CALCULATIONS ON NEGATIVE-PARITY STATES IN THE NUCLEI OF A=36-40

PHYSICAL

REVIE%

C VOLUME

23,

NUMBER 1 JANUARY

1981

Calculations

on negative-parity

states

in

the nuclei

of

A

=36

40

S.

T.

Hsieh and M.C.Wang

Department ofPhysics, National TsingHua University, Taiu/an, Republic ofChina

D.

S.Chuu

Department ofElectrophysics, National Chiao-Tung University, Hsinchu, Taiwan, Republic ofChina (Received 25June 1980j

A shell-model calculation ofthe negative-parity states in theA

=

36-40nuclei is presented. The nucleus "Cais assumed tobe an inert closed core. Active holes are restricted tothe (1d3/22$1/Qj configurations and an active

particle is allowed tooccupy the 1f,/2or2p3/2orbit. Thetwo-body effective interaction is assumed tobe the modified

surface-delta type. Theenergy spectra are calculated from aleast-squares fittothe experimental data, varying the

T

=

0and T

=

1strengths ofparticle-hole and hole-hole interactions and the three single-particle level splittings. Spectroscopic factors, E2,E3,and M1 transition rates, and two-body matrix elements are also calculated and compared with the observed values and the previous theoretical results. Thevalidity ofthe weak-coupling model is also tested.

NUCLEAR STRUCTURE

4=36-40,

calculated effective interaction, energy spectra,

spectroscopic factors and EM transition rates.

I. INTRODUCTION

For

negative-parity states in nuclei with mass

numbers A

= 36-40,

only afew calculations have

been performed. Most ofthese assumed an inert closed ~S

_core

_{with configuration}

_restricted

_to

within the (1d3/&, 1f&/,)or (ld3/g 2p3/2) space. In this paper, the nuclei with A

=

36-40

are

cal-culated within the framework of the shell model.

The nucleus Ca

is

assumed tobe an inert

core.

In actual calculations, holes

are

assumed tobe

distributed in the 1d3/2 and

2s«,

orbitals and one

particle in the 1f&/2 or 2p3/2 orbital. The

exten-sion of the model space

is

expected to obtain a

bet-ter

systematically theoretical prediction of

odd-parity states in this mass region.

The

earlier

examples of shell-model calculations

on levels of odd parity in the region of the 1d3/2

shell nuclei were performed by Goldstein and

Tal-mi and Pandya with the

restriction

that one

pro-ton

is

in the 1d3/g shell and one neutron

is

in the lf&/2 shell. Erne extended their model space to

include an arbitrary number of nucleons in the

Id3/2 shell with all inner shells considered an inert

S

core.

Twelve two-body interaction matrix

ele-ments of the

extra-core

nucleons and two binding

energies to the

core

were treated as

free

parame-ters

to

fit

sixty nuclear levels, including

even-parity states belonging to configurations in the

1d3/g shell only, of nuclei in the range

S-

'Ca.

Under this severely

restricted

(ld~/2, 1f&/2)

con-figuration, the agreements between the theoretical

and experimental odd-parity levels

are,

of course,

only possible for the very lowest

states.

Maripuu

and Hokken studied the nuclei ofA

=35,

37, and

39with residual interaction in the form of a

modi-fied surface-delta interaction (MSDI). Taking into

account the configurations of

(1d3„,

1f„,

)and

(ld3/2 2P3/2), great disagreement with the

experi-ment still

arose.

Neither the energies nor the

level order ofthe identified states of some nuclei

could be reproduced well by his calculations.

Maripuu et

al.

calculated energies and wave

func-tions for Cl, K, and Kin a model space which

included d&/&, s«&, d3/2, f&/&, and p3/2 orbitals. The two-body matrix elements used in their

Ham-iltonian were calculated from the Sussex relative

oscillator matrix elements with space truncation

effects added. In their calculation, the observed

anomalously large 32

-4&

M1 transition strength,

which cannot be understood when only assuming a

weak-configuration mixing, thus can be

correctly

predicted. The odd-parity states of nuclei with

mass numbers A

=39-41

have also been studied by

Hsieh et

al.

6

They included the complete 2s,1d

and

_1f,

2P shells in

a

first

order Tamm-Dancoff

calculation. The spectroscopic

factors

and EM

transition

rates

were obtained in reasonable

agree-ment with the experimental data. Recently, Has-per' presented a shell-model calculation on

even-and odd-parity states of nuclei in the mass region

A

=36-39.

In spite ofhaving used about the same

model space (the restriction of the number of

nu-cleons distributed in the

_2s«2

and 1fv/-2

are

differ-ent) as has been used in the present work, an inert

closed ~_Si

_core

was assumed in his work. In

ad-dition to this, only seventeen experimental levels

were included in his least-squares

fit.

However,

the reliable experimental data available now

cer-tainly include much more than this value.

Fur-thermore, he did not calculate the spectroscopic

factors

and EM transition and thus there

is

no way

to determine how good the wave functions

are.

In this work, we calculate the negative-parity

states in A

=36-40

nuclei assuming the nucleus

Ca tobe an inert closed

core.

The model

as-sumptions used in the present work

are

described

in detail in

Sec.

II.

Section IIIgives the

results.

Conclusions

are

presented in the final section.

H. ASSUMPTIONS

As indicated above, Ca

is

assumed to be an

in-ert

core.

One active particle

is

allowed in the 1f7/2 or 2p3/& orbital while active holes

are

dis-tributed in the 1d3/2 and 2sf/2 shells. The neglect

of the 2p«2 orbit

is,

we feel, the most serious constraint on our model. The omission of the 1d5,2 and 1f~/2 orbitals from the model space

is

quite reasonable because the splittings of observed

single-particle levels of 2s«&-ld5/& and

1f5/&-2p3/g

are

much larger than those of 1d3/2-2sf/2 and

2p3/)-1'/2

Since the spacing between 1f«2-1d3/g

is

comparatively larger than that between

1d3/2-2sf/ 2

it

is

reasonable to assume that only one

particle

is

distributed in the 1f7/2

or

2p3/2 active

orbital. Under these assumptions, the wave

func-tions of eigenstates can be written

as

linear

com-binations of basis states of the form

t/

=1[spit

oiTi&i,&3/2o2T2~2]Th'/A _jp

o

_T,

J,

where _j~

=1f,

/2 or 2P3/2.

