
generator
coordinate
ansatz for the wave function
(1)
IV = i CA@,) I=1
is used,
where
single-particle
lQ1) is a normalized wave functions ;
Slater
determinant
(A, Z). The normalization c;JVJJ,
built
from
normalized
of l!P) imposes
= 1
(2)
I,J=l
with
.&=det((X,4p,J)). For the model wave function (l), the Griffin-Hill-Wheeler following generalized eigenvalue equation form:
(3) equation
assumes
i%'C=EJ-C, where C is the column defined by
vector built
with the C, coefficients
the
(4) and the matrix
Xis
As for the single-particle states IX, Z), we choose Cartesian harmonic-oscillator states with different oscillator lengths for all major shells (i.e. a priori three parameters for each shell, one in each direction). It is worth noting (see appendix A) that two-body matrix elements of the Skyrme type or of the gaussian type may be analytically evaluated between harmonic-oscillator states corresponding to different oscillator lengths (with the exception, however, of the density-dependent delta force whenever this dependence is not linear). In this study the set of determinantal wave functions { @,} has been restricted to axially-symmetric states corresponding to given values of the rms and quadrupole moments. We first determine the determinant @,, corresponding in our simple particle variational space to the lowest (!&,~HI@,) energy. This Hartree-Fock-type ground-state solution is thus defined by an rms radius R, and a quadrupole moment Q,. In order to describe only monopole vibrations around QO, we use a set of
D. E. Mec$adi, P. Quenlin / Microscopic approach
optimized
@,‘s having
variable
rms radii and a quadrupole
293
moment
value fixed to
QO. For the quadrupole
mode now, the set to be considered
should
take into account
the geometrical increase of the .rms radius R when increasing the quadrupole moment Q. This is taken into account by linking Q and R through the following relations holding for an ellipsoidal liquid drop of A particles: Q = 2r;A5’3q-2’3( r =
q* -
1))
@r,,A’/3qp1/3/q,
(6)
- 1.2 fm and q is a deformation where r0 is the usual liquid drop parameter parameter equal to the semi-axis ratio in the ellipsoidal case. It should be emphasized that the previous choices for the monopole and quadrupole modes, though quite natural and of widespread use, do not guarantee a priori that we will excite exact normal modes through them [see, however, in ref. 18) where evidence is given that this seems indeed to be so]. To check this point we have performed so-called coupled quadrupole-monopole calculations in which we have taken as our set of Slater determinants to be mixed, those corresponding to a mesh in the two-dimensional (R, Q) space around the equilibrium Hartree-Fock-type Slater determinant, allowing thus for our solution the freedom of possibly mixing what we decided to be monopole and quadrupole states. In order to approximately correct for the spurious center-of-mass motion we have subtracted from the effective hamiltonian H the center-of-mass kinetic energy leading thus to the intrinsic nuclear hamiltonian (with usual notation)
(7) However, two-body
when using the Skyrme effective forces which have been fitted without part of the center of mass correction we have instead made use of
the
Solving the generalized eigenvalue equation (4) yields a set of eigensolutions { ‘k,} corresponding to eigenvalues E,,. For such states one may define usual moments for a given excitation operator 0 by m,(O)
= c lPAOl%)12(~?lrl#O
W,
(9)
where k is integer (positive, negative or zero) and ]‘k‘,) the calculated ground state. Upon assuming that this restricted set { ‘k;l} is a sufficient approximation for spanning the subspace of states that are connected to I*,,) through 0, one may evaluate the m,(O) moments through the usual sum rules. For instance, one
D. E. Medjadi, P. Quentin / Microscopic approach
294
considers
the energy-weighted
sum rule
m,(O) or the inverse
vf~a1I%~~
= H%l[O7
energy-weighted
(10)
sum rule m_,(O>=P(O).
In
eq. (11)
evaluated
P(0)
is the static
polarisability
(II) associated
with
0
which
may
be
as
where I q,,(p))
is the ground-state
solution
of eq. (4) for the constrained
H - ~0, p being a Lagrange multiplier. Finally one may also mention the m,(O) ma(O)
sum rule:
wolo21%kb).
=
hamiltonian
(13)
3. Details of the calculations 3.1. EFFECTIVE NUCLEON-NUCLEON
INTERACTIONS
Two forces have been used. On the one hand, the density-independent force Bl of Brink and Boeker 14) (two gaussians with a Wigner and Majorana exchange-operator mixture) which has been fitted to the saturation properties of a-particle and infinite nuclear matter. With this force no spin-orbit force has been considered in the calculations. On the other hand, we have used density-dependent Skyrme forces either within the SIII”) or the SkM”) and modified SkM13) parametrizations. Both the SIII and modified SkM forces reproduce very well saturation properties over the whole chart of nuclides. The major difference between the SIII and SkM or modified SkM is related to the exponent in the density-dependent delta force which is 1 for SIII and i for the two others. The latter leads to an incompressibility modulus which is consistent with measured isoscalar EO giant-resonance properties 15). 3.2. COULOMB INTERACTION
To compute matrix elements of the Coulomb interaction determinants, one may take advantage 16) of the identity 1 _= s
-2 l-1 ?r
between
two Slater
me-“2,2’4 0
2
’
P
and then reduce Coulomb matrix elements as integrals of gaussian matrix elements with respect to the range. When oscillator constants differ from shell to shell this leads to analytic formulae involving hypergeometric functions. With the exception of
D. E. Medjadi, P. Quentin / Microscopic approach
the
1s and
functions,
lp their
shells
where
evaluation
We have therefore preferred tion. We have approximated
these functions
implies
algorithms
295
can be cast into
simple
which
slowly convergent.
are rather
to evaluate the Coulomb contribution the direct contribution as /Jp~~)(r)p:l;)(r’)v*lr-
elementary
by space integra-
r’]d3rd3r’,
(15)
where ~$9 is the proton density transition corresponding to the Slater determinants @I and QJ (see appendix B). For the exchange part we have extended the current Slater approximation [which has been shown “) for a single Slater determinant to be correct up to - 5-7% in light nuclei] as
3e2~)1’3J[p~~)(1)]4’3d3~. 4N,:/lj
(@Ilv(z~+DJ)= -
to the usual proton density, Eq. (15) is exact for I = J, the density pfy corresponding whereas in this case eq. (16) reduces to the standard Slater approximation. The validity of the approximate equations (15) and (16) versus exact contributions (even for the exchange terms) is substantiated in appendix A.
3.3. CONVERGENCE
AND OTHER NUMERICAL PROBLEMS
As discussed in ref.‘), we have rejected from our basis space the subspace spanned by all eigenstates of the norm matrix corresponding to eigenvalues less than 10P3. Indeed it has appeared that a blind numerical solution of the generalized eigenvalue eq. (4) yielded “spurious” states easily recognizable by their large overlap with eigenstates
of the norm matrix
having
almost vanishing
eigenvalues.
Moreover
such “spurious” states turned out to be almost completely decoupled (in terms of the smatrix) with the physical states. It is not clear to us if these “spurious” states are connected with symmetry-breaking, such as translational invariance for instance. Indeed
matrix
elements
of the
center-of-mass
coordinates
calculated
for
these
“spurious” states have been found similar to those calculated for “physical” states. In the present work we have generally included up to 5 (7) Slater determinants for describing uncoupled vibration modes in 160 (40Ca). In the coupled case, 9 Slater determinants have been taken into account for both nuclei. The relevance of such a small basis has been checked by calculations of the monopole mode in 160 with the Skyrme SIII 11) and modified SkM13) forces including 5, 7, 9, 11 and 13 Slater determinants. The 5-determinant results agree with the ll- and 13-determinant results within (i) less than 70 keV for the energies of the three first eigenstates, (ii) less than 0.01 fm for the rms radii, and (iii) less than l-2% for the m_,, m,, m, sum rules.
D. E. Medjadi, P. Quentin / Microscopic approach
296
A similar
agreement
“spurious”
states is performed
have been norm
is found
thrown
matrix
having
between
(as stated
results above)
out by mere inspection
where
a clean
removal
of the
and the results where these states
of their overlap
with eigenstates
of the
almost zero eigenvalues.
4. Results In what follows we will present the results of calculations performed for the two pure monopole and quadruspherical 160 and 40Ca nuclei in the pure quadrupole, pole-monopole coupled cases. Two types of effective forces have been used: Brink and Boeker and Skyrme forces.
4.1. THE BRINK-BOEKER
FORCE CASE
Even though this density-independent force is not suitable for a correct description of a wide variety of nuclear properties, we have nevertheless performed extensive calculations with it for the sake of comparison with the calculations of refs. 5,6*8).Our 160 results are seen in fig. 1 to be very close to those obtained in the
C.B.A. -
a=123
-
Q-6.1
C.B.A.
R q2.92 Rd.10 -.
Q-8.2
Q=8.1 FL278
20-
7
cq N o/
Rz2.78
-Rz2.94
Rz2.57
-,Rz2.66
“! P
Rz2.5
E=105.7
-R.256 A
E=106.5
Ed055
- k2.66
B
E.106.5
E=105.7 E-1068 C
Fig. 1. Spectra of pure monopole (A), pure quadrupole (B) and monopole-quadrupole coupled (C) excitations for the 160 nucleus calculated with the Bl Brink-Boeker effective force. Our results are compared with those of ref.s) (S.Z.) and ref.5) (C.B.A.). Intrinsic quadrupole moments Q (in fm2) and rms radii R (in fm) are also given along with ground-state energies E (in MeV) and the transition matrix element p2 of the text (in fm2) between the ground and first excited states for the monopole vibration mode.
D.E. Medjadi, P. Quentin / Microscopic approach
291
more refined above-mentioned approaches. For the pure monopole case where our intrinsic solutions are directly comparable with the angular-momentum-projectedbefore-variation solutions of ref. 5), the agreement for the first 0” excitation energy is quite good. It is even more so when comparing with the hyperspherical formalism results of ref. ‘)_ Moreover, the transition matrix element of the operator p2 = Cf=“=,ri’ is about the same in all three approaches. For the quadrupole mode a direct comparison between our results and those of ref. 5, is not straightforward since our simple approach yields only intrinsic-state giant resonances. However, the two calculated spectra are in fair agreement up to the two-phonon states. When coupling quadrupole and monopole vibrational modes in the spherical 160 nucleus the one-phonon states are found to be roughly unaffected with respect to the uncoupled corresponding states as far as the p2 transition matrix elements and the energies are concerned. This is indeed not the case for higher excited states. In all situations where the structure of the nuclear states obtained in ref. 5, may be directly qualified in terms of simple intrinsic phonons, the agreement between monopole diagonal matrix elements R is found to be rather satisfactory. In the discussion of figs. 1 through 6 we will consistently mention not only the first but also the second excited monopole state. However, as we will see later in subsect. 4.3, the first excited state exhausts most of the m, sum rule, for instance. It should then be clear that by “second” monopole state we mean a kind of two-phonon state in a quasi-harmonic well in the monopole variable that we have chosen and loot a state that would carry a substantial amount of the monopole strength.
