Collective bands of the positive parity states inf7/2 shell nuclei

Collective bands of the positive parity states in fT/z shell nuclei

D P AHALPARA

Physical Research Laboratory Ahmedabad 380 009 MS received 16 December 1977; revised 23 January 1978.

Abstract. Collective bands of the positive parity states in odd-A fT/~ shell nuclei are described in the framework of deformed Hartree-Fock theory by projecting states

--1 ~n-t-l~

from lowest energy intrinsic states with d312. I7/2 ] one hole configurations. In the calculation empirical (d3/~-fTt~)2 effective matrix elements have been used to test the tacit assumptions of the Bansal and French model.

Keywords. Collective bands; positive parity states; empirical effective interaction;

Bansal-French model.

1. Introduction

Several odd-A f~/~ shell nuclei have displayed collective bands of low-lying positive parity states (Tabor et al 1975a, b; Nann et al 1975, 1976; Poletti et al 1976; Alford et al 1971; Blasi et al 1970, 1971, 1973; Kownacki et al 1973; Toulemonde et al 1974, 1976; Adachi and Taketani 1973; Eichler and Raman 1971; Haas and Taras 1974; Haas et al 1975 a, b; Chowdhary and Sen Gupta 1974; Schulz and Toulemonde 1974; Taras et al 1974; Samuelson et al 1977, Maheshwari et a11971 ; Tarara et al 1976; Noe et al 1974). These states often being populated through neutron and

- 1 proton pick-up reactions on even-A .fT/¢ shell nuclei have been regarded as d3/2 hole states. From a simple harmonic oscillator shell model point of view the states resulting from promotion of a nucleon from a fully occupied lds/~ orbit to 1 f~/~

orbit would occur at ~oJ energy, which for nuclei near 4°Ca is ~-~12 MeV. But sur- prisingly enough, many of the positive parity states in odd-A f7/2 shell nuclei occur at very low excitation energy, as low as 0"01 MeV for the first excited 3/2 + state in 45Sc.

There have been basically three different kinds of attempts in studying the energy systematics o f t h e hole states in fT/a shell nuclei namely the shell model (Bansal and French 1964), the phonon particle coupling model (Mittig et al 1974) and the rotation particle coupling model (Styczen et al 1976). In this paper we have considered the shell model approach and have attempted to examine the internal consistency between the recently determined empirical effective interaction in (da/~--fT/2) ~ space and the Bansal-French (BF) shell model description of the hole state systematics.

The earliest attempt to study the energy systematics of the hole states in f~/a shell 399

400 D P Ahalpara

nuclei is by Bansal and French (1964). In the (da/2--f71z) 2 model space they consider a°Ca have closed (da/2) s structure and assigned

wave function to the lowest 3/2 + states. In the BF model the hole states having - 1

dominant components of 2sl/2 or lgg/2 single particle states do not arise. Their emphasis is only on the energy systematics of d - ! hole states. They employed a 3/2 monopole isospin dependent particle-hole interaction of the form V t j ~ - - a + b t I • tj.

The energies of (f~/z)n and (.f7/2) ~+1 nucleons needed for calculating the excitation energy of the 3/2 + hole states relative to the 7/2- ground states were obtained from the empirical binding energies. By varying the two parameters a and b of the particle hole interaction they were able to successfully reproduce the trend of lowest 3/2 + state energies in Se and Ti isotopes.

The BF model was later used by various authors. Zamick (1965) showed how the strong isospin dependence of particle hole interaction leads to some low lying multi- hole states in s-d shell and f7/., shell nuclei. Sherr et al (1974) have shown that the model is quite successful in reproducing the energies of 3/2 ~ states throughout .f7/2 shell. Bernstein (1972) has given a qualitative description of many of the low lying positive parity states in aJCa by considering multi-hole states in (da/a-f~/2) n space.

Sherr and Bertsch (1975) have shown that the BF model is quite successful in account- ing for the Coulomb displacement energies of particle-hole states in many light nuclei.

The BF model involves two simplifying assumptions: (1) They treat the lowest 3[2 + states in oddfT/9, shell nuclei as single hole states whereas the experiments indicate that the lowest 3/2 ÷ states are not single hole states but are ' band h e a d ' states of collective band of positive parity states. (2) In determining the hole state energies the empirical values of the binding energies of (fT/a) n configuration relative to 4°Ca were used. The tacit assumption of this procedure is that the effective interactions between the fTl~. nucleons alone shall be able to reproduce exactly the empilicat binding energies of these nuclei.

