Projected Hartree-Fock spectra in 1f-2p shell nuclei

1.D.1

I

Nuclear Physics A U 0 (1968) 472--480; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

P R O J E C T E D HARTREE-FOCK SPECTRA IN lf-2p S H E L L NUCLEI S. B. K H A D K I K A R and M. R. G U N Y E

Tata Institute of Fundamental Research, Bombay, India Received 31 October 1967

Abstract: The projected Hartree-Fock technique is used to calculate energy spacings and the electromagnetic properties of even-isotopes of Ca, Ti, Ni, Zn and some odd-mass isotopes of Cu. The energy spacings are found to be smaller than the experimentally observed ones, while the (B(E2)~) quadrupole transition reduced matrix elements compare fairly well with the experimental values.

1. Introduction

It is well known that many low-lying levels in the nuclear spectra could be obtained by projecting states of good angular momentum from an intrinsic axially deformed single determinental state 1). This intrinsic state was shown to be a self-consistent single-shell Hartree-Fock (HF) solution in the lp shell as well as in the 2s-ld shell 2, 3) nuclei. The purpose of the present work is to test whether the H F states in the lf-2p region obtained by the axially H F calculation could be used as intrinsic states from which properties of some low-lying levels could be explained. No effort is made to fit the results by extensive variation of parameters. The cases considered are even-mass isotopes of Ca and Ti with 4°Ca as the core and even isotopes of Ni, Zn and oddmass isotopes of Ca taking 56Ni as the inert core. We use the Rosenfeld mixture with Yukawa type radial dependence for the twobody interaction. This force has been tested in the 2s-ld shell region s) and found to give fairly good results there. Recently shell-model calculations with a similar force have been performed 6) for Ni isotopes and found to give fairly good results. It is known that the collective vibration 7) has a dominant role in the excitation of low-lying levels of these isotopes, and we do not expect very good agreement here with experimental results from this type of calculation. For better agreement, "band-mixing" calculations would be necessary, in which case the intrinsic state could be a linear combination of several determinants. But these calculations would be rather lengthy as far as the computer time is concerned. We also test the wave function with other probes such as electric quadrupole and magnetic dipole moments and the transition probabilities. 2. Formalism

The mathematical formulation for the projection of good angular momentum states and for calculating the electromagnetic properties of such projected states has 472

HF SPECTRA

473

been developed in ref. s). We shall avoid a detailed formulation here, however, for the sake of completeness we quote some results. The energy E~ of a state IJ, M, K ) projected from a state I~, K), where c~denotes quantum numbers other than K (the band quantum number), is given by the following expression: E~ = (~, KIHP]d~, K)/(~, KIeWIt, g ) ,

(1)

where H is the Hamiltonian of the system, and the operator P~ is given by P~ = (J+½)

fo

sin 0 dO d[~.~(O)e-~°J'.

(2)

We drop the index c~ hereafter. Also the reduced matrix element of a tensor T~ is given by (apart from normalization)

(IKIITZIIJK) = ~ (JAK-I~#IIK)(KIT~P~_u, ~[K),

(3)

where P~_g, K is given by j

P~-,. r = (J+ ½)

io

•

sin 0 dO d:r-,, r(O)e-'°s'.

(4)

The evaluation of the matrix elements containing one projection operator needs the matrix element (g[o e-'°J~lg), where O is a one-(two-)body operator. This cart be done either by introducing a complete set of one-(two-)particle-one-(two-)hole states 3) between the two operators or by expressing the matrix element in terms of the overlaps of one-(two-)hole states *' 5) depending upon the nucleus under consideration (as the number of particlehole states will, in general, be different from the square of the number of hole states). 3. Calculations and results In the H F calculations, we use the two-body interaction v(1, 2) as a Rosenfeld exchange mixture with Yukawa radial dependence e-r12/a V(1, 2) = V0 ~(zl" z2)(0.3+0.7 a 1 "a2) - - ,

rl2/a

(5)

where the range parameter a = 1.37 fro. Using the standard method for the calculation of the H F wave function and the projection technique briefly discussed earlier, one finds the spectra and the expectation values of the other physical operators of the nuclear system. In all the following calculations we have taken the harmonic oscillator parameter as h~o = 11.51 MeV. In order to study the dependence of the calculated

474

S. B . K H A D K I K A R A N D M . R . G U N Y E

3 --

2+

-

-

4+

>

4 -I"

24%.

4 ~-

24% 3 2 % _ _ 2

o

_

_

_

o

÷

z5%

(a)

2-t32%

2+

-

2-I-

-

-

-

~-

o+ z 5 % _ o

(b)

+

(c)

_ _ o

+

_

(d)

_

o

+

~e)

Fig. 1. Energy levels of 4~Ca: (a) ExperimentaI spectrum, (b) calculated with V0 = 45 MeV, (c) calculated with 110 = 51 MeV. The probability of a state J~ in the intrinsic H F state is shown in percentage figures on the left of the corresponding level. Energy levels of 44Ca: (d) experimental spectrum, (e) calculated spectrum.

