on the non-pairing aspects of the t = 1 interaction

1 1.c I Nuclear Physics A192 (1972) 625-640; @ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfibn without written permission from the publisher

ON THE NON-Pace ASPECTS OF THX T = 1 ~TE~~TI~N

S. K. SHARMA and K. H. BHATT

Physical Research Laboratory, Naorangpura, rihmedabad-9, India

Received 22 December 1971

Abstract: Emphasis is generally @aced on the pairing nature of the T = 1 interaction, while at the same time considering the 7’ = 0 interaction to play an important role in producing deformation. Contrary to this, we show that the T = 1 component of the effective interaction in the 2p-lf shell has a muhipole character almost similar to that of the T = 0 component. However, some evidence is presented which indicates a slightly greater pairing tendency of the T = 1 component. The quadrupole dominance of the interaction in both the T = 0 as we11 as the T -= 1 channets is demonstrated. it is then shown that the enhanced pairing effects in the Zp-lf shell are largely due to a somewhat unfavourable sequence of experimental single-particle energies. Further, it is found that the Pauli principle has a strong effect on the magnitude of the deformation that an interaction can produce.

1. Introduction

It has usually been felt that the T = 1 component of the two-body interaction has a pairing character with a tendency to spherify nuclear shapes, while the T = 0 component, on the other hand, has a defo~ation-producing tendency. In general, one observes that the T = 1 interaction is characterized by large matrix elements between J = 0 paired states, and the T = 0 interaction has large matrix elements between J # 0 states. Thus, in an approximate manner, the T = I component of the effective interaction appears to resemble the conventional pairing interaction - an interaction which exists only between identical nucleons in opposite, time-reversed states and which makes them scatter very strongly from one J = 0 paired state to another. These features of the T = 1 matrix elements, together with the fact that nuclei with only neutrons as valence m&eons do not display rotational properties and can be described to a good extent in the seniority scheme, have led to the assumption that the T = 1 component of the effective interaction is of the pairing type and has a spherifying tendency. Further, these assumptions regarding the nature of the T = 1 interaction have also resulted in the feeling that the T = 0 and the T = 1 components of the interaction compete with each other in producing deformation and that the multipole nature of the T = 1 component is quite different from that of the T = 0 component.

With the conjecture that the assumptions regarding the role played by the T = 0 and the T = 1 components in producing deformation may not be correct, we have made an attempt to examine their deformation-producing tendencies in~vidually.

The deformation-producing character of an interaction can best be brought out by examining the various multipole moments of the self-consistent HF field generated

625

626 S. K. SHARMA AND K. H: BHA’IT

by that interaction. Thus the difference in the multipole character of the T = 0 and T = 1 interactions will be reflected in the differences in the multipole moments of the HF states they give rise to. For example, an interaction with a greater pairing tendency should give rise to larger values of the higher-multipole moments. On the other hand, one can make an attempt to investigate the pairing or spherifying character of an interaction by allowing for the mixing of the HF states with pair-excited configura- tions. This is because the lowest-order corrections to the HF state, which necessarily are through the admixture of 2p-2h type determinants, are due to the two-body residual interaction left after the extraction of the one-body HF potential. This residual interaction consists mainly of the pairing components of the interaction. Thus the component (T = 0 or T = 1) which gives rise to a larger admixture of 2p-2h states may be considered to have a more dominant pairing character.

To illustrate these ideas, we have chosen to analyse the multipole character of the T = 0 and T = 1 components of the effective interaction obtained by Kuo and Brown ‘) for the 2p-lf shell. In sect. 2 we give a brief outline of the HF method, and in sect. 3 the results obtained by carrying out deformed HF calculations with different components of the two-body interaction are discussed. It is shown by carrying out several theoretical experiments that the T = I interaction has about the same multipole character as the T = 0 interaction and that both have a dominant effective quadrupole-quadrupole component.

Through the work of Elliott “) one knows that a quadrupole-quadrupole two-body interaction would give rise to an intrinsic state with maximum possible deformation only if the single-particle levels are arranged in a 1(Z+ 1) sequence. Keeping this in mind, it has been demonstrated 3, “) by making a series of conventional shell-model calculations of the nucleus 44Ti that the development of rotational features in the beginning of the 2p-lf shell is governed to a large extent by the sequence of single- particle energy levels. Thus it has been shown that lowering the p-state below the f-state results in the development of a near SLJ(3) coupling scheme in 44Ti. This interplay between the one-body and the two-body parts of the Hamiltonian in producing deformation is studied in the HF formalism in sect. 4.

In sect. 5 it is shown that if the single-particle levels are chosen to have a Z(1+ 1) energy sequence, the T = 1 interaction gives rise to maximally deformed intrinsic states even for identical particles. We have also examined the effect of the Pauli principle on the magnitude of deformation produced by the interaction. Sect. 6 con- tains some concluding remarks.