Recently empirical effective interaction matrix elements in the (d3/z--fTtz) 2 configura- tion have been determined by Erskine et al (1971) and Shert et al (1971). These matrix elements have been deduced from the observed spectra of two particle like nuclei with the implicit assumption that the d3/~ orbit is completely filled in a°Ca. These effective interaction matrix elements are therefore appropriate to verify the validity of the above two basic assumptions of BF model. The empirical (f7/2) 2 interactions are quite successful (McCullen et al 1964) in explaining the energies of ground state band o f collective states in f7/2 shell nuclei. It has been found (Lawson 1961 and Lawson and Zeidman 1962) that the structure of ground state bands of these nuclei are well approximated by projecting them from the lowest energy intrinsic states.

In the present paper we have used the empirical (d3/2--fT/z) ~ effective interaction matrix elements to study the energy systematics of hole states in 41, 48Ca ' 43,45,47,49Sc '

~5, 4ri,i, 4~,4a,49,sx V and 61Mn using the projected Hartree-Fock method in the (da/~-- f~/~.)" model space. The hole states are projected from the lowest energy intrinsic state of the ___d~/-12 (fv~) "+1 configuration. This procedu#e directly gives a collective band of states in contrast to the single hole states obtained in the BF model.

We note here that the empirical effective interactions are defined in the (d3tz--f7/z) ~ configuration and hence we ignore the 2st/~ and lgp/2 single particle orbits in the shell model basis space. Our aim here is not to study the detailed structure of the positive parity states of the (fP--gp/2) shell nuclei. Rather we are interested in examining how successful are the recent empirically determined effective matrix elements in the (da/2--fT/2) 2 space in describing the energy systematics of the low-lying positive parity states o f the fT/~ shell nuclei in the (d3/z--fT/2) n configuration space. Since the transition rates and electromagnetic moments are sensitive to the detailed structure of wave functions we do not expect them to come out in the present limited configura- tion space.

In section 2 we discuss the present model. The calculated spectra of the hole state bands are compared with experimental ones in section 3.

2. Model

2.1. Model space and Hamiltonian

We consider the model space consisting of lda/~ and lfTiz as active orbits as is consist- ent with the empirical effective interactions defined from the low-lying spectra of two particle like nuclei near 32S and 40Ca. It should be mentioned here that from the results of MBZ calculation (McCuilen et al 1964) for f7/2 shell nuclei in (f7/~)"

model, the need to include other (fp) shell single particle orbits was felt. However at the same time their model is quite successful in describing the energies of the ground state band of collective states of nuclei not very close to 4°Ca. Similarly the energy spectra of the collective band of positive parity states in the ( d ~ ~" n+ l~ J 7/2 J configura- tion space may be well reproduced by the effective (dal2--fTl~) ~ empirical interaction.

The Hamiltonian in this model space is completely defined by the single particle energies ~d3/~ and ~tT/~ of the da/~ and f7/2 orbits and the set of effective two body interaction matrix elements ( d3~21 V[ d3~ 2 )JT, ( d3/~f~/2 [ V I ds/2fT/~ ) J T and (fT/~ ~ [ v [.f7/2 a )JT. The proton and neutron single particle energies in ds/~. orbit E~d3i 2 and %d312 are taken from the empirical binding energies (Wapstra and Gove 1971) of 83CI and 33S relative to 32S. By taking the single particle energy separation ea$: eaa/2- c$7/2:3"5 MeV, as inferred from the observed 7/2- single particle state in 3zC1 (Erskine et al 1971), the proton and neutron single particle energies ~,o'~/~

and ~$7/2 in f7/3 orbit are also defined. The coulomb energies Ec(d~), Ec(f z) and E~(df) for proton pairs in d31 ~, fTlz and d3/a--fT/~ states are taken similarly for empirical binding energies.

Recently the effective (ds/z--f7/~) ~ two-body interaction matrix elements have been deduced empirically by Erskine et al (1971) and Sherr et al (1971) from the binding energies and low lying levels of two particle like nuclei near z~S and 4°Ca. As will be shown later these interactions do not reproduce the binding energies of f7/~ shell nuclei well. Hence after adjusting the eentroids of the interactions to yield ground stage binding energies of ~°Ca and fw2 shell nuclei reasonably well we have used the resulting empirical interactions to study the hole state systematics in .fT/~ shell nuclei.

