1 .E.1 :1.E .7

Nuclear Physics A303 (1978) 313-332; ©North-fitolland Publishing Co ., AmsterdamNot to be reproduced by photoprint or microSlm without written permission from the publisher

STUDY OF ISOBARIC ANALOGUE RESONANCES IN f-p SHELL NUCLEI :A UNIFIED-MODEL DESCRIPTION

K. HEYDE, M. WAROQUIER, P. VAN ISACKER and H . VINCX

Laboratorium voor KernJysica, Proeftuinstraat 86, B-91XX1 Gent, Belgium

S. GALÈSInstitut de Physique Nucléaire, 91406 Orsay, France

and

V. PAARPrirodoslovno-matemaüéki jakultet, University of Zagreb

andInstitute "Rudjer Bo .~kovic"', Zagreb, Yogasisals

Received 7 March 1978

Abstract : Spectroscopic factors as well as parentage coefficients for collective J; = 2; , 3; and 4; stateshave been obtained for N = 29 nuclei and s9Ni via (s, d~) reactions through isobaric analogueresonances (IAR). In the framework of a unified-model calculation, treating the one-neutron corecoupling (N = 29) and three-neutron core-coupling as well as pure three-neutron shell-modelcalculations (3'Ni), a description can be given for most of the observed phenomena (energies,spectroscopic factors, parentage coefficients on the J; = 2; , 3; , 4; levels) . Some discrepanciesremain, especially with respect to the description of the splitting of the 2p, ~z neutron single-particlestate . A compàrison with earlier calculations for the N = 29 nuclei is carved out. With respect tos9Ni, the three-particle luster effeéts are pointed out and an extensive comparison between bothtypes ofcalculation and with experiment is made.

I. IIItrOdlletlll~

The spectroscopic properties of the low-lying states in the N = 28 + 1 neutronnuclei have been extensively studied through neutron stripping experiments ' -')and by investigation of their y-decay scheme using particle-y coincidences tech-niques ' " a, a - t z) .These experiments have established the spin and the parïties of the low-lying

states which contain a large amount of the neutron particle strength correspondingto the 2pß, 2pß, lf~ and lg~ orbitals located above the shell closure at N = 28.However, a small fraction of the 2dß neutron particle strength has been observed

in these nuclei around 4 MeV excitation energy . Although our knowledge on thelevel scheme and on the fragmentation of the neutron single-particle strength inthese nuclei should be considered satisfactory, this information is limited in almostall cases to the first term of the wave function ofthese nuclear levels (neutron coupled

313

314

K. HEYDE et al .

to the .P` = 0+ ground state) . Moreover, the spectroscopic factors for the 1 f~ andlg~. states deduced from neutron stripping experiments at low bombarding energyshould be confirmed using different reactions where the matching conditions favourlarge angular momentum transfer .

Thereforé the study ofthe population and the particle decay ofthe isobaric analogresonances (IAR) of the low-lying states in theN= 28 +1 neutron nuclei could leadto useful and complementary spectroscopic information on the properties of theparent nucleus.These studies have been performed recently 'a-1 s) using the (T, d) reaction on

various doubly even targets (48Ca, s °Ti, ssCr, s4Fe and seNi) in f-p shell nuclei toselectively excite highly IAR. The investigation ofthe proton decay ofthese IARhasbeen carried out by means of the sequential (T, dp) reaction i s-1 s) . The angulardistributions of the emitted deuterons from the (z, d) reaction have been measuredusing the 25 MeV 3He beam delivered by the Tandem MP accelerator of the Institutde Physique Nuclea-ire d'Orsay and a split-pole spectrograph . The angular corre-lations of the decaying protons from the (T, dp) process have been observed incoïncidence using geométry II of Litherland and Ferguson ie) with 0° detection ofthe deuterons ") .The existence of a simple relation between the single-particle strength of a analog