2. The method of deformed orbitah for the Hartree-Fock calculation

The Hamiltonian of the system consisting of one-body and two-body parts in second quantized form is

T = 1 INTERACTION 627

Mere the states \E> are one-particle states of any arbitrary representation {say, the harmonic oscillator ftmctions ijr~.>), .sclp is the matrix element of the one-body part of the Hamiltonian and

is the antisymmetrized matrix element of the two-body interaction. In the HF method 5), the intrinsic ground state of the nucleus is considered to be a

single determinantal state @uF such that @nF gives a minimum for the expectation value of H. The variational equation

G(@lHl@> = 0 (2)

gives rise to the eigenvalue equation

where h is a one-body operator defined as

The occupied orbitals IA) are expanded in a basis 1 jm) of eigenstates of the spherical harmonic oscillator in the 2p-lf ‘shell:

Id> = G &Jjm). j

Here c;~ are the expansion coefficients and the states IA> have been assumed to have axial symmet~. Since h itse~depends on its eigenvectors iA>, the problem has to be solved self-consistently by an iterative method. One starts with an initial guess for IA), calculates h, diagonalizes to get a new set IA>, and so on, until the total energy

converges to a constant value. The difference in energy between the highest occupied and the lowest unoccupied

single-particle deformed orbitals is called the Hartree-Fock energy gap G. The summation in eq. (3) is carried out over the occupied orbitals. In order to

obtain the best zero-order product wave function for the ground state one chooses the lowest-A orbitals resulting from eq. (2) as the occupied states in each iteration. One can also obtain the minimum-energy determinantal wave function with a specific prescription regarding the occupied deformed orbitais by following a “tagged” self- consistent procedure. This involves imposition of the particular restriction cornering the occupied orbitals in each iteration before one proceeds to calculate the summation implied in eq. (3).

628 S. K. SHARMA AND K. H. BHATT

3. Deformation-producing tendency of the T = 1 interaction

The effective two-body interaction that we have employed is specified by the set of matrix elements given by Kuo and Brown. The effective one-body operator is specified by four single-particle energies taken from the 41Ca spectrum. The values for the single-particle energies are (in MeV): s(f%) = 0.00, s(pg) = 2.10, .s(f*) = 6.50 and s(p+) = 3.90.

In order to bring out the deformation-producing tendency of the T = 0 and T = 1 components of the two-body interaction we have carried out single-shell deformed HF calculations in 2p-lf shell space using (i) the full interaction, (ii) the T = 0 part of the interaction and (iii) the T = 1 part of the interaction. We have then compared the deformations obtained in each case with experimental single-particle energies as well as with all the single-particle energies switched off. The results are discussed here. They were briefly reported in ref. “).

In table 1 we give the results of HF calculations for the nucleus 44Ti carried out by using the experimental energies for the single-particle states. The first column in table 1 specifies the interaction or the parts of interaction that we have employed.

TABLET

The multipole moments of the HF state for 44Ti with different components of the interaction

Interaction <QOZ> <Q04> <Qo6>

T=O+T=l 20.00 67.20 79.20 T=O 15.17 47.80 61.48 T=l 14.35 44.80 59.08

&(T=Q+T=l) 14.70 46.16 60.70 2(T= 0) 20.25 67.80 80.49 2(T == 1) 19.70 66.40 78.21

Single-particle energies taken from expsriment were used. The moments <Qoh> are in units of 6” where 6 is the harmonic oscillator constant.

It is seen that with the full interaction the (2:) Qz and Qg moments for the HF state for 44Ti are 20 b’, 67.2 b4 and 79.2 b6 respectively, where b is the harmonic-oscillator parameter (b2 = h/mco). When all the T = 1 matrix elements are set to zero and the T = 0 interaction alone is used, the quadrupole moment for the HF state which results in this case is 15.17 b2. On the other hand, when the T = 0 interaction is com-

pletely switched off and the T = 1 interaction alone is used, the resulting HF state has a quadrupole moment (Qi) = 14.35 b2. The Qz and Qg moments obtained with the T = 0 interaction alone are also similar to those obtained with the T = 1 interaction alone.

Thus the T = 0 as well as the T = 1 interaction seem to have strikingly similar deformation-producing character. However, it is to be noted that when either the T = 0

2’ = 1 INTERACTION 629

component or the T = 1 component is turned off, the deformation of the HF state decreases. This decrease is to be expected because the two-body interaction tries to mix the various single-particle states in order to produce the deformed field and in doing so it has to compete with the energies of spherical single-particle orbits. Using only one of the two T-components of the interaction by switching off the other, in a way amounts to reducing the overall strength of the interaction by roughly a factor of 2 and a weaker interaction will then naturally produce a smaller deformation. Fol- lowing the same reasoning, one can argue that if the matrix elements of the full (T = 0 + T = 1) interaction are divided by a factor of 2, the resulting weakened interaction should lead to a HF state with a deformation approximately of the same order as given by the T = 0 or T = 1 interaction alone.