The Hamiltonian in this space has the form

H=H~+Hhh+Hp ~

Here Hh, represents the effective two-body inter-action between the particle in the 1f7/2 or 2p3,2

orbital and the holes in the 1d3,& and 2s«2

orbitals.

The term H» represents the two-body effective in-teraction between the holes in the 1d3/2 and

_2s«,

orbitals. Hp represents single-particle energies

for

active orbitals which were chosen initially to

roughly approximate the observed single-particle

spectra

of the masses

39,

40, and

41.

The initial

splitting between single-particle levels was taken

from Gerace and Green, who treated ground-state

correlation energy in Ca explicitly.

In this study, the two-body residual interactions

between particle-hole and hole-hole

are

assumed

to be of the modified surface & _type

V~/

—

—

4vA

r

6(Q

s)

+

B

r,

where Ap,A&

are

the corresponding interaction

strengths

for

T=O,

T=1

states,

and Bp Bg

are

the

corrected

terms to the diagonal matrix

ele-ments for T

=

_{0, T}

=

_1,

respectively. Four

inter-action strengths Ap,h Agh, Aphh and A&hh, four

parameters Bp~, B&~, Bp», and

_Bf»

and three

sylittings ofsingle-particle levels ofd3/2 sf/2,

f&/& d3/2, -and

p„,

-d„,

were allowed to vary in a least-squares

fit

to the observed energy

spectra.

For

the selection of energy data, in principle we

included all the available low-lying states with

re-liable

4'

assignments up to the point that the

first

level with uncertain

4'

assignment appeared.

Ex-ceptions will be discussed individually. Thus a

total of 56 experimental energy levels were

includ-ed in the least-squares calculation to determine

the eleven parameters mentioned above. The

overall root-mean-square deviation

is

0.

30MeV.

where N and Z

are

the total numbers ofneutrons

and protons in the nucleus. The calculated

inter-action strengths for Hhh were used to reproduce the

even-parity

states.

In the calculation ofground

states,

only h-h interactions

are

considered. Since only the energy spacings relative to the ground states

are

concerned in our calculation,

it

is

hoped that the adjustable single particle energy

may absorb the discrepancy due to the lack of

2p-2h interaction. Table I presents the calculated

and observed binding energies of the ground

states,

with the nucleus Ca assumed tobe

a

core.

The

calculated excitation

spectra

together with the

ob-served ones

for

nuclei with A

=36-40

are

shown

TABLE

I.

Experimental and calculated binding ener-gies ofthe ground states E~inMev by assuming Ca asacore.

Nucleus +exp

%ev)

36K 36Ar »Cl 36C_I

"s

3+ 2 3+ 2 0+ 0+ 2 033

-1.

84

-6.

39

-1.

08

-13.

10

-7.

12

-2.

15

-2.

57

-2.

44

-6.

15

-1.

09

-13.

09

-6.

41

-2.

16 III. RESULTS A. Energy levels

TIle excitation encl gles of the negative-pal lty

states were made tofit the observed energy

spac-ing relative to the ground

states

of nuclei with A

=36-40.

The ground-state energy

is

defined as

Eg

—

—

—

([E//(N/ Z)

—

Es(

Ca)]

CALCULATIONS

ON

NEGATIVE-PARITY

STATES

IN

THE.

.

.

iO-

)-(i)

.

4-(i) 2 ()) 5 {)) 2 (i) 3-()) 5 ())

.

2 ()) x I 3-()) 4-()) -3-(i) -4-(t) 3 2 3

2:

I ₄ 2 exp. caI.

FIG.

1.

Experimental and theoretical energy spectra for the 4=40nuclei.

in

Figs.

1-5.

Experimental data

are

taken from

Endt and Leun and Baumann et

al.

' Levels with

an

asterisk are

included in the least-squares

fit.

A

first

calculation reproduced quite well the

level sequence in the individual

masses.

It,

how-ever, exhibited a consistent shift of the

'

Ca

spec-trum by about 600keV to lower excitation energies.

This

is

due to the

fact

that Ca

is

not avery good

closed shell. To remove this discrepancy we con-sidered the effect of ground-state correlations to

our model calculation. Since the ground-state

correlations

are

stronger in Ca, we added

0.

6 MeV to _{the gap} parameter for

Ca.

Thus, the gap

for Ca turned out to be

0.

6 MeV larger than the

others. This

is

in agreement with the results of

Hsieh et

al.

6 _{and Hubbard and Jolly. '}

The energy spectra of odd-parity levels for A

=40

nuclei

are

presented in

Fig.

1.

The

agree-ments between the calculated and the observed

en-ergies

are

reasonably good for most of the levels, especially the 5& and

3,

for T

=0

and the 4&, 3,, 2,, and 22 for T

=1.

In our calculation, the

first

1

state was not included in the fitting and the

agree-ment between the calculated and the observed

val-ues for this state

is

unsatisfactory. The reason for the large discrepancy can be seen from the

earlier

shell-model calculation by Hsieh eI.

al.

Their results showed that the wave function of the

1,

state is composed of only 30ip of the

_(P»„

d»,

') and

_(P»„s»,

')

configurations, and the

triangular rule forbids the

_(f»„d»,

')

or

(f,

&„

s»,

')

to be the component of the 1 state. In

addition, the effect due tothe spurious state on

the 1 state of

"'Ca is

rather large. Therefore, it

is

beyond the model space considered in this

work and has to be excluded in the least-squares

fit.