pair of levels permits useful comparison of the deduced spectroscopic factors meas-ured in two different experiments. In general the proton resonance scatteringreactions have been employed to populate IAR in f-p shell nuclei 1e-23) . However,IAR. with high angular momentum (l = 3 and 4) are hardly observable in protonresonance scattering due to the high Coulomb +centrifugal barrier in thep+C(C fortarget) system . In addition, these IAR located at high excitation energy (around15 MeV in °9Sc and 7 MeV in s9Cu for example) are embedded in a high density ofT~ states with low angular momentum (l = 0, 1) and the analysis of the protonelastic and inelastic scattering becomes difficult due to the lack of selectivity of thesereactions. Therefore our overall study of the IAR in f-p shell nuclei via the (T, d)reactions and the subsequent DWBA analysis of these unbound levels (using aGamow function as form factor for the transferred proton aa) have lead to theidentification of about 10 to 15 IAR in each final nucleus and to the detenmination ofthe neutron single-particle component of the parent levels in °9Ca, siTi, ssCr, ssFeand s9Ni .Ageneral agreementbetween the spectroscopic factorS' deduced from theDWBA

analysis of the unbound IAR and the corresponding number Sa, v from neutronstripping experiments ' " '), has been obtained .

In addition to the !-value determinations and the single-particle component, thestudy of the sequential (T, dp') reaction in a suitable geometry 's - i s) permits furtherinvestigations of the proton decay of the IAR. This method has been shown toprovide information on the spin of the IAR and gives directly the branching ratioP`(p;)lT where P(p~) and Tare respectively the proton partial width via channel i and

IAR IN f-p SHELL

3l5

the total width of the IAR. The inelastic partial width P(p~ deduced is thereforerelated to the amplitude of the neutron coupled to target excited states, and thusinformation on the wave function ofthe parent level not studied in (d, p) experimentsresults .We emphasize that without interference effects and background problems which

always arise in the analysis of (p, p') data, these core-excited components can bedetermined. In spite of low statistics due to the coincidence set-up, branching ratipsfor inelastic decay as low as r(p,)lT 10- z could be investigated . Themain limita-tion in this mass region for a precise determination ofthese components is due to therelatively low energy of the emitted proton for target excited states . Forexample inthe case of the first significant component of the 1 g~ single-neutron configuration,these levels are known to have a large ~3 ~ ® 2pt) component in their wave function,but the corresponding decay to the 3i state in N = 28 nuclei has only been observedin the case of the lgt IAR in °9Sc, whereas the J" _ ~+ IAR in siV, ssMn, ssCoand s9Cu show only weak ~2 i ® 2d t~components due to the lower excitation energyof the IAR in these nuclei .Thecombined results ofthese two kinds of experiments, (T, d) and (i, dpi), provide

a complete set of spectroscopic data concerning the analog states of the low-lyinglevels of 49Ca, siTi , s3Cr, ssFe and s9Ni nuclei . Therefore the resulting level schemeand amplitudes of various components of the wave functions in f-p shell nuclei arebelieved to be a severe test for calculations andcould lead to interesting comparisonwith a systematic description of these nuclei in terms of unified-model calculations(sects . 2 and 3) .

2. The model2 .1 . DESCRIPTION OF THE UNIFIED MODEL

In view of the coherent set of experimental data on the N = 29 nuclei obtained viaIAR decay studies (elastic and inelastic), we will try to describe the data within aunified-model approach in which the single-particle motion (extra neutron) will becoupled to the core states of the underlying doubly even nuclei (Z = 20, 22, 24, 26) .The Hamiltonian describing the interacting system,

will not be discussed at length because earlier studies have given all the necessaryformulae ss-ze). If we idealise the core nuclei to harmonic vibrational nuclei, weoversimplify considerably the actual nuclei, although the lowest .P` = 2+ and 3 -levels are connected with the J` = 0 + ground state in all cases by well enhanced( 10W.u.) B(E2) and B(E3) values a9 " 3~. This is due to the fact thatJi = 2i and 3ilevels are notjust composed of the valence proton (lf~)" configuration, but also coreexcitations for protons through the Z = 20 core, i.e . s-d -. f-p, and for neutronsthrough theN = 28 core, i.e . 1 f~ -" (1ft, 2pß, 2pß), occur, thus giving rise to collec-

316

K. HEYDE et al.

tive states with enhanced B(E~ .) values s'). By meansofthis unified=model approach,it will be possible to treat globally the N = ~9 nuclei whereas earlier calculationswere either very fragmentary az-ss) or had different underlying assumptions in themodel description ae-sa) . The unified-model calculation will provide the possibilityto test not only the single-particle component but also collective configurationsobserved in the IAR studies, such as 12 ; ® 2pß) (Jx = ~-) and 13~ ® 2p~i,12i ® ld~), 12i ® lg ti (J~ _ ~+ and ~+). In the particular case of S9Ni, we performboth three-particle-core coupling s9) and three-particle shell-model calculations[ref. 4o-az)] because in this nucleus, three neutrons outside the 36Ni core nucleusinduce particular single-particle correlations (cluster effects) . Details will be discussedin subsect . 3.3 .