The results presented in the fourth row of table 1 show that this is indeed the case. Here the interaction obtained by dividing all the matrix elements of the full interaction by a factor of 2 is labelled as $(T = 0 + T = 1). Conversely, when either the T = 0 or the T = 1 matrix element is doubled, the deformation of the resulting HF state is quite similar to that obtained with the full interaction a shown in the fifth and sixth rows in table 1. These interactions are labelled as 2( T = 0) and 2( T = 1).

In a deformed HF calculation, the deformation that a two-body interaction with a given strength can generate depends on the energies of single-particle states. Our prima- ry aim here is to compare the deformation-producing tendencies of the T = 0 and T = 1 components of the interaction. We would therefore like to eliminate the effect of the single-particle energies. For this purpose we now make all the single-particle energies equal. In this case the deformation of the self-consistent field depends only on the multipole nature and not on the strength of the two-body interaction which generates it.

TABLE 2

The multipole moments of the HF state for 44Ti with the full two-body interaction as well as only the T = 0 and 2’ = 1 parts of it, keeping the single-particle energies degenerate

Interaction <Qo'> <Qo4> <Qo6>

T=O+T=l 23.18 78.79 86.95 T=O 23.25 78.54 85.08 T=l 23.16 78.90 87.84

The deformations produced by the various interaction with degenerate single- -particle energies are compared in table 2. One is at once struck by the similarity of all the deformations obtained with the full interaction and with its T = 0 and T = 1 ,components.

The slightly larger values of the Qt and Q8 moments produced by the T = 1 interaction may be attributed to its greater pairing tendency.


3.1. THE RELATIVE STRENGTHS OF THE T = 0 AND T = 1 COMPONENTS OF THE INTERACT10 N

We have just discussed the similarity in the multipole character of the T = 0 and

T = 1 interactions. Here we discuss the differences in the effective strengths of these

two components of the interaction.

The strength of an interaction is reflected in the HF energies and the energy gaps.

With degenerate single-particle energies only the HF energies and the HF energy gaps

depend on the strength of the interaction and are proportional to it. The structures of

HF orbitals are independent of the strength of the interaction and depend only on its

multipole character. With non-degenerate single-particle energies the HF energies

and the HF energy-gaps are not linearly dependent on the strength of the interaction.

In addition, the structure of the wave functions is also determined by the strength as

well as the multipole character of the interaction.

In table 3 the structures of the deformed orbitals resulting from the different com-

ponents of the interaction in the presence of the experimental single-particle energies

are given. It is to be noted that the changes in the overall strengths of different inter-

actions manifest themselves in the corresponding HF energies and HF energy gaps in

a somewhat non-linear way. The depletion of the strength of the interaction is seen to

result in a smaller deformation and significantly larger fz components in the wave

function.

TABLE 3

The structure of the occupied orbitals with k = 4 for the 44Ti HF state, the HF energy E and the HF energy gap G, for different components of the two-body interaction

Interaction k cg c$ CAk 2 c+k E (MeV) G (MeV)

z-=0+2,= 1 H 1 0.83 -0.24 -0.44 0.23 -9.35 3.33 T=O 4 0.95 -0.11 -0.27 0.10 -4.72 1.13 T=l 4 0.96 -0.09 -0.24 0.09 -2.54 0.73

&(T = O+T = 1) 9 0.96 -0.10 -0.25 0.09 - 3.62 0.83 2(T = 0) 4 0.82 -0.24 -0.45 0.23 -11.75 3.97 2(T = 1) 4 0.84 -0.25 -0.43 0.22 -6.95 2.64

The coefficients cj* in the expansion jk) = II,cjkljk> are listed. Experimental single-particle

energies have been used in all cases.

TABLE 4

The structure of the occupied orbitals with k = 4 for the 44Ti HF state, the HF energy and the HF energy gap with the full interaction, its T = 0 component and its T = 1 component

Interaction k C$ CL’< il C+” c*-l E (MeV) G (MeV)

T=O+T=l 4 0.603 -0.468 -0.531 0.368 -16.32 5.15 T=O B 0.604 -0.455 -0.536 0.374 - 9.35 3.96 T=l _$ 0.599 -0.478 -0.531 0.363 - 6.97 2.19

All single-particle levels were degenerate.


As seen from table 4 where the results ~o~espondi~g to degenerate since-particle energies are given, a comparison of the HF energies resulting from the T = 0 component alone with those resulting from the T = 1 component alone provides an esti- mate of the relative overall strengths of the two T-components. Thus it is seen that the strength of the T = 0 component is about 1.34 times that of the T = 1 component.

3.2. DEFORMATIONS FOR OTHER NUCLEI IN THE 2p-lf SHELL

It might be felt that the similarity in the deformation-producing tendencies of the 2’ = I and T = 0 interactions is due to the simple N = 2 structure of the nucleus 44Ti. Moreover since 44Ti is also likely to have the rn~irn~ qua~Llpole moment per nucleon i6 the 2p-lf shell, the quadrupole component of the interaction might play a more dominant role in the nucleus 44Ti than in other nuclei.