The wave functions of the

first

level for most

of

4

for T=0and T=1

are

almost the pure states

of

(f,

&„d,

&, ')with an intensity larger than 86'fo.

One important point that has to be mentioned here

is

that even for the lowest lying state,

e.

g.,

3,

,

the intensities of

(f

&„s,

&, ') and

(P»„d»,

') are

still important and cannot be neglected. This

seems tojustify the necessity of the inclusion of

the

s,

&,and

P,

&, configuration space at

least.

The calculated and observed energy spectra for

A

=39

nuclei

are

presented in

Fig. 2.

Our

calcu-lation reproduced quite well the level sequence and

most of the level spacings for the nuclei in this

mass number. The level at

8.

89 MeV was

tenta-tively assigned as

J'

=

(—,

—

2

),

T

=

—, (Ref. 14);the

fitted & state at

8.

77 MeV favors a

J'

of 2

.

Su-garbaker et

al.

investigated the low-lying

high-spin states in K by using the 'Ca(d, n) K

reac-tion. In order to reproduce both the 'Ca(d,

o}

K

and

K(o,

cv') SKreaction da,ta, they concluded that

the spin assignment for the

5.

72 MeV level in K

has to be a -J' of '2

.

Our calculated '2 state at

6.

01MeV gives strong support to their suggestion.

The importance of the component

(f»„d»,

,

')

con-tained in the wave'functions for the yrast states in

this mass number

decreases

appreciably. The

in-tensitiesofthe components(f,

_&„d»,

',

s»,

'}, (P»„

d»,

'),

and

_(p»„d»,

',

s,

&,

')

for

theenergylev-els

in this mass number increase appreciably

com-pared with those for A

=40.

This fact manifests

again the necessity ofenlarging the model space.

Figure 3 shows the calculated and observed en-ergy levels for the nuclei with A

=

38.

The level

sequence and spacing

are

reproduced well. The

s tates at

3.

42,

12.

32, and

12.

42 MeV

are

uncer-tainly assigned at

J'=(4,

5,

6) for

T=0,

and

4'

=(1,

2, 3) and (2,3,4) for

T=2

states,

respec-tively. Ou. calculated

6,

T

=0

state at

3.

49 MeV

seems to favor a 6 state at

3.

42 MeV. The

cal-culated

1,

T

=

2state at

12.

29 MeV strongly

man-ifests that the state at

12.

32 MeV

is

a 1

state.

The observed level at

12.

42 MeV has a theoretical

2J

(2T)

2J

(2T)

J(T)

J

(T)

7(3)

9—

,(3-II)(3) A5,7,9) (3) -3 (3) (5,7,9) (3) 5-(3) (3) II-(3) 3 (3) 9-(3) -7 ₍₃₎ '5 ₍₃₎ I2 (2,3) (2) 32-4) (2) ,(0,I)-(2) ;(I-3)-(2) 3-(2) (2) 3-(2) 5 (2) 0 (2) -4 (2) 2 (2) (2)

,

4-(2) ,5-(2) 3 (2) 3 (2) 3 (3) 3 (3)

0

6—

LU 7 (3) (9, I3)-(5, 7)-II 5 9 7 (3) l3 il 'lI 5 7 9 '5 /7 ilI

5:

'9

0

K LLI

5

5-(I) (I-3)-(I) 3 (I) 5-(I) 4-(I) 3-(I) (I-5) (4-6)-(2 )

.

0

-2:

I ;(2, 4)-3 5 (I) 3-(I) 5-(I) 3(I) 4-(I) 3 2 exp. caI. exp. A=

39

ca

I.

A=38

FIG.

3.

Experimental and theoretical energy spectra

for the A=38 nuclei.

FIG.2. Experimental and theoretical energy spectra

for the A=39nuclei.

Aarts et

al. '

studied the high-spin states of

'

Ar

with the Cl(n,py) Ar reaction. Unambiguous

spin-parity assignments of

J'

=

7,

9,

and

11

to

the Ar levels at

7. 51, 10. 17,

and

11.

61 MeV

were obtained. The levels of 9 and 11

are

beyond

our model space, and the calculated

J'=7

level

at

8.

49MeV does not agree as well as the other

states.

This discrepancy may be improved by

en-larging the model space. Most ofthe states in this

mass number have rather strong mixing. There

are

only

a

fewstates with anearly pure

_(f,

_&„d,

_&,

')

configuration.

Figure 4 presents the experimental and

calculat-ed energy spectra for the nuclei with A

=37.

Ow-ing to the severe lack of definite spin assignments,

we only calculated the energy levels lying below

4.

7 MeV for T =~2. Below this energy, all

calcu-lated states have counterparts in the experimental

level scheme. Most of the levels included in the

least-squares fit agree very well with the observed ones except for the ~&, T

=

& state, which shifts

from the observed value of

3.

52 MeV with a large

discrepancy of

0.

53 MeV. The energy levels

ex-cluded in the least-squares

fit are

also not

repro-duced well. However, the calculated

_2,

T

=

₂

state a,t 4, 63 MeV and '2', T

=

& state at

9.

54 MeV

are

in excellent agreement with the experimental

counterparts. Baumann et

al.

' studied the deex-citation of high-spin states in Cl via the

Al(

F,

2p), Al( C,2p), and S(n, p) reactions.

A new yrast level at

7.

02 MeV with

J'

=P

was

determined from a

recoil

distance experiment.

Our theoretical counterpart for this state

is

cal-culated at

8.

39 MeV. The reason for this

dis-crepancy may be the

restriction

of our model

space.

The mixings of most ofthe levels for this

mass number

are

stronger than those for the A

=

38-40

nuclei.

The calculated and observed energy spectra for

fit-CALCULATIONS

ON

NEGATIVE-PARITY

STATES

IN

THE.

.

.

