PH YSICAL RE VIE% C VO LUME 17,'Ã UMBER 2

Effective interactions, Coulomb displacements, and separation energiesfor E = 50 and X = 28 isotones*

S. T. Hsieh and M. C. %'ang

Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, Republic of China

D. S. Chuu and C. S. BanDepartment of Electrophysics, ¹tional Chiao Tung University, Hsinchu, Taiwan, Republic of China

{Received 14 April 1977)

Effective interactions for X = 50 and S = 28 isotones are obtained by fitting the low-lying excited levels.

Coulomb displacements and separation energies are deduced directly from the difference of observed and

calculated binding energies.

NtjCLEAH STHUCYUHF, += 50 and &=28; calculated effective interactions, de-duced Coulomb displaceIQGIlts RI1d Sepal"ation eIM1"gi68.

It is well known that the Coulomb displacementener'gy exists only between the binding energiesof the ground states for two neighboring mirrornuclei, whereas the J' a,ssignments' and the energyspacings of the excited states remain almost thesame. Similar situations exist in the isobaricanalog states in medium and heavy nuclei. In fact,the effective interaction adopted in the calculationof energy spectra by the conventional shell modelis Qot presumed to represent the kind of nucleonbeing studied, . The distinction between the p-p,p-'Pl) OF FE-s 1QteFactlons exists 1Q the CGQslderatloQof different 1soballc spin among two-body matrixelements of the effective interaction.

Recently, Ball et ak. ' and Gloeckner and Ser-duke' calculated the N = 50 isotones by treating"Sr as an inert closed shell. Lips and MeElli-strem' investigated the X= 28 isotones by shell-model calculation with the Ca nucleus assumedto be an inert core. All these studies employedthe least-squares fit in their calculation of bindingenergies and treated the two-body matrix elementsas adjustable parameters. Since the experimentalvalues of Coulomb displacement energies are notcomplete for N = 50 and N = 28 isotones, the ob-served binding energies, adopted in the above works,still include the effect due to Coulomb repulsion.

The purpose of this paper is to discuss the dif-ferent aspects which exist in the shell-model cal-culation of binding energies by the least-squaresfit when binding energies of the ground states areincluded in the fitting (henceforth referred to asmodel I) or excluded from the fitting (referred toas model II). These may be analyzed from the fol-lowing three considerations: (i) reliability of theeffective interaction obtained from the fitting, (ii)explanation of the Coulomb displacement and sep-

aration energy, and (iii) reasonableness of the ob-tained wave functions. For N= 50 isotones, directcomparison between calculated p-p and n-n inter-actions is not available. However, the reactionmatrix elements for the 1f-2p shell nuclei' pro-vide us with the necessary information for com-parison. From the above comparisons, the effectof Coulomb repulsion may be investigated. ForN = 28 isotones, the investigations (i) and (ii) canbe reached by direct comparison between the cal-culated two-body matrix elements for p-p 1nter-action and those for n-n interaction obtained byMcorory et a/,

" for Ca isotopes.Ball et al. ' and Gloeckner and Serduke' calcula-

ted the X= 50 isotones by assuming "Sr as an in-ert core with Z —38 protons being distributed inthe 2P, ~, and 1g,&, shells. Two single-particleenergies for 2p, &

and 1g,~, orbits and nine two-body matrix elements are treated as adjustableparameters to fit to the binding energies of theground states and low-lying excited energies fornuclei of X= 50, 39~Z ~ 44. Both of these studiesobtained almost the same results.

Therefore, it is only necessary to compare theresults obtained in the present study with one ofthe above studies. In our calculation, "Sr is as-sumed to be an inert core and the model space isstill assumed to be 2Py/ and 1g,&, orbits. In themodel I calculation, 6 binding energies of theground states and 49 excitation energies, exceptthe 3 state ln Mo and the exc1ted 0 state 1Q

"Ru, are included in the least-squares fit. In themodel II calculation, we include only the excitationenergies in the least-squares fit. The resultingrms deviations are 0.100 MeV for model I and0.085 Mev for model II. The calculated two-bodymatrix elements and single-particle energies for

EFFECTIVE INTERACTIONS, COULOMB DISPLACEMEWTS, . . .