In order to verify that the similarity of T = 1 and T = 0 interactions is not restricted to the nucleus 44Ti alone, we have carried out HF calculations for a number of nuclei in the 2p-lf shell including some with neutron excess, using the full interaction as well as only its T = 0 and T = 1 components. The results are presented in table 5. In all the cases we have kept the single-particle energies degenerate.

TABLE 5

The various multipole moments, the HF energies and the HF energy gaps for some nuclei in the Zp-11 shell with the full interaction as well as its T = 0 and T = 1 components

Nucleus Interaction <Qo'> <Qo'> <Qo9

‘Wa T=l

46Ti T=O+T= 1 T=O T=l

“*Cr T=O+T= 1 T=O T=l

WZr T=OO+T=X T=O T=l

j2Fe T=O+T=l T=O T=I

56Ni T=Oi_T=l T=O T=l

17.52 29.67 -08.57 (18.00) (59.30) (120.00) 29.23 69.23 32.81 29.39 68.50 30.17 29.07 69.61 35.31

(30.~) {99.04) (195.00) 35.24 59.24 -19.74 35.39 59.02 -21.30 35.00 59.60 -16.69

(36.00) (118.60) (240.00) 41.50 26.32 -36.74 41.56 25.16 -37.72 41.46 26.16 -34.78 (42.00) (247.17) (481.25) 47.34 38.81 - 123.70 47.31 38.82 - 124.62 47.29 38.78 - 120.23

(48.00) (157.73) (330.00) 47.73 -04.83 -41.54 47.70 -04.93 -41.19 47.75 -05.19 -41.21

(48.00) (166.73) (322.50)

-4.98 0.27

-23.25 4.52 - 12.77 2.09 -10.49 2.15

-35.72 0.15 -21.02 0.02 -14.94 0.42

- 57.20 0.88 -33.66 0.24 -23.57 0.63

-68.62 5.62 -42.34 3.20 -26.32 2.26

-94.18 2.46, -59.86 0.93 -34.33 1.53

0.52 0.05 0.47

0.15 0.02 0.42

1.46 0.12 1.28

5.62 3.20 2.26

2.46 0.93 1.53

Here E is the HF energy and Gp (G,) is the HF energy gap for the proton (neutron) single-particle WF spectrum. The entries in brackets are the nmximum possibl2 values for 2’” pole moments ~<Q~~>max for each nucleus. All shell-model singie-particle states were degenerate.


Again a striking similarity of all the multipole moments produced by both the T-components of the interaction is seen. Some small but noticeable differences between the deformations produced by the two interactions are seen in the values of the Qg moments. The largest differences are for 46Ti and 48Cr. It is to be noticed here that for all the nuclei considered here, the Qz and (2: moments obtained with the T = 1 interaction alone are consistently slightly larger (more positive) than those obtained with the T = 0 interaction alone, and to this extent it appears that the T = 1 interaction has a slightly greater pairing tendency than the T = 0 component.

3.3. THE DOMINANT QUADRUPOLE-QUADRUPOLE CHARACTER OF THE T = 0 AND 7 = 1 COMPONENTS OF THE INTERACTION

A two-body interaction which consists predominantly of an effective quadrupole- quadrupole component should in a HF calculation tend to yield an effective one-body HF potential which is predominantly a quadrupole field. Thus the HF orbits are expected to have a good overlap with the W(3) orbits, i.e. the eigenstates of the single-, particle quadrupole operator in a major shell. In the appendix we have given the ten single-particle states $(v, E) of the 2p-lf shell in a Qi field (here v = (1,) is the expectation value of the component of the angular momentum along the symmetry axis, and E is the eigenvalue of the intrinsic quadrupole moment in units of b2. The level scheme in a quadrupole well is given in fig. 1. The maximum possible quadru-

Fig. I. Single-particle states in a Q 02 field, E = <Qo2), in units of b2 where b is the harmonic

oscillator parameter.

pole moment in the 2p-lf shell for four particles is thus obtained when the occupied deformed orbit with a four-fold degeneracy is identical to 4(0, 6), the value for the Qi moment in that case being 24b2.

The results for the Qg moments of HF states for 44Ti as presented in table 3 show that in each case the value of the quadrupole moment is almost the maximum possible (24b’) and this clearly brings out the fact that the full interaction, as well as its T = 04 and T = 1 components individually, possess a dominant effective quadrupole-quadrupole component. The quadrupole dominance of the two T-components of the interaction is further evident from the values for the quadrupole moments for various nuclei as given in table 5. In a pure (2: field the levels shown in fig. 1 would tend to fill in order of decreasing E = (Qg). Thus in the nucleus 44Ti the four particles fill the 40 = 4(0,6) state and the next two neutrons in 46Ti can go into either the 4+ 1 or the 4 _ 1 orbit and in both cases the maximum possible quadrupole moment is


30b’. Similarly the maximum values of the quadrupole moment for the nuclei 48Cr, 52Cr and 52Fe would be 36b2, 42b2 and 48b2. The values given in table 5 are almost the maximum possible for all the nuclei with either of the two T-components of the interaction. In marked contrast to the quadrupole moments, note that the Qt and Qg moments of the nuclei are significantly smaller than their maximum possible values.

