### Shell-model

### calculations

### of

### one-hole states

in### the

nuclei### of

A### =41

### —

### 43

### D.

### S.

Chuu and### C. S.

HanDepartment

### of

Electrophysics, National Chiao Tu-ng University, Hsinchu, Taiwan, Republic### of

China### S.

### T.

HsiehDepartment

### of

Physics, National Tsing Hua University, Hsinchu, Taiwan, Republic### of

China### M. M.

King YenDepartment

### of

Nuclear Engineering, National Tsing Hua University, Hsinchu, Taiwan, Republic### of

China (Received 17 February 1982)The one-hole states ofA

### =41

### —

43nuclei are calculated with amodel space spanning the (f7/2p3/2)" (d3/2si/2) ' configurations. The two-body effective interaction is assumed tobe the modified surface-delta type. Energy spectra and spectroscopic factors are calculated and compared with the observed values. Satisfactory results are obtained.

NUCLEAR STRUCTURE A

### =41

### —

43,calculated effective interaction, energy spectra, spectroscopic factors.### I.

INTRODUCTIONThe nuclei in the

### lf-2p

shell have long been### of

in-terest to both experimental and theoretical nuclear physicists. Extensive studies have been done' 2_{for}

the normal parity states

### of

the nuclei in this region.### For

the non-normal parity levels, however, the situ-ation isless well understood. This paper will give a systematic study### of

the one-hole states in the mass region### of

### A=41

### —

43 within the framework### of

theshell model. The particles are assumed

### to

be distri-buted in the 1_{f7/2 and}

_{2p3/2}orbitals and one hole is

assumed tobe in the 1d3&z or2s&&q orbital.

A least-squares fit calculation on the non-normal parity states in

### lf-2p

shell nuclei has been per-formed by Dieperink and Brussard within the description### of

the (lf7/2)"(ld3/2) ' configuration. In their results, only the low-lying energy levels can be reproduced and the calculated transition rates are not in agreement with the observed values. This discrepancy isdue to the fact that the configuration space they used is too small. Later, Hsieh et### al.

### '

calculated the non-normal parity states for thenu-clei in the

### A=39

### —

41 region by expanding themodel space to include the complete

### (lf-2p)

shelland single-hole states in the (2s,ld) shell. Some dis-tinct improvements were obtained. The spectro-scopic factors and electromagnetic (EM) transition rates obtained were in reasonably good agreement

with experimental data.

### For

the heavier massre-gion

### A=42,

the extended model space has not yetbeen employed. Lawson and Muller-Arnke have studied the magnetic moment

### of

the### (3/2)

i+_{state in}Scand found that this state does not have apure

### (f

7/2) (d3/2 ) ' configuration. The lifetime andmagnetic moment can only be accounted for

simul-taneously by extending the model space to include about

### 10% of

_{the (f7/2) (si/2)}' admixture. The importance

### of

the s~&z hole state can also be seen inthe spectrum

### of

potassium isotopes; the### l=0

hole strength steadily decreases in energy with increasing mass. From the### /=1

single-particle strengthob-served for levels around 1 MeV in Ca, it is clear that the contribution

### of

the_{p3/2}orbital for nuclei

### of

A &42 is also important. Johnstone recently re-ported calculations for the one-hole states in

potas-sium isotopes

### (A=40

### —

46) using a model space based on the### (f

7/2~P3/2 )"(d3/2,$1/2)configuration. The matrix elements

### of

the residual interaction are treated as free parameters which are determined by aleast-squares fit tothe observeden-ergy levels; good agreement is achieved in his re-sults. The works mentioned above encourage us to reinvestigate the one-hole states in the mass region

### of

A### =41

### —

### 43.

A better systematic prediction is ex-pected### if

we expand the original model space (lf7/2,ldll/2 ') to include the active particle orbit 2p3&z and the active hole orbit 2s&&z.### It

is hopedthat so doing, and considering only one-particle, in-stead

### of

multiparticle, excitation from the 1d3/p oron the energy level calculation is expected to be small. The reason is as follows. The spurious states are distributed in the space within laic@

### of

excita-tion, and our model space contains only part### of

them. Furthermore, the intensities### of

the low-lyingstates are rather concentrated in certain components

### of

the basis states. Thus, the effect### of

spurious states is negligible except in the calculation### of

### El

transition rates, which weare not interested in.### II.

