Microscopic description of low-lying states in even Ge and Se nuclei

Nuclear Physics A483 (1988) 317-347 North-Holland, Amsterdam

M I C R O S C O P I C D E S C R I P T I O N OF LOW-LYING STATES IN EVEN Ge AND Se NUCLEI A. PETROVICI 1"2, K.W. SCHMID 2, F. GR(IMMER 3, Amand FAESSLER 2 and T. HORIBATA 2'4

i Institute for Physics and Nuclear Engineering, Bucharest, Romania 2 Institut fiir Theoretische Physik, Universitiit Tiibingen, FRG 3 Institutfiir Kernphysik, Kernforschungsanlage Jiilich, FRG 4 Department of Physics, Tokyo Metropolitan University, Tokyo, Japan Received 14 December 1987 Abstract: The Hartree-Fock-Bogoliubov approach with spin and number projection before the variation

extended to the description of nonyrast states (EXCITED VAMPIR) is used to describe simultaneously many properties of the 68"7°'72"74Geand 72"74Se nuclei. Low-lying spectra, neutron and proton occupation numbers for the spherical-basis orbitals, E0 and E2 transitions, and quadrupole moments are calculated and a fairly good agreement is obtained with experiment. The analysis of the EXCITED VAMPIR mean fields reveals that a strong mixing of prolate and oblate intrinsic quasiparticle determinants is responsible for the complex behaviour of the nuclei in A = 70 region.

1. Introduction

Since many years, the nuclei of the A = 7 0 mass region have been studied intensively in experiments and theoretical investigations. Special attention has been paid to the structure of low-spin states in even Ge and Se nuclei or, more generally, to the nuclei around the N = 40 region, because of the puzzling features of their low-energy spectra. The low energy of the first excited 0 ÷ state, the strong dependence of this energy on the mass number, and also several discontinuities observed in systematic studies of several even-even nuclei by means of (d, 3He), (p, t) and (t, p) reactions have been interpreted as possibly indicating that the nuclei in this mass region undergo a shape transition and that some of them may also display a dynamical shape coexistence. Unfortunately, a microscopic insight into the structure of Ge and Se nuclei by exact shell-model calculations is severely hindered because of the large number of valence nucleons that are distributed among many singleparticle levels. Those of the theoretical studies 1-1o), which tried to explain the interesting behaviour of low-lying levels in Ge and Se nuclei in the framework of the usual shell model, had therefore to be carried out in rather restricted configuration spaces, in the other investigations several collective models, and also models involving an interplay of particles and collective excitations were used. A clear understanding of all the known facts, however, is not yet possible. Thus a parameter-free microscopic description of the special behaviour of the low-lying states in even Ge and Se nuclei could be very useful. 0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

318

A. Petrovici et al. / Microscopic description

Recently we have successfully applied a self-consistent description of non-yrast states, the so-called E X C I T E D V A M P I R approach i1), to the energetically lowest four 0 ÷ states in 5°Ti. The basic idea of this model is to obtain optimized mean fields for each state of a given spin separately. In a chain of variational calculations for projected determinants starting from the lowest state for given angular m o m e n t u m I and then stepping up from one state to the next lowest one for the s a m e / , with the current test wave function always being constrained to be orthogonal to all the solutions already obtained, one can describe states of very different structure. Therefore this approach should also allow one to describe the shape coexistence, which is suggested by the data in the Ge and Se nuclei. One even could go so far and say that the complex situation encountered in the A = 70 region is a challenging testing ground for this theoretical model. In the present investigation we shall therefore try to describe the low-lying states in the 68'7°'72'74Ge and 72'74Se nuclei and by this also to learn something about the effective nucleon-nucleon interaction in this mass region. In sect. 2 we briefly summarize the E X C I T E D V A M P I R approach, discussed in detail in ref. 1~). Sect. 3 contains then the results of the calculations and a comparison with the experiment. Finally, conclusions are presented in sect. 4.

2. The EXCITED VAMPIR procedure The m a n y nucleon hamiltonian with an N N interaction renormalized for a finite model space in a given mass region, consisting of general one- and two-body terms is assumed to be given by:

I2I =Y~ t(ik)C~Ck+1 Y~ v(ikrs)c~c~Gcr. ik

(1)

ikrs

This hamiltonian is represented with the help of creation {c~, Ck, ÷ • • -}~t and annihilation {ci, Ck. . . . }M fermion operators corresponding to our model space defined by a finite M-dimensional set of orthonormal spherical single-nucleon states {li), Ik),...}M. The first step to the ansatz of our m a n y - b o d y wave function is to define a Hartree-Fock-Bogoliubov (HFB) quasiparticle basis + d )=-Y, (A,,,(d)c~+ B~,,(d)c,) as( i

a,,( d) =-~ ( B*,( d)c~ + A*~(d)c,) .

(2)

i

The quasiparticle vacuum then has the form

,q(d))=(~ a~(d)),O),

(3)

where 10) denotes the particle vacuum. I f a particular vacuum Iq(0)) of the form (3) is known, then any other H F B vacuum Iq(d)) which is nonorthogonal to Iq(0)> can

A. Petrovici et aL / Microscopic description

319

be represented using the generalized Thouless' theorem 19) as Iq(d))= c( d) exp{ ~<~ d~,~a+~(O)a+~(O)}[q(O))

(4)

c( d) =-(q(O)lq( d)) .