As an interesting manifestation of the quadrupole dominance of the T = 1 interaction, it is seen that even for 44Ca one gets an intrinsic state having a quadrupole moment which is very close to the maximum possible value (18b2) for four identical particles, two of which occupy an orbit similar to $0 and the other two particles OCCU-

py an orbit similar to either orbit $+ 1 or 4 _ 1. The quadrupole dominance of the Kuo-Brown f-p shell interaction was also shown

in ref. ‘).

3.4. THE HEXADECAPOLE FIELD IN THE 2p-lf SHELL

As we have already seen, the structure of single-particle deformed orbitals resulting from the HF calculations appears to be mainly governed by the quadrupole field. The one-body HF potential generated by the two-body interaction clearly consists of quadrupole as well as hexadecapole fields but the latter appears to be comparatively small. This might be due not only to a large quadrupole-quadrupole and a small hexadecapole-hexadecapole component in the interaction, but also to the fact that in a HF procedure the field effects are picked out mainly from the quadrupole- quadrupole part of the interaction, and the hexadecapole-hexadecapole component, even if it is large, would not be expected to contribute significantly to the one-body HF potential. However, the hexadecapole-hexadecapole and the @ - q6 components of the interaction may contribute more significantly to the pairing effects of the interaction responsible for the instability of the HF states.

It is interesting to examine the effect of the interplay between quadrupole-quadrupole and hexadecapole-hexadecapole components on the structure of single-particle deformed orbitals for nuclei in 2p-lf shell. This effect can be brought out by a comparison of the structures of single-particle states in pure quadrupole and hexadecapole fields which would be produced by the quadrupole-quadrupole and the hexadecapole- hexadecapole components of the interaction individually. These states (see appendix) reveal that the lowest v = 0 and v = + 1 states in the quadrupole field have large over- laps with those of the hexadecapole field, As a result the nuclei in the beginning of the 2p-If shell have large Qi as well as large Qz moments. On the other hand, the wave functions of higher orbits in the hexadecapole field are quite different from those of the quadrupole-field orbits. Hence the heavier nuclei cannot have large Qz moments together with large Qg moments. This result is clear from table 5.

In a way, therefore, as far as the HF field is concerned the quadrupole-quadrupole and the hexadecapole-hexadecapole components of the interaction appear to be in phase with each other for the nuclei in the beginning of the 2p-lf shell and are somewhat out of phase with each other for the nuclei in the rest of the shell.

634 S. K. SHARMA AND K. H. RHATT

Significant departures from rotational behaviour in some nuclei in the 2p-lf shell have recently been demonstrated - by carrying out Hartree-Fock-Bogoliubov calculations - to be related to the increased importance of pairing,correlations in this shell “). One knows that the ne~~tro~ excess in 2p-lf shell nuclei gives rise to degeneracies at the Fermi surface which tend to favour the onset of pairing correlations. But as the considerations of the dynamics at the Fermi surface due to neutron excess does not, by itself, seem to provide a complete explanation for enhanced pairing effects, one might be led to believe that the two-body interaction in the Zp-lf shell has a larger pairing component_

As we have already seen in sect. 3, this is not actually the case and the two-body interaction appears to be predominantly of the quadrupole-quadrupole type. The higher multipole components in the interaction do not seem to be significantly large. This is quite evident from the fact that, as seen in table 5, the values of the quadrupole moments obtained with degenerate smgle-particle energies are almost the maximum possible, whereas the values for hexadecapole and Qg moments are much smaller than their maximum possible values.

An explanation for the .absence of ro~tion~ features, because of an apparent enhancement of pairing correlations, therefore lies in considering the fact that as pointed out earlier, a rotational spectrum results with a quadrupole-quadrupole two-body interaction onZy if one has a 1(1+ 1) spectrum for single-particle energies, a situation quite unlike the experimental sequence of energies for single-particle states in the 2p-If shell.