4 ~

_

3

_

6

8%

6+

20%

4+

÷

-_

_

4

+

4+ >

2 m

m

4

÷

t0%

8+

29%

.... 6 +

19%

6+

31%

4+

2+

54% 30%

Im

_

_

2

4+

+ 32 %

22%.

0 --

_

_

0

(a)

2+

2+

"r

2. +

2 "r

5°/°

0 "r

(b)

0 "r"

(c)

I0%

O "I-

(d)

0 4- 21%

(e)

O~

(f)

Fig. 2. Energy levels of 46Ti: (a) Experimental spectrum, (b) calculated spectrum. Energy levels of ~STi: (c) experimental spectrum, (d) calculated spectrum. Energy levels of 5°Ti: (e) Experimental spectrum, (f) calculated spectrum. The percentage figures on the left of a level J= represenl its probability in the H F state.

HF SPECTRA

475

results on the strength V o, we carried out the calculations o f 42Ca spectra for V o = 45 M e V and V0 = 51 MeV. The results are shown in fig. 1, it is clear f r o m it that the spectrum is not sensitive to the strength parameter Vo. In the rest of our calculations, we have taken V o = 51 MeV. The calculated energy spectra o f 44Ca and 46'48'5°Ti isotopes are shown in figs. 1 and 2, respectively. I n the above calculations of Ca and Ti isotopes, the single-particle levels (in M e V ) are taken as f~ = 0.0, f~ = 6.0, p~ = 2.0 and p~ = 4.5. These are approximately the experimental singleparticle levels f r o m the 41Ca spectrum. There is appreciable strength o f the 4 + level in the H F states of 42'44Ca, but experimentally the situation is not clear. F o r the 46'48Ti isotopes, the calculated spectra are squeezed by a factor o f ½ as c o m p a r e d with experimental results. The H F state contains appreciable strength up to the J = 6 level. The agreement between the calculated and experimental spectra is m u c h better for 5°Ti (see fig. 2). Table 1 shows the B(E2) T values calculated for the 0 ~ 2 transition. F o r the calculation o f electric quadrupole m o m e n t s Q and electric quadrupole transitions E2, we have taken two sets of values for the effective charges for protons (ep) and neutrons (e,). It is seen that ep = 2e and en = e gives g o o d agreement with the experimental values o f 46'4gTi. This shows that long-range correlations are included in this procedure of effective charges, while the p o o r agreement with energy spectra shows that all the short-range correlations are n o t taken into account. TABLE 1

The set Of single-particle energies used for nuclei with the f~ shell closed Set I

~ MeV

~ MeV

0

½ MeV

0

3

II

0

0.78

1.08

III

0

0.80

1.60

TABLE 2

Comparison of the experimental and calculated ground-state electric quadrupole and magnetic dipole moments Ground-state magnetic moment (n.m.) Nucleus

nSCu nsCu

calculated T2.92 ±3.10

Ground-state quadrupole moment (e. b)

exp.

ep = 1.5, en = 0.5 calc.

n-2.226 -I-2.38

0.10 0.10

ep = 2, en -- l calc. --0.157 -- 0.157

exp. 0.16 0.15

The results o f the calculation o f energy spectra o f Ni, Ca and Z n isotopes are shown in figs. 3-5, respectively. We have shown the results for those cases for which the

476

S.B.

KHADKIKAR

AND

M.R.

GUNYE

H F solutions were deformed so that we could project the relevant states. F o r example we considered three sets of single-particle energies as shown in table 2. The first set is used by Kisslinger et al., while the last set is the single-particle energies used in recent calculations 6). 3 --

4+ _

0.3% 2

.

.

.

.

.

_

4+

4 -I-

--

2+

>

21% _ 44%

_

4+ 2 +

_ _ 2 + 5 8 % _ _ 2

I__

- - 0

+ 35%

(o)

0+42%

(b)

+

0+

.....

i

(c)

0+

28%

4 Jr

39%

2 +

~/I

(d)

l 0 +

(e)

Fig. 3. Energy levels of 58Ni: (a) some of the experimental levels, (b) calculated spectrum with singleparticle energies in set I, (c) calculated spectrum with single-particle energies in set III. Energy levels of 6°Ni: (d) some of the experimental levels, (e) calculated levels. 2--

7*/_-

9,,~-

7/z-z l°"

____

7/2-

v/z-"

~/~

__½-

3/z-

__3/z-

5/~-

zo*/~

5/z-"

23./.__Y/z-

30%

t / e -

.