(5)

with

Here d is an antisymmetric parameter matrix, and a~(0) are the quasiparticles of the state Iq(0)). As in ref. 11), for simplicity, we shall enforce conservation of partity and isospin projection as well as axial symmetry on the HFB transformation (3). Using these approximations, the parameter matrix d takes the form d~

=

8(¢r~, ~r~)8(r~, r~)8(m~, -m~)d~ a

(6)

with/~ denoting the time-reversed partner of the quasiparticle state/3. To get physical states we restore the broken symmetries of the HFB-type quasiparticle determinants as described in detail elsewhere 12.13). For this purpose we use the number projectors

t~(N~)=-~-~ f~ dC~,exp {i~,( Ng- ~')}

(7)

with No 1 = Zo and No+l = No, and the operator

^ P(IM, K) - 2I+1 8¢r2

f( 4=) daO~r(a)/~(O)

(8)

with /~(D) being the usual rotation operator and D~K(O) its representation in angular momentum eigenstates 2o), which projects from the I3 = K component of the symmetry-breaking "intrinsic" wave function onto a state with total spin I and z-component Iz = M in the laboratory frame 21). We now follow the VAMPIR procedure 14-17)to get the most general wave function for a given angular momentum I and particle numbers for the protons Zo and the neutrons No from the vacuum (4) under the above approximations Ig'(d)) = I¢~(d))((4)(d)l ¢'(d))) -'/2

(9)

l~(d))- P( IM ; O)Q. (No)Q( Zo)lq( d)) .

(10)

with

Because the HFB variational problem is solved after spin and number projection, the state Igt(d)) and the projected determinant I~(d)) obviously depend on the considered quantum numbers/, M, Zo and No. These are left out in the formulas to ensure a more convenient notation.

320

A. Petrovici et al. / Microscopic description

The minimization of the energy functional for the test wave function (9), E,[d] = ( ~ , ( d ) l / ~ Iqtl(d)) = ( @ l ( d ) l / ~ l @,(d)) (~b,(d)J q),(d)) '

(ll)

together with the diagonalization of the intrinsic quasiparticle energy matrix H ~1 as defined in ref. 14) yields then the VAMPIR solution for the yrast state of a specified angular momentum and the desired particle numbers Iqt,) = Iq)l)/3,,.

(12)

To produce the wave function for the first excited state of the same spin the EXCITED VAMPIR approach uses the ansatz Iq'2(d)) = [4,,)/3,2 + I4~2(d))/322

(13)

and requires orthogonality with respect to 1 ~ ) and normalization of J~,_). From the Schmidt orthogonalization procedure one obtains /322 = [( O2(d )[ ~2( d )) - l( q~,Iq~2(d ))12(( q~,Iq~l))-'] -,/2,

(14)

/3,2 = -((q~ll q~,))-'( ~1l q~2(d))/322•

(15)

Applying the variational principle to the test wave function (13) the parameter matrix d for the second determinant is determined. This procedure can be generalized for higher excited states. Assume that the first n orthonormal solutions for a given angular momentum have already been obtained. The test wave function for the (n + 1)th state has then the form: n

[~.+~(d))= E

[~j)/3jn+l-~'l(~n+l(d))/3n+l

n+l

(16)

j--I

with ft.+, .+, = [ ( O.+,( d)l~b.+,( d))-O++,( d )A-lO.+,( d ) ] -'/2 , /3j.+, = -[a-~o.+~(d)]j/3.+~ .+,,

j = 1 , . . . , n,

(17) (18)

where O.+~(d) is the n-dimensional column vector [O.+,(d)]j=(Ojl~.+l(d)),

j=l,...,n,

(19)

and A the (n x n) overlap matrix Ao = (¢'i[ ~j),

i,j = 1 , . . . , n

(20)

of the in general nonorthogonal projected determinants. The variational equations for the (n + 1)th state of a given angular momentum are obtained by the minimization of the energy functional: E.+,[d] = (~.+,(d)lI2II ~ . + , ( d ) ) .

(21)

A. Petroviciet al. / Microscopicdescription

321

This yields a system of equations

OE[d][

=0,

(22)

which has to be complemented with the diagonalization of the H ~ matrix belonging to the vacuum [q(d,,+O). Finally, the lowest m wave functions of the E X C I T E D VAMPIR procedure in a m-dimensional configuration space [~b~)=

11Pj)fj~= j=l

E I@,)/30f~, j=l

a=l .... ,m

(23)

i=l

and the corresponding energies E~,(a = 1 , . . . , m) are obtained by diagonalizing the residual interaction ('/',IHI~) between the orthonormal states ]qt i)

( H - E~)f = 0

(24)

f + f = 1.

(25)

with

The use of the variational principle for each state I~i) minimizes the residual interaction between the different m solutions and creates the optimal m-dimensional basis for the lowest m states of a given angular momentum. In spite of the symmetry restrictions we imposed on the HFB transformation the resulting projected determinants may have a completely different structure from each other. The essential degrees of freedom are automatically selected. Furthermore, the E X C I T E D VAMPIR approach can describe strong configuration mixings between states of very different structure (e.g., shapes or pairing properties).