@AR
B.
s.2.
AC.
R~3.48
E=263.3
E--264.4
Fig. 2. Same as fig. 1, but for the pure monopole vibration mode in the 40Ca nucleus. compared with those of ref.‘) (B.), ref. ‘) (S.Z.) and ref.6) (A.C.).
Our results
are
D. E. Medjadi, P. Queniin / Microscopic approach
298
A similar study has been undertaken for the monopole vibrations of @Ca. In fig. 2, the calculated one-phonon state is found slightly too high in energy as compared with the results of refs. 6,8) [see also the RPA result of Blaizot ‘)I. For this nucleus also, the p2 transition matrix elements obtained in the various approaches are again in fair agreement.
4.2. THE SKYRME
FORCE
CASES
In view of the good results obtained in our simple approach as compared to more refined calculations using the Brink-Boeker force, we have extended our study to more predictive effective forces, namely the Skyrme effective forces, in the parametrization known as SIII, SkM and “modified SkM”11-i3). Our results for the uncoupled monopole and quadrupole modes are seen in fig. 3 to compare very well with the RPA results of Blaizot ‘), the hyperspherical formalism results of Navarro 9, and the generator coordinate results of Flocard and Vauthe~n 7)_The latter calculations do not incorporate the Coulomb interaction but include a number of Slater determinants which is larger than ours (143 instead of 5-9). The main difference between these calculations lies in the fact that we allow more flexibility for the variation of the Slater determinants. To be specific, for an axially-symmetric 160 solution, the choice of definite values of r and Q (see eqs. (6)) would determine completely the Slater determinant in the approach of ref. 7, whereas in our approach we are left with two “effective” variational parameters (i.e. four parameters bound by two conditions). The good reproduction of RPA results with a basis size as small as ours assesses the relevance of limiting the number of determinants while increas-
s
RJ.04 B.
F.V.
Cb23.8
Q=23.8 Ffa8l
z&o_-- FM.62
F. V.
F.V
B.
N.
Q41.5
_
ML 2 _
Rx2.83
%2.90 -
0=12.3
M3.1
3
I Ezl26.8
-Rd.73
-Rae2
Rs2.58
Rd.65 --
s2.69
M
Fk2.63 --
RG.62
Ed28.0
E=l2ZO
E=l2L5
E=lZCIO
151271
E=12zO
E-126.9
E=l2Xl
A
B
C
Fig. 3. Same as fig. 1, but for the SIII effective force. Our results are compared with those of ref. ‘) (B.), ref.‘) (F.V.) and ref.‘) (N.).
299
D. E. Medjadi, P. Quentin / Microscopic approach
1 E=33o.~
-
-
-
E=3412
EG33l.5
E-8
Fig. 4. Same as fig. 2, but for the SIII effective force. Our results are compared ref. ‘) (F.V.) and ref. 9, (N.).
( with those of ref. ‘) (B.),
ing the number of relevant variational parameters. More generally, adjusting more carefully each Slater determinant allows one to separate “true” correlation effects from a mere adjustment of the mean field to internal deformation constraints. Similar conclusions could be drawn for the 40Ca nucleus, as seen in fig. 4 for the monopole mode. Finally one may mention the agreement of our results with the adiabatic analysis of TDHF calculations 18) using the same force where the following resonance energies were found: 30.4 and 22.2 MeV for the monopole and quadrupole modes in 160. The above-discussed comparison of our results with those of other microscopic approaches is gratifying in so far as it demonstrates their relevance. Its interest however is merely theoretical for the monopole mode since it deals with an effective force (Skyrme SIII) where parameters have been fitted before knowing the constraints imposed on the incompressibility modulus’5) by the discovery of giant isoscalar monopole resonances 19). It is therefore particularly interesting to present the monopole and quadrupole vibrational modes of 160 and @Ca for effective forces belonging to the Skyrme-type parametrization, which are better conditioned in this respect. Such a force, the SkM force proposed in ref. 12), has been shown 13) to need a slight improvement in order to reproduce accurately the saturation properties and the fission barrier heights. In our opinion, the resulting force, the so-called modified SkM force (quoted here as SkM*), yields the best, to date, overall reproduction of nuclear properties in the P-stability valley for Skyrme-type effective forces. The
D. E. Megiadi, P. Quentin / Microscopic approach r
E ev)
SkM* R.33 0
SkM
Hyd.
SkM
SkM*
SkM*
SkM
Rz3.09
Rz3.03 -
Q=25.0
Q=23.2
0~24.8
Qz22.4 -
R.2.89 Qz12.1
Q-11.7
Q=13.0
_R=2.86 0~12.4
-R3.2.69
-R=2.65
--Rz2.69
R12.66
E1129.8
E.126.3
R&i. 04
l-l-
E=126.1
Ez129.8 A
E=126.1 8
E1130.0 C
Fig. 5. Same as fig. 1, but for the SkM and SkM* effective forces. Our results are compared the hydrodynamical approach (Hyd.) of ref.l’).
results obtained for 160 are reported compared in the SkM force case with results are comparable for the SkM within simple sum-rule approaches20) pressibility modulus and the effective
with those of
in fig. 5. For the monopole mode, they are the hydrodynamical approach of ref. “). The and SkM* forces, which can be understood as related to the fact that both the incommass are identical for the two forces (see an
extensive discussion of this point in subsect. 4.3). The only difference lies in the increase of the surface tension term of the liquid-drop energy formula reflecting itself in the calculated decrease of the binding energy (126.1 MeV for the SkM* force to be compared with the experimental value 127.6 MeV, the difference being due to the severe limitation of the variational space in the present calculation). As observed for the Brink-Boeker and SIII forces, the coupling of the monopole and quadrupole vibrational modes does not alter significantly in 160 the energy spectrum or wave functions of the ground and first excited states. On the contrary (see table 1 and also fig. 6), for both the SIII and SkM* forces, coupling the two collective (quadrupole and monopole modes) results in a significant change of the solutions in “aCa. The effective amount of mixing may be measured by the reduction of the transition matrix elements for the Q (p2) operator between the ground and the first excited state of the quadrupole (monopole) type. While this reduction is not apparent for 160, it is found to be - 20% for both quadrupole and monopole matrix elements in “eCa as seen in table 1. This trend seems consistent with recent experimental evidence21). Further discussions about the change in the coupling pattern between 160 and %a will be made in subsect. 4.3.
D. E. Medjadi, P. Quentin / Microscopic approach
301
TABLE 1
Comparison between coupled and uncoupled solutions in r6 0 and a Ca nuclei calculated with the SkM* effective force I60
Dominant mode quadrupole
‘first [Meal
QfirstPm*1 monopole
(first]QP) [fm*] ‘first PW
Qfirst[fm*I (first]p*P)
[fm*]
‘+OCa
uncoupled
coupled
uncoupled
coupled
19.9 12.1 30.1 24.2 2.90 18.9
20.1 13.0 30.0 24.1 2.89 18.9
17.6 12.7 68.5 18.7 3.48 38.9
16.0 - 6.3 56.7 21.6 3.49 32.1
The first-excited-state energy E,,, of quadrupole (monopole) character is given along with the quadrupole (monopole) moment of this excited state. The corresponding transition matrix element to the ground state for the quadrupole (monopole) operator is also shown,
Even though, as we have discussed above, the experimental quadrupole strength distribution appears rather fragmented’*), the current estimation of its centroid energy at - 22 MeV [ref.23)] (- 18 MeV [ref.24)]) for the 160 (@Ca) nucleus is in fair agreement with the SkM* and SkM results. It should be noted that for the spherical nuclei under consideration here, it is not necessary to take into account explicitly axially-asymmetric modes, that are equally described by symmetric modes 25).
E (Me
r
40 I
20-
o-
-
Rz3.48
Qz22.7
Rz3.48
0~12.7
Rd.42
L-EL336.0 A
R-3.48 Qz12.1
fk3.49
Rz3.42 E=3363
B
0~3.16
Rz3.43 Ez336.5
C
Fig. 6. Same as fig. 5, but for the @Ca nucleus with the SkM* effective force.
D. E. Medjadi, F. Quentin / Microscopic approuch
302
4.3. SUM RULES
From the solution of the generalized eigenvalue equation (4), one may evaluate the moments m,(O) of any operator 0 upon assuming that the finite set of calculated eigenstates span reasonably well the space of physical states connected by 0 with the ground state. Indeed the existence of sum rules like those given in eqs. (lo), (11) and (13) provide a good test for the validity of such an assumptiont. Nevertheless, before summing the different contributions to the moments m,(O) one should reject the spurious states as described in subsect. 3.3. In table 2 we have reported the values of the three moments m-t, m0 and ml obtained with the operators C$ tri2 and E$. lqi in the monopole-quadrupole coupled calculations when using the Bl, SIII, SkM and SkM* forces for the 160 nucleus [our results for the SIII force and the moments m_, and m, are consistent with those of ref. 26)]. Table 3 displays the same moments for the ‘%‘a nucleus with the SkM* force. These moments are compared with the expectation values of eqs. (10) and (13) for m, and m,,. The moments m_, are compared with the polarisability resulting from the solution of eq. (4) where the hamiltonian H is replaced by the constrained TABLE 2
Moments of the strength function for the monopole ( p2) and the quadrupole (Q) isoscalar operators in the I60 nucleus calculated for the Bl, SIII, SkM and SkM* forces defined in the text
Force Bl
Operator P2
Q SkM*
P’
Q SkM
P” Q
SIXI
P2 Q
m-1 [MeV-’ . fm4] 15.4 (15.5) 23.6 (23.6) 15.0 (15.0) 44.8 (44.8) 14.1 (14.1) 41.8 (41.8) 9.1 (9.1) 37.3 (37.3)
[Me? 356 (374) 633 (661) 367 (370) 902 (904) 352 (355) 865 (870) 285 (292) 808 (812)
bz ]
0.844 (0.842) 1.70 (1.68) 0.913 (0.900) 1.81 (1.80) 0.892 (0.902) 1.79 (1.81) 0.912 (0.861) 1.75 (1.72)
As an alternative length unit we have used the bi12 for ml moments. In parentheses we have reported on the lower line the expectation values (or polar&abilities for m-i) whose equality with corresponding moments constitutes a sum rule.
t The demonstration of the theorem energy-weighted sum rule) is apparently
of appendix C in ref.‘) (concerning not applicable to our case.
the exact exhaustion
of the
D.E. Medjadi, P. Quentin / Microscopic approach
303
TABLE 3 Same as table 2, but for the a Ca nucleus
Operator
m-3 [MeV3 fm4]
P2
0.218
Q
0.934
m-1 [MeV-’ . fm4] 78.3 (78.3) 271 (271)
and the SkM* force
[Me? 0.156 (0.179) 0.480 (0.525)
b*]
3.27 (3.80) 8.86 (7.59)
[Met’. 74.7 176
b’]
[Me:‘.