402 D P Ahalpara

2.2. Intrinsic states

It has been shown earlier that the ground state band of collective states projected from the lowest energy intrinsic states have large overlaps with corresponding states in an exact shell model calculation (Lawson 1961, Lawson and Zeidman 1962). It is therefore reasonable to associate the low-lying band of collective states in f7/2 shell nuclei with the states projected from deformed intrinsic one hole states. We therefore construct the determinantal 'intrinsic states for f7/~ shell nuclei by filling valence nucleons in the lowest available deformed orbits ~fl of the dz/2 and f7/2 orbits in a Q8 z field. These lowest intrinsic states correspond to a completely filled d31 ~ orbit with remaining nucleons filling the fTl~ orbit. The one hole intrinsic states are then obtained by promoting a proton or a neutron from k : 3 [ 2 + orbit of the dz/2 state to the lowest unfilled orbit in f7/2 state.

In the BF model the lowest J = 3 / 2 ÷ states in odd Z nuclei arise due to the excitation o f a d312 proton to f712 state. Whereas in even Z nuclei the hole sates resulting from neutron excitation are lower in energy. It is expected therefore that the hole states in o d d Z nuclei may be projected from a single proton hole intrinsic states but for the even Z nuclei it may also be necessary to consider intrinsic states involving neutron excitation. In contrast to the proton excited intrinsic states the neutron excited intrinsic states do not have a definite isospin. In this case a linear combination of proton and neutron excited intrinsic states isnecessary to form good isospin intrinsic states. We illustrate this by showing typical one hole intrinsic states of odd Z a n d even Z nuclei.

2.2.1. Odd Z nuclei: In figure 1 is shown the proton excited intrinsic state of 46Sc.

The (fT/z) 8 part of the state has isospin To= 1 and the total isospin is T = T 0 + 1/2=3/2.

2.2.2. Even Z nuclei." Figure 1 also shows the ' neutron excited ' intrinsic states of ~Ti. None of the three intrinsic states has definite isospin but they can combine to form (]'~l~)eTo configuration with To=0 and 1 leading to T = T o : k l / 2 = l / 2 states.

K

~ 2

) ~v, i .

v •

AT,.t A *

4 3/2 ,1/;~ "~ . •

p 45Sc n K

A A • •

3/z v ~ v

I ^T

• • ~ • A A * ' • A A *

ert..~/~--.. - T . . T • - • . •

45Ti

Ftgm-e 1. The proton excited and neutron excited intrinsic states in 45S¢ and 4~Ti respectively.

For the 41'4~Ca, ~'45'47'49Se, 45'47Ti ~7'48'49'51V, and *ZMn nuclei studied in the present model we have followed the above scheme in forming the intrinsic states.

Good angular momentum states were then projected from these intrinsic states following the standard technique (Warke and Gunye 1967; Ripka 1968).

2.3. Constraints on interactions

In defining the ( da/2 ~ Iv I dalz ~) J T a n d ( da/~fT/~ I v I da/zf~l~) J T empirical effective interaction matrix elements from the low lying levels of a~'3sCI and 4°Ca,s~S has to be regarded as 'core'. The (fv/22 [ v l.f~l~3 ) JTeffective matrix elements were obtained fromthe low-lying levels of aZSc by considering a°Ca as 'core'. In this Talmi approach of deducing two-body empirical effective matrix elements from the binding energies and low-lying levels of two-particle like nuclei the off diagonal matrix elements of the form ( da/~ a ] v ]fT/~ ~ ) JT do not arise and the effects of these matrix elements on the energies of multi-particle states are already absorbed in the diagonal effective two- body matrix elements.

These tacit assumptions made in defining the empirical effective two-body matrix elements put certain constraints on the matrix elements. We now check whether these assumptions are fulfilled by the empirical effective matrix elements.

