A microscopic calculation of the collective ground-state correlations in the Fe-Ni-Zn isotopes

Nuclear Physics A384 (1982) 179--1g9 © North-Holland Publishing Company

A NIICROSCOPIC CALCULATION OF THE COLLECTIVE GROUND~STATE CORRELATIONS IN THE Fe-Ni-Zn ISOTOPFS M . GIROD Sernlce de Physique Neutronique et Nudfaire, Centre d'Etudes de Bruyères-le-Chatel, France

and P: G . REINHARD

t

Institut fur 1CernPhysfk, Unioersit6t Mainz, i7-6500 Mainz, Federal Republic of Genntmy

Received 21 April 1981 (Revised 11 December 1981) Abstract : The collective hamiltonian for the quadrupole surface vibrations has been calculated microscopically for the Fe, Ni and Zn isotopes using axially symmetric deformed Hartreei3ogoliubov states which are evaluated with the Gogny force and quadrupole constraint. The mi~pic wave function of the wrrelated ground staté is described as a linear superposition of the wave functions of the deformed Hartreo-l3ogoGubov basis with a weight given by the ground state solution of the collective Schri3dinger equation. This correlated ground state is then used to evaluate the isotopic differences of the charge density and the rms radii . The results are compared with experimental data from electron scattering .

1. Introdaction The introduction of density-dependent eûective forces was a great success in the microscopic description of the nuclear ground state. Since then one has been able to reproduce in a Hartree-Fock or Hartree-Bogoliubov calculation the binding energies, level spacings and radii for awide range ofnuclei throughout the periodic table, all with the same force' ). However if one looks at more detailed quantities, one realises that a pure Hartree-Bogoliubov description is insufficient . For example, it is hard to obtain from a pure Hartree-Bogoliubov calculation a charge density which is as smooth as the experimental one. The same happens for the isotopic differences of nuclear radii where the deformed Hartree-Bogoliubov calculations produce strong fluctuations

t

Supported in part by Gesellschaft für Schwerionenforschung, contrail no . 06 MZ 709. Heisenberg awardee . 179

1SO

M. Girod, P.-G. Reinhard lMicroscopic calculation

quite different from the smooth pattern of the experimental results. To describe those erects one needs to include some correlations in the description of the ground state. It has been shown in a recent study of the isotopic differences of nuclear charge distributions that ground-state correlations indeed are important, in particular those correlations which come from the surface quadrupole vibrations ~). In that previous study a semiphenomenological approach was used taking the parameters of the collective surface vibrations from experimental data . In the present paper we aim at a more microscopic calculation of the correlated ground state where the parameters of the collective motion are obtained from the same microscopic force as was used for the Harttee-Bogoliubov ground state ; in practice we are going to use the Gogny force a " a), The calculations are done for the isotopes of Fe, Ni and Zn for which very accurate experimental data are available s). The calculation proceeds in three steps : First we evaluate the collective potential with a constrained Harttee-Bogoliubov calculation where we use a quadrupole constraint and the density-dependent and finite-range force proposed by Gogny s . 4); from the deformed wave functions we obtain also the collective mass using an approximate Inglis cranking formula. Second, from collective mass and potential thus obtained we set up the collective hamiltonian and determine the ground state solution of the corresponding collective SchrBdinger equation . The resulting collective wave function together with the microscopic basis states from the deformed Harttee-Bogoliubov calculation form the microscopic representation of the collectively correlated ground state. Finally we evaluate the expectation values ofsome measuring operators, namely the charge distribution andthe rms radius, for that correlated ground state ;that is done within the gaussian-overlap approximation to the generator coordinate method (GCMG where we represent the correlated state as a folding of the microscopic basis wave function with the collective ground-state solution and include some correction for the collective zero-point fluctuations contained in the basis states. This altogether yields a consistent microscopic picture of the ground state including those correlations which come from the surface quadrupole vibrations . The outline of the paper is as follows : In sect . 2 we introduce the force and the constrained Harttee-Bogoliubov method used. In sect . 3 we explain the collective hamiltonian and the solution of the collective Schrôdinger equation. In sect. 4 we evaluate the expectation values of measuring operators within the gaussian-overlap approximation to the GCM . Finally in sect . 5 we present the results and compare them with experimental data from electron scattering S).