b’]
1984 3943
hamiltonian H - ~0. It has been found that the quadrupole (monopole) moments are vanishing when computed with excited states assumed to represent pure monopole (quadrupole) vibrations. This result indicates that our variational calculations in the uncoupled monopole or quadrupole cases may indeed be interpreted in terms of pure monopole or quadrupole vibrations. On the other hand, the agreement between moments and the corresponding expectation values (or polarisabilities) is indeed excellent. It is to be noted that in most of the calculations, the m, sum rule is almost saturated with the first excited (one phonon) state. For instance in 160 the monopole m. moment is due for - 97% to the transition between the one-phonon and the ground states with the SIII force as well as with the SkM* force. Taking stock of the excellent fulfilment of the sum rules in the m _ 1, m,, m, cases, we have also computed the moments m_3, m,, m3, without checking them against sum-rule results. An example of such results is given in table 3 for the 40Ca nucleus with the SkM* Let us define
force. the energies
E,:
Ek= imk/mk-2. According to ref.20) following conditions:
such
energies
satisfy
(due
07) to the
Schwarz
inequality)
the
We will compare now such energies with those calculated in our generalized eigenvalue problem. The energies E3, E, and E_, have been displayed in table 4 for the monopole mode, calculated with both the SIII and SkM* forces, in 160 and 40Ca. The first-excited-state eigenenergy is denoted by E,. One sees that for all cases under study in this table, there is a spreading of the energies E, which corresponds to a rather large distribution of the strength function for the p2 operator, which has also been observed in self-consistent RPA calculations using same or similar effective forces27). Indeed knowledge of the m,, m, and m2 moments yields the actual variance u of this distribution (equal in the gaussian case to - 0.43 times the
D. E. Medjadi, P. Quentin / Microscopic approach
304
TABLET First-excited-state
energies of the monopole
type in l6 0 and 4o Ca calculated effective forces
with the SIII and SkM*
SkM*
R E, (uncoupled) E, (coupled) EB
2.63 30.9 30.1 37.1 34.2 31.7 31.2
@pprox.)
E3
El E -1
3.43 27.7 27.6 28.6 31.2 28.6 28.2
160
40Ca
2.69 24.2 24.1 29.9 27.6 24.7 24.4
3.43 18.7 21.6 23.4 24.6 20.4 19.0
The energies E, (in MeV) obtained in both coupled and uncoupled calculations are compared with an approximate value calculated with the ground-state radius R (in fm). Energies Ek (with k = - 1,1,3) defined in the text are also reported.
full width at half maximum).
One finds 20)
+p,‘. Again through
due to the Schwarz the moments
m_,,
inequality
one may also provide
09)
an upper
bound
for u
m, and m3 by2’)
(20) For the SkM* calculations we have found u (a,,) equal to 5.1 MeV (6.1 MeV) in 160 and 6.4 MeV (6.9 MeV) in @Ca. The observed increase in the width when going from 160 to “eCa is consistent with our previous discussion concerning the coupling of quadrupole and monopole modes. The energy E, of our solution is closer in all cases to E, or E _ 1 than to E,. This can be interpreted in terms of a collective path [see e.g. ref. 28)] for our solution that is closer to a constrained Hartree-Fock path than to a scaling path. Indeed it has been shown in the harmonic limit that, whereas E, corresponded to the collective energy obtained by a scale transformation of a Hartree-Fock equilibrium solution 29), for a family E _1 in turn was obtained by performing adiabatic TDHF calculations of Slater determinants obtained in Hartree-Fock calculation under a constraint on the collective operator (here p2) [refs. 30,31)]. On the other hand, it is known32) that our generator coordinate method should provide an upper bound for E _1 in the harmonic limit. For such a comparison, however, one should use the uncoupled
D. E. Mea)&,
P. Quentin / Microscopic approach
305
generator coordinate (i.e. pure monopole) solutions. As seen in table 4, our values of E, (uncoupled) fail to fulfill this condition by less than 0.5 MeV. This may be due to either the presence of anharmonicities or to small residual uncertainties in our calculations (for instance, when evaluating strength-function moments) due to the very small number of Slater determinants being mixed. In ref. 20) the energy E, for the breathing mode is approximated by
(21) where m is the nucleonic mass, R the ground-state radius and KA a universal value of the incompressibility modulus in finite nuclei, given roughly (in MeV) as a function of the nuclear-matter incompressibility modulus K,,,, by KA = K,.,.-
630
+ Y>
(22)
( y being the power of the density dependence of the effective force). The result given in eq. (21) generalizes the result of Pandharipande3’) using the nuclear-matter K,,,, modulus. From eq. (22) and since K,.,, depends also linearly34,35) on the positive quantity y, one sees that the approximate value of E, given by eq. (21) varies roughly as the square root of an increasing linear function of y. We have numerically evaluated the approximate breathing-mode energies E, discussed above, taking the ground-state radius R as given in our calculations. The KA value resulting from eq. (22) is 229.4 MeV (155.9 MeV) for the SIII (SkM*) force, since the K,,,. values are 355.4 MeV [ref.“)] and 216.7 MeV [ref.13)] for the two forces considered. As seen in table 4, the approximate values E, are found far higher than the exact ones. This is partly due to the already discussed dispersion of the strength function. However, these values are found also rather far from the energies E, which they are supposed to approximate. This discrepancy is mostly related to the fact that eq. (22) for K, is only an average (over the number of nucleons A) value, overestimating2’) the actual value of m3 by about 2% in 40Ca and significantly more in 160. Let us consider now the results concerning the quadrupole mode displayed in table 5. The different E, (k = -1,1,3) energies in the 160 case are very close to each other. Indeed in the calculation using the SkM* force, the width u (a,,) is found equal to 0.6 MeV (0.9 MeV). Our calculated widths are consistent with the results of the RPA calculations of ref. 27). The dispersion of the strength function is larger in 40Ca (with SkM* force u = 5.1 MeV and amax = 5.4 MeV) which is no surprise in view of the already discussed coupling between monopole and quadrupole modes. As was the case for the monopole mode, our generator coordinate solutions correspond rather well indeed to a constrained Hartree-Fock adiabatic path. Adiabatic TDHF calculations have been performed36) according to the method presented in ref. 28) which is free from convergence problems in the
306
D. E. Medjadi, P. Quentin / Microscopic approach TABLE 5 Same as table 4, but for quadrupole excited-state energies SkM’
EQ (uncoupled) E. (coupled) EQ (approx.) E3
El E-1
20.9 21.6 26.4 21.8 21.7 21.6
19.4 19.1 19.5 20.2 19.3 19.2
160
@Ca
19.9 20.1 25.9 20.2 20.1 20.1
17.6 16.0 19.1 21.1 18.1 17.0
particle-hole space, contrary to the approach of ref. 30). In the harmonic limit these calculations using the SIII force yield a value of - 21 MeV for the first-excited-state energy E, which is very close to what we actually obtain here. The inequality between our generator-coordinate-method energies E, and the energy E_ 1 (in the uncoupled case) is no more fulfilled here that it was in the monopole case, probably for the same reasons. Suzuki 37) using the so-called self-consistent condition of Mottelson 38) proposed, years ago, that the quadrupole resonance energy Eq should be given for a zero-range force by E,=
&ho.
(23)
This formula has been further improved by incorporating finite-range alently velocity de endence) effects in the effective force by multiplying
(or equivthe r.h.s. of
eq. (23) by i--) m/m*, where m* is the effective mass in nuclear matter and m the nucleonic mass 20,39). A s was the case for the pocket formula of eq. (21), this approximation for EQ is in fact an approximation to E,. Taking the standard value for tto [ref.@)], Ao = 41A-‘/3
MeV 9
(24
mass values in unit of m as 0.76 [ref. ‘l)] (0.79 [ref. 13)]) for the SIII (SkM*) force, one gets as estimates of EQ, according to eq. (23), the values listed in table 5. They are found indeed to be reasonable estimates for 40Ca but rather poor for a nucleus as light as 160 . The latter may be understood by the fact that for very light nuclei. the liquid-drop-type concepts used to derive eq. (23) are of course partially inappropriate. and the effective
4.4. TRANSITION
DENSITIES
As shown in appendix B, from the solution of the generalized eigenvalues eq. (4), one may compute the transition density pn(rl, r[) between the ground and n th
D. E. Medjadi, P. Quentin / Microscopic approach
excited
307
states as (25)
Hydrodynamical approximations have been proposed for the diagonal part of the transition density between the ground and first excited states. For the quadrupole mode, Tassie41)
has proposed PlW
where p(r) symmetrical that
a ‘r,(i)d,
Q(r) 3
is the usual one-body density for the ground state of the spherically nucleus. For the monopole mode, Werntz and ijberall 42) have assumed
p,(r)
a 3p(r)
In eqs. (26) and (27) the proportionality
+Z$+.
constant
(27) has been chosen in such a way that
/pl(rW(r)d3r = WP>~
(28)
r:Xlf’ 3: *I
2-
l-
o-
) __/’
-1 L
3
Fig. 7. Transition density between the ground and first excited state for the monopole vibration mode in I60 calculated with the SkM* force. The dashed lines correspond to the results obtained within the Wemtz-ijberall approximation4*).
D.E. Medjadi, P. Quentin / Microscopic approach
(f mV3)
Fig. 8. Same as fig. 7, but for the quadrupole vibration mode. The transition density has been evaluated for the angle 0 = fr in Y,,(e, cp), see eq. (26). The dashed line corresponds to the results obtained within the Tassie approximation41).
As seen in figs. 7 and 8, both approximations reproduce fairly well the results of our microscopic calculations. This is, as well known, related to the fact that the axial quadrupole and monopole generalized sum rule of Kao and Fallieros43,44) is exhausted
by the one-phonon
4.5. COLLECTIVE
states.
WAVE FUNCTIONS
As suggested in ref.‘), we have considered as collective wave functions g the result of the application of the Ml/* operator (square root of the norm operator) on the generator coordinate weights f. Such wave functions are reported in fig. 9 for the quadrupole and monopole modes in 160 and in fig. 10 for the monopole modes in 40Ca. In all cases the Skyrme SkM* effective force has been used. One easily notices in these figures the standard 0-, l-, 2-phonon gross trends of such wave functions, attesting thus the quasi-harmonic character of such vibrational modes. This is of course only roughly true as we will discuss now. In figs. 1, 3, 5 and 6 we have assigned to pure quadrupole or monopole states, as well as to monopole-quadrupole coupled states of a quadrupole type, a quadrupole moment value which corresponds of course to an intrinsic quadrupole moment and not to a moment in the laboratory frame. As was found in earlier calculations5,‘) these values are not equal to zero as would have been expected for an ideal vibrator
309
D. E. Medjadi, P. Quentin / Microscopic approach
u.-l'"~/~
; !
’
t 2.5
;’
\
\
i’ !'