2.3.1 ( da/zzl v] d3/~ a ) JT matrix elements: Since the ( f ,lz 2 Iv If~/z ~) J T empirical matrix elements were deduced by assuming a closed da/28 sphelieal structure for 4°Ca, the ( ds/~ ~ I v I d8/2 ~ )JT matrix elements should in turn reproduce the empirical binding energy of 4°Ca. However the empirical effective matrix elements of Ers- kine et al (1971) overbind ¢°Ca by 12"5 MeV. The energy of the closed (ds/~)aj= 0 configuration is determined by the centroid of the ( d8/2 z [ v[ d3/z 2 ) J T matrix ele- ments. It is therefore necessary to shift the centroid energy so as to exactly fit the empirical binding energy of 40Ca.

2.3.2 ( da/efTla[vlds/afT/a ) J T m a t r i x elements: After fitting the binding energy of 40Ca the next step is to cheek whether the empirical ( dda f~/a I v I ds/a fva ) JT matrix element fits the observed binding energy of 4aCa. The binding energy of ~xCa relative to 4°Ca is defined by two parametem: the neutron single particle energy

~,sT/z and the centroid E(2)a of the ( ds/a fT/z I v I d3~ ~ f~/a ) JT matrix elements.

~aCa J 0 C a = , , / 7 / a + 8~((2)a.

Using the empirical (d3,sf~l~l v!d3/¢f~l~) JT matrix elements we find that 4zCa is underbound, by 0-21 MeV. This small departure was taken care of by making the centroid energy of multiplet attractive by 0"03 MeV.

2.3.3. (f~71~ [ v [.fz7/z) ,IT matrix elements: In the BF model these matrix elements are expected to fit exactly the binding energies of the (f7/2) n nuclei relative to *°Ca.

This assumption plays an important role in the success of the BF model. It is found however that the binding energies of these nuclei obtained with the empirical matrix elements are consistently too high. The departure from empirical binding energy for a (Z, N) nucleus increases with Z or N. We shifted the centroid energy of the ( p v / z I v [f27/~) dT matrix elements by 0"19 MeV to improve the fit to the empirical binding

404 D P Ahalpara

Table 1. Root mean square deviations in MeV for the calculated binding energies of various isotopes infT/~ shell nuclei.

Modified empirical

Empirical interaction

interaction (Centroid shift=0"19 MeV)

Ca 3.35 0.27

Sc 4"45 0'94

Ti 5-31 1.54

V 7-58 2.34

Cr 11.40 3-I5

Mn 16.00 6-01

T a b l e 2 .

(in MeV)

%raat 2 = - 2.28 ev a3ts= - 8 ' 6 4 Ec(d~)=0.41

T

1 1 1 l 1 1

0 0 0 0 0 0

The empirical effective interaction matrix elements used in the calculation

EdfO=0"55

c ~ $ 7 / ~ = 1 "22 evf~/ =--5"14 Ec( d f ) = 0 . 23 d (d~/~)~ (d,l~-f,I..) (f7/2)"

0 --2"51 -- --2"98

2 0.04 -- 3"0 -- 1 "39

3 - - -- I "74 u

4 - - --1-64 --0"17

5 - - --2-09 - -

6 u - - 0.26

1 --1"87 - - --2"37

2 - - --0.44 - -

3 --2"36 --0"19 --1'49

4 - - 0 . 8 1 - -

5 - - 0"05 -- 1"47

7 - - - - --2"36

energies

I t is seen t h a t even w i t h t h e shift in t h e e e n t r o i d o f t h e (f7/2) 2 m a t r i x e l e m e n t s t h e d i s c r e p a n c y b e t w e e n t h e c a l c u l a t e d a n d e x p e r i m e n t a l b i n d i n g e n e r g i e s is q u i t e large c o m p a r e d t o the s t r i n g e n t r e q u i r e m e n t o f t h e B F m o d e l . A n a t t e m p t was m a d e t o fit t o t h e e x p e r i m e n t a l b i n d i n g energies t o t h e a c c u r a c i e s r e q u i r e d by t h e B F m o d e l b y v a r y i n g a l l t h e (fT/~) ~ m a t r i x e l e m e n t s . T h e r o o t m e a n s q u a r e d e v i a t i o n was r e d u c e d t o 0"1 M e V . H o w e v e r m a n y o f t h e m a t r i x e l e m e n t s were f o u n d to b e u n p h y s i c a l l y large. I t is difficult t o o b t a i n a g o o d fit t o t h e o b s e r v e d b i n d i n g energies b y a s m a l l v a r i a t i o n o f t h e m a t r i x e l e m e n t s a r o u n d t h e e m p i r i c a l values.