3. Results and discussions

For our calculations in the A = 70 mass region we used a 4°Ca core and took as active single-particle orbits the lp~/2, lp3/2 , 0t"5/2, 0t"7/2, 2S~/2, ld3/2, ld5/2, 0g7/2, and 099/2 proton and neutron levels into account. For neutrons in addition the 0h11/2 state was included. As single-particle energies for this basis the values 0.040, -0.270, 0.300, -0.560, 0.513, 0.730, 0.157, 0.756 and 0.029 for the protons and -0.070, -0.332, 0.130, -0.690, 0.356, 0.614, 0.079, 0.797, -0.043 and 0.530 for the neutrons were chosen. All these values are in units of hoJ and their order corresponds to the order of the levels given above. The effective two-body interaction was based on a microscopically calculated Brueckner G-matrix for nuclear matter, derived from the Bonn OBEP 18) nucleonnucleon interaction. These matrix elements were slightly renormalized in a phenomenological way in order to describe the deformation and pairing behaviour in the region correctly. A global renormalization was obtained adding two shortrange central gaussians (both of 0.707 fm range) with strengths Vpp = - 5 0 MeV in

322

A. Petrovici et al, / Microscopic description

the T = 1 pp channel and Vnn 40 MeV in the T = 1 nn channel and a spin-orbit :

gaussian o f 0.5 fm range with strength VLs(T = O, 1) = - 1 5 0 0 MeV. The onset o f the deformation was influenced introducing a monopole shift of - 0 . 1 1 5 MeV to the

T = 0 matrix elements (fg9/2, I T = 0 IG[ fg9/2, I T = O) with f being either the f7/2 o r t"5/2 orbit. Using this renormalized hamiltonian we applied the E X C I T E D VAMPIR (EXVAM) approach to some low-lying states in 68Ge, 7°Ge, 72Ge, 74Ge and 72Se, 74Se nuclei. Figs. 1-6 present the calculated and the experimental spectra of these nuclei for the investigated spin states. The maximum number of A-nucleon E X C I T E D VAMPIR configurations for one angular momentum was five. For each angular momentum we show as many states as variational solutions of eq. (16) had been constructed by the EXVAM procedure. The HFB-type projected determinants of eq. (10) have been obtained using different trial vacua: we started from an oblate and a prolate

~ie

a

6.0 6

-4 4,o

8--

(6)

2 o

6 A.. o 4

IAJ

Z 0 4C 1-

2.O

AI

o k.

2

o

2-

J

0

0 EXP

TH

Fig. 1. Comparison of the theoretical and experimental tow energy spectra for 7°Ge.

A~ Petrovici et aL / Microscopic description

i

l

'

I

I ' I o~

'

o

o~

I'

'

I

oo~

323

i

I

'

I

c~o

o

r-

4) ¢9

o

....... ,

~.I

I

I '~I

~I

,

AOlfl I AgI:I~ N3

I

'

"I

.....i.....

~I,

e~o

o4

I ~I

n ~I

, ......~I

I

I ........... I

r~

NO I.LVII3X~

I ........... i

I

' o

co

o

ll

¢,q

o

~O

o

, AeN / AOHgNg

NOI JLVJ.I:)X3

qol

'~

U

"G

324

A. Petrovici et aL / Microscopic description

sO

-

~Ge 0

q,o

2 2

>. =E

at)

o w

o

2

nr

itU

2D

0 2

•

2

~

Z

o F-

0

o.__o

o

o EXP

TH

Fig. 4. Same as in fig. 1, but for nucleus 74Ge.

trial wave function for the first guesses of the [q (d)). Furthermore, we used also less and more deformed zero-order approximations to initiate the minimization of the energy (11) with respect to the Bogoliubov HFB transformation. As numerical minimization procedure the so-called BFGS method was used, which is described in detail e.g. in ref. 5~). One obtains VAMPIR wave functions, which are orthogonaiized to each other, in various local minima. The A = 70 region is a good example to illustrate the power o f this E X C I T E D VAMPIR method since one expects due to the various gaps in the Nilsson spectrum minima at rather different deformations. As a first state in a successively created optimal basis o f A-nucleon configurations the most bound from the projected determinants so produced is taken. For the next states of the form (16) the variational principle guides us to choose the appropriate (n + 1)th projected determinant. Diagonalizing the residual interaction within the lowest m variational states one obtains the best possible description o f the m lowest states of the considered spin value, which can be achieved by rn projected HFB type quasiparticle determinants. Of course, increasing the number of successive solutions m, we improve the structure o f any wave function of the type (23), since the diagonalization of the residual interaction between many more orthogonal states introduces more correlations in the final EXVAM wave functions. This is illustrated in figs. 7, 8 for the 0 ÷, 2 +, 4 + states in 72Ge and for the 0 +, 2 + states in 745e,

A. Petrovici et al. / Microscopic description

325

~Se 4.O

I

lb.

- -

o ~e ttl

zZO w Z

o I-

2

2 0

w m

~.0

0

O EXP

TH

Fig. 5. Same as in fig. 1, but for nucleus 72Se.

~'Se 4.._o

,o

=P

u

:E >. ¢3 E UJ Z uJ

3.0 2

-

Z _o __

0

(~) (o)

o

2

2

_

I,I-

m X .1

;~ 1

O

0 2

0_.0

2

O

,,0 EXP

TH

Fig. 6. Same as in fig. 1, but for nucleus 74Se.