2. The microscopk basis The microscopic dynamics is described by a hamiltonian consisting of the kinetic energy and the Gogny force Dl . The Gogny force has three contributions : a finite-

M. Girod, P.-G. Reinhard l Microscopic calculation

181

range two-body interaction, a zero-range density-dependent part and a ! ~ s force V = ~ (W,+BiPQ-H,Pt-M,PQPs)exp(-(rl-rz)z/~z) t~i,z

+t°(1+xoPe)P°(~{ri +rz))S(ri - rz)

+iwts(eI +QZ)~ (vl-vZ)xa(rl-rZ)(o~-oz)

(2.1a)

with the parameters as given in ref. °), namely Pi =

0.7 fm,

Pz =

1.2 fm,

Wl = -402.4 MeV,

Wz = -21.30 MeV,

MeV,

Bz = -11.77 MeV,

Bl = -100 H, _

-40.62 MeV,

Hz =

M1

-23.56 MeV,

Mz = -68.81 MeV,

=

t° = 1350 a=

~,

MeV ~ fm 6,

37.27 MeV,

x° =

(2.1b)

1,

W~ = 115

MeV ~ fms.

Due to its finite range this force is probably the most realistic one from the presently known effective forces ; in particular it allows a consistent Hartree-Bogoliubov treatment because the matrix elements fall offsufficiently fast at large energy separation. Due to its gaussian form the force is particularly suited for calculations in a harmonic oscillator basis. The algebra for evaluating the matrix elements of the force and the Hartree-Fork hamiltonian is rather tedious ; it is outlined in ref. s) and has found its application in the so-called Gogny code whose axial and triaxial version we have used for the numerical calculations 6''). Since we want to describe not only one single Hartree-Bogoliubov ground state but also some surface quadrupole vibrations about it we have to construct a whole set of collectively deformed wave functions. This set is generated by the constrained Hartree-Bogoliubov equations where b means variation with respect to the single-particle wave functions or the occupation amplitudes of ~45a= ~, $ z° is the quadrupole moment ¢z° = rz Yzo(9, tp), and yo is adjusted to obtain the desired deformation 4n _ azo - ~~°`=o 13ARz ~zo 1 45~=0 ~. (R is the nuclear mass radius R = 1 .2 fm ~ A}.) The constraint

(2.3) Qzo

yields axially

182

M. Girod, P.-G . Reinhard l Microscopic calculation

symmetric deformations . In order to obtain triaxial deformations, i.e. those where aZZ $ 0, we add sometimes a further constraint yZ~zz+~z-z) where ¢ zz = rZ YZZ(B, ~P). From these solutions of the constrained Hartree-Bogoliubov equation we obtain a preliminary collective potential (2.4) The mass parameter is evaluated with cranking in the Inglis approach 8), i.e. neglecting the residual interaction while inverting the linear response equation. The same approximation (i.e. neglecting the residual interaction) is used for the matrix elements bf the cranking operator : (ilô~~lj) _ (11dtz IIU(Qt+elf where ~4z ,, = Qz4~c/3ARz, and where li) and Ij) are the quasiparticle states in Id;~ and ei, el are its quasiparticle energies . Thus we evaluate the mass parameter as -

~ (il~z,~lj)(il~z .li)

(2 .5)

A derivation of the collective hamiltonian within the gaussian-overlap approximation (GOA) to the GCM shows that the potential (2.4) is not yet the true collective potential 9. '°). It has.to be corrected for the. zero-point energies coming from the quadrupole fluctuations in the basis wave packets Idi Qi . This yields approximately V(azo) _ ~(azo) -

(2.6)