2

J

(2T)

2J (2T)

le-lO— (5,I39)(3)(3) II (3)

,

9 (3) 5 (3) S-(3) 7'-(3) I3 (3) 5 (3) (3) g2@,4)(2) (I,ZP)-(2) (I,2,3)(2) 3 (2) (2) -2 (2) -I-(2) (2) 4 (2) 3 (2)

0

X LU

4—

0

exp. goal 7 y9 col.

3:

₉ ~3 ~S II ~7 N₅ 9

l2—

0

lP Z Ul 2 (I)

8—

.

6 (I) -(O,I,2, 3)(I) -4-(I) -(I,2)(I) -4(I)PI) x()2P~(') 2,3) ()) 4 () 5-_—I) I) 2-(I) rs 4

5:

2 0 (I) /_{4 (I)} ~l (I)

i

2-(I) X2 ()),5 0) 4 (I),3-(I) (I) 3 (I) 4 5 ~I -3 w2 4 5 3

FIG.

4.

Experimental and theoretical energy spectra for the A=37 nuclei.

exp. coI.

ting for this mass number

is

worse than the others.

One level

for

which the agreement

is

not good

is

the

first

excited 2 level at

4.

97 MeV. The

calcu-lated energy spacing value gives alarge

discrepan-cy of

0.

78MeV compared to the observed one. In

fact,

the fit to this state

is

the worst ofall the

fit-tings. The mixings of different components for the

energy levels

are

strongest among the nuclei

con-sidered in this work.

In conclusion, our calculated energy levels

agree, in general, reasonably well with the

ob-served ones

for

either the level spacings or the

level sequences. The importance of the

s«,

and the P3/g orbits

is

manifested in the large intensities

Fl_f

comprised in the (f'f/3 sf/3 d3/3) and (p3/3 s(/3,

d3/-3) components. The intensity of the configura-tion with n&&3 for the s&/& orbit

is less

than 10%

for

all levels. The mixing between different

com-ponents

increases

as the mass number A

decreas-es.

For

even-mass nuclei, the root-mean-square

deviation for each mass number

decreases

as the

mass number A

increases.

The same conclusion

is

obtained for odd-mass nuclei. The rms for

in-dividual mass numbers A

=36-40

are

0.

403,

0.

251,

0.

346,

0.

204, and

0.

280 MeV, respective-ly.

For

individual mass numbers, higher isospin

states,

in general, give

a

smaller

root-rnean-A =

36

FIG.5. Experimental and theoretical energy spectra

forthe A=36nuclei.

square deviation. The only exception

is

the A

=40

nuclei, which gives rms

=0.

263 MeV for

T=O,

and

0.

294 MeV for

T=1.

Although we did not remove the spurious states

due to the

c.

m. motion, the effect of the spurious

states on the energy level calculation may still be

negligible. The reason

is

as follows: The

spuri-ous states

are

distributed in the space with 1@~

excitation, and our model space contains only part

of them. Furthermore, the intensities of the

low-lying states

are

rather concentrated in some

com-ponents of the basis

states.

Thus, the effect of

spurious states

is

negligible, except in the

calcu-lation of energy levels of Ca and

E1

transition

rates.

For

the two T

=

0,

J

=

1 states of4

Ca, the

1~state contains 12%and the 1&state contains 33%

of the spurious

state.

For

the other states in A

=

36-39,

the effect of the spurious state

is

smaller

than 10/~.

B.Spectroscopic factors

Table IIshows the spectroscopic

factors

of Ca

TABLE

II.

The experimental and theoretical spectroscopic factors of Caand Kfor the

l=1and l=3stripping reactions on 39K.

Nucleus E'"~(MeV) theo exp'

40Ca 4'K 3 5 1 2 3 3 2 4 3 2 4 3 2

3.

74 449 5.61 5.90 6.03 6.29 6.58 6.75 7.66 7.69 8.42 8.55 0

0.

03

0.

80

0.

89

0.

42

0.

25

1.

00

1.

00

0.

33

0.

99

0.

01

0.

58

0.

16

0.

01

0.

99 0.96

0.

90

0.

05

0.

86

0.

08

1.

00

0.

96

0.

90

0.

05

0.

86 0.08

1.

00 0.49, 0.51, 0.01, 0.02

0.

65, 0.73, 0.76, 0.78, 0.03, 0.05, 0.19, 0.20, 0.04, 0.05, 0.31, 0.40, 0.11,0~16, 0.36,

0.

50 0.05,

0.

19, 0.60,

0.

97,

0.

79,

1.

17 0.06, &0.11 0.56, 0.58, 0.02 0.69,

0.

84, 0.90 0 90 0.02 0.92 0.02b 0.81b

0.

60,

0.

71

0.

76, 0.91, 0.07 0.22, 0.06 0.47, 0.17, 0.84,

1.

22

1.

31

0.

36 0.61, 0.71

0.

21, 0.23 0.20

1.

22,

1.

33 0.64,

1.

0 0.91,

1.

18

'

_{Reference 10.} Reference 18.

stripping reactions on K.

For

the

T=0

states,

we selected 8 levels with reliable

J'

assignments.

Among them, most of the states

are

in good

agreement with the experimental ones. In general,

the T

=1

states

are

calculated better than those

for T

=

_{0, especially for} the neutron stripping

re-action on K, and this

is

consistent with the

re-sults obtained in the calculation on energy levels.

The spectroscopic

factors

of Kand Ar for the

l

=1

and l

=3

stripping reactions on Ar listed in

Table

III.

Our calculated values

for

this mass

number agree very well with the observed ones.

One state for which the agreement

is

not as good

as the others

is

the & state for T

=

0at

3.

02 MeV.

The calculated value of

0.

28

is

overestimated

com-pared to the observed value of

0.

02. The level at

3.

06MeV of Ar

is

tentatively assigned to

4'

=(

j

~

~);

our calculated spectroscopic factor for

the & state agrees very well with the experiment.

Table IVpresents the calculated and the observed

spectroscopic information of Ar and Clfor the

l

=1

and l

=3

stripping reactions on

'Cl.