326

L

A. Petrovici et al. / Microscopic description

~Ge ~O

>

~_o

~o

o1

z

P

~.O

F

O+--states

2÷I st ates

~+-- states

Fig. 7. The lowest 2, 3, 4, 5 solutions including the diagonalization of the residual interaction for 0÷, 2+, 4+ states in 72Ge. The dimension of the EXVAM configuration spaces increases successively going from the right to the left. In the most right column for each spin the lowest prolate and oblate projected quasiparticle determinants are indicated. The lowest of the EXVAM energies for 0+ states calculated with residual interaction in the largest configuration space (five in this case) is set to 0 MeV. All other energies are given with respect to this zero point. respectively. In the m o s t right c o l u m n o f e a c h b u n c h o f states the lowest p r o l a t e a n d o b l a t e first p r o j e c t e d d e t e r m i n a n t s are p r e s e n t e d . Then, g o i n g f r o m the right to left the E X V A M results with r e s i d u a l i n t e r a c t i o n are given for a basis i n c l u d i n g an i n c r e a s e d n u m b e r o f solutions. As o n e c a n see f r o m the plots a d d i n g one state to a p r e v i o u s c o n s t r u c t e d basis m o r e c o r r e l a t i o n s are i n t r o d u c e d in the a l r e a d y p r o d u c e d states o f a given spin, a n d so t h e s e b e c o m e m o r e b o u n d . N e v e r t h e l e s s , the c o n v e r g e n c e is a c h i e v e d fast e n o u g h , as it s h o u l d be, a n d we d o n o t n e e d t o o big c o n f i g u r a t i o n s p a c e s to get a g o o d d e s c r i p t i o n o f the first few l o w e s t states for a given a n g u l a r m o m e n t u m . W i t h two to five c o n f i g u r a t i o n s d e p e n d i n g on the m a x i m u m n u m b e r o f e x p e r i m e n t a l l y k n o w n states for a given a n g u l a r m o m e n t u m , o n e o b t a i n s results with the E X C I T E D V A M P I R ( E X V A M ) w h i c h seem to yield a g o o d d e s c r i p t i o n for the structure o f the different states. To get an i d e a a b o u t the structure o f the E X C I T E D V A M P I R wave f u n c t i o n s for l o w - s p i n states in the i n v e s t i g a t e d G e a n d Se i s o t o p e s we d i s p l a y in t a b l e 1 the a m p l i t u d e s f ~ f r o m (23), the e x p a n s i o n coefficients flij from (16) a n d the o v e r l a p m a t r i x o f the p r o j e c t e d d e t e r m i n a n t s (20) for the lowest five 0 +, 2 +, 4 + states in 72Ge. The o b l a t e o r p r o l a t e n a t u r e o f e a c h E X V A M p r o j e c t e d d e t e r m i n a n t is t r a n s p a r e n t f r o m t a b l e 2. H e r e we

A. Petrouici et aL / Microscopic description

327

~Se 0

"'l Lo

I~I.0

O

I,U

P

P o

m

~O

O+-- states

~--

states

Fig. 8. Same as in fig. 7, but for 0 + and 2 + states in 74Se.

Ge .,

o,

~o

~

isotopes

04

o~

(o,)

(o5) o,

(o,) _

03

O3

04

o,

3,0

-

0-~

O3

03

02

2.0

Oi

o.o

e, hO

O,

~.o

EXP

04

TH

O,

EXP

04

TH

O,

EXP

o,

TH

Oi

EXP

TH 74

MASS NUMBER A Fig. 9. Comparison o f theoretical and experimental energy trends for the 0 + states in the investigated Ge isotopes.

A

A

•~ ~ o

A

~o. o ~ • O.

~ r=

. N. . .

o o o

II A

',~" ~

o

',d-

~

o

o

o

~

"~

I

I

I

~

o

o~ ~

~

I A

o

I

I

+

A

A

A

II

A

I

.< h,

,~

A oe,l

II

II

I

--

A

I

I

@

e-

> o

A

~,o

o

II

I I

oO ~ 0 0 ~

A

I I I

d

I

d

0

d

d

"d"

d

~ ° ° ~

..0

m

~

x

~.~

I I

< A oe-i

Ndddd I

-I

c~ ~ c ~ o o ~ ; ; ;

I I / I

A o--

ddddd I

~.~

I

I

-I

I

~.~ .~

~

A. Petrovici et al. / Microscopic description

329

TABLE 2 The type of the intrinsic deformation of the projected quasiparticle determinants IO~) successively created by the EXCITED VAMPIR procedure for each spin a)

6,8Ge36 OPpOO 2opopo

7°Ge38 Oopopo 2oopop 4oopop 6opop 8opopp

72Ge40 Oopopo 2oopop 4oopop 6opp 8PP

74Ge42 Ooppo 2oppop

~2Se38

74Se40

Opopo 2opo

0 p°p° 2oop

a) The lowest energeticallydeterminant is specified by o(p) for oblate (prolate) in the first place of the sequence going from the right to the left.

give for all calculated spin states in the Ge and Se chains the sequence of projected HFB determinants 10~) built by the E X C I T E D VAMPIR procedure, and used to construct the configuration space for each set of quantum numbers (/, Zo, No). The notation 2 °°p°p has to be read in the following way: the lowest (i = 1)lO~) for spin I = 2 has a prolate character, the second is an oblate one, the third a prolate again, the fourth and the fifth are oblate quasiparticle determinants. As one expects these determinants are nonorthogonal and this can be read from the overlap matrices given in table 1. The mixture between the different Schmidt-orthogonalized variational principle states 1~ ~) and also the EXVAM states I• ~) before the orthogonalization can be found in table 1, too. The final mixing is obtained after the diagonalization of the residual interaction <~gl/~l~f>, and we give the amplitudes for the first five 0 ÷, 2 +, 4 ÷ states in 72Ge, to visualize a typical configuration mixing produced by the E X C I T E D VAMPIR approach for this shape coexistence region. In contrast to the application in ref. n), where the structure of the first four 0 ÷ states was obtained by the EXVAM procedure, here the mixture between the orthogonalized configurations is rather strong even in the lowest states of a given angular momentum. Often a strong mixing of oblate and prolate quasiparticle determinants in the final wave functions is obtained. Also determinants characterized by the same sign of the deformation, but different absolute values of the quadrupole moments are mixed. This can be easily seen not only from the structure of the wave functions illustrated in table 1, but also from some information presented in figs. 10-15. In these figures we show for all EXVAM 0 ÷, 2 ÷ and 4 ÷ states in 68"7°'72'74Ge and 72'745e nuclei the intrinsic quadrupole moments for both neutrons (circles) and protons (squares) for the ith projected determinant 10~) and for the 2 + and 4 + states the corresponding values calculated from the spectroscopic quadrupole moments of the physical states. Due to the strong mixing of prolate and oblate projected determinants in the final wave functions the intrinsic quadrupole moments are often drastically reduced with respect to those of the projected nonorthogonal determinants. Moreover, even the sign of the intrincic quadrupole moment for a state may appear changed with respect

A. Petrovici et aL / Microscopic description

330

>.