/
\\
\
.! '\ \
,/
I
i
\,
\ , ‘\ \
i
,
i
\
\
; 3.0 I
,. i
-40 ;
i
i
\
\
!
, r lfml
I i i i i i
\ \
\
\
\
\
\ \
'\ ' 'x_ _./
\ \ \
i i i
\
\
i
-0.2.-
i i '! -0.4_'\ -.
i i !'
.' ,' .I'
i i i
\ \
i
\
!
\
\ 0 ‘\
, ’ \
\
Q, Ifti
‘! \ ! I
’ 40
i
\ \ \
i i i i
\\
\
\
\ \ \ \
\ \
\ \
\
\
,.*,I
i i i i
'\
--.
Fig. 9. Collective wave functions g defined in the text for the monopole (1.h.s.) and quadrupole (r.h.s.) vibration modes in 160 calculated with the SkM* effective force. The ground, one-phonon and two-phonon wave functions correspond to the solid, dashed and dot-dashed lines. The units for g are case. In the monopole case, the arrow indicates fm-’ in the monopole case and fm-* m the quadrupole the location of the rms mass radius for the lowest uncorrelated wave function.
around a spherically-symmetric
equilibrium state. This trend by itself reflects the
influence of some anharmonicities present in our solutions and is also illustrated for instance in fig. 9 where one sees that the “quasi” 1-phonon and 2-phonon states are not exactly symmetrical with respect to Q = 0. 5. Conclusions By studying
two isoscalar
collective
modes for a variety of effective
forces
(including the best available within the Skyrme class of parametrizations) we have proven that our simple generator coordinate approach was accurate enough to yield, without too much numerical effort, a reasonable reproduction of more involved RPA or equivalent microscopic approaches. The calculations of ref. ‘) are in many respects very close to ours. The present approach, however, uses a much smaller number of Slater determinants for mixing by the generator coordinate integration. On the other hand, it allows much more flexibility for the adjustment of each Slater determinant and includes within a very good approximation the direct and exchange Coulomb contributions. As it stands, our approach which might easily be gener-
D.E. Medjadi, P. Quentin / Microscopic approach
310
Fig. 10. Same as fig. 9, but for the monopole
vibration
mode in 40Ca
alized to other modes, provides a useful tool in the perspective parameter adjustment of the nucleon-nucleon effective force. It is a pleasure to acknowledge fruitful Morand. One of us (D.E.M.) is also grateful
of pushing
forward
a
discussions with Y. Abgrall and B. to the Director of the Ecole Normale
Suptrieure for having authorized his leave of absence and the members of the Laboratoire de Physique warm hospitality extended to him. The fruitful stay Theoretical Division of the Los Alamos National
from Algiers and to B. Bonnier Theorique in Bordeaux for the of the other author (P.Q.) at the Laboratory during the second
semester of 1984 is also gratefully acknowledged. Finally, we thank the referee for having led us to substantially enlarge the discussion of the sum-rule properties of our solutions. Appendix A CALCULATION
Starting
OF ENERGY
MATRIX
from 3-dimensional
ELEMENTS
harmonic-oscillator
states
D. E. Me&d,
P. Quentin / Microscopic approach
311
and separating the relative and center-of-mass motion as r=&(5-r2),
R=(R,,R,,R,),
with
one may define generalized Moshinsky coefficients (CXstanding for x, y, z): (~n(wa)N,(D,)ln,l(w,l)n,*(wd,)> =
+q),(nl_q)!(na,_q),q [N,!na!na~!na~!11’2XC(-)q(n,_n a
.
.
a
.
where
In the above equation and throughout this appendix the summation over q is limited by the condition that all the arguments in the factorials should be 2 0. The overlap between two basis states is factorizable for each direction x, y and t. For one direction one gets (ml(w!)lm(w))
= (1 +(_l)“+“‘)
x
T22q[
:;+;l’“‘-‘“2[
[~41’4]~1~‘1]“2 [(w + wt)2”+“.+l]l/2 :;“:I’“““[
$yJq
Gaussian interaction
Its matrix elements are also factorized. One gets for the x-direction, for instance, ,+I(
wJ)
le-2x2/P2In&))
= (1 +(-l)“+“‘)
[ m!m~!]“2[ p/3,]1’4 2m+m’+l( p + p’ + 4/#)]
1’2 (m-q)/2
where p = mu/h.
312
D. E. Medjadi, P. Quentin / Microscopic approach
Skyrme interaction. Let us define some quantities (a standing for x, y, z): & = (P&$‘*, --P= (pxPvB,)1’3, A(m)=
(1 +(-l)~)(-l)ffl’z(~~)!(~!/2~+2)1’2.
(i) a-force. One gets
X4% (the
-
w(%+M~,+2)
notation cy,ff + 1, ff + 2 referring to circular permutations). force. One gets
(iii) e2& + S?
(n:(w:)n;(w;)n:(w:)I~*S(~)+S(r)V2ln,(w,)n,(w,)n,(w;)) a (iv) Spin-orbit force. One gets for the matrix elements of the x-component instance) of the L-operator
(for
(n:(w:)n~(~~)n:(w:)IL*ln,tw,)n,(cl?,)n,(w.)) =2i(~/~)X[(PIP,)“‘Jn;n,A(n;-l)a(n:)A(n,)A(n,-1) - (&~.J’*/G&:)~(%
- I)+,-
(u) reality-dependent S-force. Its cont~bution approximated very well? by
I)~(%)]~(~:)~(%). to the matrix element
.FIJ
is
where prJ is the transition density defined in appendix B and y the exponent of the density dependence of the &force. Coulomb interaction. We have used two methods, one exact for the lightest nuclei, the other approximate for heavier nuclei. The exact methods consists in reducing t As seen for a y = 1 Skyrme force (SIII) where exact calculations delta force are possible.
in terms of an equivalent
three-body
313
D. E. Medjadi, P. Quentin / Microscopic approach TABLE 6 Approximate (upper line) and exact (lower line) Coulomb interaction matrix elements between the Slater determinants @, and GJ
I=1
J=l
J=2
J=3
J=4
J=S
15.43 14.96
13.7 12.90 14.27 13.89
10.93 11.06 13.57 13.27 13.73 13.39
8.73 9.13 12.25 12.13 13.14 12.86 13.25 12.95
5.26 6.05 9.19 9.46 10.86 10.89 11.98 11.84 12.50 12.27
I=2 I=3 I=4 I=5
Coulomb matrix elements in term of gaussian matrix elements integrated over the range (see eq. (14) of the text). Obviously direct and exchange terms are treated on the same footing in this case. In the approximate method, the direct Coulomb interaction contributes to ;X;, as shown in eq. (15). For axially-symmetric wave functions this contribution is proportional to
with E(k)=l”/2J1-kZsinZBdb’. 0
The exchange
Coulomb
interaction
is approximated
as given in eq. (16). For 160
we have computed for a given physical Slater determinant superposition (corresponding to the mixing of 5 determinants yielding monopole vibrations with the Skyrme SIII force) both the exact and approximate Coulomb (including direct and exchange contributions) matrix elements ( @,lVclQJ). The results (see table 6) differ 2-3% for the diagonal term (where the only physical source of difference is the by Slater approximation) and by some tens of keV for the non-diagonal term. Appendix B TRANSITION
DENSITIES
The transition defined as
density
from the ground
state
I!P”) to the excited
state
19”)
is
314
D.E. Medjadi, P. Queniin / Microscopic approach
which gives in terms of the Slater determinants wave functions q;, cp,“)
@,, QJ (built with single-particle
Defining &,(I, J) as the minor associated to the element (m, n) of the matrix (q$lqi) the transition density pIJ may be written as
References 1) See e.g. Nguyen Van Giai and H. Sagawa, Nucl. Phys. A371 (1981) 1; J.P. Blaizot, Phys. Reports 64 (1980) 171, and references therein 2) D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1102; J.J. Griffin and J.A. Wheeler, Phys. Rev. 108 (1957) 311 3) B. Jancovici and D.H. Schiff, Nucl. Phys. 58 (1964) 678 4) D.M. Brink and A. Weiguny, Nucl. Phys. A120 (1968) 59, and references therein 5) E. Caurier, B. Bourotte-Bilwes and Y. AbgraJl, Phys. Lett. 44B (1973) 411 6) Y. Abgrall and E. Caurier, Phys. Lett. S6B (1975) 229 7) H. Flocard and D. Vautherin, Phys. Lett. 55B (1975) 259; Nucl. Phys. A264 (1976) 197 8) M. Sotona and J. Zofka, Phys. Lett. 57B (1975) 27 9) J. Navarro, Phys. Lett. 62B (1976) 22 10) K.V. Shiticova, Nucl. Phys. A331 (1979) 365 11) M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 12) H. Krivine, J. Treiner and 0. Bohigas, Nucl. Phys. A336 (1980) 155 13) J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. H%kansson, Nucl. Phys. A386 (1982) 79 14) D.M. Brink and E. Boeker, Nucl. Phys. A91 (1967) 1 15) J.P. Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys. A265 (1976) 315 16) P. Quentin, J. de Phys. 33 (1972) 457 17) C. Titin-Schnaider and P. Quentin, Phys. Lett. 49B (1974) 397 18) P. Bonche, H. Doubre and P. Quentin, Phys. Lett. 82B (1979) 5 19) N. Marty er a/., in Proc. Int. Symp. on highly excited states in nuclei, Julich (1975) p. 17; D.H. Youngblood et al., Phys. Rev. Lett. 39 (1977) 1188 20) 0. Bohigas, A.M. Lane and J. Martorell, Phys. Reports 51 (1979) 267 er al., Phys. Lett. 130B (1983) 9 21) S. Brandenburg Phys. Rev. Lett. 39 (1977) 922, and B. Morand, E. Caurier and B. Grammaticos, 22) Y. Abgrall, references therein 23) M.N. Harakeh et al., Nucl. Phys. A265 (1976) 189 24) F.E. Bertrand, Ann. Rev. Nucl. Sci. 26 (1976) 457 Nucl. Phys. A346 (1980) 431 25) Y. Abgrall, B. Morand, E. Caurier and B. Grammaticos, 26) K. Goeke and B. Castel, Phys. Rev. Cl9 (1979) 201 27) K.F. Liu and Nguyen Van Giai, Phys. Lett. 6SB (1976) 23 and P. Quentin, Phys. Rev. C21 (1980) 2060 28) M.J. Giannoni 29) J.M. Martorell, 0. Bohigas, S. Fallieros and A.M. Lane, Phys. Lett. 6OB (1976) 313 30) K. Goeke, Phys. Rev. Lett. 38 (1977) 212
D. E. Medjadi, P. Quentin / Microscopic approach 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44)
315
D. Vautherin, Phys. Lett. 69B (1977) 393 K. Goeke, A.M. Lane and J. Martorell, Nucl. Phys. A2% (1978) 109 V.R. Pandharipande, Phys. Lett. 31B (1970) 635 L. Zamick, Phys. Lett. 45B (1973) 313 A. Mekjian and L. Zamick, Phys. Reports 95 (1983) 324 M.J. Giannoni and P. Quentin, Phys. Rev. C21 (1980) 2076 T. Suzuki, Nucl. Phys. A217 (1973) 182 B.R. Mottelson, in The many-body problem, Les Houches lectures (Wiley, New York, 1958) p. 283 M. Golin and L. Zamick, Nucl. Phys. A249 (1975) 320 A. Bohr and B.R. Mottelson, Nuclear structure, vol. 1 (Benjamin, Reading, Mass., 1969) L.J. Tassie, Austr. J. Phys. 9 (1956) 407 C. Wemtz and H. Uberall, Phys. Rev. 149 (1966) 762 E.I. Kao and S. Fallieros, Phys. Rev. Lett. 25 (1970) 827 See e.g. P. Ring and P. Schuck, The nuclear many-body problem (Springer, New York, 1980) pp. 335-339
A441 (1985) 291-315 Publishing Company
A SIMPLE MICROSCOPIC APPROACH FOR THE STUDY OF GIANT RESONANCES D.E. MEDJADI L.aboratoire de Physique Thkorique*, Bordeaux, France and Ecole Not-male Supt+ieure***, Algiers, Algeria and P. QUENTIN Laboratoire de Physique Theorique**, Bordeaux, France and Theoretical Division, Los Alamos**‘,
USA
Received 9 July 1984 (Revised 5 February 1985) Abstract: A simple approach for the description of monopole and quadrupole giant resonances of light nuclei in the framework of the generator coordinate method is proposed. Various effective forces, including recent Skyrme force parametrizations, have been used. Excitation energies, transition matrix elements and densities, sum rules and collective wave functions are presented and discussed.