To see this crucial role played by the single-particle energy spectrum in producing deformation, we apply Elliott’s “) theory of nucleons interacting in a single major shell with a monopole-plus-qua~upole interaction. As Elliott shows, the low-lying eigenstates of such a system can be classiged according to the eigenstates of the SU(3) Casimir operator,

C = -xQ - Q+3xL2, (6)

where Q = xi qi is the total-mass quadrupole operator of the nucleus, x is a constant and L is the angular momentum of the system. The angular momentum L com- mutes with C and therefore the Hamiltonian

H = f&,--x&+ Q = &4-C-3@, (7)

where He is the oscillator potential, will give rise to a rotational L(L+ 1) spectrum. Rewriting the expression (7) as

where ci is the SU(3) Casimir operator for the ith nucleon, it becomes obvious that a given two-body interaction which is of the quadrupole-quadrupole type will give rise


to a rotational band of collective states belonging to a unique SU(3) representation if

the single-particle levels have a Z(Z+ 1) spectrum. The expression (8) also reveals the

fact that the I(I+ 1) spread of single-particle levels is directly proportional to x, the

strength of the quadrupole-quadrupole interaction.

As an application of these general arguments to the particular case of the 2p-If

shell and to observe the effect of single-particle energies on the deformation, and the

development of rotational behaviour resulting from the two-body interaction, we have

carried out HF calculations with various sets of single-particle energies. Again we

choose 44Ti as a test nucleus for carrying out these theoretical experiments.

In table 6 we have given the values of various multipole moments for 44Ti HF

states resulting from full interaction and different sets of single-particle energies.

TABLE 6

Effect of the single-particle spectrum on the various multipole moments for the 44Ti HF state

<Qo4> <Qo6>

(a) ev. 20.00 61.22 79.34 (b) ef-ec = 0 23.18 78.79 86.95 (c) zs-Ep = 1 23.96 72.92 62.54 (d) .s-zP = 2 23.27 58.72 35.82

<Qo’>ma. 24.00 79.48 150.00

The multipole moments are given for the cases when (a) the experimental single-particle spectrum is used, (b) all the single-particle states are taken to be degenerate, (c) the p-states are degenerate and 1 MeV below the f-states, and (d) the p-states are degenerate and 2 MeV below the f-states. The maximum possible values for various 2A pole moments in the p-f shell are given in the last row.

TABLE 7

Effect of the single-particle spectrum on the occupied HF single-particle orbitals, the HF energy E and the HF energy gap G for 44Ti

(4 em. + 0.83 -0.24 -0.44 0.23 - 9.35 3.33 (b) &r-s, = 0 0.60 -0.47 -0.53 0.37 -16.32 5.15 (c) &r-Ep = 1

: 0.51 -0.39 -0.63 0.43 -14.28 5.32

(d) Q-E,, = 2 4 0.39 -0.29 -0.72 0.49 -12.94 5.16

With the spin-orbit interaction switched off, an increase in the quadrupole moment

is observed as the p-orbit is taken below the f-orbit. The maximum possible value

of the Qg moment for the 44Ti state, which is 24.0b2, is obtained when the p-doublet

is 1 MeV below the f-doublet.

The single-particle orbits, the HF energies and the HF energy gaps corresponding

to different sets of single-particle energies are given in table 7. One notices here that

the HF energy gap is 5.32 MeV when the energy separation between the f- and p-


doublets is 1 MeV whereas it is only 3.33 MeV when the experimental single-particle energies are used. As the onset of pairing correlations is very crucially governed by the energy gap in the HF single-particle spectrum, it becomes obvious that the unfavourable sequence of single-particle levels will contribute significantly to the increased pairing effect in the 2p-lf shell.

5. The effect of the Pauli principle on deformation

One of the main reasons for the lack of proper emphasis on the deformation-producing character of the T = 1 component of the effective interaction has been the observed fact that the nuclei with only identical particles outside a closed core, such as the Ca isotopes, exhibit seniority spectra and are not deformed “).

We have already seen that both the T-components appear to possess about the same multipole character. Thus as seen in sect. 3, even for 44Ca, in which only the T = 1 interaction is operative, one gets almost the maximum possible deformation for four identical particles when all the single-particle energies are degenerate. The absence of rotation-like features is, therefore, not to be considered as a reflection of the pairing character of the T = 1 interaction. In this section we attempt to show that the fact that large deformation results when both protons and neutrons are present is largely a consequence of the Pauli principle.

As we have seen earlier, the quadrupole moment obtained for the normal 44Ti HF state in which the two neutrons and the two protons occupy the k = +$ and k = -4

orbits is 20.0b’. With the same set of single-particle energies we find that a normal HF state for 44Ca, in which two neutrons occupy k = ++ orbits, has a moment 10.66b2 (see table 8a, first row).

In order to bring out the effect of the Pauli principle on the deformation for four nucleons outside the 40Ca core, we try to treat the nucleus 44Ti on the same footing as the nucleus 44Ca as regards the prescription concerning occupied orbits. Thus we perform a tagged HF calculation in which we try to obtain a minimum-energy de-

TABLE 8

The multipole moments for a 44Ti HF state, with restrictions on occupied proton and neutron single-particle orbitals (a tagged HF state), as well as a normal 44Ca HF state with (a) experimental set of single-particle energies, and (b) with all single-particle states degenerate

Occupied orbits <Qa=> <Qo4> <Q06>

(a) 44Ca exp. ii, 33 10.66 13.72 -16.71

44Ti exp. ii, 35 10.83 9.94 -28.13

(b) 44Ca degenerate ii, 35 17.52 29.67 -08.57

44Ti degenerate Ii, 33 17.77 28.71 - 12.50

<Qo”>mx 18.00 59.30 120.00


te~n~n~~ wave function for 44Ti with protons and neutrons ~stributed in various ways in the orbits with k = &$ and k = ++, as shown in fig. 2.