2.2. PARAMETERS AND TRUNCATION

In determining the necessary parameters of the unified model, single-particleenergies, coupling strengths (~z, g3) and phonon energies (ficoz, ~tco 3 ) occur. In orderto find good starting values for a fitting procedure in determining the neutron single-particle energies, Hartree-Fock calculations have been performed by Beiner "a) fortheN = 29 nuclei and S 9Ni (see fig. 1) .

O

TK

2D

OA

2d~,zi

UNBOlMO

2pß

~Ca S~Ti ~r 55Fe S~Ni ~Ni

Fig. l . Hartree-Fork energies for the neutron single-particle states, i.e . 2pais, 2p1~=, lfs ~ 2 , lg9iz, 2d,�(3s, ~ z ), in the N = 29 nuclei and in s9Ni, calculated with the energy density approximation .

IAR IN f-p SHELL

317

Starting from these values, variations were allowed to determine the best fit values,

by the requirement of minimizing the expression

8

aZ

â 4A

!_

2 .

I~-.SexP I Z + ~ IEth-EexPl2i

<

<i

i

PARAMETER___-- P-N SHIFT A'j

..

\

v

2PY~ L-~____

as~a siT~ s3~~ ssFe

(2.3)

Fig. 2 . Thebest fit single-particle energies d~as well as the energy shifts dj, calculated with the Kuo-Brownproton-neutron interaction, in going from `9Ca(Z = 20, N = 29) towards s°Fe(Z = 26, N = 29) .

with the sum over the JR = ~i , -~~ , ~~ , ~i , fir; and ~i levels. These values d~ aregiven in fig. 2, together with the single-particle energies obtained by taking the 49Caenergies and calculating the variation induced by the residual proton-neutron inter-action, when filling the lf~ proton orbital. Therefore we calculate

d~

4(2j+ 1) ~(2J+ 1)<J1f~ ; JI V~j1f~ ; Ji- 6~ (2.1+ 1)(2p~1f~; J~ V~2p~1f~ ; Ji,(2.4)

giving the variation in neutron single-particle energy after adding two protons, if alinear filling of the lf~ proton orbital in going from 49Ca towards °'Ni is assumed.

31 8

K. HEYDE et al.

Therenonmalised Kuo-Brownmatrix elements aa) have been used with respect to the48Ca core (G+G 3p _ th). Thus, the proton-neutron interaction readily accounts forthe overall trend in the best-fit d~ values . The phonon energies i'tco z , >~tco3 are takenfrom the experimental excitation energies for the Ji = 2 ; and 3i levels, whereasthe coupling strengths ~z and ~s were determined respectively from the B(E2) andB(E3) values z9,

a° ) and using the harmonic vibrator relation

~x -

rôr

4ZeRô

~~3tco z ~

(2.5)

Thevalues of~i ; ~3, itu~ z and ficU 3 are given in table 1 .

T~aLe 1

The phonon energies (fuuz , tao s ) and coupling strengths (~z, fia) for quadrupole and octupole vibrations,respectively (units are MeV)

The truncated model space is limited at Ei = 15 MeV up to which all con-figurationst ~I(NgRq , N°R °)R,j; J11~ with

(i) three quadrupole phonons ;(ü) two octupole phonons;(iii) mixed one-quadrupole-one-octupole phonon states, are considered.For the nucleus S9Ni, where three-particle configurations become important in

determining the level scheme below Ei = 2 MeV, we have performed either :(i) A pure {[(lf~r;(2p~);=]J,2(2p.t~;; J} shell-model calculation with ~t n t = 3. As

residual interaction either a modified surface delta interaction (MSDI) °i) or theadjusted (ASDn surface delta interaction ao,az) was used .