!

~Ge

i.~

l.e

~ 1.6

1._6

~" 1.k

1./,

1.2

1.2

1,0

1.0

3OO

~ -a.o ~

Z

-~0

;I z ,,10

;1

(~

0-4

wave lunctionl [EXCITED VAMPIR ]

neutr~s

Ik-4

-JO0

prot~s

: :

~

0

-100

-100

-ZOO

?, z,-200

D 0

Fig. 10. The intrinsic neutron and proton quadrupole and hexadecapole moments of the lowest five projected quasiparticle determinants obtained by the EXVAM procedure for the 0 + and 2 + states in 7°Ge and the intrinsic quadrupole moments obtained from the corresponding spectroscopic value for the E X C I T E D V A M P I R wave functions calculated with residual interaction (full symbols are used). No effective charge has been used. The average neutron and proton pairing gap, defined in ref. ts), is given also for the above specified determinants. Squares refer to the proton, circles to the neutron, open symbols are for projected determinants.

331

A. Petrovici et al. / Microscopic description

Symbols defined as for

7OGe 1,._66

~

< ~,.

1.~ 1.2

~o ~

0

z

w

too

o

o -1oo

uu

re

0_.

o

zO ,,,

-'1oo re"

-200

p,

,,

,o,

p,

Fig. 11. Same as in fig. 10, but for calculated 0+, 2+ and 4+ states in 72Ge.

to that for the corresponding dominant determinant of the ith variational solution as it is the case for the second, third and fourth 2 + states in 7°Ge and the second 2 + state in 72Ge. The nature of each projected determinant included in the calculations for a set (/, Zo, No) of states, which was labelled only by o (oblate) or p (prolate) in table 2, is pictured in figs. 10-15 by the intrinsic quadrupole and hexadecapole moments and the average pairing gaps. The details for these calculations are explained elsewhere 15). The mentioned quantities are given for both neutrons and protons. In all the cases under investigation the nature of the ith state (16) is mainly dictated by that of the ith produced EXVAM projected determinant in a successively created optimal basis. The shape coexistence in the Ge isotopes is corroborated by our results. A tendency from an almost pure oblate 0 ÷ ground state in 68Ge to a strongly

A. Petrovici et al. / Microscopic description

332

>

eSGe

1-~

I

1.2


1.O

1o_.0

~.o

(jOloo

~.

z_*,

100

o

o

--loo

.r-2oc

-200 Symbols defined as for

~E 100

i:E_0

,~

7OGe

A,A

0__

z_.o -~__oo ,o,

-200

o Fig. 12. Same as in fig. 10, but for calculated 0 + and 2 ÷ states in 6aGe.

mixed prolate-oblate one in 'e"4Ge is predicted. For I # 0 states this will be seen directly from the theoretical spectroscopic quadrupole moments, which will be discussed later on. For the ground states of the studied Se nuclei a tendency to an increasing prolate mixing going from 72 to 74 isotopes can be observed. For the excited 0 ÷ states both Ge and Se isotopes manifest a strong prolate-oblate mixing. In fig. 9 we plot the energies of all the calculated and experimentally known 0 ÷ states in the 68Ge, 7°Ge, 72Ge, 74Ge isotopes. From here and also from the spectra given in figs. 5, 6 for the Se nuclei we can see that the experimental trend for both the Ge and Se chains are fairly well reproduced by the theoretical calculations with the main discrepancy being that the first excited 0 ÷ state in 7°Ge stays below the 0F state in 72Ge in the calculations.

A. Petrovici et ai. / Microscopic description

333

"Ge

1.8 1.6

<~

1.2

I~)

1.0

300 •e E

too

oO

0

z_., n- 0 I.-L

-ZOO

Symbols defined as for

70 Ge

-300

zo~

u~--

z_~.loo e,

p,

o, p,.

Fig. 13. Same as in fig. 10, but for calculated 0+ and 2+ states in 74Ge.

More information about the structure of the yrast and non-yrast states can be obtained from the spherical occupations of both neutron and proton orbitals illustrated in figs. 16-21 and in table 3. In figs. 16-17 we give the significant occupation numbers for the first four 0 ÷ and 2 + states in 7°Ge and 72Ge nuclei obtained from five EXVAM variational solutions. As we can see the important proton orbitals for the 0 + states in both the 7°Ge and the 72Ge isotopes are the lp3/2 , 0f5/2, 0f7/2 and 0g9/2 levels, while on the neutron side the orbitals with changing, occupation from o n e 0 + state to another ~ e the lp,/2, lp3/2, 0g9/2 levels and for 72Ge the 0t"5/2 state,

A. Petrovici et al. / Microscopic description

334

+=

,,f

~r

2


<5_

E

AelN/dVO ONIEI IVd 3OV~3AV

mj.a/J.N=qNOIN =r'IOdVO=~QVX3H ~JJS/1N31NOIN =I-IOdf'IEIQVFID OlSNII:iJ.NI 31SNIt:I/NI

v; L~

¢; .=_

oal

ooj

ol

o~1 .,-~1 N

d.