THEMODELIn this calculation, we consider the one-hole states

### of

### A=41

### —

43 nuclei; the nucleus Ca is as-sumed to be the core. The active nucleons are dis-tributed in the lf7/p and 2p3/2 orbitals and oneac-tive hole is allowed in the 1d3/2 or2s&/2 orbital. In

order to obtain a model space

### of

manageable size,we neglect the 2p~/2 orbital. This should not

intro-duce significant errors since the _{2p&/2} lies about 4
MeV above the lf7/2 orbital. Also, according to
the Kuo-Brown matrix elements, the _{f7/2} p'[/2

in-teraction is very weakly attractive. The omission

### of

the lds/2 and 1_{f5/2 orbitals from the model space}is

quite reasonable because the splitting

### of

theob-served single-particle levels

### of

2s~/2-1d_{5/2}and

1

### f

5/2-2p3/p are much larger than those### of

ldll/g-2s)/2 and 2p3/2 lf7/2 Within this model space we diagonalize an effective one-body plus

two-body Hamiltonian. We identify the resulting

eigenvalues with observed energy levels and use the eigenvectors as wave functions to calculate various observables.

The effective Hamiltonian in this space has the orm

### H =Ho+Hpp+Hgp

### .

Here Ho is the Hamiltonian

### of

the single particle in the effective field### of

the core. The single-particleenergies for active orbitals were chosen initially from the observed single-particle spectra

### of

the masses### 39,

40, and### 41.

H&& represents the effective two-body interactions between the particles in the1_{f7/2 and} _{2p3/2} orbitals, while II~~ represents the
effective interaction between the hole in the 1d3/2

or 2s&/2 orbital and the particle in the lf7/2 arid 2pq/2 orbitals. The two-body effective interactions

are assumed tobe the modified surface delta form

VJ.

### =4rrAr(Q,

_{J}

### )+Br,

### T=0,

### 1,

With this prescription we carry out a least-squares fit calculation on the energy spectra

### of

### '"

_{Ca,}

### '

'_{K,}

_{and}

_{Sc. For}

_{the input }

experi-mental data, we include in principle all the available

low-lying levels with reliable

### J

assignments up tothe point that the first level with an uncertain

### J"

assignment appears. Forty-three energy levels are included in the calculation. The eight interaction strengths (AO,A~,BO, B~)zz and (AO,### A„BO,

B~)~~and the ld&/2-2s&/2, lf7/2-ld&/2, and 2ps/2-id&/r

single-particle energy spacings are varied until the discrepancy between the calculated and observed

en-ergy levels is minimized. The overall rms deviation is

### 0.

26 MeV. The best-fit interaction strengths and single-particle energy spacings are listed in Table### I.

The center-of-mass spurious states have not been re-moved in the calculation.

A. Energy spectra

TABLE

### I.

The particle-particle and hole-particle in-teraction strengths and the single-particle energy spacings (inMeV). hp### —

1.73### —

0.57### —

0.03 0.02 Bp 2.20### —

2.02### —

0.66### —

0.32 1tg3/2 2s~/z 1f7/2 1d3/2 2p3/2 183/2 1.68 5.71 6.66The calculated energy spectra produced by the

hest fit parameters together with the observed

values are shown in Figs. 1

### —

### 3.

The experimental data are taken from Endt and Leun. Those levelsthat are included in the least-squares fits are marked with an asterisk. The excitation energies

### of

the one hole states were made to fitthe observed en-ergies relative to the ground state energies

### of

the nuclei with### A=41

### —

### 43.

Since we are only con-cerned with the energy spacings relative to the ground states in our calculation, itis hoped that the adjustable single-particle energies may absorb the discrepancy due to the effect### of

neglecting the three-hole configuration.### 6

4 I72 I_{7/2}3 I 3/2 7/2 5/2,'/2 I:/2 5/g Ii/2 lg~ i/2

### ~

I_{5/2}5/2 I/2 I 2 9/2 3/2 7/2 I/2 5/2 i

_{5/&}5/2 7/2

### a.

13/2 5/2 I_{l/2}gp ~/2

_{il/2}

### ~

7/2 l I/ i/2 N_{I/}7/2 9/2 3/2 I 5/2 &/2 I$2 5/g I/2 I

_{I/2}9/2 7/2 5/2 —-&/2 (I/2) 3 I3/2 9/2 -5/2 7/2 I

_{/12}3/2 -7/2 41/ I/2 3/2 2 3/2

### 0

### 0

### Ex

### p.

### Ca

### l.

### Exp.

### 4l

### FIG.

1. Experimental and theoretical even parity energy spectra fortheA### =41

nuclei.The energy spectra

### of

even parity levels for### 2=41

nuclei are shown in### Fig. 1.