(ü) A three-particle core-coupling calculation s9) considering the 2p .ß, 2pß, lf~,lg~ and2dß configurations . In this case, only quadrupole vibrations were consideredin building the basis states denoted as I((lt,jz)Jtz,j3) Jtzs, NR ; J1li~ .An extensive comparison of both (i) and (ü) will be made in discussing in more

detail the results in subsect. 3.3 .The spectroscopic factors for one-nucleon transfer will also be calculated . In the

N = 29 nuclei, the stripping spectroscopic factor is given by

St.1(J~) = I~,~(~~ (~p,1J~1za~.r

(2.6)

t Here NQ , RQ (No , R o) denote the number and angular momentum of quadrupole (octupole) phononsrespectively, R the total collective angular momentum andj stands for the single-particle wnfguration(n, h .J)~

49~ s'Ti ssCr °sFe

~2 3 .83 1 .55 1 .43 1 .41~W3 4 .51 4.42 4.56 4.56~X z 0.75 1 .25 2.00 1 .75S3 1 .25 1 .25 1 .00 1 .00

where c; denotes the amplitudes of the configurations in the expansion of the wavefunction IJ~M% . In the case of'9Ni, stripping proceeds from the nucleus SsNi, andtherefore we need a description of the J~ = 0+ ground state as

Io; i = ~ al((.liJi)Jiz~N~R~~O)I(lilz)Jiz,N~R~ ;~)'

(2.7)~o,l

If the wave function describing states IJ;M~ in S9Ni can be written as

I JiM% _ ~ b,(((Î,Jz~lz~J3)Jlz3, NR ; J)I((.ÎlJz~lrl3~lz3~ NR ; JM~,

(2.8)iorF

the spectroscopic factors are calculated as

IAR IN f-p SHELL

319

Si.1(Oi ~ Ji) = I< S9N1(J~)Ilaij lls8lvi(0; ) >IZ

Sj.J

~

(2.9)2J + 1

1 a1(Uijz~iz~ N~R~ ; 0)b;((U lJz~ 12+~3)J 123+ NR ; J){o~}((2j+1x2R+1)

X ~(v1~2~ 12+~3~12311 a1,jIIv1~2r 12~SRR'aNN'S R' .Ji= I Z Sj.J'

(2.10)

Here {O;} - {ji,ji, J~ Z, N', R'} and {Oi} --- {jl , j2, J12 , j3, J,z3, N, R} denote allquantum numbers, summed over in the initial and final state, respectively .

3. Results3.1 . LEVEL SCHEMES, SPECTROSCOPIC FACTORS: SYSTEMATICS

In figs. 3-6, the general results are collected for 49Ca, s 'Ti, 53Cr and SSFe re-spectively. In each case, the experimental spectrum as resulting from the IAR study,nuclear levels established via other means 1-' z) as well as the spectroscopic factorsS' deduced from experiment 13 - 1s) are given.

In the calculated level schemes, the corresponding spectroscopic information isalso indicated .Some general remarks on systematic ef%cts in comparing theory with experiment,

can be made :(i) The spectroscopic factor for theJ; _ ~~ level is in nearly all cases reproduced

with a value too small compared with experiment .(ü) A second, important fragment of the 2p# neutron level is (except for 49Ca)

experimentally observed at high excitation energy (S' x 0.50 and 3 S Ex ~ 3.5MeV) which is not reproduced theoretically . This situation becomes worse withincreasing charge Z. In s 1Ti the spectroscopic factor SZp~�(i2) = 0.47 agrees rather

320

K. HEYDE et al .

30

20

t0

aol_ 3.~as~=°°~__-~

Fig. 3 . Comparison of experimental (left-hand part) and calculated (right-hand part) level schemes for` 9Ca . The angular momenta are indicated with double the physical value 2J. Levels marked with anopen (filled) circle have been analysed in the elastic (inelastic) (r, d~) scattering via IAR and thecorresponding spectroscopic factors S;~( J; ) ( 5p, v , tables 2 to 6) are indicated on the particular levels .