1. Introduction A fully microscopic treatment of giant resonances is currently available in terms of self-consistent RPA calculations using effective nucleon-nucleon interactions ‘). Another possible approach consists in performing calculations within the framework of the generator coordinate method*) whose connection with RPA was established long ago 3), in particular in the gaussian-overlap approximation case4). Calculations along these lines using the most successful effective interactions to date are not easy to perform. They are indeed almost restricted, so far, to light nuclei within a rather limited variational space for single-particle wave functions 5-7). Alternative yet equivalent approaches (for the giant monopole mode only) make use of the hyperspherical harmonic approximation8-‘0). * Equipe de Recherche associ& au CNRS. Postal address: Laboratoire de Physique Theorique, Universitt de Bordeaux-I, Rue du Solarium, Gradignan, France. ** Postal address: Ecole Normale Superieure, Vieux Kouba (Algiers), Algeria. *** Postal address: Group T9, MS B279, LANL, Los Alamos, NM 87545, USA. * Permanent address. 291
33170
D.E. Medjadi, P. Quentin / Microscopic approach
292
It is the aim of this paper to present a nonexpansive yet accurate enough to investigate the giant-resonance properties of current effective-interaction
approach parame-
trizations, especially those corresponding to recently proposed Skyrme-type forces 11-13). 0 ur approach, which corresponds to a simple version of generator coordinate calculations, is presented in sect. 2, whereas calculational details are discussed in sect. 3. Sect. 4 is devoted to a survey of our results, and some conclusions are drawn in sect. 5.
2. Description of the approach A discretized
generator
coordinate
ansatz for the wave function
(1)
IV = i CA@,) I=1
is used,
where
single-particle
lQ1) is a normalized wave functions ;
Slater
determinant
(A, Z). The normalization c;JVJJ,
built
from
normalized
of l!P) imposes
= 1
(2)
I,J=l
with
.&=det((X,4p,J)). For the model wave function (l), the Griffin-Hill-Wheeler following generalized eigenvalue equation form:
(3) equation
assumes
i%'C=EJ-C, where C is the column defined by
vector built
with the C, coefficients
the
(4) and the matrix
Xis
As for the single-particle states IX, Z), we choose Cartesian harmonic-oscillator states with different oscillator lengths for all major shells (i.e. a priori three parameters for each shell, one in each direction). It is worth noting (see appendix A) that two-body matrix elements of the Skyrme type or of the gaussian type may be analytically evaluated between harmonic-oscillator states corresponding to different oscillator lengths (with the exception, however, of the density-dependent delta force whenever this dependence is not linear). In this study the set of determinantal wave functions { @,} has been restricted to axially-symmetric states corresponding to given values of the rms and quadrupole moments. We first determine the determinant @,, corresponding in our simple particle variational space to the lowest (!&,~HI@,) energy. This Hartree-Fock-type ground-state solution is thus defined by an rms radius R, and a quadrupole moment Q,. In order to describe only monopole vibrations around QO, we use a set of
D. E. Mec$adi, P. Quenlin / Microscopic approach
optimized
@,‘s having
variable
rms radii and a quadrupole
293
moment
value fixed to
QO. For the quadrupole
mode now, the set to be considered
should
take into account
the geometrical increase of the .rms radius R when increasing the quadrupole moment Q. This is taken into account by linking Q and R through the following relations holding for an ellipsoidal liquid drop of A particles: Q = 2r;A5’3q-2’3( r =
q* -
1))
@r,,A’/3qp1/3/q,
(6)
- 1.2 fm and q is a deformation where r0 is the usual liquid drop parameter parameter equal to the semi-axis ratio in the ellipsoidal case. It should be emphasized that the previous choices for the monopole and quadrupole modes, though quite natural and of widespread use, do not guarantee a priori that we will excite exact normal modes through them [see, however, in ref. 18) where evidence is given that this seems indeed to be so]. To check this point we have performed so-called coupled quadrupole-monopole calculations in which we have taken as our set of Slater determinants to be mixed, those corresponding to a mesh in the two-dimensional (R, Q) space around the equilibrium Hartree-Fock-type Slater determinant, allowing thus for our solution the freedom of possibly mixing what we decided to be monopole and quadrupole states. In order to approximately correct for the spurious center-of-mass motion we have subtracted from the effective hamiltonian H the center-of-mass kinetic energy leading thus to the intrinsic nuclear hamiltonian (with usual notation)
(7) However, two-body
when using the Skyrme effective forces which have been fitted without part of the center of mass correction we have instead made use of
the
Solving the generalized eigenvalue equation (4) yields a set of eigensolutions { ‘k,} corresponding to eigenvalues E,,. For such states one may define usual moments for a given excitation operator 0 by m,(O)
= c lPAOl%)12(~?lrl#O
W,
(9)
where k is integer (positive, negative or zero) and ]‘k‘,) the calculated ground state. Upon assuming that this restricted set { ‘k;l} is a sufficient approximation for spanning the subspace of states that are connected to I*,,) through 0, one may evaluate the m,(O) moments through the usual sum rules. For instance, one
D. E. Medjadi, P. Quentin / Microscopic approach
294
considers
the energy-weighted
sum rule
m,(O) or the inverse
vf~a1I%~~
= H%l[O7
energy-weighted
(10)
sum rule m_,(O>=P(O).
In
eq. (11)
evaluated
P(0)
is the static
polarisability
(II) associated
with
0
which
may
be
as
where I q,,(p))
is the ground-state
solution
of eq. (4) for the constrained
H - ~0, p being a Lagrange multiplier. Finally one may also mention the m,(O) ma(O)
sum rule:
wolo21%kb).
=
hamiltonian
(13)
3. Details of the calculations 3.1. EFFECTIVE NUCLEON-NUCLEON
INTERACTIONS
Two forces have been used. On the one hand, the density-independent force Bl of Brink and Boeker 14) (two gaussians with a Wigner and Majorana exchange-operator mixture) which has been fitted to the saturation properties of a-particle and infinite nuclear matter. With this force no spin-orbit force has been considered in the calculations. On the other hand, we have used density-dependent Skyrme forces either within the SIII”) or the SkM”) and modified SkM13) parametrizations. Both the SIII and modified SkM forces reproduce very well saturation properties over the whole chart of nuclides. The major difference between the SIII and SkM or modified SkM is related to the exponent in the density-dependent delta force which is 1 for SIII and i for the two others. The latter leads to an incompressibility modulus which is consistent with measured isoscalar EO giant-resonance properties 15). 3.2. COULOMB INTERACTION
To compute matrix elements of the Coulomb interaction determinants, one may take advantage 16) of the identity 1 _= s
-2 l-1 ?r
between
two Slater
me-“2,2’4 0
2
’
P
and then reduce Coulomb matrix elements as integrals of gaussian matrix elements with respect to the range. When oscillator constants differ from shell to shell this leads to analytic formulae involving hypergeometric functions. With the exception of
D. E. Medjadi, P. Quentin / Microscopic approach
the
1s and
functions,
lp their
shells
where
evaluation
We have therefore preferred tion. We have approximated
these functions
implies
algorithms
295
can be cast into
simple
which
slowly convergent.
are rather
to evaluate the Coulomb contribution the direct contribution as /Jp~~)(r)p:l;)(r’)v*lr-
elementary
by space integra-
r’]d3rd3r’,
(15)
where ~$9 is the proton density transition corresponding to the Slater determinants @I and QJ (see appendix B). For the exchange part we have extended the current Slater approximation [which has been shown “) for a single Slater determinant to be correct up to - 5-7% in light nuclei] as
3e2~)1’3J[p~~)(1)]4’3d3~. 4N,:/lj
(@Ilv(z~+DJ)= -
to the usual proton density, Eq. (15) is exact for I = J, the density pfy corresponding whereas in this case eq. (16) reduces to the standard Slater approximation. The validity of the approximate equations (15) and (16) versus exact contributions (even for the exchange terms) is substantiated in appendix A.
3.3. CONVERGENCE
AND OTHER NUMERICAL PROBLEMS
As discussed in ref.‘), we have rejected from our basis space the subspace spanned by all eigenstates of the norm matrix corresponding to eigenvalues less than 10P3. Indeed it has appeared that a blind numerical solution of the generalized eigenvalue eq. (4) yielded “spurious” states easily recognizable by their large overlap with eigenstates
of the norm matrix
having
almost vanishing
eigenvalues.
Moreover
such “spurious” states turned out to be almost completely decoupled (in terms of the smatrix) with the physical states. It is not clear to us if these “spurious” states are connected with symmetry-breaking, such as translational invariance for instance. Indeed
matrix
elements
of the
center-of-mass
coordinates
calculated
for
these
“spurious” states have been found similar to those calculated for “physical” states. In the present work we have generally included up to 5 (7) Slater determinants for describing uncoupled vibration modes in 160 (40Ca). In the coupled case, 9 Slater determinants have been taken into account for both nuclei. The relevance of such a small basis has been checked by calculations of the monopole mode in 160 with the Skyrme SIII 11) and modified SkM13) forces including 5, 7, 9, 11 and 13 Slater determinants. The 5-determinant results agree with the ll- and 13-determinant results within (i) less than 70 keV for the energies of the three first eigenstates, (ii) less than 0.01 fm for the rms radii, and (iii) less than l-2% for the m_,, m,, m, sum rules.