O_

c~.

AalN/dV90NII:IIVd 3DV~::IAV

._=

~1 ~1 ol ~,1 ~,1

~uje/1N=IINOIN :I"IOdVO3QVX3H ~m~/1N=IINOIN 3"IOdFIEIQVND OISNII:I/NI OISNIt:l ILNI L~

A. Petrovici et aL / Microscopic description

335

8

3~ e~

o

~'~

o

t ' , ~ t ",-

P-r'.,.

r . . . - t ,~

t , ~ . t ".~

t'.~t'.-

t',-t'~

~,~ . ~

0

" 0

< ¢'.,le.l

e.le.l

m m

m m

e4eg~

eg, eg~

~'~

aa

,.o e~

• -~

~

o e~

.o.

~'~ aS

© 4~

4~

"--2.

4 ~

~

~-'~ ~ ~ ~

e2~

~

44

"--2.

+-"~ +~.~, o~ o~

~

4~

~

g=

"--2.

.. .. +~'~ ~ .~

,-

~

0

336

A. Petrovici et al. / Microscopic description full lines: neutrons dashed lines: protons

--

f12 ~ - - ~ _ . _ . ~ s t"h

7.O m

u~

U,J -

f5/2

°

~

_ a

Z 0 g.O

r--. ¢J 0

3~

ff5/2 ~

",,

f5/2

~ ......

io f5¢~ ¢ ' /

Pl/'J. /

,/

/

/

.....

io:"

i~

_.._

d

.....

i°I

i°1

i~I,

14

12~

iz~

Fig. 16. Significant neutron and proton occupation numbers of the spherical-basis orbitals for the lowest four 0 ÷ and 2 + states in 7°Ge. Full lines are for neutrons, dashed lines for protons. The results have been calculated from the full wave functions (23).

too. The trend and also the value of the number occupations for both neutrons and protons are different in 7°Ge than in 72Ge and so it becomes once again clear that the structure o f 0 ÷ states is not the same in these two isotopes. In other words, the underlying prolate-oblate shape mixing is changing. A similar behaviour can be seen in figs. 16, 17 for the lowest four 2 + states in these two isotopes. However, for the 2 + states the occupation numbers do not vary so much from one 2 ÷ state to another in a given nucleus. The large effects o f the orthogonalization and of the diagonalization o f the residual interaction show up clearly in an analysis o f the neutron and proton occupation numbers displayed in table 3. Two successive lines o f this table contain the significant occupation numbers for neutron and proton orbitals for some selected a = i states I~b~) ofeq. (23) and the corresponding projected determinants I~[) in the Ge isotopes. Numbers are presented for the first and second 0 ÷ states in 68Ge and 7°Ge, and also for the first and second 2 ÷ states in 72Ge. More about the similarities and the differences concerning the structure of a particular state in a chain o f isotopes can be learned from the trend of the proton and neutron spherical occupation numbers given in figs. 18-21 for the first three 0 ÷ states and the first two 2 ÷ states in all the investigated nuclei. We can also easily

A. Petrovici et aL / Microscopic description

337

full lines: neutrons

~Ge

dilhld

lines : :rotons

so j

~

5.d

s.o

•.og ~ ~ f ~ ...... f rd~ ° -

..I

. . . .

s

~._ - - ' -

~.o

f

~ -

I°:

_

. ."-. .

i~

*,

~.

I "

s :__ -_. . . . .

i°;

"

io,~

~

l~"

~

i=;

oz

i=; "

i ~"

Fig. 17. Same as in fig. 16, but for 72Ge.

compare these properties in a chain of isotopes. We start with an analysis of the occupation probabilities of the proton levels for the 0 + states in the Ge isotopes. Large differences in the occupation numbers can be seen for the lp3/2, 0f5/2 and moderate changes for the 0f7/2, 0g9/2 orbitals. The total occupation probabilitity for the lp3/2 and 0f5/2 orbitals is almost constant in the range 3.6-3.8 and only accidentally lowers to 3.2. The corresponding values deduced from the (d, 3He) reactions 22) o n 7°'72'74'76G¢ are very close to 3.6 for all these Ge isotopes. A drastic increase of the 0f5/2 occupations is realized at the cost of the lp3/2 orbit or vice versa, but the sharp variation of the occupation occurs for the different states at different points o f the isotopic chain: between N = 38 and 40 for the ground state, between N = 40 and 42 for the first excited 0 + state. For the third 0 + state two drastic changes are observed: first between N = 38 and 40 and a second one, in an opposite direction, between N = 40 and 42. Therefore, by a simultaneous analysis of the three lowest 0 + states in the 68-74Ge isotopes we will not find a special point, which would

A. Petrovici et al. / Microscopic description

338

r; g~

,) / w

w

•

•

°° ~I .~..=

¢_ =

o

,,

¢o

,o_

t,

l'

l

I

o

I

I

o

1

o

3

e~

r~

o

q

o.

S=I'IOIINVd,/NOIIVdnO00

o NOIONd

A. Petrovici et al. / Microscopic description

339

l,

I= c~

o

¢=

.I

-.. ".~\

\'\.

..o oo"

°ii

.=_

o,)

j!

I ~,,4

"'-.