The theoretical spectroscopic factors are also indicated (only values up to S Z 0.01) .

well with experiment ' 3- t s ), but occurs too low in excitation energy . In s 3Cr andssFe the fragmentation amongst the theoretical JR = ~- levels grows . This is oneofthe main failures of the unified-model description. Through the specific multipole-multipole coupling mechanism between collective and single-particle degrees offreedom in the unified-model description as-28), it seems impossible to obtain atthe same time a high-lying Jt = ~z level (3 < 4 Mew with a large spectro-scopic factorS2p,�(~z) ~ 0.50 .(iii) The'J, _ ~i and ~i levels are well reproduced and for the spectroscopicfactors, rather good agreement with experiment results. Both states contain as themost important configuration the ~3i ® 2p~~ configuration .The above-mentioned remarks (i) and (ü) equally apply to earlier studies within

the unified-model approach where, however, only quadrupole-phonon excitations'were considered 32-sa) [supplemented by the one-octupole-phonon states in ref. as)]and neither the lg} nor the 2dß. neutron excitations were considered . In these calcu-lations, also theItco 2 (fico3) values)were taken as free parameters and sometimes differconsiderably from the J, = 2i and 3~ excitation energy of the underlying nucleus.

~9G

5 .0 3 -17,9) 7 -

151`(,,3,

~' 5`_

S9~ ~r~ ;_

ao -s9' ~-3-

5 ~o .n ,-~ .255_

zg

120

L0

IAR IN f-p SHELL

321

51~AS

15,7,9l

-5_17,9)" ~r

(5,7,91" ~~~51 _

157~1

17,9f~~~

I5~1,

~~\ 31

515

\ `A219~

\17i

\

131" -o5" ~°

~3

9

15~7)-

i~~57_

~~_ .44

QO

3" ~-85 __ .87

3_

Fig. 4. Same as fig . 3, but for s'Ti.

Shell-model calculations have also been performed in theN= 29 nuclei 36-38) whereeither the (lf~)-"(lf~2p~2p~)' configurations 36 ) or the different (lf~)-"'(lf~2p~2p~p2configurations, with the restriction n, S 3, are considered 3' " 3e). These calculationsalso rely on a kind ofweak-coupling mechanism between the proton and the neutronconfigurations through the microscopic proton-neutron interaction. The results ofrefs . 36.3') are on the average not much better then our calculations although,through a microscopic (lf~)- "' proton configuration, the core system is describedbetter and therefore the coupled configurations give a better agreement in excitationenergy with experiment than our results do . One interesting point in the calculation

32 2

K. HEYDE et al.

4.