The agreementbetween calculated and observed energies is

reason-ably good for most

### of

the levels.### For 'Ca

ourcal-culafed

### E

### =(3/2)+

rotational band based on the### 2.

01 MeV### (3/2)+

level is similar to the resultsob-tained by Hsieh et

### al.

using the complete (1### f

2p)"(2sld) ' configuration space. This seems to— 3,5 (2i3) (2-4) f, ~-3)

### (o-3)

5_{(o-33}

3
1'
4
2
5
--0
### Exp.

### Ca l.

_{Exp.}

_{Cal.}

### 42

### FIG.

2. Experimental and theoretical odd parity energy spectra for theA### =42

nuclei.indicate that the inclusion

### of

the_{d5/q}

### ',

_{f5/2,}and

p~~2 orbitals is not necessary.

### For

some lower-lying levels, the intensities### of

(s~/2### f

7/p) and(s]/2 p3/2) are rather strong.

### For

example, the cal-culated (1/2)~+### (T=3/2)

state is dominated by the(s)/2 'f7/2 ) configuration. This seems to justify

the necessity

### of

the inclusion### of

s&~2 and p3/2con-figuration space at least. The four high spin states populated in the

### K(a,

d)### 'Ca

experiment' can bewell reproduced in our model. The lowest

### (11/2)+,

### (13/2)+, (15/2)+,

and### (17/2)+

levels are given at### 3.

### 39,

### 3.

83,### 3.91,

and### 5.

11MeV, and are observed at### 3.

37,### 3.

92,### 3.

83, and### 5.

22 MeV, respectively.### For

### 'K,

the calculated### (3/2)

t state gives a largediscrepancy

### of

about### 0.

6MeV compared to the ob-served one. In fact, this is one### of

the worst fittings in this mass region. The level observed at### 1.

56MeV has been suggested" to be### J

### =(1/2)+.

Our calcu-lated (I/2)z+ state at 1.67MeV supports thispredic-tion.

Figure 2 shows the results for the odd parity states

### of

nuclei with### A=42.

### For

Ca, most### of

thelevels are reproduced reasonably well, except the 3~

state. The calculated energy value for this state is displaced about

### 0.

49MeV from the observed value. The level at### 4.

67 MeV is tentatively assigned as### J

### =(3

### —

4)### .

The fitted 34 state at### 4.

56 MeV seems to favor a 3 state at### 4.

67 MeV.### For

### K,

-'/z-'Xz)
i_{3/z}
i)/z
(3/z,s/z)
i_{s/z}
s/z
('/z 7/z)
9/z
l_{i/z}
S/2
(3/z~s/z)
ih
7/z
2
5/2
3/2 (3/z,s/z)
7h
3/z
s/z _{3/z}
(3/z,s/z)
3/z

### Ex

### p.

C### a

### l.

### Ex

_{p.}

C### al.

### Cal.

43 4_{3qa}

FIG. 3. Experimental and theoretical even parity energy spectra forthe

### 2

### =43

nuclei.most

### of

the calculated states have counterparts inthe observed level scheme. The states at

### 0.

68,### 0.

### 79,

### 0.

84,### 1.

19,and### 1.

26 MeV are uncertainly assigned as### J

### =(0

### —

### 3),

(0### —

### 3),

### (l

### —

### 3),

(2### —

### 4),

and (2### —

3)### .

Our calculated states at### 0.

54,### 0.

72,### 0.

84,### 1.01,

and### 1.

20 MeV seem to suggest that these fivelevels are

### 0~,

1&,### 33,

### 34,

and 35 states, respective-ly.The energy spectra for the even parity states

### of

nuclei with A### =43

are presented in### Fig.