D. E. Medjadi, P. Quentin / Microscopic approach
296
A similar
agreement
“spurious”
states is performed
have been norm
is found
thrown
matrix
having
between
(as stated
results above)
out by mere inspection
where
a clean
removal
of the
and the results where these states
of their overlap
with eigenstates
of the
almost zero eigenvalues.
4. Results In what follows we will present the results of calculations performed for the two pure monopole and quadruspherical 160 and 40Ca nuclei in the pure quadrupole, pole-monopole coupled cases. Two types of effective forces have been used: Brink and Boeker and Skyrme forces.
4.1. THE BRINK-BOEKER
FORCE CASE
Even though this density-independent force is not suitable for a correct description of a wide variety of nuclear properties, we have nevertheless performed extensive calculations with it for the sake of comparison with the calculations of refs. 5,6*8).Our 160 results are seen in fig. 1 to be very close to those obtained in the
C.B.A. -
a=123
-
Q-6.1
C.B.A.
R q2.92 Rd.10 -.
Q-8.2
Q=8.1 FL278
20-
7
cq N o/
Rz2.78
-Rz2.94
Rz2.57
-,Rz2.66
“! P
Rz2.5
E=105.7
-R.256 A
E=106.5
Ed055
- k2.66
B
E.106.5
E=105.7 E-1068 C
Fig. 1. Spectra of pure monopole (A), pure quadrupole (B) and monopole-quadrupole coupled (C) excitations for the 160 nucleus calculated with the Bl Brink-Boeker effective force. Our results are compared with those of ref.s) (S.Z.) and ref.5) (C.B.A.). Intrinsic quadrupole moments Q (in fm2) and rms radii R (in fm) are also given along with ground-state energies E (in MeV) and the transition matrix element p2 of the text (in fm2) between the ground and first excited states for the monopole vibration mode.
D.E. Medjadi, P. Quentin / Microscopic approach
291
more refined above-mentioned approaches. For the pure monopole case where our intrinsic solutions are directly comparable with the angular-momentum-projectedbefore-variation solutions of ref. 5), the agreement for the first 0” excitation energy is quite good. It is even more so when comparing with the hyperspherical formalism results of ref. ‘)_ Moreover, the transition matrix element of the operator p2 = Cf=“=,ri’ is about the same in all three approaches. For the quadrupole mode a direct comparison between our results and those of ref. 5, is not straightforward since our simple approach yields only intrinsic-state giant resonances. However, the two calculated spectra are in fair agreement up to the two-phonon states. When coupling quadrupole and monopole vibrational modes in the spherical 160 nucleus the one-phonon states are found to be roughly unaffected with respect to the uncoupled corresponding states as far as the p2 transition matrix elements and the energies are concerned. This is indeed not the case for higher excited states. In all situations where the structure of the nuclear states obtained in ref. 5, may be directly qualified in terms of simple intrinsic phonons, the agreement between monopole diagonal matrix elements R is found to be rather satisfactory. In the discussion of figs. 1 through 6 we will consistently mention not only the first but also the second excited monopole state. However, as we will see later in subsect. 4.3, the first excited state exhausts most of the m, sum rule, for instance. It should then be clear that by “second” monopole state we mean a kind of two-phonon state in a quasi-harmonic well in the monopole variable that we have chosen and loot a state that would carry a substantial amount of the monopole strength.
@AR
B.
s.2.
AC.
R~3.48
E=263.3
E--264.4
Fig. 2. Same as fig. 1, but for the pure monopole vibration mode in the 40Ca nucleus. compared with those of ref.‘) (B.), ref. ‘) (S.Z.) and ref.6) (A.C.).
Our results
are
D. E. Medjadi, P. Queniin / Microscopic approach
298
A similar study has been undertaken for the monopole vibrations of @Ca. In fig. 2, the calculated one-phonon state is found slightly too high in energy as compared with the results of refs. 6,8) [see also the RPA result of Blaizot ‘)I. For this nucleus also, the p2 transition matrix elements obtained in the various approaches are again in fair agreement.
4.2. THE SKYRME
FORCE
CASES
In view of the good results obtained in our simple approach as compared to more refined calculations using the Brink-Boeker force, we have extended our study to more predictive effective forces, namely the Skyrme effective forces, in the parametrization known as SIII, SkM and “modified SkM”11-i3). Our results for the uncoupled monopole and quadrupole modes are seen in fig. 3 to compare very well with the RPA results of Blaizot ‘), the hyperspherical formalism results of Navarro 9, and the generator coordinate results of Flocard and Vauthe~n 7)_The latter calculations do not incorporate the Coulomb interaction but include a number of Slater determinants which is larger than ours (143 instead of 5-9). The main difference between these calculations lies in the fact that we allow more flexibility for the variation of the Slater determinants. To be specific, for an axially-symmetric 160 solution, the choice of definite values of r and Q (see eqs. (6)) would determine completely the Slater determinant in the approach of ref. 7, whereas in our approach we are left with two “effective” variational parameters (i.e. four parameters bound by two conditions). The good reproduction of RPA results with a basis size as small as ours assesses the relevance of limiting the number of determinants while increas-
s
RJ.04 B.
F.V.
Cb23.8
Q=23.8 Ffa8l
z&o_-- FM.62
F. V.
F.V
B.
N.
Q41.5
_
ML 2 _
Rx2.83
%2.90 -
0=12.3
M3.1
3
I Ezl26.8
-Rd.73
-Rae2
Rs2.58
Rd.65 --
s2.69
M
Fk2.63 --
RG.62
Ed28.0
E=l2ZO
E=l2L5
E=lZCIO
151271
E=12zO
E-126.9
E=l2Xl
A
B
C
Fig. 3. Same as fig. 1, but for the SIII effective force. Our results are compared with those of ref. ‘) (B.), ref.‘) (F.V.) and ref.‘) (N.).
299
D. E. Medjadi, P. Quentin / Microscopic approach
1 E=33o.~
-
-
-
E=3412
EG33l.5
E-8
Fig. 4. Same as fig. 2, but for the SIII effective force. Our results are compared ref. ‘) (F.V.) and ref. 9, (N.).
( with those of ref. ‘) (B.),
ing the number of relevant variational parameters. More generally, adjusting more carefully each Slater determinant allows one to separate “true” correlation effects from a mere adjustment of the mean field to internal deformation constraints. Similar conclusions could be drawn for the 40Ca nucleus, as seen in fig. 4 for the monopole mode. Finally one may mention the agreement of our results with the adiabatic analysis of TDHF calculations 18) using the same force where the following resonance energies were found: 30.4 and 22.2 MeV for the monopole and quadrupole modes in 160. The above-discussed comparison of our results with those of other microscopic approaches is gratifying in so far as it demonstrates their relevance. Its interest however is merely theoretical for the monopole mode since it deals with an effective force (Skyrme SIII) where parameters have been fitted before knowing the constraints imposed on the incompressibility modulus’5) by the discovery of giant isoscalar monopole resonances 19). It is therefore particularly interesting to present the monopole and quadrupole vibrational modes of 160 and @Ca for effective forces belonging to the Skyrme-type parametrization, which are better conditioned in this respect. Such a force, the SkM force proposed in ref. 12), has been shown 13) to need a slight improvement in order to reproduce accurately the saturation properties and the fission barrier heights. In our opinion, the resulting force, the so-called modified SkM force (quoted here as SkM*), yields the best, to date, overall reproduction of nuclear properties in the P-stability valley for Skyrme-type effective forces. The
D. E. Megiadi, P. Quentin / Microscopic approach r
E ev)
SkM* R.33 0
SkM
Hyd.
SkM
SkM*
SkM*
SkM
Rz3.09
Rz3.03 -
Q=25.0
Q=23.2
0~24.8
Qz22.4 -
R.2.89 Qz12.1
Q-11.7
Q=13.0
_R=2.86 0~12.4
-R3.2.69
-R=2.65
--Rz2.69
R12.66
E1129.8
E.126.3
R&i. 04
l-l-
E=126.1
Ez129.8 A
E=126.1 8
E1130.0 C
Fig. 5. Same as fig. 1, but for the SkM and SkM* effective forces. Our results are compared the hydrodynamical approach (Hyd.) of ref.l’).
results obtained for 160 are reported compared in the SkM force case with results are comparable for the SkM within simple sum-rule approaches20) pressibility modulus and the effective
with those of
in fig. 5. For the monopole mode, they are the hydrodynamical approach of ref. “). The and SkM* forces, which can be understood as related to the fact that both the incommass are identical for the two forces (see an
extensive discussion of this point in subsect. 4.3). The only difference lies in the increase of the surface tension term of the liquid-drop energy formula reflecting itself in the calculated decrease of the binding energy (126.1 MeV for the SkM* force to be compared with the experimental value 127.6 MeV, the difference being due to the severe limitation of the variational space in the present calculation). As observed for the Brink-Boeker and SIII forces, the coupling of the monopole and quadrupole vibrational modes does not alter significantly in 160 the energy spectrum or wave functions of the ground and first excited states. On the contrary (see table 1 and also fig. 6), for both the SIII and SkM* forces, coupling the two collective (quadrupole and monopole modes) results in a significant change of the solutions in “aCa. The effective amount of mixing may be measured by the reduction of the transition matrix elements for the Q (p2) operator between the ground and the first excited state of the quadrupole (monopole) type. While this reduction is not apparent for 160, it is found to be - 20% for both quadrupole and monopole matrix elements in “eCa as seen in table 1. This trend seems consistent with recent experimental evidence21). Further discussions about the change in the coupling pattern between 160 and %a will be made in subsect. 4.3.
D. E. Medjadi, P. Quentin / Microscopic approach
301
TABLE 1
Comparison between coupled and uncoupled solutions in r6 0 and a Ca nuclei calculated with the SkM* effective force I60
Dominant mode quadrupole
‘first [Meal
QfirstPm*1 monopole
(first]QP) [fm*] ‘first PW
Qfirst[fm*I (first]p*P)
[fm*]
‘+OCa
uncoupled
coupled
uncoupled
coupled
19.9 12.1 30.1 24.2 2.90 18.9
20.1 13.0 30.0 24.1 2.89 18.9
17.6 12.7 68.5 18.7 3.48 38.9
16.0 - 6.3 56.7 21.6 3.49 32.1
The first-excited-state energy E,,, of quadrupole (monopole) character is given along with the quadrupole (monopole) moment of this excited state. The corresponding transition matrix element to the ground state for the quadrupole (monopole) operator is also shown,
Even though, as we have discussed above, the experimental quadrupole strength distribution appears rather fragmented’*), the current estimation of its centroid energy at - 22 MeV [ref.23)] (- 18 MeV [ref.24)]) for the 160 (@Ca) nucleus is in fair agreement with the SkM* and SkM results. It should be noted that for the spherical nuclei under consideration here, it is not necessary to take into account explicitly axially-asymmetric modes, that are equally described by symmetric modes 25).