We first discuss the results obtained with experimental single-particle energies. We notice from the results presented for various multipole moments in various cases

OCCUPANCY

(P wn)

<Go’>

<G>

Ii,i;li Ii;35 Ki3 13;;:

20.00 IO.83 12.31 9.58

67-20 9.94 WI6 6.60

- 79.20 -26*13 --13*61 -29-24

Fig. 2. The multipole moments for some KF states with restrictions regarding the occupied proton and neutron orbitals. The occupied proton and neutron orbitals have been sbown in each case by indicating the proton by x and the neutrons by 0. The label (11, 33) implies that the two protons

are in orbits with k = &$ and the two neutrons are in orbits with k = kg.

in fig. 2 that, significantly, the quadrupole moment obtained when the variational calculation is carried out with the protons restricted to the k = fr 3 orbit and the two neutrons restricted to the k = L-4 orbit is only 10.83b2. This is strikingly similar to that obtained for the normal HF state for44Ca.

These results clearly show the fact that a nucleus having identical nucleons has a smaller deforma~on is largely due to the Pauli principle - which allows only two particles in a given orbit - and should not by itself be regarded to imply any special pairing character of the T = 1 interaction.

However as mentioned earlier the T = 0 and the T = 1 interactions have different overall strengths. As a result, the orbits obtained for the nucleus 44Ca and those obtained for the nucleus 44Ti with restricted occupancies for protons and neutrons in the presence of experimental single-particle energies are slightly different. This difference is reflected in the values of the Qz and Qz moments given in table Sa. This difference is greatly reduced when the single-particle energies are madedegenerate as is clear from the results presented in table 8b. The fact that, even in the absence of single- particle energies, the Q: and Qg moments of the HF state for 44Ca are larger than those of the tagged HF state for 44Ti can now be attributed to the slightly greater pairing character of the T = 1 interackion.

It can be mentioned here that the HF state with the occupied orbits as shown in the first column in fig. 2 has both time-reversal as well as n-p symmetries whereas the occupied orbitals as given in the second column have time-reversal symmetry only.

638 S. K. SHARMA AYD K. H. BHATT

TABLE 9

The HF single-particle orbitals and the HF energies E for a normal HF state for 44Ca and for some 44Ti HF states with restrictions regarding occupied proton and neutron orbitals

Yuclcus Occupied k

orbits -_ .--

‘%a Ii-, 33 4 i:

44Ti Ii, 11 9

Ii,35 8

$

l?, i3 4 -f

-3 3

13,13 1

3

.-. .-

0.973 -0.048 0.986 -0.110

0.957 -0.106

0.970 0.039 0.981 -0.120

0.929 -0.086 0.982 0.112 0.929 0.086 0.982 0.112

0.988 0.059 0.9si --0.101 0.988 -0.059 0.987 0.101

c.2

-0.218 -0.126

--0.252

-0.238 -0.120

-0.340 -0.152 -0.340 -0.152

--- 0.144 -0.127 -0.144 ---0.127

_ .- 0.054

-2.17

0.097 -9.35

0.024 -0.152 -4.94

0.117 -4.64

-0.117

-0.020

0.020 -4.07

The occupied orbitals shown in the third and fourth columns of fig. 2 correspond to

HF states with none of these symmetries as is also clear from the wave functions

given in table 9.

6. Conclusions

The T = 1 interaction possesses many non-zero matrix elements which are not of

pairing type, that is, matrix elements which do not correspond to the scattering of two

nucleons from one J = 0 paired state to another. But the pairing caricature of the

T = 1 interaction seems to have become so well known, because of the resemblence

of certain T = 1 matrix elements to those of a conventional pairing interaction, and

also because nuclei such as Ca isotopes exhibit seniority-like spectra, that the role

of non-pairing-type T = 1 matrix elements has never been properly investigated.

This, probably, has overshadowed the deformation-producing character of the T = 1 interaction.

We have shown that the T = 1 interaction has about the same deformation-produc-

ing tendency as the T = 0 interaction and that they both have a dominant quadrupole-

quadrupole component. Hence it is clear that the observed absence of well-developed

rotational features in the 2p-If shell arises mainly because of the somewhat unfavour-

able sequence of experimental single-particle energies.

It is also made clear that it is the Pauli principle together with the experimental

single-particle energies, and not any special feature of the T = 1 interaction, which is

largely responsible for the absence of characteristic features due to deformation in the

nuclei with only identical valence particles.