The l

=3

transitions for the 3&, T

=

1state at

3.

81 MeV, the 3&,

T=

2 state at

0.

76 MeV, and the 4&,

T=

2 state

at

1.

31MeV

are

all slightly underestimated. The

l

=1

transition for the

3,

T

=2

state at

0.

76 MeV

is

overestimated.

For

the other states the

agree-ments

are

very good. Maripuu et

al.

have

calcu-TABLEIII. The experimental and theoretical

spec-troscopic factors of39K_{and Ar for} the l=1and l=3

stripping reactions on Ar.

Nucleus & J~ E'„'" (MeV) L theo exp

"Ar

2 3 2 3 2 3 2 3 2 3 2 3 2 7 2 3 2 3 2 7 2 3 2 5 2 3 2 7 2 3 (-, ,—,) 2.81

3.

02 4.08 0.0

1.

27 2.09 2.43 2.48 2.63

3.

06 3 0.82 0.60 1 0.28 0.02 1 0 53 0 30 3 0.73 0.60

0.

66 1

057 055

053

3

0.

00 0.01

0.

02 1

0.

00

0.

02 3 0 15 0 09 007 1

0.

16 0.19

0.

20 3 0.02 0.02 Reference

19.

Reference 20. Reference 21.

lated the spectroscopic

factors

of

'

Cl for neutron

stripping in a larger model

space.

Our results

are

reasonably consistent with

theirs.

CALCULATIONS

ON NEt

ATIUE-PARITY

STATES

IN

THE.

. .

527

TABLE IV. The experimental and theoretical

spec-troscopic factors of Ar and Clforthe l=1and l =3

stripping reactions on3~C1.

TABLEVI. The experimental and theoretical

spec-troscopic factors of 6Ar and C1forthe l=1and l =3

stripping reactions on Cl.

Nucleus T J~ E»" (MeV) l theo exp Nucleus T

"

(Me V) l theo exp

3sAr 3sC1 1 3 1 4 1 5 1 3 2 2 2 5 2 3 2 4 2 3 2 0 2 2

3.

81 4.48 4.59

4.

88

0.

0

0.

67

0.

76

1.

31

1.

62

1.

75

1.

98 3

0.

01

0.

19 1

0.

06

0.

01 3 0.07

0.

04 3

0.

37 0.34 3 0.32

0.

31 1

0.

01

0.

01 3

0.

83

0.

84

0.

72 1 O.P1

0.

02 3

0.

90

0.

78 P.68 3

0.

32

0.

59 0 54 1

027

009

008

3

0.

45 0.70

0.

66 3 0.05 1

0.

2P

0.

40 P.29 1 P.98

1.

10 P.

89'

3

0.

01 1

0.

43

0.

70

0.

48

"Ar

36( 1 2 5 1 3 2 3 1 5 1 4 1

(2,

3 ) 4.18 4.97 5.17 5.84 5.86

1.

95 2.47 2.52 2.81 2.90

3.

21 3

0.

17

0.

40 1

0.

30

0.

06 3

0.

67 0~52 1 0.02

0.

02 3

0.

74

0.

72 1

032

033

0.

34

0.

375' 1

0.

01 3

0.

74

0.

86" 1

0.

01 3

0.

46

0.

77 1

011

010

3

0.

82

0.

85 0.26 0 45b 1 0.30

0.

28 ' 0.20

0.

20bd 1

0.

10 ~_{Reference 22.} Reference 23. Reference 18. Reference 22~ Reference 26 For 2 state. For3 state.

Nucleus +»" (MeV) l theo exp

37A

_r

3tCl 1 2 2 f 3 2 2 1 3 2 2 5 2 2 i

i"

2 2 i 3 2 2 3 2 2 3 5 2 2 3 5 2 2 3 Q 2 2 3 ii 2 2 i3 2 2 5 2 2

1.

61 2.49

3.

52 4.40 4.44 4.63

3.

10

3.

71

3.

74 4.55 5.27 5.38 3

0.

72

0.

51

0.

76 1

060

035

044

1

0.

03

0.

23

0.

35 3

0.

00

0.

03 p.pp

0.

145~p.14b 1

0.

02

0.

02

0.

01 3

0.

13 1

0.

06 3

0.

46 1

0.

03 3

0.

12

0.

03 3

0.

01 3 0.86 3

0.

.00 3

0.

01 1

0.

10 Reference 24. Reference 25.

TABLEV. The experimental and theoretical

spectro-scopic factors of ~Ar and Cl forthe l=1and l =3

stripping reactions on 36C1. Ar and Cl for the l

=

1 and l

=

3 stripping

reac-tions on 36C1. The calculated values

for

the —,

state at

4.

40 MeV and the

$

state at

4.

44 MeV for Ar

are

zero.

The experimental values

for

these

two states

are

very small, however, these null

re-sults

are

due to the neglect of _{the pf/2 and f5'&}

or-bitals.

Therefore,

it

may be more reliable to

in-clude these orbitals in the model

space.

The other

states

are

in reasonable agreement with the ob-served ones.

For

~Cl the experimental results

are

not available. The spectroscopic information

should be obtainable from the following suitable

reactions:

(d,

p}

or (t, d}on 38Cl. Thus, we

list

also the predicted values of

'Cl

in Table V for

future comparison. Since the two excitation levels

for

4=~

differ by only

0.

03 MeV, the

spectro-scopic

factors

may be inverted.

The calculated and experimental spectroscopic

factors

of 6Ar and Cl for the l

=

1and l

=

3

strip-ping reactions on Cl

are

listed in Table VI.

For

the

first

3 state at

4.

18MeV of the nucleus Ar,

the agreement between the calculated and observed

values

is

not as good as the others. The

calculat-ed values for most of the levels

are

in good

agree-raent with the observed values with three

excep-tions: The l

=

3 stripping

for

the 2 state at

1.