### 3.

Thecal-culated energy levels in

### Sc

are in reasonably good agreement with experimental values, especially the (3/2)~+, (3/2)z+, and (5/2)~+ states. A pickup reaction' leading tothe (3/2)+& state indicates that this state may be a pure d3&2 hole state. But asmentioned above, Lawson and Miiller-Arnke have

shown that the observed smaller

### B

### (M2)

values canonly be explained

### if

some (s~/z) '(f7/z) admixture is included. Our calculation shows that the (3/2)~+ state contains about### 5% of

the s~~2 ' holecom-41K 4'Ca 3 2 3 2 3 2 3 2 1 2 1 2 1 2

### 3+

2### 1+

2### 3+

2 1+ 2### 3+

2### 1+

2 1+ 2 0.11 0.98 1.56 2.68 2.01 2.67 3.40 4.54 1.06 0.49 1.38 2.54 1.04 0.18### 44'

1.2'### 09'

30d 1.0 0.18 4.6b 1.Ob O.21b### 095

2.### 9'

4.5' 1.3' 0.5' 0.61' 'Reference 15. bReference 13. 'Reference 14. References 16and 17. 'Reference 18.ponent.

### For

Ca,the agreement between calculated and experimental energy levels is not as good as in the case### of

### Sc.

The calculated excitation energyfor the (3/2)i+ state lies

### 0.

71 MeV below the ob-served one. In fact, the fit to this state is the worst### of

all the fittings. The reason for this discrepancy may be that the interaction we used is oversimpli-fied.### For

### K,

most### of

the levels came from theCa(d, He) and Ca(t,

### a)

reactions. '## '

Our calcu-lated results are satisfactory.### B.

Spectroscopic factorsTable

### II

shows the spectroscopic factors### of

### 'K

and### 'Ca

for### l=0

and### 1=2

proton and neutron pickup reactions on Ca. Our calculated values for this mass number agree very well with the observed data. The calculated value for the### (5/2)+

states are zero due to the neglect### of

the d5&2 ' hole in themodel space.

### For 'Ca

no experimental### l=2

reac-tion has been found for the### (5/2)+

level at 2.61 MeV. Thus, the omission### of

the_{dz/2}orbital is reasonable for low-lying levels. However, the ob-served strength is

### 0.

17 for the### (3/2)+

level at### 3.

53 MeV and reaches### 0.

39 for the### (5/2)+

level at### 3.

74 MeV. Our calculated value### of

### 0.

01 for the### (3/2)+

level at

### 3.

53 MeV istoo much underestimated. Table### III

shows the spectroscopic information### of

### K

for the### l=2

pickup reaction on Ca. The main components### of

the wave functions for these fourlevels are

### 3/2,

f7/2 Ti——_{—,}

,

### Ji)T

p### J

### —

1 3TABLE

### III.

Theexperimental and theoretical spectro-scopic factors of Kfor### l=2

pickup reactions on Ca.Residual
nucleus
K
'Reference 15.
Eexp_{r}
(MeV)
0.00
0.11
0.26
0.70
theo.
0.43
0.47
1~18
1.31

### CS

exp.### '

0.45 0.88 1.1 1.4where

### J,

is mainly —,for the### J"=4

and 5 states,but is split into —, and —, for the

### J

### =2

and 3states. Our calculated value

### of

### 0.

47 for the 3 state at### 0.

11 MeV is underestimated compared totheob-served value

### of

### 0.

88. This is because the wave func-tion is spread too much in this state.The calculated and experimental spectroscopic factors

### of"

### K

and Ca for the### 1=0

and### 1=2

_{p or}n

pickup reactions on Ca are listed in Table

### IV.

The results for most

### of

the states are in good agree-ment with the experimental results.### For

Ca, the calculated value forthe### (5/2)+

state at 1.39MeV is zero. The experimental value for this state is alsovery small; however, the null result is due to the neglect

### of

the d5/2 orbital in our model. The levelat 2.27 MeV is uncertainly assigned as

### J

### =(3/2,

5/2)2+. Our calculated strength### of

the (3/2)2+ state is only### 0.

06, while the experimentyields about

### 0.

### 2.

### For

### K

the levels at### 1.11, 1.