~ .1vN/4.1~1~~

where Zw. i s the width of the gaussian overlap defined in a~later section [see eq . (4.5)] and the summation runs over all five ~, i.e. from ~ _ -2 up to p = +2 . In that approach we take into account only the kinetic zero-point energy and we neglect cross-over contributions, i.e. terms with ~ ~ v. 3. The collective dynamics The surface vibrations correspond to five degrees of freedom and can be described in the five-dimensional space of the coordinates aZV [ref. i')] . In fact while using constrained Hartree-Bogoliubov with ¢ZO as constraint for axial deformations, and ¢zz +~z _ z for triaxial ones, we are already working in the so-called intrinsic frame, in which the five aZZ are transformed to the intrinsic deformation a2 ° , aZZ and the three Euler angles b l, aZ, S 3 . We go one step further and transform the intrinsic aZo and aZZ to the so-called ß, y representation lz) which is given by a2 ° = ßcosy and aZ Z = ~ßsiny. In that frame the collective hamiltonian for angular momentum J = 0 states reads ß~ou - 2M

~~ ß ~ aß + ßz sin 3y ây

(sin 3y)

~YI +

V~, Y~

(3.1)

M. Gtrod, P.-G. Reinhard / Microscopic calculation

183

where we have neglected the rotational kinetic energy because we are interested only in the ground state with J = 0 and we have replaced the ß, y dependent matrix Mß..(ß, -y) by an average diagonal mass tensor Ma,,,,~eQe = 1~5~,. This is justified because the cranking mass comes out to be rather smooth as a function of ß and y and almost diagonal. In fact we have carried out some calculations with a ß, y dépendent mass and found only negligible differences in the results. For a proper handling of the geometry ofthe quadrupole vibrations near sphericity we actually need the five-dimensional dynamics expressed in the a2~ or the ß, y, S  S2, S3 frame. However the treatment of the y degree of freedom can be greatly simplified because the F~Ni-Zn isotopes have a rather smooth potential in the y-direction 1°). Thus we can approximate the potential by a polynomial expansion in eos3y up to first order :

v(ß, y} = 2[v(ß, o~ }+ v(ß, ~°)] + 2[v(ß, o°}- v(ß, 60°}] cos 3y. (3.2} There the (ß, y = 0°) and .(ß, y = 60°) are the axially symmetric configurations, i.e. we interpolate the potential in the full triaxial plane from the potential V,~,,,(a2o) at the axially symmetric broundaries where V(ß, y = 0°) = V~t,,(~ and V(ß, y = 60°) = V~ra,(-ß) . We have counterchecked the approximation (3.2) by evaluating a triaxiai point at y = 30° and ß = 0.15 and found good agreement within ±0.2 MeV. As solution of the Schrôdinger equation with the collective hamiltonian (3.1) we obtain the collective ground-state wave function in the five coordinates ß, y, S SZ, S3. The dependence on the Euler angles is trivial, namely a constant which corresponds to J = 0 with the angular function Dôo. Thus it remains the ß, y dependence which again can be expanded in orders of cos 3y up to first order ~P~ou(ß, Y) _

Po(ß)+cpl(ß) cos 3y.

~

(3.3)

This approximation isjustified becausetheterm proportional to cos3y in the potential is very small. We have checked the second-order correction and found a negligible contribution . 4. Evaluation of meawring operators Now in the third and final step we build the microscopic representation ~~) of the collective ground-state vibrations ~P~u and evaluate the expectation value of some measuring operator Â. This will be done in a similar approximation as was used to derive the collective hamiltonian, namely within the gaussian-overlap approximation (GOA) to the generator~oordinate method (GCM). Within the GCM the microscopic representation of the collective state ~~~ is obtained as the superposition