95

MeV, the 3 state at

2.

47 MeV, and the 4 state at

2.

81MeV of Cl

are

slightly underestimated.

cal-TABLEVII. The experimental and theoretical reduced width ofEMtransitions fornuclei ofA=36-40. Nucleus

J

~& Ex& (MeV)

E~

(MeV) E~

Nucleus

J

~& (Me V)

E~

I

(10

"

eV) (MeV) exp~ theo

Ca 3 374 0+ _g.

s.

9.7 2.2 6 E3 Ar 3

3.

81 p+ _{g.S. 5.9}

_1.

4 E3 5 4.49 4 5.61 2 6.03 3 5

3.

74

1.

7

3.

74 4.6

3.

4 4.49

1.

3

6.

3

3.

74 2.6

0.

48 5.6 44.5

1.

1 4.1 7.3 6 E2 4 M1 5 E2 4 M1 4 M1 38Cl 4 4.48 5 4.59 3

0.

76 4

1.

31 3 3 4 2 2

3.

81 3.81 4.48 gas. g.ST

3.

0

3.

5

3.

1 2.1 8.1 61.3 4.3 10.0

0.

15 0.035 5 M1 E2 M1 E2 3 6.29 5 4.49 2.0 9.9 0.013 3 E2 0.081 4 E2 3

0.

67 0.76 94 2.2

1.

6

0.

017 4 M1 4'K ₃

_0.

_{03 g.}

_s.

_1.

1

0.

85 7 M1 3

1.

62 2 _{g. S.} 5.6 16.5 M1 2

0.

80 5

0.

89

"Ca

-' _2.8p 2 4 3+ 2

0.

03 g.S. g.S.

1.

6 &1.4 5.4 5.3

1.

2

1.

0

1.

9 12.8 8.2

3.

7 7 E2 4 M1 6 E2 7 E3 3?Ar — _2.49 2

3.

19 9 2 5 3 4 7 2 7 2 0.67

0.

76

1.

31

1.

61

1.

61

9.

9 8.4

1.

5 5.7

1.

6 5.9 12.9 32.8 0.42

1.

6 5.3 0.48 M1 M1 39K 7w 2 9 2 2.81

3.

60

3.

94 ii 2 4.08 2 4.13 7 2 4.52 9 2

"Ar

-' _2.09 2 3+ 2 3+ 2 7 2 7 2 9 2 3 2 7 2 9 2 11 2 7 2 g.S. g.S. 2.81 2.81

3.

60

3.

02 2.81

3.

60

3.

94 g.S. 6.2 4.6

1.

6

3.

2 2.9

1.

6

1.

5

3.

0 8.5 &4.9 5.1 0.10 50.4 2.6

1.

0 19.5 2.9 0.63

11.

8 7.5 14.2 21.7 7 E3 6 E3 6 M1 6 E2 5 E2 4 M1 2 M1 3 M1 4 M1 3 M1 5 E2 3027 5 2 3 2 7 2

3.

52

3.

53

3.

71 ii 2 4.02 9 2 3 2 3 2 7 2 9 2 5 2 7 2 9 2 7 2 9 2

1.

61 2.49 2.49

1.

61

3.

19

3.

27

1.

61

3.

19

1.

61

3.

19 6.4 4.7

1.

2 29.6

1.

8

1.

4 6.8 2.3 4.5

1.

8 2.1 4.3 0.82 2.3 26.8

0.

49

1.

5 10.4 3.6

3.

1

1.

4 7.0 481.0 M1 E2 M1 M1 E2 3 2 7 2 2.43 2.48 39A

_r

— _2.63 2 2.76 2 3 2 ? 2 3 2 2 5 2 3 2 5 2 7 2

1.

27 g.S.

1.

27 g.ST 2.09 5.1

1.

6 4,7 2.3

11.

6 4.9 16.5

0.

63 0.34 4 M1 4 E2 4 M1 3 M1 4 M1 2.09 goS~

1.

9

1.

2 2.8 4 M1 3 M1

1.

27

3.

8 265.0 5 M1 Cl ₂ 9 2

3.

10 4.01 4.55 11 2 5.27 13 2 3+ 2 3+ 2 7 2 7 2 9 2 9 2 ii 2 g.S. g.ST

3.

10

3.

10 4.01 4.01 4.55 9.4 6.2 3.8 5.9 4.0 &9.4

1.

9 &9.4

1.

9

1.

6 5.8

1.

4

0.

053 6 10.3 9-8 344.0 E3 E3 M1 E2 M1 E2 2.65 11 2 3 2 7% 2

1.

27 g.S. 2.6

1.

7 &1.6 6.6 10.6 10.7 4 E2 3 M1 4 E2 6Ar 3 4.18 2 4.97 5 5.17 p+ 3 g.S. 4.18

1.

97

1.

3

1.

7 2.0

3.

0

0.

23 6.2

0.

062 5

1.

4

1'

1 E2

CAI CUI ATIONS ON

NEGATIVE-PARITY

STATES

IN

THE.

.

.

TABLE VII.(Continued)

E;

E&

r

(10"eV)

Nucleus

J;

(MeV) J~ (Me V) exp theo n

E& E& I

(10"

eV) Nucleus J'; (MeV) J& (MeV) exp theo n

3 4.18

1.

0 3 586 3

418

20

4 5.90 3 4.18

1.

3

3.

3 C1 3 2.47 2

1.

95 4.5 2.8 8 E2

15

4 M1

14

3 M1

40

5 E2

3.

5 4 36gl _4.0 5 2.52 2+ _g.

_s.

_1.

₂ 0 22 7 E2

0.

030 7 E3 5.6

51

9 E2 4 2.81 2

1.

95

1.

6 415.0 9 E2 5 252

63

126

5 M1

'

_References 10a~d 27.

culated spectroscopic

factors

for the 3&states of

even-even nuclei with the observed ones

are

not

as good

as

the other states with the exceptions of

the l

=3

stripping of Ca and the l

=1

stripping of

Cl.