E (Me
r
40 I
20-
o-
-
Rz3.48
Qz22.7
Rz3.48
0~12.7
Rd.42
L-EL336.0 A
R-3.48 Qz12.1
fk3.49
Rz3.42 E=3363
B
0~3.16
Rz3.43 Ez336.5
C
Fig. 6. Same as fig. 5, but for the @Ca nucleus with the SkM* effective force.
D. E. Medjadi, F. Quentin / Microscopic approuch
302
4.3. SUM RULES
From the solution of the generalized eigenvalue equation (4), one may evaluate the moments m,(O) of any operator 0 upon assuming that the finite set of calculated eigenstates span reasonably well the space of physical states connected by 0 with the ground state. Indeed the existence of sum rules like those given in eqs. (lo), (11) and (13) provide a good test for the validity of such an assumptiont. Nevertheless, before summing the different contributions to the moments m,(O) one should reject the spurious states as described in subsect. 3.3. In table 2 we have reported the values of the three moments m-t, m0 and ml obtained with the operators C$ tri2 and E$. lqi in the monopole-quadrupole coupled calculations when using the Bl, SIII, SkM and SkM* forces for the 160 nucleus [our results for the SIII force and the moments m_, and m, are consistent with those of ref. 26)]. Table 3 displays the same moments for the ‘%‘a nucleus with the SkM* force. These moments are compared with the expectation values of eqs. (10) and (13) for m, and m,,. The moments m_, are compared with the polarisability resulting from the solution of eq. (4) where the hamiltonian H is replaced by the constrained TABLE 2
Moments of the strength function for the monopole ( p2) and the quadrupole (Q) isoscalar operators in the I60 nucleus calculated for the Bl, SIII, SkM and SkM* forces defined in the text
Force Bl
Operator P2
Q SkM*
P’
Q SkM
P” Q
SIXI
P2 Q
m-1 [MeV-’ . fm4] 15.4 (15.5) 23.6 (23.6) 15.0 (15.0) 44.8 (44.8) 14.1 (14.1) 41.8 (41.8) 9.1 (9.1) 37.3 (37.3)
[Me? 356 (374) 633 (661) 367 (370) 902 (904) 352 (355) 865 (870) 285 (292) 808 (812)
bz ]
0.844 (0.842) 1.70 (1.68) 0.913 (0.900) 1.81 (1.80) 0.892 (0.902) 1.79 (1.81) 0.912 (0.861) 1.75 (1.72)
As an alternative length unit we have used the bi12 for ml moments. In parentheses we have reported on the lower line the expectation values (or polar&abilities for m-i) whose equality with corresponding moments constitutes a sum rule.
t The demonstration of the theorem energy-weighted sum rule) is apparently
of appendix C in ref.‘) (concerning not applicable to our case.
the exact exhaustion
of the
D.E. Medjadi, P. Quentin / Microscopic approach
303
TABLE 3 Same as table 2, but for the a Ca nucleus
Operator
m-3 [MeV3 fm4]
P2
0.218
Q
0.934
m-1 [MeV-’ . fm4] 78.3 (78.3) 271 (271)
and the SkM* force
[Me? 0.156 (0.179) 0.480 (0.525)
b*]
3.27 (3.80) 8.86 (7.59)
[Met’. 74.7 176
b’]
[Me:‘.
b’]
1984 3943
hamiltonian H - ~0. It has been found that the quadrupole (monopole) moments are vanishing when computed with excited states assumed to represent pure monopole (quadrupole) vibrations. This result indicates that our variational calculations in the uncoupled monopole or quadrupole cases may indeed be interpreted in terms of pure monopole or quadrupole vibrations. On the other hand, the agreement between moments and the corresponding expectation values (or polarisabilities) is indeed excellent. It is to be noted that in most of the calculations, the m, sum rule is almost saturated with the first excited (one phonon) state. For instance in 160 the monopole m. moment is due for - 97% to the transition between the one-phonon and the ground states with the SIII force as well as with the SkM* force. Taking stock of the excellent fulfilment of the sum rules in the m _ 1, m,, m, cases, we have also computed the moments m_3, m,, m3, without checking them against sum-rule results. An example of such results is given in table 3 for the 40Ca nucleus with the SkM* Let us define
force. the energies
E,:
Ek= imk/mk-2. According to ref.20) following conditions:
such
energies
satisfy
(due
07) to the
Schwarz
inequality)
the
We will compare now such energies with those calculated in our generalized eigenvalue problem. The energies E3, E, and E_, have been displayed in table 4 for the monopole mode, calculated with both the SIII and SkM* forces, in 160 and 40Ca. The first-excited-state eigenenergy is denoted by E,. One sees that for all cases under study in this table, there is a spreading of the energies E, which corresponds to a rather large distribution of the strength function for the p2 operator, which has also been observed in self-consistent RPA calculations using same or similar effective forces27). Indeed knowledge of the m,, m, and m2 moments yields the actual variance u of this distribution (equal in the gaussian case to - 0.43 times the
D. E. Medjadi, P. Quentin / Microscopic approach
304
TABLET First-excited-state
energies of the monopole
type in l6 0 and 4o Ca calculated effective forces
with the SIII and SkM*
SkM*
R E, (uncoupled) E, (coupled) EB
2.63 30.9 30.1 37.1 34.2 31.7 31.2
@pprox.)
E3
El E -1
3.43 27.7 27.6 28.6 31.2 28.6 28.2
160
40Ca
2.69 24.2 24.1 29.9 27.6 24.7 24.4
3.43 18.7 21.6 23.4 24.6 20.4 19.0
The energies E, (in MeV) obtained in both coupled and uncoupled calculations are compared with an approximate value calculated with the ground-state radius R (in fm). Energies Ek (with k = - 1,1,3) defined in the text are also reported.
full width at half maximum).
One finds 20)
+p,‘. Again through
due to the Schwarz the moments
m_,,
inequality
one may also provide
09)
an upper
bound
for u
m, and m3 by2’)
(20) For the SkM* calculations we have found u (a,,) equal to 5.1 MeV (6.1 MeV) in 160 and 6.4 MeV (6.9 MeV) in @Ca. The observed increase in the width when going from 160 to “eCa is consistent with our previous discussion concerning the coupling of quadrupole and monopole modes. The energy E, of our solution is closer in all cases to E, or E _ 1 than to E,. This can be interpreted in terms of a collective path [see e.g. ref. 28)] for our solution that is closer to a constrained Hartree-Fock path than to a scaling path. Indeed it has been shown in the harmonic limit that, whereas E, corresponded to the collective energy obtained by a scale transformation of a Hartree-Fock equilibrium solution 29), for a family E _1 in turn was obtained by performing adiabatic TDHF calculations of Slater determinants obtained in Hartree-Fock calculation under a constraint on the collective operator (here p2) [refs. 30,31)]. On the other hand, it is known32) that our generator coordinate method should provide an upper bound for E _1 in the harmonic limit. For such a comparison, however, one should use the uncoupled
D. E. Mea)&,
P. Quentin / Microscopic approach
305
generator coordinate (i.e. pure monopole) solutions. As seen in table 4, our values of E, (uncoupled) fail to fulfill this condition by less than 0.5 MeV. This may be due to either the presence of anharmonicities or to small residual uncertainties in our calculations (for instance, when evaluating strength-function moments) due to the very small number of Slater determinants being mixed. In ref. 20) the energy E, for the breathing mode is approximated by
(21) where m is the nucleonic mass, R the ground-state radius and KA a universal value of the incompressibility modulus in finite nuclei, given roughly (in MeV) as a function of the nuclear-matter incompressibility modulus K,,,, by KA = K,.,.-
630
+ Y>
(22)
( y being the power of the density dependence of the effective force). The result given in eq. (21) generalizes the result of Pandharipande3’) using the nuclear-matter K,,,, modulus. From eq. (22) and since K,.,, depends also linearly34,35) on the positive quantity y, one sees that the approximate value of E, given by eq. (21) varies roughly as the square root of an increasing linear function of y. We have numerically evaluated the approximate breathing-mode energies E, discussed above, taking the ground-state radius R as given in our calculations. The KA value resulting from eq. (22) is 229.4 MeV (155.9 MeV) for the SIII (SkM*) force, since the K,,,. values are 355.4 MeV [ref.“)] and 216.7 MeV [ref.13)] for the two forces considered. As seen in table 4, the approximate values E, are found far higher than the exact ones. This is partly due to the already discussed dispersion of the strength function. However, these values are found also rather far from the energies E, which they are supposed to approximate. This discrepancy is mostly related to the fact that eq. (22) for K, is only an average (over the number of nucleons A) value, overestimating2’) the actual value of m3 by about 2% in 40Ca and significantly more in 160. Let us consider now the results concerning the quadrupole mode displayed in table 5. The different E, (k = -1,1,3) energies in the 160 case are very close to each other. Indeed in the calculation using the SkM* force, the width u (a,,) is found equal to 0.6 MeV (0.9 MeV). Our calculated widths are consistent with the results of the RPA calculations of ref. 27). The dispersion of the strength function is larger in 40Ca (with SkM* force u = 5.1 MeV and amax = 5.4 MeV) which is no surprise in view of the already discussed coupling between monopole and quadrupole modes. As was the case for the monopole mode, our generator coordinate solutions correspond rather well indeed to a constrained Hartree-Fock adiabatic path. Adiabatic TDHF calculations have been performed36) according to the method presented in ref. 28) which is free from convergence problems in the
306
D. E. Medjadi, P. Quentin / Microscopic approach TABLE 5 Same as table 4, but for quadrupole excited-state energies SkM’
EQ (uncoupled) E. (coupled) EQ (approx.) E3
El E-1
20.9 21.6 26.4 21.8 21.7 21.6
19.4 19.1 19.5 20.2 19.3 19.2
160
@Ca
19.9 20.1 25.9 20.2 20.1 20.1
17.6 16.0 19.1 21.1 18.1 17.0
particle-hole space, contrary to the approach of ref. 30). In the harmonic limit these calculations using the SIII force yield a value of - 21 MeV for the first-excited-state energy E, which is very close to what we actually obtain here. The inequality between our generator-coordinate-method energies E, and the energy E_ 1 (in the uncoupled case) is no more fulfilled here that it was in the monopole case, probably for the same reasons. Suzuki 37) using the so-called self-consistent condition of Mottelson 38) proposed, years ago, that the quadrupole resonance energy Eq should be given for a zero-range force by E,=
&ho.