.

.

.

(,a)

3/2- 37%

(b)

(c)

36%

~/U

(d)

Fig. 4. (a) Experimental spectrum of 68Cu. (b) calculated 68Cu levels by projection from a K ~ HF state, (c) experimental spectrum of 65Cu, (d) 65Cu levels, calculated by projection from a K ~ ~HF state. The percentage figures on the left of a level J= show the probability of the corresponding state in the HF state. The results of the H F calculations were f o u n d to be very different for sets I a n d l l I a n d very little different for sets II a n d III. The single-particle energies of sets I I a n d I I I tended to give spherical H F states for almost all cases considered except, peculiarly enough, for 58Ni a n d for one of the Z n isotopes. I n the case of C u isotopes, we o b t a i n e d consistently spherical solutions for the lowest H F state w h e n we used sets II a n d III.

HF SPECTRA

477

We present below the results obtained from set (I) of single-particle energies. In this case also we found the 6°Ni H F solution to be spherical. Except this case and the unexplained ½- first excited state of Cu isotopes, the agreement for energies is good. We feel that another " b a n d " K = ½-, may start from the second I = ½- state, i.e. we have to project the I = ½- state from another H F solution. The I = ~ - state from this band, we hope, does not mix very much with the I = ~- state from K = { " b a n d " which we have calculated. We may note that we do get the reversal of ~ and ½- states in the case of 65Ca levels as, probably, the experimental values require. 4 m

10% :3

6+

__

4+

27%

>2-:E

2 9 % _ _ 4

-

-

2+

2+ 50%

0 4-

(a)

4 4 ° A ~

+

2+

2+

17%

0 ~-

(b)

0+

{c)

15%

0+

(d)

Fig. 5. Energy levels of 64Zn: (a) experimental spectrum, (b) calculated spectrum. Energy levels of enZn: (e) experimental spectrum, (d) calculated spectrum. The percentage figures on the left of a level Y= show the probability of the corresponding state in the HF state. The electric quadrupole moments and the magnetic dipole moments of the 63Ca and 6SCa ground states show good agreement with the experimental values when we take the effective charges for proton % = 2 and for neutron en = 1. The magnetic moments are smaller than the Schmidt value of 3.75 n.m. But they are still larger than the experimental values (see table 2). The B(E2)T values calculated for the same set of effective charges show good agreement with the experimental values for all the cases considered (see table 3). In fig. 5 we show the results for 64,66Zn" One observes in the H F states of these nuclei a considerable amount of 4 + and 4 +, 6 + states, respectively. Experimentally these levels are not found. The 0-2 separations and the B(E2)T for 0 ~ 2 transitions show good agreement with the experimental values.

478

s.B. KHADKIKAR AND M. R. GUNYE

TABLE 3 The experimental and calculated electric quadrupole transition reduced matrix elements B(E2)~' Transition No.

Nucleus

B(E2)~' in the units of e ~ • b z

Jl

J~

calculated ep = 1.5, en = 0 5

calculated % =2, en= 1

1 2 3

4~Ti 48Ti 58Ni

0 0 0

2 2 2

0.016 0.015 0.0034

0.038 0.026 0.013

4 5 6 7 8 9

e~Cu eaCu esCu eSCu S~Zn eeZn

~ _~ ~ ~ 0 0

-~z ~ ~ 2 2

0.0053 0.0037 0.0033 0.0034 0.028 0.025

0.017 0.012 0.011 0.010 0.076 0.066

exp. 0.056 0.070 0.10 :~0.025 0.029 0.053 0.034 0.028 0.083 0.087

TABLE 4 The H F energy (El_IF), H F quadrupole moment (QHF)-- in units o f b ~ (where b = 1.9 × fm), energy of the projected 0 + ground state and the gaps for neutron and proton orbitals Gap (G) (MeV) No. 1 2 3 4 5 6 7 8 9

Nucleus

EHF(MeV)

Eg.s.(MeV )

ZaNi e~Ni e~Zn 66Zn 4eTi 4STi S°Ti ~Ca 44Ca

-- 0.93 -- 2.78 --17.73 -- 19.82 --13.23 --15.65 --17.25 -- 0.60 -- 1.49

-- 1.73 -- 3.34 --18.99 --21.20 --14.64 --16.50 --18.53 -- 1.58 -- 2.58

QHF/b ~

+ 6.43 + 5.77 --15.65 -- 13.90 +24.95 +23.10 --11.47 + 6.07 + 9.06

protons

neutrons

--1.4 1.4 3.7 3.1 1.37 ---

0.90 0.14 0.96 0.22 1.12 0.82 2.73 0.50 0.49

Here single-particle set (IIl) is used for Nos. 1 to 4 and the strength of the two-body force (V0) is 51 MeV. T a b l e 4 s h o w s t h e H a r t r e e - F o c k e n e r g i e s (EHF), t h e e n e r g y o f t h e p r o j e c t e d g r o u n d s t a t e (Eg.s.) a n d t h e g a p ( G ) b e t w e e n t h e o c c u p i e d a n d t h e u n o c c u p i e d H a r t r e e - F o c k o r b i t a l s as w e l l as t h e i n t r i n s i c m a s s q u a d r u p o l e