The overall results show that the calculated

value for higher isospin states

are,

in general,

better than those for lower isospin

states.

This

fact

manifests that our model space

is

more

suit-able

for

the higher isospin

states.

For

lower

iso-spin

states,

an enlarged model space

is

probably

necessary.

C. Electromagnetic transition rates

We calculated

E2, E3,

and M1 transition

rates

to provide a more sensitive test of the calculated

complete wave functions. The reason we did not

consider the

E1

transition

rates

is

that the

spuri-ous states were not removed in the energy level calculation. Thus itwould be meaningless to con-sider the

E1

transition

rates.

For electric

multi-pole transitions, effective charges ofe~

=1.

5 and

e„=0.

5

are

assumed, which

are

consistent with the results obtained by Wilkinson, and the radial integrals

are

evaluated using harmonic oscillator

wave functions, giving

a

total rms radius

as

de-termined from electron scattering with k~

=10.

6

MeV. The ground-state transitions in Ar,

'Cl,

and Ar

are

obtained with the wave functions of

Sec.

IIIA; for Ca

g.

s.

, Ca, and K, apure doubly closed shell-model state and a hole in 1d3/2

and 2sf/g

are

taken.

For

M1 transitions, the

free

gyromagnetic

factors are

used.

Table VII shows the calculated and experimental

electromagnetic transition

rates.

The calculated results

are

in reasonably good agreement within

a

small factor for most of the transitions.

For

the

electric

multipole transitions, even we used large

values ofeffective charges

for

the proton and

neu-tron, most of the calculated results

are still

smaller than the experimental data. These dis-crepancies may be improved ifan enlarged model

space

is

assumed.

For

A

=39

nuclei, we have

two E3 transitions for Ca and Knuclei; the

cal-culated results

are

all in reasonable agreement

with the experimental ones.

For

the M1 transitions, much theoretical work has been performed previously. ~ Our

predic-tions are in uniformly good agreement with the

ob-served values. Inparticular, the observed

enhance-ment ofthe M1 transition in Cl between 3& and4&,

which cannot be accounted for by weak

particle-hole admixtures,

is

predicted in reasonable

agreement with the observed data in our model.

For

the other transitions, where comparison with

previous works

is

possible, the agreement

is

also

reasonable. Our calculated M1 transition

rates

are

slightly

larger

than the experimental data.

This defect could probably be improved if the

ef-fective magnetic moments and a larger model

space

are

taken into account.

D. Validity ofthe weak-coupling model

Ellis

and Engeland have formulated

particle-hole states by coupling in

J

and T eigenfunctions resulting from diagonalization of (sd}"&and p"c.

The main assumption

is

that the correlations

be-tween particles in the same major shell

are

of

pre-dominant importance, and the p-h interaction can

be treated as a small perturbation. The p-h states were selected by combining eigenfunctions obtained

by solving the n~ particle problem in the sd shell

and the 12-n& hole problem in the _p shell. The

negative-parity levels with A

=

16-19

were

repro-duced well with experiment.

In order tostudy how good the weak-coupling

model

is,

we used our hole-hole interactions to

calculate the wave functions for the low-lying pos-itive-parity states within the (d3&~,

s,

~}

shells.

These wave functions were then coupled by a parti-cle in thefq/&

or

p3/2 shell. The degree of the

overlapping between these resultant wave functions

and the wave functions for odd-parity

states

ob-tained in this work were then calculated. Our

TABLE VIII. Wave functions for the negative-parity states of Ar and Clgiven interms ofthe (6/2,d3/2~ andp'3/2 orf7/2

»gl P'=&) 3+ i+ 5+ 7+ 3' 2i 2i 2i 2_i 2 2 2 2

»s

p'=')

3+ f + g+ ~ 3+ r+ Total 2i 2i 2i 22 22 3_Ar 0 3i 47 34 1 2 0 2i 69 1 14 0 _5i 74 18 6 0 1i 32 31 8 3 0

3,

35 21 7 6 2 11 20 17 97 99 98 93 0

4,

16 52 11 1 52 2 14 6 49 13 10 2 85 36Cl _{1 2i 75 4 6} 1 5i 82

]4

1 1 3i 30 10 27 1 1 22 31 1 20 3 2 15 1 3 26 1 9 98 99 96 94

higher for most of the states except the (T,

J')

=

(~,&2) and (r~,_$&)states for 'Cl and _(0,5,) state

for

Cl.

The overlappings

for

all second excited

states of each (T,

J')

decrease

slightly.

For

illus-tration, Table VIII shows the overlappings between

the non-normal parity state wave functions ofA

=36

and the wave functions obtained by combining

the low-lying positive-parity state wave functions

ofA

=35

and one nucleon in thef7&&

or

p3~& shell.

The positive-parity levels of the A

=35

nuclei

are

in the sequence of the experimental data. Our

re-sults show that

it

is

reliable to apply the

weak-coupling model to the calculation of negative-parity

states

ofA

=35

and 34nuclei.

E. Effective interactions and two-body matrix elements

As mentioned in

Sec. II,

we have four interaction

parameters Ap, A&h„Bp&, and B&&in the

hole-particle interaction, four parameters Ap» Af»,

+p», and B&hh in the hole-hole interaction, and

three splittings of single-particle levels of

d3/p-s(/2, fg/g d3/2 and PB/2 d3/& to be-determined in

the least-squares

fit.

The searched values for these 11parameters

are —

1.

15, —

0. 31,

—

_1.

_52,

0.

04,

—

0. 91,

—

_{0. 91,}

—

_1.

_25,

_0.

34,

1.

58,

6.

13,

and

7.

93 MeV, respectively. The hole-hole

inter-action strengths have been used to reproduce the

normal-parity ground

states.

The remarkable

agreement between the calculated and observed

ground state energies as shown in Table

I

indicates

the reasonableness ofthe use of the modified

sur-face-

~interaction.