TABLE I. Comparison of the calculated two-bodymatrix elements (in MeV) for E= 50 isotones. Slngle-particle energies of 2P ~g2 and 1g~y2 states are listedin the last Wo rows.

2 jg 2jp 2 j3 2 j4 J Our

1 09 09 4

59 0

2

68

P ~ j2 single-particle energy

g~y2 single-particle energy

-0.4840.9010.6900.175

—1.719-0.603

0.1640.5080.570

-7.094

-6.255

-0.7190.9070,392

—0.008—1.937-0.896-0.098

0.1280.318

0.00.877

—0.2490.650

—1.147-0.852-0.259—0.037

0.094

model I are consistent with those obtained byBall et al. ' Table I li.sts the calculated two-bodymatrix elements. Column 2 of TaMe I shows the two-body matrix elements calculated by Ball et al. , andcolumn 3 lists our results with model II. Column4 gives the results obtained from the reaction ma-trix elements. ' The single-particle energies arelisted in the last two rows. Since we do not con-sider the blndlQg encl"gles of the ground states lnthe least-squares fit for the model II calculation,e(2p, i,) is set etiual to zero Our .calculated twobody matrix elements agree reasonably well. withthose of Kuo and Brown. ' Comparison betweencolumn 2 and column 3 shows the main differencesbetween the two-body matrix elements of Ballet al. , and those of the present study are in thediagonal elements. The overall average of thedifferences for these eight diagonal elements is .

0.265 MeV, which can be considered as the Cou-lomb interaction energy for a pair of.protons inthe (2P,i„lg, ~,) model space.

The binding energies of the ground states calcu-lated with the parameters in model II are listed inin column 3 of Table II. Column 2 shows the cor-

responding observed values. The differences be-tween them are given in column 4. This differencewill be interpreted as due to the Coulomb interac-tion energies since these matrix elements are notincluded in the parameter obtained in model II. Ifc i.s the Coulomb interaction of a pair of protons inthe 2p, g, -lg, i, shell, then the total Coulomb inter-action energy of g extracore protons can be writ-ten as

( )n(n —1)

2

The calculated values of e in column 5 are almostconstant, varying between 0.222 and 0.277 MeV.The average value of 0.255 MeV is in almost per-fect agreement with the mean Coulomb displace-ment energy, 0.265 MeV, obtained above. Theseparation energies, which can be calculatedf

lorn

S(n) =S(1)—(n —1)c —[E {n)—E (n —1)]

are listed in column V, where S(1) is assumed tobe the experimental value of 89Y, i.e. , 7.067 MeV,and Es(n) is taken directly from our calculatedbinding energies. Column 6 shows the correspond-ing experimental separation energies. The calcu-lated separation energies are found to be in verygood agreement with the observed ones.

It is well known that in shell-Inodel calculations,if various effective interactions are employed with-in the same model space, the most realistic inter-actloQ will ploduce the most conf lguratlon mlxlngof eigenvectors among the components. For N= 50isotones, the eigenvectors obtained by excludingthe binding energies of the ground states from theleast-squares fit are found to possess more con-figuration mixing. For instance, the eigenvectorfor the ground state of "Mo in model I calculation1S

while that obtained with model II is

TABLE II. Comparison of observed and calculated binding energies, Coulomb displace-ment energies, and the separation energies (in MeV) for N=- 50 isotones.

NucleusBinding energy

Calc. Diff.

Coulombdisplacement

Calc.