In the calculation of the hole state band spectra we have used the empirical (f~/22) matrix elements with the centroid shifted by 0"19 MeV to reduce deviations in calcu- lated binding energies compared to experimental ones. These are listed in table 2 together with the empirical single particle energies.

3. Results

We compare the calculated and experimental spectra of Ca, Sc, Ti, V and Mn isotopes in figures 2 to 7. The spectra are shown relative to the first excited 3/2 + states.

The energies of the 3/2+ states relative to 7/2- ground states axe also shown in the figures.

The calculated excitation energies o f 3/2+ states are off by about 2 MeV in contrast to the good fit obtained by the BF model. This is because the empirical particle-hole

Figure 2.

>

o 2

I y l l / ~ (II-;Yl

I I I7

l?

I& I ( ' ~ " ) II

• 01

to II

Ill-I?)

! .,J~

m •

( t i l l I II

13 - - 1' ,, •

I •

(11) I

? . . •

I

11 4 " ; . $ I . ~ $ 3 . ~ • O'gO •

2J P..all. l i p . 2J ~ l i e .

4 ICG 4 ~ A

(see caption in p. 408)

=E 2

¥1glm'e 3.

II =

, l ~ ) - -

Ill) =

sm - -

r I

i - -

~ ~J.2 ~ o,_.~

2 J Coic l i p

i $ - -

2J C~c e z p

45Sc (see caption in p. 408)

i

? - -

S - -

2J Celt eql

4 7 S c

3,7..._~" 3 2.3_' 2J r . ~

19So

P.--5

406 D P Ahalpara

4

>. 2

s 2-s...~ s o ~__~

~J CalC Isp.

4 5 T i

Figm'e 4. (see caption in p. 408)

>- 2

=E

4 1 ~

t B )

I - -

0 5 z--!-' .s o - z *

Z4

¢ 0 1 ¢ l i p

4 7 V

Figure 5, (see caption in p. 408)

S q - -

1 ^{s --}

f --

f ~

Fism'e 6. (see caption inp. 408)

, $ m

D ~

D - -

I b m r m

2J Col= a l p ,

4 7 T i

I I g

r I r m

Sm

~ m O-T,5

2J Cole. ell~

L9 V

f ~

6 " - -

S - m

l z D S ~

V m

2,eE 24 Celo. e l l .

: [

Z 02

$ - -

2 J Ce*e

5 t M n e -

e l ;

Figure 7. (see caption in p. 408)

interaction is more repulsive compared to BF values. By shifting the centroids of particle-hole interaction to BF values we find that the excitation energies of 3/2 + states are reduced but still do not fit the experimental values. For nuclei away from 4°Ca the deviations between the empirical binding energies of (f7/2) n and (f7/2) "+1 configurations relative to 4°Ca and the ones given by empirical interaction are large. Hence we do not expect the energies of 3/2+ states to be fitted with better accuracy.

For the 3/2 ÷ bands in odd-A fTla shell nuclei we find that the energy spectra for nuclei away from 4°Ca are fairly well reproduced by the empirical interaction. For nuclei near 4°Ca the deformation effects are large and the empirical interactions do not reproduce these effects. This was also reflected in the MBZ calculation (McCullen et al 1964) for (fv/2) n nuclei using empirical (f~/a) ~ effective interaction. Also the effects of the intruder states arising from multiparticle-hole states are likely to be much larger for nuclei neat 4°Ca.

To summarise then the collective bands of positive parity states in higher mass odd- ,4 f~/2 shell nuclei are fairly well reproduced by the empirical (da/S --f~/~) 2 effective interaction. However it fails to reproduce the excitation ¢negies of 3/2 ÷ states.

Acknowledgements

The author wishes to sincerely express his gratitude to Dr K H Bhatt for suggesting the problem and guiding throughout the course of work. Valuable discussions with Dr S B Khadkikar and Dr D R Kulkarni and with Dr A K Dhar in the earlier phase of work are thankfully acknowledged. Thanks are also due Mr C S R Murthi for providing computer code of least squares fit.

References

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408 D P Ahalpara

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Figures 2-7. Comparison of calculated and experimental energies of collective bands of d81s -t hole states. The spectra are drawn relative to the band head states. The figures near these states indicate their excitation relative energies to ground states.