In order to test the reasonableness of our single-particle level splittings, let us compare our

re-sults with those of

Erne.

To make the comparison

possible, we have to make the following

transfor-mation:

6(f7/2) 6(d3/2) E(fv/2) f(/f3/g)+

_k

Q

(2T+

1)(2

I+

1)(d3/gf7/2I Vld3/gf'j/g)r~

T,Z

(2T

+

1)(2

J

+

1) (d,

(,

'

IVId,(2')

r~,

where

e(j)

and K(j)

are

the single-particle energies

assuming Ca and S to be

cores.

After using the

values ofErne for F(

j)

and hvo-body matrix

ele-ments, we get e(f,/2) —e(d3/g)

6.

97MeV compared

with

6.

13MeV for

ours.

Therefore, the

consis-tency

is

satisfactory. In order to provide

a

more

sensitive test of our calculated interaction

I

strengths, we also calculated the two-body matrix

elements

for particle-particle

and particle-hole.

Table IX

lists

the two-body matrix elements in the

f7&&and d3&2

orbits.

The values listed in the

col-umn entitled Erne

are

those obtained by Erne in

his work of calculating the odd-parity levels with

CALCULATIONS

ON

NEGATIUE-PARITY

STATES

IN

THE.

. . 531

TABLEIX. Two-body matrix elements in the_f7h and d&/2 orbits.

Particle-particle (MeV)

T

J

Present Ernid MSDI

Particle-hole (MeV) Present Emd MSDI 0 2 3 4 5 1 2 3 4 5

-5.

46

-2.

84

-2.

40

-3.

19 Q.Q4

-0.

0 7 0.04

-0.

33

-3.

65

-1.

85 1077

-2.

90

0.

38

-0.

08 0.92 0.43

-3.

70

-2.

10

-1.

83 2032 0.70 0.54

0.

70 0.19

-3.

62

-1.

90

-0.

87

-2.

21

-0.

35

-0.

32

-0.

22

-1.

11

-1.

02

—

_1.

31

-0.

87 2023

1.

99

1.

46

1.

02 2.22

-1.

85

-2.

87

-1.

18

-1.

92

1.

58 0.66 0.56

1.

35

-1.

22

-1.

86

-1.

59

-2.

51

1.

26

0.

78

0.

51

1.

22

0.

78

0.

26

0.

01

-0.

94 2.12

0.

93

0.

66

1.

62

results obtained by Glaudemans et

al.

"

The data

listed in the column G

are

those obtained by Osnes

and Kuo. The four

sets are

rather similar,

ex-cept the T

=1

matrix elements for

particle-parti-cle

and T

=

0matrix elements for particle-hole in

column G have different signs from the other three

sets.

Our results for p-p

are

more attractive and

those for p-h

are

more repulsive than those

ob-tained by Erne and Glaudemans et

al.

However,

in general, our calculated two-body matrix

ele-ments for p-p and p-h in the f&/, and d3/g orbitals

are

reasonably consistent with the previous works.

IV. CONCLUSIONS

A modified surface-delta effective interaction

has been used in a least-squares fit to energy

lev-el data of negative-parity states in the A

=36-40

nuclei. The calculated energy spectra

are

in

rea-sonable agreement with the observed values; the

levels for odd-mass nuclei

are

in very good

agree-ment with the experimental values. Some levels

with model analogs, however, deviate by about

500keV from the observed ones. This

is

probably

due to the neglect of the other single-particle

or-bits.

Some very low-lying

states

contain large

inten-sities

of the components of (/7/2/sf/g)/ (p3 d/32/

/}

2/ fl$ Tlg

(/7/ 2 sf/2p d3/ 2), and (p3/2 sj/2 d3/ 2)

configura-tions. This seems to justify the necessity of the

inclusion of the s&/2- and p3/2- orbits in the

config-uration space at

least.

The root-mean-square

de-viation for energy-level fitting in each mass

num-ber

decreases

as the mass number A

increases

for either the even mass nuclei or the odd mass nuclei.

Higher isospin

states,

in general, give a smaller

rms deviation for individual mass numbers. This

fact

shows the structure of higher isospin states

may be simpler than that oflower isospin states

and thus can be accounted for satisfactorily by our

model

space.

The spectroscopic

factors

are

in good agreement

with the experimental data. The l

=1

and l

=3

strippings for the 3&state ofeven-even nuclei do

not agree

as

well as the others. This discrepancy

may be improved by taking into account the

octu-pole vibration and/or the enia.rgement of model

space.

The calculated EM transitions

are

in reasonable

agreement with the observed values. Our

calcu-lated E2 and E3 transition

rates are,

in general,

slightly underestimated. The predicted M1

transi-tion

rates,

on the contrary,

are

slightly

overesti-mated for most of the transitions. These

discrep-ancies may be reduced, provided that an enlarged

model space

is

taken into account and the effective

magnetic moments

are

used.

%e

also tested the validity of the weak-coupling

model by calculating the overlappings between the

calculated wave functions forthe non-normal parity

states of mass number A in this work and the wave

functions obtained by combining the low-lying

posi-tive-parity state wave functions with mass number A—1 and one nucleon in thef7/2-

or

the p3/2—_shell.

The wave functions for the low-lying

positive-pari-ty state

are

reproduced by the hole-hole

interac-tions obtained in our calculation. Our results

show that the degree of overlapping

is

larger than

90f& for most of the

states.

This provides a

suit-able approach to study the structure of the nuclei

for A

=34

and 35with the weak-coupling model.

In conclusion, the nuclear properties observed

in the A

=36-40

nuclei can be explained quite well

by assuming a modified surface-delta type

two-body interaction, provided that the model space

is

enlarged to include the 2s&/2, 1d3/» 1f7/pp and

2p3/2

orbits.

ACKNOWLEDGMENT

This work was supported by the National Science

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