184

M. Girod, P.-G. Reinhard / Microscopic calculation

where -the collective superposition function f~a is related to the collective wave function Win by an unfolding of the GCM norm operator,f~ = Î- ~~p~, where 1 is defined by (Îj)(a) _ ,(dsa'(~~I~s')i(d ). Here and in the following derivation it is convenient to switch to the frame of the five az~, coordinates because the GOA used in the further evaluation has its most simple form in that representation'z " 13) . After having worked out all the formal steps we will transform the result back to the ß, y frame. Furthermore we are using a vector notation a = (az _ z, az _ 1 , azo, az ,, azz) as a shorthand in the following. We are interested in the ground-state expectation value < SPIÂI+~) of some measuring operator Â. With the GCM representation of I`~i we obtain <`Pldl`p) = ds ad'all~ôu(aK~sldl~a'>hu(d), J

(4.2)

where all the microscopic information is carried by the GCM overlap kernel (~,IÂI~a.) . In general this overlap is rather tedious to evaluate . However one knows that for the hamiltonian overlap a good approximation is provided by the GOA; we expect that this holds also for more general overlaps, at least for operators which measure nuclear bulk properties, as e.g. the density or the rms radius . In secondorder GOA we write p. v <~aldl~s'> - <~al~,'><~aldl~a>,

(4.3) (4.4)

where (4 .5) The GOA width ~,~ is evaluated similarly to the cranking mass M~, in eq . (2.3) but with one power (e,+ej) less, i .e. t .~

(Qi+e~) z

(4.6)

In the Â-overlap we have skipped the kinetic contribution ~(a'-aK~,lâQÂô,.I~~.) (a'~c) ; it is assumed to be negligible because  is a local operator (containing no momentum dependence). Using the standard-techniques ofthe GCM in theGOA we end upwith the approxi-

M. Girod, P:G. Rtinhard / Microscopic calculation

18 5

mate expression for the expectation value ~)

(4 .7)

~~a~Â~~s/J~ Ô w where .~ is an average norm tensor similar to the average mass 1Gr in eq. (3.1). Finally we transform this to the ß, y frame and obtain

<
J

ß°dßd(cos

3y)~~~ou(~, v)1 2 {A~oc(~, Y)

(sin 3Y) I ß,4 ~ + Aß(ß, Y)~ ~ (4.8) \ ôß ôß ß s~ ôY ~YJ where A,«(ß, y) is the ß, y overlap of the rotationally invariant operator

and k(a) is the operator of a rotation about the three Euler angles S1 , 8Z, E3. For the Aro,(ß, y) we make the same approximation as for V(ß, y) and tp~u(ß, y), namely the A,a(ß, y) _ <~a~Â,~~~Qi is evaluated at y = 0° and y = 60° and interpolated up to first order in cos 3y similar to eq. (3.2). The above formulae can be applied to any local measuring operator. In the following we use in particular the operator for the charge density at r, namely  = p(r) _ ~e8(r-~ri), and the operator for the square radius, namely  = R~, _ ~r,2/Z. For Rrm, the rotational projection (4 .9) has no effect ; for p(r) it produces the radial density proa(~r~) which is p(r) integrated over all angles . 5. Results snd discussion The first step in the theory is to set up the collective hamiltonian by constrained Hartree-ßogoliubov and cranking . In fig. 1 are shown the resulting potentials V(aZa) (which are corrected for the zero-point fluctuations) for the isotopes of Fe, Ni and Zn. Obviously for all isotopes we obtain a smooth potential which leads to very soft anharmonic vibrations about a spherical shape rather than a rotation about a well developed intrinsic deformation. The potential in the y-direction is very soft justifying the expansion up to first order in cos3y [see eq . (3.2)]. The masses are also very smooth andjustify the use of an average constant and diagonal mass parameter. The~ second step is the solution of the collective Schrödinger equation. As a

186

M. Girod, P: G. Reinhard / Microscopic calculation EtiFa IM~/l

6G

L,

Sßw SG~ 66 ~

~Ni

6ß~

56~

61~

6GK 58~

~

~nwj

navj

tMavl

..5 _ .5

Fig. l. The potential energies for ß-motion (including Dero-point energy corrections) for the isotopes of Fe, Ni and Zn .