90gr

@Mo

Tc~au

-1.2890.6190.2193.1824.0087.904

-1.511-0.213—1.256

0.5630.0922.435

0.2220.8321.4752.6193.9165.469

0.2220.2770.2460.2620.2610.260

8.3565.1597.4674.1046.2413.171

8.3235.2597.3454.2286.2633.194

l0'&,

~= 0.581

l lg, g,', v = 0) —0.016l 1g,(,', g = 4)

+ 0.814l 1g,i,', 2P, i, '& .

Alt ough the difference betw~e~I0'&i and 10'&„

is not large, it still may produce a considerableeffect on the transition rate.

The existence of the states & at 1.507 MeV andat 1.745 MeV for the 89K nucleus shows that the

"Sr nucleus is actually not a good core for thecalculation of .N = 50 isotones. This difficulty maybe solved by enlarging the model space to includethe 2P,~, and lf, ~, orbits. HoweVer, the calcula-tion becomes more complicated with such consid-eration because, in addition to the particle-part-icle interaction, one must take into account theparticle-hole and hole-hole interactions in the en-larged model space. However, if the ground-state binding energies are not included in the cal.-culation, one may assume the pseudonucleus &ooSn

to be an inert core, and therefore, only the hole-hole interaction has to be considered in this sit-uation.

The effective P-P interaction is the sum of nucle-ar and Coulomb contributions. It is to be expectedthat the Coulomb interaction, because of its longrange, should have two-body matrix elementswhich depend rather weakly on J. Since the Cou-lomb interaction contributes very little to off-di-agonal matrix elements, the ma, in effect of theCoulomb interaction is to add a constant to the di-agonal elements. For example, in the calculationof N = 50 isotones, the difference between the off-diagonal elements (p, /2'l p'lg, /, '&~=, for model 1

and model II is only a negligible value of 0.006MeV as shown in Table. I. Furthermore, our two-body matrix elements for X= 50 isotones are fittedto the experimental energy levels. Therefore, theeffects of Coulomb interaction on the off-diagonalmatrix elements and the J dependence of the di-agonal elements are automatically included in ourresults. - The negligible effect of Coulomb inter-action on the off-dia, gona. l elements ensures thesimilarity between the eigenvectors obtained byBall et al. and this study. A slight difference ex-ists between these results because the eigenvec-tors obtained in our calculation with model II pos-sess more configuration mixing.

In the calculation of Ã= 28 isotones, we assumethe "Ca nucleus to be an inert core with Z —20protons being distributed in the 1f,/„2P, /„ lf, /„and 2P, /. orbits. We allow only one proton tojump to a higher orbit from the 1f7/2 shell A two-range central-plus-tensor potential, proposed bySehiffer and True, ' is assumed in the practicalcalculation. Harmonic oscillator wave functionsare employed with the oscillator constant beingfixed at v = 0.96 )&~ z/3 fm 2 where ~ = 50

TABLE III. Interaction strengths (in MeV) for &= 28,22» Z~28 nuclei. Column I and column II show the cal-culated results with model I and model II, while columnST represents those obtained by Schiffer and True. Thelast three rows are the single-particle energies (s.p.e).CSE, CTO, and TTO mean the strengths of centralsinglet-even, central triplet-odd, and tensor triplet-oddcomponents respectively. 4

Range ors.p,e ST

CTQ

TTO

ShortLong

Short

& (2p3/2)~ (2P y/2)~(1f 5/2)

—102.9533.74

—269.86120.94

3.093.944.38

—57.7713.53

—250.92105.47

—20.94

3,224.004.23

—49.3215.47

—155.8262.06

—6.10

radial dependence of the effective interaction isassumed to be of the Yukawa type, and the inter-action ranges are taken directly from Ref. 6. Theinteraction strengths of the. presumed two-rangeforce and the single-particle energies of 2p, /„lf, ~„and 2P, ~, with respect to the lf, &, orbit aretreated as free parameters. In the model I calcu-lation, 28 low-lying excited states in addition to6 binding energies of the ground states for nucleiX = 28, Z = 22-28 are included in the least-squaresfit and yield an rms deviation of 260 keV. In thecalculation with model II, the. 6 binding energiesare excluded from the lea.st-squares fit, and therrns deviation is 221 keV.