55,and

### 2.

67 MeV (not shown in### Fig.

3) are alltenta-tively assigned to be

### J

### =(3/2,

### 5/2)+.

The good agreement### of

the calculated and experimentalTABLE IV. The experimental and theoretica1 spectroscopic factors of

### 'K

and Ca for### l=0

and### l=2

pickup reactions on Ca.Residual nucleus 43K 43Ca 5 2 5 2 5 2 5 2 5 2 3 2 3 2 1+ 2

### 3+

2### 3+

2 ]+ 2### 3+

2### 1+

2 EexP {MeV) 0.0 0.56 1.55 2.45 0.99 1.96 theo. 4.05 1.40 0.29 0.01 2.92 1.18 268 1.4' 0.27' 2.4d 1.0d### CS

exp.### 38

1.4b 0.60' 0.38b 33' 16e 3.8' 2.2' 0.35' 0.09' 'Reference 19. Reference 13. 'Reference 14. Reference 20. 'Reference 17.strengths strongly supports the

### (3/2)+

assignment for levels at both### 1.

11 and### 1.

55 MeV. This is con-sistent with the result obtained in the calculation### of

the energy levels. The calculated value for the

### (3/2)+

level at the 2.67MeV state is### 0.

02,which ismuch smaller than the observed value

### of

### 0.

### 47.

### To

further test the wave functions obtained in our model, we calculated the### EM

transition ratesusing the experimental gamma energies. In general, the agreement between the calculated and observed

values issatisfactory.

C. Comparison ofour model space with the simple

### (f

7/2, d3/2 ors~/2 ')configurationIn order to test the significance

### of

our model space, we calculated the intensities### of

the com-ponents### of

our wave functions which are seen in the coupling### of

the### s,

/2 or d3/2 hole to the simple f7/3 structure.The calculated intensity

### of

the (f7/3 d3/2### ',

or s~/3 ')component increases, in general, as the massnumber decreases.

### For

an individual mass number, higher isospin states, in general, give a larger inten-sity### of

the component (f7/2 d3/3 ol sf/2### ').

Ourresults also show that the intensity

### of

the com-ponent (f7/3 Sf/3 ) is small compared to that### of

the component (f7/2 d3/3 ) One important point

that has to be mentioned here is that even for cer-tain low lying states, such as (1/2)& and (1/2)2 states in

### 'K,

the### (1/2)l,

(5/2)2, and (3/2)3 states in 'Ca, the### (1/2)l

state in Ca, and the (1/2)~ and (3/2)3 states in### K,

the intensity### of

(f7/q,s~/2 ')islarger than

### 32%

and cannot beneglected.The intensities

### of

the component (f7/2, d3/3### ',

ors~/3 ') contained in the wave functions for the

yrast states are larger than

### 77%,

in general. Excep-tions are the### (9/2)l, (11

/2)~, and### (13/2)l

states in'Ca; the

### 3,

and 1~ states in Ca; and the (3/2)~,### (1/2)l,

(5/2)&, (7/2)&, (9/2)&, and### (11/2)l

states in### Sc.

All### of

them are smaller than### 47%.

This man-ifests the fact that the simple model cannot account for the lower isospin states, and the inclusion### of

the p3/2 configuration space seems to be necessary inthis mass region.

Figure 4shows atypical result for the calculation

### of

### 3=42

nuclei.### For

### K,

the intensities### of

the compo~e~t (f7/33d3/3 —1 ol s]/2### —

1)for the low ly-ing states are larger than 90%%uo. Therefore, thesim-ple model is suitable to account for the level struc-tures

### of

this nuclei. However, for 4Ca, theinterisi-ties are smaller than 80%%uo. Especially, the 1&, 3&,

and 34states give only

### 22%,

### 46%,

and### 43%

intensi-ties### of

the component_{(f7/3}d3/3

### ',

or sl/2### ').

Furthermore, larger intensities### of

the component### (f

7/3 $ ]/2 ') are also exhibited in the 2~ and 3&states

### of K,

and the 32,33,and 34states in Ca.In conclusion, the simple model is only able to account for most

### of

the low lying states### of

nuclei with a small mass number, and is only suitable for the calculation### of

higher isospin states for individu-al mass numbers.### For

nuclei with a large massnumber, or for states with lower isospin, enlarging

the model space becomes more important.