countercheck for the collective solution we compare the spreading width of the quadrupole operator a~ in the collective ground state with the experimental ßz values ta, is): ßz is evaluated from the B(E2) value and corresponds to ßi = ~I<~laz~l2i i12~ It is approximately equal to the spreading width d za = ~,  I<~laz~l2 )I z if we assume that the fast 2 + state exhausts the a z~, excitation modes. The result is given in table 1 . It shows that the theory produces smoother tendencies than the experiment . That may be caused in part by the incomplete exhaustion through the (21) states and in part by the rather uncertain B(E2) values . But altogether TABLS 1

Comparison between theoretical ß: value (from (~_~ of the correlated ground state) and experimental ß= value [from B(E2 ; 0; y 2 ; )]

ßz tbeor ~x ~P ßs theor ßs exp

saFe

ssFe

saFe

saNi

e° Ni

bzNi

0 .231 0 .18

0.229 0.23

0 .226 0 .26

0 .226 0.18

0 .222 0 .20

0.225 0.19

6"Ni

6"~II

66Zn

6a~II

~°Zn

0.212 0.16

0 .230 0 .244

0.216 0.226

0 .193 . 0 .205

0 .186 0 .229

M . Girod, P .-G. Rcinhard / Microscopic calculation P 1 fm'3 )

187

~Ni

Fig . 2. A comparison of the charge density of e°Ni for spherical Hartree-liogoliubov, dashed curve, correlated ground atau, full curve and experiment (arrows) .

the absolute magnitude of the spreading width agrees fairly well with experiment . (That is already a success compared to many other microscopic calculations, like Hartree-Bogoliubov with a pairing-plus-quadrupole force, or the Nilsson model with the Strutinsky prescription .) An interesting point is the influence of the collective ground-state correlations on the charge distribution p~ for one single isotope. In fig. 2 we compare the radial charge density from pure Hartree-Bogoliubov with the one from the fully correlated ground state and with experiment 3). We see that the correlations soften the fluctuations in the pure Hartree-Bogoliubov density and that the correlated density comes closer to the experimental one. A more sensitive test of the correlations is provided by the isotopic and isotonic differences. In fig. 3 we compare the differences of the charge rms radius for three cases : first from the one deformed Hartree-Bogoliuvov state which has the minimal energy in the potential energy surface, second from the collectively correlated ground state and third from experiment S). We see that the pure Hartree-Bogoliubov ARrmslmfl 50~ 40

ARrois Iwf)

o ExQerinent

100

a+ Defar~d HFB

0

0

30 s<

20

0

. IiFB * GSC

0

,50 O

10

a

N: :wo n.u 3a.~t n-.~c3c.3e x»3e 3e-ae Fe

Ni

Zn

e

0

0

_

.

i

0

s~, sa~ ~ ecK

~H d°ri ~z~ ~m

Fig. 3 . A comparison of the isotopic differences of R,m , for the isotopes of Fe, Ni and Zn, where Q = experiment, ~ = wrrelated ground state and / = deformed Hartreo-Bogoliubov.

(88

M. Girod, P.-G . Rehhard l Mkroscoplc calculation

results differ substantially from the correlated ones. The reason is that in the rather flat potential energy surfaces (see fig. 1) the deformation minima are rather insignificant and that therefore their position can change drastically from one isotope to the next . Thus it is hazardous to consider one particular deformed HartreeBogoliubov state as the representative of the ground state ; this leads to strong fluctuations in the rms radii as seen in fig. 3. The conelations, on the other hand, explore the collective potential for a whole range of ß and y ; this smoothens the fluctuations and leads to a fair agreement with experiment . However in the case of the isotonic differences, i:e. from Fe to Ni and from Ni to Zn, the deviation from experiment remains still quite large ; this effect has not yet been understood.

Fig. 4 . A comparison of the isotopic differences of the charge density from fig . 2 .