0

..,Io

,,

\\-'-----Z

',

'7-I

'--..~,._Z:-----.~

I~I

~,,,

S:il OIlHVd/NO

/i

~'I,,,

t-QI,,

IJ,VdnO00

NOI::I.LFI3 N

,,°4j,

',\~!

t~l

A. Petrovici et al. / Microscopic description

340

EXCITED

VAMPIR

a._c

full lineS: Ge isotopss dashed lines: Sa isotopes

zo

~/

~. ¢ a , . / ~ ' ~ ~"

I

..__-------

_ i.co

--

~p . . , " * ' ~ - - - - ~ ~ t.

' •

~o_ Gs

ss

168

70

I'r,

72

.......... I,,

74

'

Gs Se

68

70

7~

I .....................

Fig. 20. S a m e as in fig. 18, but for the lowest t w o 2 + states.

correspond to a drastic change in the proton structure. The theoretical results for the proton number occupations for the ground state in the Ge isotopes are in a good agreement with the experimental results obtained from (d, 3He) transfer reaction on 7°-74Ge targets 22). It should be noted here, that from (t, p) and (p, t) reactions 23-28) a structural change around N = 40 is observed, which sometimes is stressed as an evidence for a structural transition in Ge isotopes occurring between N = 40 and 42, but on the other side it is considered that the data from Coulomb excitations 29,30) clearly reveal a change of nuclear properties between N = 38 and 40. Information about the proton occupation changes in the lowest three 0 + states can be extracted from fig. 18 also for the two investigated 72Se and 74Se isotopes which correspond by the number of neutrons to the 7°Ge and 72Ge isotopes, respectively. We see immediately the large similarity for the occupations in the

341

A. Petroviei et al. / Microscopic description

2~

EXC,,~D VA.~,.

2;

8,0

8....~C

full lines: Ge isotopes deMted

hnes: Se isotopes

°

/ I II

f~ j

g

|

i

1

.-J

~"~ .....

:

-

.

-:/.

. 0.@

168

So

70

7"2

I¢~

174

74

68

70

~

74,

I

Fig. 21. Same as in fig. 18, but for the neutron occupations of the lowest two 2 + states.

ground states of the two Se nuclei and the small differences for the second and the third 0 + states. This microscopic picture obtained by the EXCITED VAMPIR procedure agrees with the experimental results from (p, t) reactions 31,32). A survey of the properties of even-even nuclei in the Ge region 33) obtained from transfer reactions concludes also that there is no striking evidence in Se for a dramatic change in the ground-state structure. Our prediction for 72Se and 74Se is also in

A. Petrovici et a L / Microscopic description

342

agreement with the results obtained for the proton occupancies in the even Se ground states via the (d, 3He) reaction 34), but we intend to extend our investigation to some other Se isotopes to study also the trend in the Se chain. Using a similar plot the neutron occupation numbers for the first three 0 ÷ states are displayed in fig. 19. On the neutron side drastic variations could be seen for the g9/2 and f5/2 occupations, but significant changes appear also for the P3/2 and P]/2 levels in both the Ge and Se nuclei. These orbitals could make a large contribution to the 2 neutron-transfer cross sections. Finally both proton and neutron occupations for the first 2 ÷ states in all the investigated Ge and Se nuclei are displayed in figs. 20 and 21. On the proton side, clearly, the large variations observed in the analysis of the 0 ÷ states, cannot be seen here. The explanation resides in an almost equal contribution of the prolate and oblate projected determinants to the wave functions for all 2 ÷ states in each investigated nucleus. Further information about the structure of the states and mainly about the prolate-oblate interplay can be extracted from fig. 22 and table 4, where we give the spectroscopic quadrupole moments for some 2 ÷ states in all the nuclei under investigation. The electromagnetic properties have been calculated using one and the same effective charge e, = 0.25 and ep = 1.25 for all nuclei and states. From a comparison of the theoretical quadrupole moments for the first 2 ÷ states and the available experimental data illustrated in fig. 22 it becomes clear that our prediction for the underlying microscopic structure of these states is very good. By a strong mixing of intrinsic large deformed prolate and oblate projected quasiparticle determinants we can obtain very small quadrupole moments for the physical states. The results given in the first column of table 4 support the idea that a transition from more oblate to more prolate deformed configurations can be seen going from 68Ge to 72Ge and also from 7 2 S e to 74Se. The trend seems to be the same going from 72Ge

E

• Th. • Exp.

c

Ge isotopes

Se isotopes

E o E

iI

o

!

IF

~8

70 I

I

72

I

t 72

74-

I

I

I

Fig. 22. C o m p a r i s o n of the t h e o r e t i c a l s p e c t r o s c o p i c q u a d r u p o l e m o m e n t s for the first 2 + states in the i n v e s t i g a t e d G e a n d Se n u c l e i a n d the c o r r e s p o n d i n g a v a i l a b l e d a t a ( E X P ) . The effective c h a r g e is e n = 0.25 a n d e r = 1.25.

A. Petrovici et al. / Microscopic description

343

TABLE 4 S p e c t r o s c o p i c q u a d r u p o l e m o m e n t s in units o f e fm 2 c a l c u l a t e d for s o m e 2 + states in 6s'7°'72'74Ge a n d 72"748e nuclei

6aGe 7°Ge 72Ge 74Ge 725e 7ase

2~

21

2;

28.9 -1.6 - 14.4 -6.8 33.9 - 15.3

-29.1 - 15.0 - 16.6 3.9 43.7 2.4

14.4 20.8 -0.2 -29.7

The effective charges were e n = 0.25 a n d ep = 1.25. The e x p e r i m e n t a l d a t a for the first 2 + states are g i v e n in fig. 22.