(23)
This formula has been further improved by incorporating finite-range alently velocity de endence) effects in the effective force by multiplying
(or equivthe r.h.s. of
eq. (23) by i--) m/m*, where m* is the effective mass in nuclear matter and m the nucleonic mass 20,39). A s was the case for the pocket formula of eq. (21), this approximation for EQ is in fact an approximation to E,. Taking the standard value for tto [ref.@)], Ao = 41A-‘/3
MeV 9
(24
mass values in unit of m as 0.76 [ref. ‘l)] (0.79 [ref. 13)]) for the SIII (SkM*) force, one gets as estimates of EQ, according to eq. (23), the values listed in table 5. They are found indeed to be reasonable estimates for 40Ca but rather poor for a nucleus as light as 160 . The latter may be understood by the fact that for very light nuclei. the liquid-drop-type concepts used to derive eq. (23) are of course partially inappropriate. and the effective
4.4. TRANSITION
DENSITIES
As shown in appendix B, from the solution of the generalized eigenvalues eq. (4), one may compute the transition density pn(rl, r[) between the ground and n th
D. E. Medjadi, P. Quentin / Microscopic approach
excited
307
states as (25)
Hydrodynamical approximations have been proposed for the diagonal part of the transition density between the ground and first excited states. For the quadrupole mode, Tassie41)
has proposed PlW
where p(r) symmetrical that
a ‘r,(i)d,
Q(r) 3
is the usual one-body density for the ground state of the spherically nucleus. For the monopole mode, Werntz and ijberall 42) have assumed
p,(r)
a 3p(r)
In eqs. (26) and (27) the proportionality
+Z$+.
constant
(27) has been chosen in such a way that
/pl(rW(r)d3r = WP>~
(28)
r:Xlf’ 3: *I
2-
l-
o-
) __/’
-1 L
3
Fig. 7. Transition density between the ground and first excited state for the monopole vibration mode in I60 calculated with the SkM* force. The dashed lines correspond to the results obtained within the Wemtz-ijberall approximation4*).
D.E. Medjadi, P. Quentin / Microscopic approach
(f mV3)
Fig. 8. Same as fig. 7, but for the quadrupole vibration mode. The transition density has been evaluated for the angle 0 = fr in Y,,(e, cp), see eq. (26). The dashed line corresponds to the results obtained within the Tassie approximation41).
As seen in figs. 7 and 8, both approximations reproduce fairly well the results of our microscopic calculations. This is, as well known, related to the fact that the axial quadrupole and monopole generalized sum rule of Kao and Fallieros43,44) is exhausted
by the one-phonon
4.5. COLLECTIVE
states.
WAVE FUNCTIONS
As suggested in ref.‘), we have considered as collective wave functions g the result of the application of the Ml/* operator (square root of the norm operator) on the generator coordinate weights f. Such wave functions are reported in fig. 9 for the quadrupole and monopole modes in 160 and in fig. 10 for the monopole modes in 40Ca. In all cases the Skyrme SkM* effective force has been used. One easily notices in these figures the standard 0-, l-, 2-phonon gross trends of such wave functions, attesting thus the quasi-harmonic character of such vibrational modes. This is of course only roughly true as we will discuss now. In figs. 1, 3, 5 and 6 we have assigned to pure quadrupole or monopole states, as well as to monopole-quadrupole coupled states of a quadrupole type, a quadrupole moment value which corresponds of course to an intrinsic quadrupole moment and not to a moment in the laboratory frame. As was found in earlier calculations5,‘) these values are not equal to zero as would have been expected for an ideal vibrator
309
D. E. Medjadi, P. Quentin / Microscopic approach
u.-l'"~/~
; !
’
t 2.5
;’
\
\
i’ !'
/
\\
\
.! '\ \
,/
I
i
\,
\ , ‘\ \
i
,
i
\
\
; 3.0 I
,. i
-40 ;
i
i
\
\
!
, r lfml
I i i i i i
\ \
\
\
\
\
\ \
'\ ' 'x_ _./
\ \ \
i i i
\
\
i
-0.2.-
i i '! -0.4_'\ -.
i i !'
.' ,' .I'
i i i
\ \
i
\
!
\
\ 0 ‘\
, ’ \
\
Q, Ifti
‘! \ ! I
’ 40
i
\ \ \
i i i i
\\
\
\
\ \ \ \
\ \
\ \
\
\
,.*,I
i i i i
'\
--.
Fig. 9. Collective wave functions g defined in the text for the monopole (1.h.s.) and quadrupole (r.h.s.) vibration modes in 160 calculated with the SkM* effective force. The ground, one-phonon and two-phonon wave functions correspond to the solid, dashed and dot-dashed lines. The units for g are case. In the monopole case, the arrow indicates fm-’ in the monopole case and fm-* m the quadrupole the location of the rms mass radius for the lowest uncorrelated wave function.
around a spherically-symmetric
equilibrium state. This trend by itself reflects the
influence of some anharmonicities present in our solutions and is also illustrated for instance in fig. 9 where one sees that the “quasi” 1-phonon and 2-phonon states are not exactly symmetrical with respect to Q = 0. 5. Conclusions By studying
two isoscalar
collective
modes for a variety of effective
forces
(including the best available within the Skyrme class of parametrizations) we have proven that our simple generator coordinate approach was accurate enough to yield, without too much numerical effort, a reasonable reproduction of more involved RPA or equivalent microscopic approaches. The calculations of ref. ‘) are in many respects very close to ours. The present approach, however, uses a much smaller number of Slater determinants for mixing by the generator coordinate integration. On the other hand, it allows much more flexibility for the adjustment of each Slater determinant and includes within a very good approximation the direct and exchange Coulomb contributions. As it stands, our approach which might easily be gener-
D.E. Medjadi, P. Quentin / Microscopic approach
310
Fig. 10. Same as fig. 9, but for the monopole
vibration
mode in 40Ca
alized to other modes, provides a useful tool in the perspective parameter adjustment of the nucleon-nucleon effective force. It is a pleasure to acknowledge fruitful Morand. One of us (D.E.M.) is also grateful
of pushing
forward
a
discussions with Y. Abgrall and B. to the Director of the Ecole Normale
Suptrieure for having authorized his leave of absence and the members of the Laboratoire de Physique warm hospitality extended to him. The fruitful stay Theoretical Division of the Los Alamos National
from Algiers and to B. Bonnier Theorique in Bordeaux for the of the other author (P.Q.) at the Laboratory during the second
semester of 1984 is also gratefully acknowledged. Finally, we thank the referee for having led us to substantially enlarge the discussion of the sum-rule properties of our solutions. Appendix A CALCULATION
Starting
OF ENERGY
MATRIX
from 3-dimensional
ELEMENTS
harmonic-oscillator
states
D. E. Me&d,
P. Quentin / Microscopic approach
311
and separating the relative and center-of-mass motion as r=&(5-r2),
R=(R,,R,,R,),
with
one may define generalized Moshinsky coefficients (CXstanding for x, y, z): (~n(wa)N,(D,)ln,l(w,l)n,*(wd,)> =
+q),(nl_q)!(na,_q),q [N,!na!na~!na~!11’2XC(-)q(n,_n a
.
.
a
.
where
In the above equation and throughout this appendix the summation over q is limited by the condition that all the arguments in the factorials should be 2 0. The overlap between two basis states is factorizable for each direction x, y and t. For one direction one gets (ml(w!)lm(w))
= (1 +(_l)“+“‘)
x
T22q[
:;+;l’“‘-‘“2[
[~41’4]~1~‘1]“2 [(w + wt)2”+“.+l]l/2 :;“:I’“““[
$yJq
Gaussian interaction
Its matrix elements are also factorized. One gets for the x-direction, for instance, ,+I(
wJ)
le-2x2/P2In&))
= (1 +(-l)“+“‘)
[ m!m~!]“2[ p/3,]1’4 2m+m’+l( p + p’ + 4/#)]
1’2 (m-q)/2
where p = mu/h.
312
D. E. Medjadi, P. Quentin / Microscopic approach
Skyrme interaction. Let us define some quantities (a standing for x, y, z): & = (P&$‘*, --P= (pxPvB,)1’3, A(m)=
(1 +(-l)~)(-l)ffl’z(~~)!(~!/2~+2)1’2.
(i) a-force. One gets
X4% (the
-
w(%+M~,+2)
notation cy,ff + 1, ff + 2 referring to circular permutations). force. One gets
(iii) e2& + S?
(n:(w:)n;(w;)n:(w:)I~*S(~)+S(r)V2ln,(w,)n,(w,)n,(w;)) a (iv) Spin-orbit force. One gets for the matrix elements of the x-component instance) of the L-operator
(for
(n:(w:)n~(~~)n:(w:)IL*ln,tw,)n,(cl?,)n,(w.)) =2i(~/~)X[(PIP,)“‘Jn;n,A(n;-l)a(n:)A(n,)A(n,-1) - (&~.J’*/G&:)~(%
- I)+,-
(u) reality-dependent S-force. Its cont~bution approximated very well? by
I)~(%)]~(~:)~(%). to the matrix element
.FIJ
is
where prJ is the transition density defined in appendix B and y the exponent of the density dependence of the &force. Coulomb interaction. We have used two methods, one exact for the lightest nuclei, the other approximate for heavier nuclei. The exact methods consists in reducing t As seen for a y = 1 Skyrme force (SIII) where exact calculations delta force are possible.
in terms of an equivalent
three-body
313
D. E. Medjadi, P. Quentin / Microscopic approach TABLE 6 Approximate (upper line) and exact (lower line) Coulomb interaction matrix elements between the Slater determinants @, and GJ
I=1
J=l
J=2
J=3
J=4
J=S
15.43 14.96
13.7 12.90 14.27 13.89
10.93 11.06 13.57 13.27 13.73 13.39
8.73 9.13 12.25 12.13 13.14 12.86 13.25 12.95
5.26 6.05 9.19 9.46 10.86 10.89 11.98 11.84 12.50 12.27
I=2 I=3 I=4 I=5
Coulomb matrix elements in term of gaussian matrix elements integrated over the range (see eq. (14) of the text). Obviously direct and exchange terms are treated on the same footing in this case. In the approximate method, the direct Coulomb interaction contributes to ;X;, as shown in eq. (15). For axially-symmetric wave functions this contribution is proportional to
with E(k)=l”/2J1-kZsinZBdb’. 0
The exchange
Coulomb
interaction
is approximated
as given in eq. (16). For 160
we have computed for a given physical Slater determinant superposition (corresponding to the mixing of 5 determinants yielding monopole vibrations with the Skyrme SIII force) both the exact and approximate Coulomb (including direct and exchange contributions) matrix elements ( @,lVclQJ). The results (see table 6) differ 2-3% for the diagonal term (where the only physical source of difference is the by Slater approximation) and by some tens of keV for the non-diagonal term. Appendix B TRANSITION
DENSITIES
The transition defined as
density
from the ground
state
I!P”) to the excited
state
19”)
is
314
D.E. Medjadi, P. Queniin / Microscopic approach
which gives in terms of the Slater determinants wave functions q;, cp,“)
@,, QJ (built with single-particle
Defining &,(I, J) as the minor associated to the element (m, n) of the matrix (q$lqi) the transition density pIJ may be written as
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