64Ni

to

62Ni, casts

as in

A comparison of theory and experiment for the differences of the radial charge densities of 6`Ni ~ 62Ni is shown in fig. 4. We see that for differences of densities the effects of the ground-state conelations are much more pronounced than for the density ofone nucleus alone (see fig. 2). Obviously the ground-state correlations lead to a better agreement with experiment, particularly in the more important outer region . 6. Concludon

In this paper we have presented a consistent calculation of the nuclear ground state which goes beyond a Hartree-Bogoliubov description by including the groundstate conelations due to the collective surface quadrupole vibrations. For the microscopic calculations we use the Gogny force Dl. The collective potential energy surface is obtained from constrained Hartree-Bogoliubov with a quadrupole constraint and the collective mass is obtained from an approximate Inglis cranking ; the . collective potential is corrected for the zero-point fluctuations of the quadrupole moment in the Hartree-Bogoliubov states . From the collective Schrödinger equation thus obtained we determine the collective wave function for the ground state. The

M. Girod, P.-G. Reinhard / Microscopic calculation

189

microscopic wave function of the correlated ground state is generated as a linear superposition of the deformed Hartree-Bogoliubov states with a weight which is given by the collective wave function unfolded by the GCM norm operator. Finally we evaluate the operator expectation values from the collective wave function together with the microscopic information contained in the deformed HartreeBogoliubov basis ; this is done within the gaussian-overlap approximation to the generator coordinate method. In particular we apply this to the operators of the radial charge distribution and the charge rms radius . The results for the charge density and the rms radii and for the isotopic differences of both is compared with the results of electron scattering experiments for the isotopes of Fe, Ni and Zn. It comes out that the collective potential energy surfaces are flat, showing no pronounced minima, and that the masses are smooth functions of the deformation; thus we consider these nuclei to be anharmonic vibrators rather than rotâtors with a well-developed intrinsic deformation. We find that the inclusion of the collective ground-state correlations has an important influence on the charge distribution and an even stronger influence on the isotopic differences of charge distributions and rms radii. The correlations improve the theoretical results towards the experimental ones, e.g. without the ground-state correlations the rms radii differences show unnaturally strong fluctuations . Altogether the results show a fair agreement with the data. We want to point out that this agreement is obtained by a straightforward procedure outgoing from a given force; there is no need (nor any freedom) to fit any parameter anew. The authors would like to thank Prof. D. Drechsel for his constant interest on the subject and for many helpful comments. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

H. Flocard, Proc. Int . Conf. on nuclear self-constant fields, Trieste, 1975 P.-G. Reinhard and D. Drechsel, Z. Phys. A290 (1979) 85 D. Gogny, Nucl. Phys . A237 (197 339 J. Recharge and D. Cogny, Phys . Rev. C21 (1980) 1568 E. B. Skiera, E. T. Ritter, R. B. Perlons, G. A. Rinker, L. K. Wagner, H. D. Wohlfahrt, G. Friche and R. M. Steffen, Phys . Rev . C14 (1976) 731 ; H. D. Wohlfahrt, O. Schwentker, G. Fricke, H. G. Andresen and E. B. Skiera, Phys . Rev. C22(1980) 264 J. Recharge, M. Girod and D. Gogny, Phys. Lett. SSB (1975) 561 M. Girod and B. Grammaticos, Phys . Rev. Lett . 40 (1978) 361 D. R. Inglis, Phys . Rev. 103 (1956) 1786 P.~. Reinhard, Nucl . Phys. A252 (197 133 M. Girod and B. Grammaticos, Nucl . Phys. A330 (1979) 40 J. Eisenberg and W. Greiner, N~lear theory, vol. 1 (North-Holland, Amsterdam, 1970) A. Hohr, B. MotteL4on, Mat. Fys. Medd. Dan. Vid. Selsk. 27, no . 16 (1953) P.-G. Reinhard, Z. Phys . A285 (1978) 93 Nucl. Data, sect. B, vol. 1 (New York, 1966) R. Neuhausen, Nucl. Phys. A282 (1977) 125