to 74Ge, if we look to the amplitudes of different configurations in the wave functions, but at the same time we can see from figs. 11 and 13 that the intrinsic quadrupole moment of the first projected determinant is larger in 72Ge than the corresponding one in 74Ge. To understand, if the decrease of the absolute value for the spectroscopic quadrupole moment of the first 2 + state in 74Ge with respect to the corresponding value for 72Ge is a clear indication for a transition to spherical shapes starting with74Ge, it would be useful to make calculations also for 76Ge. Analysing the predicted results presented in table 4 we can conclude that the relatively small absolute values for the spectroscopic quadrupole moments of the lowest two or three 2 + states in the investigated nuclei suggest that a strong prolate-oblate configuration mixing still exists in the structure of the second and the third 2 + states. In table 5 we present the calculated B(E2) values and compare them with experiments, although the available data for the investigated low-spin states are still very limited. The experimental results are taken from refs. 35-43) and refs. 44-5o) for the Ge and Se isotopes, respectively. Using the same effective charges for all Ge and Se nuclei we predict B(E2; 21-~01)'s which indicate an increased collectivity with increasing N, but for 7 ° G e the theoretical value is by a factor 4 smaller than the experimental result. If we trust the present experimental result for 41-->21 transition the calculated value in 7°Ge is by a factor 3 larger than the unusual small measured B(E2). The corresponding values in 72Ge agree very well. Small values for the crossover B(E2; 22-->01) transitions are predicted, but for 7°Ge the value is still too large in comparison with the measured quantity. G o o d agreement between theoretical and experimental values can be seen for B(E2; 22-->21), and it could be improved if the experimental B(E2) would include B(M1) admixtures. As for the strong collectivity revealed by the experimental B(E2; 02 ~ 21)'s, this comes out also from theory (but still too small or too large by a factor of 2-2.5) excepting the corresponding value for riSe, which is totally wrong. The large value of B(E2;

A. Petrovici et al. / Microscopic description

344

o ~d

~

t~

o~

r-.

o

o

o

-H44~d

o 0

e~

ua 0 ¢'4 t~

o

[.-

ua

H o

0 o

II 0

,4. Petrovici et aL / Microscopic description

345

• Th. " Exp

~._.~

Ge Isotopes

Se isotopes

~J "d 5 ~o

O0 68 I

170

72

I

I

I

I

Fig. 23. Comparison of the theoretical E0 transitions p(E0; 0~-~ 0~) in the investigated Ge and Se nuclei and the corresponding available data (EXP). The effective charge is e. = 0.25 and ep = 1.25.

02-> 22) in 72Se is very well reproduced, the moderate one in 7°Ge is also well predicted, but for the corresponding very small experimental value in 72Ge the theory gives a far too large result. We predict also B(E2)'s for 42->22 and 42->41 transitions in 7°Ge and 72Ge. Here only the B(E2; 42-->22) in 7°Ge is measured and this value agrees with the corresponding theoretical result. We have also calculated the E0 transitions and we give a comparison with the experimental data for E0 transition matrix elements p(E0; 02->01) in fig. 23. The theory predicts strong E0 transitions as they are displayed by the available data for the investigated Ge and Se isotopes.

4. C o n c l u s i o n s

In the present paper we have applied the EXCITED VAMPIR approach 17) to some low-spin states in the Ge and Se isotopes. This method allows by projecting angular momentum and proton and neutron numbers before the variation from a Hartree-Fock-Bogoliubov solution to define for each state an optimal selfconsistent field. We have shown here that such calculations are also feasible in large singleparticle model spaces. Our main aim in the present investigation was to find a realistic microscopic description of the complicated structure of low excited yrast and non-yrast states in the A = 70 region. The results obtained for 68'7°72'74Gg and 72'74Se nuclei provide a new microscopic understanding of the shape transition and shape coexistence supposed to be responsible for the strange behaviour of low-spin states (I ~<4) in these nuclei. The analysis of the intrinsic properties of the projected HFB-type quasiparticle determinants used to build the optimal basis of the A-nucleon configurations for each spin revealed the nature of the underlying structure of the wave functions. A variable mixing of prolate and oblate projected determinants is able to produce the trend observed in the experimental spectra for states of spin 0 ÷ and 2+. Even more, the calculated spectra for 7°Ge and 72Ge look good also for higher spins up to 6 +. The 8+ states, however, are not yet correctly described, and also for

346

A. Petrovici et al. / Microscopic description

the 68Ge a n d 7aGe nuclei still large r o o m for possible i m p r o v e m e n t s does remain. The m i x i n g o f more or less d e f o r m e d prolate a n d oblate d e t e r m i n a n t s p r o d u c e s p r o t o n a n d n e u t r o n spherical o c c u p a t i o n n u m b e r s , which agree with the available data o b t a i n e d from transfer reactions. O f course, we have still to extend o u r i n v e s t i g a t i o n also to o t h e r G e a n d Se isotopes in order to confirm these findings. We have o b t a i n e d good results for the E0 t r a n s i t i o n s B ( E 0 ; 0~--> 0~) a n d for the spectroscopic q u a d r u p o l e m o m e n t s . As for the B ( E 2 ) probabilities the m o d e l gives good a g r e e m e n t with most o f the available data, b u t there are still discrepancies in some cases. N a t u r a l l y o u r results d e p e n d o n the choice o f the effective i n t e r a c t i o n , which we use for o u r m o d e l space. It is clear that we have still to learn m u c h more a b o u t the i m p o r t a n t aspects of this i n t e r a c t i o n in A = 70 region a n d to find a better r e n o r m a l i z ation of the h a m i l t o n i a n in o r d e r to i m p r o v e o u r description. The p r e s e n t results p r o v i d e j u s t the i n c e n t i v e for further m i c r o s c o p i c a l investigations in this mass region.

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