A6 -1_5

~~~~ ~>5'S.

3g'~~---~~1_g'113

~3ll\

~5_I1

9_5'

1 ~3

1

7

Fig. 5 . Same as fig . 3, but for'~Cr.

of ref. a'), is the fact that for the (1ft2p.t2p~)Z(lf.~)-"~ configuration space, a low-lying Jx = ~- state results at Ei 1 .5 MeV, as has been observed experimentally(with small stripping but a large pick -Lp spectroscopic factor) . Also high-spin statescan be generated easily . In ssFe(s3Cr), a J~ _ ~-, ~-,~- (i-) sequence results. Thisis probably due to the neutron 2p-lh microscope coupling mechanism establishinga dJ = 1 deformed band structure . The general difficulties discussed under (i) and(ü) however remain also in these shell-model calculations, causing a real problemfor the understanding of the selective splitting of 2pß strength .

ô 3A~n- II ]6 5_

j-~ I 7'5

W`I

ZO 9_

5'v

n- 1_

~ `..Q6 3-W5" --~ \

.

vvv

103_

5"

15,71--0--13

Irl7_

1.0 5 _-o----_67 ,' S

. .45 1 _

3" ^ .77- ~~__ _ 3

IAR IN f-p SHELL

323

3-~~9.

~r-3'~s-

7_

-n-x 1 _

~~ s_7_

3

as

3.2. DETAILED DISCUSSION OF `9Ca, siTi , ssCr AND ssFe

35

r

s--~ffi

1_

_

~8

~i

~I)L 3 --{y--62 -- -.79

Fig. 6. Same as fig, 3, but for ssFe .

We intend discussing in more detail the different N = 29 nuclei, especially con-cerning the comparison of thé experimentally determined SP, v . inelastic spectrosopicfactors with parentage on the Ji = 21, 3~ and 4i levels .3.2 .1 . The nucleus 49Ca. In this nucleus, for the J; _ ~~, 2 and ~i levels, collective

components from the J; = 2i and 3i states have been obtained. Comparison withthe calculations is made in table 2. Good agreement results for this particular nucleus.

ssFe5' yo--~

\

4DH s - �.n

9'l1f~

18G1 ~

13f--o------~1131

1l51

19,~n1

isr~->` I>~f

5W

/

3 -b .10_9_

11 8`==.5 .-_o--_~~

W 1- -0 .07 \

32 4

K. HEYDE e~ al.

Levels in ` 9Ca, analysed in the inelastic (t, d~) scattering through IAR towards collective J; = 2 ;,3i , 4i levels in the core nucleus

In all cases, the experimental and theoretical energies (MeV), experimental and theoretical parentagecoefficients (SP, P.) as well as the important configurations are given .

SPhP.

T~s~ 2

T~ai.e 3

Teece 5

13; ® 2Pa/,>

Same as table 2, but for °°Fe

0.070.240.010.010.67

J; ~

Same

E~°

Trace 4as table 2, but for s3Cr

Config. Sp .aPp~ SP .u,°'

}3 2.66 2.28 121 ® 2P3n) 0.06 0.01121 ® 2Pln) 0.08 0.22

~; 3.72 3.56 12; ® gds/z) 0.07 0.0414i ® gds/z) 0.15 0.0112i ® lg9/z) 0.0713; ® 2P3/z) 0.37

}z s 4.14 4.04 12i ® ~sn) { 0.09 ~ 0.14

12i ® 1gq/z) 0.03

J; E;° E~ Contig. SP °v.Sihh. v'

ii 3.814 3.58 12 ; ® gds/z) 0.04 0.0312i ® 1B9/z) 0.07 0.0613, ® 2Pan) 0.37

4.028 2.74 12i ® 2Pi/z) 0.02 0.01I 2i ® 2Pa/z) 0.0112~ ® Ifs/z) 0.88 0.49

~i 4.463 3.97 12 ; ® 3s,/z) 0.0112; ® gds/z) 0.22 0.0212; ® 1B9/z) 0.12

J; E;'°

Same as

E;h

table 2, but for s'Ti

Config. Sô`p .

~s 2.136 2.14 12i ® 2Pa/z) 0.1012i ® 2Pi/z) 0.08

~i 3.759 3.75 12 ; ® gds/z) 0.0412i ® 189/z) 0.19

J; E~'° E~ Config . SP ô. Sp°.

$i 3.586 3.72 12; ® 2pj/z) O.SSt0.18 0.673.991 4.48 12i ®2P3/z) 0.13 f 0.05 0.26

~i 4.010 4.26 13; ® 2pa/z) 0.4 t0.2 0.72

IAR IN f-p SHELL

325

In this case, the core nucleus 4aCa seems to have well-developed vibrational J* = 2;and 3 ~ states .3.2.2. The nucleus s i Ti . In this particular nucleus, the splitting of the 1 f,~ strength

seems to be well reproduced although at Ex = 2.95 MeV, a level with S1 Ps~=(~s) = 0.31results, which is not observed experimentally . Also, the Jx = ~z level is reproducedat too low an excitation energy, although

S2n3~z(zz ) = 0.06 is in rather good agreementwith the experimental value of 0.09. Values for SP, r . for the J;` = 2 ; level in s°Tihave been obtained experimentally and are compared in table 3 for the JR = ~zand ~i levels .3.2.3 . The nucleus ss Cr . Again, as in siTi, the experimental lf~ fragmentation is

described rather well theoretically ( JR = ~~. . . , a levels) . The spectroscopic factorSlf~(~i) = 0.36, however, is only 50 ~ of the experimental value. Here we wouldlike to remark that global errors on the values of S' and Sp , n . are of the order ofx 40 ~ [refs . is-is)] . In making the sum rule, a value ~t<sMwsir�,(zi) = 1 .41results, exceeding the theoretical value by about 40 ~. In this nucleus, Sp , n . inelasticspectroscopic factors have also been determined for the Jx = 2i and 4i states inszCr . Comparison with the unified-model calculations is made in table 4.3.2.4 . Thenucleus s sFe . For the2pß strength, a selective splitting into twoimportant

fragments (SiP,~2 = 0.48 and 0.84) occurs, exceeding the theoretical sum rule signifi-cantly . Also at E_ ~ 1 .5 MeV, a second J~ _ ~- level results which is probably dueto particle-hole excitations through the N = 28 closed 1 f~ neutron shell (see alsosubsect. 3.1) .

In this nucleus too, near E_ ;., 4.0 MeV, a J* _ ~+ level is excited with an impor-tant~ingle-particle component (Sis9~~ = 0.57, S`i~ = 0.45), a trend which grows inimportance with increasing proton number Z. Thereason - the lowering ofthe single-particle energy d ls9~z-has been discussed in some detail in subsect. 2.2. .Parentagecoefficients for some important J` _ ~+, ~+ and ~- levels on the collective stateJ; = 2i m sage, are given in table 5.

3 .3 . THREE-PARTICLE CORE-COUPLING AND SHELLrMODEL CALCULATIONS

Here we study s9Ni separately because the three-neutron cluster çomplicates thetheoretical calculations . We will discuss the results ofcalculations under (i) and (ü)(see subsect . 2.2) in some detail .

In the pure shell-model calculation, a simultaneous calculation for doubly evenand odd-even Ni and Cu nuclei was performed (s' -6'Ni and se-es~). The effectivetwo-body interaction was obtained from the MSDI and from a least-squares fit tothe experimental binding and excitation energies (ASDI) a° ~ °~) . The averagedeviation between 100experimental and calculated energies, d - ~N , I~=°-Etb I Z/N,is 0.14 MeV (MSDI) and 0.08 MeV (ASDI) respectively a~. The neutron single-particle energies were also treated as. parameters giving, in the global fit, the valuesdl f ,~z = 0.81 MeV and dZn,~~ = 1 .08 MeV. The calculation with MSDI and ASDI

326

K. HEYDE et al.

Y

Y

ZWZ4QHXW

2A

5.0

4

Loi

r

3

S9NiMSDI ASDI EXPER~£NT 3P-CORE

9'- /

ii

G,~i ., i.~ :~~~. %

3=

5"%%%/ v

13 5"

;'a~ ~; i. ,i

2

MiwY LEVets

7 ___ Itl~

1"~\ !r\~\

~~

9"\\ _

3

~8w_ I_

1_

5~~

3

r~3j5~'

w lel

_ 0~72 \_~~~r \` -~~I`

~ S

.76

--

.55

3-

59 Y~S

.01 ~~5'I_

_02 3 -~1_

S" T-~1~~ 17,51"~~

5_5-~- \

S~ ~~

7_

~a_~.n ~~T \`% \\ 9_

W /

\