Our aim here has been to compare the deformation-producing tendencies of the


T F 0 and T = 1 parts of the effective interaction. This is conveniently and unambi- guously done by comparing the structures of the NF intrinsic states generated by the two interactions.

We have not attempted to investigate the pairing nature of the T = 0 and T = 1 components of the interactloa. However, our results do indicate that the T = I interaction has a slightly larger pairing tendency as compared to the T = 0 ~rn~onent. Our viewpoint is that, just as the deformations contained in the HF potential provide a good criterion for examining the deformation-producing tendency of an interaction, one can examine the pairing character of an interaction in a simple way by comparing the admixtures of the Zp-2h states to the HF states produced by the T = 0 and the T = I components separately, that is, by finding the corrections to the intrinsic HF states due to pairing effects of the T = 0 and the T = 1 types separately. This approach is obviously not a self-consistent one. A self-consistent method. of investigating the pairing tendencies of the T = 0 and the T = 1 components individually would be, therefore, to compare the structures of the Hartree-Fock Bogoliubov (HFB) intrinsic states - various multipole moments and. the occupation probabilities, etc. - produced by the two interactions.

The onset of pairing correlations depends criticahy on the sharpness of the Fermi surface, that is, the degeneracies at the Fermi surface. fn many cases one finds that the degeneracies at the Fermi surface are large mainly because of the HF states being constrained to correspond to an axially symmetric self-consistent field. In such eases, it would be incorrect to examine the pairing tendency of the interaction by including pair&g effects in the HFB methods. This is because the large degeneracy at the Fermi surface, which is somewhat artificial in nature in the sense that it vanishes when the HF field is allowed to have a more general shape such as an ellipsoidal one, would lead to “spurious” pairing effects overshadowing to a large extent the actual pairing effects due to the pairing tendency of the interaction, Therefore, one has first to obtain the proper HF field before treating the inclusion of pairing correlations in the HFB framework in making an attempt to examine the pairing tendency af the interaction. This problem is under investigation.

It is a pleasure to thank Prof. S. P, Pandya and Dr. 3. C. Parikh for reading the manuscript. We would also like to thank Prof. G. E. Brown fcr asking some questions which have contributed to the discussion of the comparative study of the pairing aspects of the interactions.

Appendix

kl. SINGLE-PARTKLE STATES IN A QUADRUFCXE FIELD

We give here the ten single-particle states @(v, 8) of the 2p-If shell in a Qz field (here v = (2,s is the expectation value of the component of the orbital angular

640 S. K. SHARM AND K. H. BHATT

momentum along the symmetry axis and E is the eigenvalue of the intrinsic quadrupole moment in units of b2):

4, = cp(<L>, s),

40 = 4(0,6) = Jf(&fo -$P,),

9*1 = 4(&l, 3) = &2&I-I%,)>

9*2 = 4(+2,0> =f*2,

4b = $qo, 0) = Jf(J~fo+&o),

$;I = 4(+1, -3) = j$(f*1+2PF,),

413 = 4(13, -3) =f*3.

A.2. SINGLE-PARTICLE STATES IN A HEXADECAPOLE FIELD

The ten single-particle states of the 2p-lf shell in a Q$ field are the following (here .s4 is the eigenvalue of the hexadecapole operator):

k = $(v = <U ~4 = <Q%>>, $,, = $(0,19.870) = 0.8039f,-OS946p,,

tiil = $(+l, 9.781) = 0.7359f,,-0.6771p,,,

$13 = $(-t-3,4.500) = fj_-‘+3,

I+I&~ = $(+I, -8.281) = 0.6771fi,+0.7359p,,,

$*2 = 4qf2, -10.500) =f*2,

+& = $(O, - 10.870) = 0.5946f0 +0.8039p0.

References

1) T. T. S. Kuo and G. E. Brown, Nucl. Phys. All4 (1968) 241 2) J. P. Elliott, Proc. Roy. Sot. A245 (1958) 128, 562 3) K. H. Bhatt and J. C. Parikh, Phys. Lett. 24B (1967) 613;

M. Harvey, Proc. Int. Conf. on properties of nuclear states, Montreal, 1969, p. 313 4) K. H. Bhatt and J. B. McGrory, Phys. Rev. C3 (1971) 2293 5) I. Kelson and C. A. Levinson, Phys. Rev. 132 (1963) 2189 6) S. K. Sharma aod K. H. Bhatt, Phys. Lett. 36E (1971) 550 7) D. R. Kulkarni and K. I-I. Bhatt, Nucl. Phys., to be published 8) K. R. Sandhya Devi, S. B. Khadkikar, J. K. Parikh and B. Banerjee, Phys. Lett. 32B (1970) 179 9) I. Talmi, Rev. Mod. Phys. 34 (1962) 704

on the non-pairing aspects of the t = 1 interaction

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