### III.

CONCLUSIONIn this paper we have presented a systematic

study on the one hole states for the nuclei

### of

### ef-intensity

### of

(### f7+

l.### 0-0

Oa- 4- 5-3- 3 3 0.4-3w 3- 0,2-0.0 0.0_{0.}8 I 4.2 I 4.6

### E

### (MeV)

5.0### FIG.

4. The intensities ofthe component (f7/2 d3/2 ',ors~/2 ')forthe states in Kand Ca.fective interaction, the calculated energy spectra are in reasonably good agreement with the experimental data, and some definite predictions are made on the spin-parity assignment for afew low-lying levels

### of

ambiguous experimental spin-parity determination in### K,

Ca, and### K.

We have assumed an inertCa core and allowed active s&/l d3/2 f7/2 and

p3/2 orbitals. The omission

### of

### f

s/z, pl/2, and ds/2orbitals from the model space will not yield signifi-cant errors because they lie at about

### 4

to 6 MeVaway from the f7/2 level. The contributions

### of

the p3/g and s&/2 orbitals are, however, ratherimpor-tant even for some low-lying states. This clearly in-dicates that any shell-model calculation which at-tempts to successfully account for the non-normal parity levels

### of

the range### of

nuclei with mass A&40 will have touse amodel space spanning at least the s]/l d3/2### f

7/2 and p3/2 single-particle orbitals.We have also calculated the spectroscopic factors

to test the wave functions obtained. The calculated spectroscopic factors are in good agreement with

the experiment. The I

### =2

pickup reactions on Cafor the 3& state

### of

### K

do not agree as well as theothers, because the wave function

### of

this state is spread too much. This discrepancy may be im-proved by enlarging the model space. The model space is tested by comparing the intensities### of

the picture### of

coupling sl/2 and d3/l holes tothe simple### f

7/2 structure. Our results show that in this massregion, the p3/2 and s&/2 orbitals seem tobe

neces-sary.

In conclusion, the one-hole states for /I

### =41

### —

43 nuclei can be well explained within the conventionalshell model by using a model space

### of

(f7/2 lp)/(2d3/2 s]/2) ' configurations ~This work was supported by the National Science Council

### of

the Republic### of

China.### J.

### B.

McGrory, Phys. Rev.C8,693 (1973)~D.H. Gloeckner,

### R.

D. Lawson, and### F. J.

D.Serduke, Phys. Rev.C 7, 1913 (1973).D.Banerjee and G. Oberlechner, Phys. Rev. C 7,2437

(1973).

4A.

_{E. L.}

Dieperink and P. ### J.

Brussard, Nucl. Phys.A106,177(1968).

5S.

### T.

Hsieh,### K. T.

Knopfle, and G.### J.

Wagner, Nucl. Phys. A254, 141(1975).### R. D.

Lawson and A. Miiller-Arnke, Phys. Rev. C 16, 1609 (1977)~~P.M. Endt and C.Van der Leun, Nucl. Phys. A310, 1 (1978).

~I.P.Johnstone, Phys. Rev. C22, 2561(1980).

### R.

### R.

Scheerbaum, Phys. Lett.### 618,

151(1976).H. Nann, W.S.Chien, A. Saha, and

### B.

H.Wildenthal, Phys. Rev. C12, 1524(1975).### 'D.

### F.

Beckstrand and### E. B.

Shera, Phys. Rev. C3, 208(1971).

G.Mairle and G.

### J.

Wagner, Z.Phys. 251,404(1972). '_{P.}

_{Doll,}

_{G.}

_{J.}

_{Wagner,}

_{K. T.}

_{Knopfle,}

_{and}

_{G.}

_{Marie,}

Nucl. Phys. A263, 210(1976).

~4R.Santo,

### R.

Stock,### J.

H. Bjerregaard, O. Hansen, O.Nathan,

### R.

Chapman, and S. Hinds, Nucl. Phys.A118, 409 (1968).

### J.

L.Yntema, Phys. Rev. 186, 1144(1969).U.Lynen,

### R.

Bock,### R.

Santo, and### R.

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'

_{J.}

_{Rapaport,}

_{W.}

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_{Dorenbusch,}

_{and}

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