~~~ `~

.13 I~~

_ 0 .q~J

3

Fig . 7 . Same as fig. 3, but for' 9Ni . In this nucleus, three-particle core~oupling calculations with differentparameters (sets A and B) (see subsect . 3.3) and also three-particle shell-model calculations with

both the MSDI and ASDI are compared .

_ 5-~~\\v

1e ~~

.lb

1- .62_ 5~ :72 _

_ .68 _

3 .63 .86

are compared with the experimental data and spectroscopic factors for neutronstripping in fig . 7. Good agreement below E_ ~ 2 MeV results in a somewhat betterfit for the ASDI .The calculation for s9Ni has also been performed by coupling three neutrons in

the 28-50 valence shell to the quadrupole vibration. This is considered as an approxi-mation to a very large (and practically impossible) shell-model calculation involvingprotons and neutrons in many shells, but with a simple residual force : pairing plusquadrupole. Such asituation~is approximated by three valence shell neutrons (cluster),interacting only via the residual pairing force, and emitting or absorbing phonons dueto the particle-vibration interaction 39) . For s9Ni we take the neutron 2p ß, lf~, 2p ßsingle-particle energies from experimental s'Ni level positions : dtf~ = 0.77,d2P~i~ = 1 .11 ; the dts,iz level position is taken to be the same as the energy of theJi = ~i state with a large spectroscopic factor in the Z = 28+1 (6sCu) nucleus :dt~q z = 2.5 MeV. A ~~2d t single-particle state from the 50-82 higher shell is includedin t~Ie calculation of the J" _ ~+ states ; its .position is estimated to be dZd} = 4.5MeV. The effective phonon energy ficoZ = }(E(2i , s6Ni)+E(2i , seNi)) = 2.08MeV was taken ; this may partly include the s6Ni core renormalization owing to the

IAR IN f-p SHELL

327

TABLE 6

The spectroscopic factors 5,.~~ for the three-particle core-coupling calculation in s9Ni

Values for three diflferent coupling strengths ~= are compared, except for the J` _ $;, Z and }; levels.The symbol < means, a value less then 0.01 . Experimental values are also given.