The interaction strengths are listed in Table III.The third column, with the heading I, shows thecalculated results with model I while the fourtheolurnn, with the heading II, shows those obtained

with model II. The interaction strengths obtainedby Schiffer and True' in their calculation of two-body matrix elements for the nuclei in the vicinityof doubly-closed shells are Iisted i.n the last col-umn. The last three rows show the single-particleenergies of 2P,~„ lf, ~„and 2P, &, referring to the

lf, ), orbit. Comparison of the last three columnsshows that the results obtained by- the calculationwith model II give better agreement with theSchiffer and True results than those obtained withmodel I.

The spectroscopic factor of the low-lying sta$esof "Sc observed by Henning et al. ' in the "Ca("N,"C)"Sc reaction suggests that the centroidof the single-particle strengths of (2p, &,) and

(1f,,~,) a.re 3.32 and 4.25 MeV, respectively. Ourcalculated single-particle energy is in rather

'good agreement with the observed values for

EFFECTIVE INTERACTIONS, COULOMB DISPI ACEMEIWTS, . . .

TABLE IV. Two-body matrix elements (in MeV) of(f,y2 I &I f7)2 )~ for N= 28 isotones .Column LM liststhe results ob&ined by Lips and McRllistrem. .ColumnsI and II show our calculated results wit& models I and H,respectively. The last column gives the modified Kuoand Brown two-body matrix elements.

KB'

. 02

6

-2.29—0.47

0.420.82

-2.31—0.49

0.300.81

—2.50—0.81-0.05

0.37

-2.11—1.11—0.10

0.23

(if, &,) and in qualitative agreement for (2p, &,).In this study, the main effect of the Coulomb

displacement energy is due to the two-body matrixelements (f»' V

~f,&,')~. Table IV shows the val-ues of (f,&,

' & f,&,')~. ..,. Column 2, with theheading LM, lists the two-body matrix elementsobtained by Lips and McEllistrem. ' The columnsheaded I and II show our calculated results withmodels I and II, respectively. The last column,KB', lists the modified Kuo and Brown two-bodymatrix elements obtained by Mcorory et al. ' forthe energy spectra of Ca isotopes. Comparisonof the second and third columns shows that ourcalculated results with model I are very similarto those obtained by Lips and McEllistrem, ' whileour calculated results with model II are in goodagreement with those listed in the column headed

KB'. The overall average of the differences be-tween I and II for J= 0, 2, 4, 6 is 0.35 MeV, whichis consistent with the value 0.318 MeV obtainedby Harchol et al. ' for fitting the Cou)omb displace-ment energies. Since the size of the nucleus in-creases when the mass number increases, theCoulomb interaction for a larger nucleus is weakerthan that for a smaller nucleus. In view of thisfact, the Coulomb interaction obtained for N= 50isotones is rather consistent with that obtainedfor N= 28 nuclei.

In conclusion, it is found in the present studythat the calculation of ¹50and X=28 isotoneswith the binding energies of the ground states ex-cluded from the least-squares fit seems to bemore practical. Furthermore, more informationcan be deduced from such a calculation, e.g. ,Coulomb displacement energy, separation energy,and effective nucleon-nucleon interaction. It iswell known that it is.difficult to investigate the de-formed nuclei within the framework of the shell-model calculation. However, if the ground stateand the low-lying excited states possess the samedeformation for such nuclei, then one may not con-sider the ground state. The excitation energiescan be calculated with the conventional shell mod-el. After the interaction has been obtained, thebinding energy can be reproduced. Informationof the deformation for some nuclei can be obtainedfrom the difference between the calculated and theobserved binding energies.

*Work supported by the National Science Council of theRepublic of China.

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