moments

Qnv. F o r

i s o t o p e s t h e o c c u p i e d ~ - o r b i t a l is h i g h e r i n e n e r g y t h a n - ~ 1-3)

bands are degenerate, however the HF

the odd Cu

a n d a l s o t h e 1~) a n d

solutions for these bands are time

reversed with respect to each other therefore mixing of these bands does not add a n y t h i n g n e w a n d i t is sufficient t o p r o j e c t f r o m a n y o n e state. For other nuclei it can be noted that the amount

o f b i n d i n g p e r p r o t o n is a b o u t

7 MeV for isotones, while the neutrons have binding energy 1 MeV per particle for i s o t o p e s o f a g i v e n n u c l e u s . T h i s effect is d u e t o t h e f a c t t h a t all t h e n u c l e i c o n s i d e r e d have more neutrons than protons

and the neutron-proton

i n t e r a c t i o n is s t r o n g e r

w h e n t h e y a r e i n s i m i l a r H F o r b i t a l s , a n d t h a t t h e r e is less a t t r a c t i o n b e t w e e n n e u t r o n s

HF SPECTRA

479

alone. The behaviour of the neutron gaps is peculiar. For example the gaps between occupied and unoccupied H F states with negative deformation for 46'48'5°Ti isotopes are 0.74, 0.33, 2.73 MeV, respectively. The isotope 5°Ti has negatively deformed ground-state H F solution while 46'48Ti have positive deformation. The H F orbitals are somewhat similar to Nilsson orbitals and this behaviour could be approximately predicted from the behaviour of the Nilsson levels. The two solutions for 46Ti differ by ~ 3.3 MeV for the ground-state energy and we do not expect mixing of these bands to improve on the results. The compression of the spectrum relative to the experimental one is also found in the 2s-ld region. An investigation of the effect in that region as it is much easier to handle nuclei in the 2s-ld shell may throw light on this effect in 2p-lf nuclei. We also see that there is enormous deviation from the rotational type spectrum when there are only neutrons outside the closed shell. This is so because the method of projection is an approximation to configuration mixing calculation and it is easy to show that the method of projection takes into account seniority force more or less exactly while the pairing matrix elements connecting different (j)2 levels are taken in the first order approximation. These later matrix elements are not so much important when there are only a few particles and they occupy a single j-level predominantly. Then, the method of projection results in a seniority scheme and we get a spectrum deviating from the J(J+ 1) rule. However when the gap between the occupied and the unoccupied levels is too small, e.g. 62Ni, one must take into account the off-diagonal pairing matrix elements. This can be done by taking for the intrinsic state a Hartree-Bogolyubov state with correct number of particles projected out of it. Investigation on these lines has been taken up and will be published somewhere else. 4. Conclusion

From the overall agreement obtained above, one observes that the wave functions calculated by the projection technique explain the main properties of the low-lying levels of lf-2p shell nuclei. We see that the results are sensitive to the type of one-body potential (singleparticle energies). Also we get good deformed H F states when there are protons as well as neutrons outside the closed shells. The neutron-proton interaction dominates in producing the deformed H F potential. The H F calculation for the intrinsic states is useful for the cases when there are both neutrons and protons outside closed shells. In the case of interesting but more complicated nuclei like 55Mn this technique is expected to yield better results. We plan to carry out such calculations. All the calculations were performed on a CDC-3600 T I F R computer. We would like to thank Dr. C. S. Warke for many useful discussions and a critical reading of the manuscript.

480

S . B . KHADKIKAR AND M. R. GUNYE

References 1) 2) 3) 4) 5) 6) 7) 8)

M. Redlich, Phys. Rev. 110 (1958) 468 D. Kurath and L. Picman, Nuclear Physics 10 (1959) 313 W. H. Bassichis, B. Giraud and G. Ripka, Phys. Rev. Lett. 15 (1965) 980 C. S. Warke and M. R. Gunye, Phys. Rev. 155 (1967) 1084 M. R. Gunye and C. S. Warke, Phys. Rev. 156 (1967) 1087 Y. K. Gambhir and Ram Raj, Phys. Rev. 161 (1967) 1125 L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 9 (1960) M. R. Gunye and C. S. Warke, Phys. Rev. 159 (1967) 885