`) Ref. ") .

J;~z = 1.8

SU~ Ji)

~~ = 1.6 {z = 2.0S,~

exp`)

~i 0.55 0.65 0.47 0.76$i 0.56 0.64 0.46 1.05~i 0.39 0.53 0.23 0.72~i < < < 0.13~z 0.13 0.05 0.21 0.33~gz < < < 0.07=S3 0.07 0.02 0.03I,3 0.02 0.07 0.07 0.11134 0.02 0.01 0.0133 < < <~4 0.01 0.02 0.02~s < 0.01 0.01~3s < < <i6 < < 0.01~̀6 < < <

34 0.03 0.13 < 0.09ii 0.68 0.81~i 0.13 0.20iz 0.04 0.07

328

Comparison of amplitudes for the most important configurations in the J; _ } ;, z , }; , }, , ~~ and } ; statesresulting from a three-particle shell-model (ASDI) and a three-particle core-coupling (P-core) calculation

The J; _ },. . . .6 spectroscopic factors for different sets of single-particle energies d~, different values of thepairing strength G and different coupling strengths ~z , showing the influence oftruncating the configuration

space

Set A' : d, rsrz = 0'7T

dsnrrz = 0 .81 .') Truncation at 7 MeV.b) Truncation at 5 .9 MeV .

1C(2P3rz2P~rz)zlg9rz]} * ) 0.340.330.24

dzcrrz = 1 .11,

K . HEYDE et al.

TABLE 7

TABLE 8

IC(2P3rz)i 1 g9rz]} *

drs9rs = 2.5 and d z"sn = 4 .5 . Set B is the same except

set A set B

Ji G = 0.4, hrys = 1 .45 G = 0.4, tuoz = 2.08 G=0.5, flcuz=2.08 G=0.4,hru z =2.08

~, = 1 .8 2 .1 2 .4 2.7 1 .6 1 .8 2.0 1 .6 2 .0 1 .6 1 .8 1 .8 ') 1 .8 b)

}~ 0.420 0 .295 0.100 0.054 0.530 0.386 0.229 0.603 0.422 0.495 0.307 0.397 0 .451}z 0 .079 0 .162 0.265 0 .255 0.047 0 .124 0.214 0.004 0.066 0.067 0.183 0.107 0.057}1 3 0 .128 0 .057 0.036 0 .033 0.002 0.169 0.016 0.012 0.01234 0.070 0 .121 0 .097 0 .076 0.125 0 .034 0.001 0.004 0.134 0.101 0.087 0.038 0.004}s 0 .001 0 .001 0.001 0.001 0.013 0 .062 0.047 0.001 0.009 0.001 0.004 0.039 0.085}6 0.032 0 .035 0.021 0.015 0.028 0 .065 0.101 0.004 0.014 0.011 0.014 0.051 0.043

Configuration P-core

}i

ASDI P-core

}z

ASDI Configuration P-core ASDI

}i

0.40 IC(2P3 r z)ôlfs r s]~ ) 0 .61 0.84IC(2P3n)ô2P~ r z] }_ ) 0 .53 0 .83 0 .20 0.13 I[Ifsrs]', ~` -) 0 .31 0 .41IC(lfsrz)ô2Pirz]} _) 0 .22 0 .36 0 .40 0 .58 IC(2Prrz)ôlfsrs ]}_) 0.30 0 .30

-0.20 IC(lg9rz)ôlfsrz]} ) ~.230 .210 .40

IC( 1 fsn 2 P3rz)i 2P~rz ]} ) 0 .26 ''t iIC(2P,n)ilfsn]}) 0.20 -0.52IC(lfsrz 2P~n)z2P_an]~ ) -0.30 0 .52 IC(2 P3rz)ôl8vn]~ * ) 0 .62IC(lfsrz)i2P3rz]} ) 0 .32 I[(lfsrz)ôlg9rz]i + ) 0 .44

IC(2Prrz)ôlgvrs]i * ) 0 .32~l

IC(lfsn)ô2Parz]} _) 0 .53 0 .58IC(2Prn)ô2Pan]} ) 0.39 0 .31 }i

0.43 0.71IC(lgerz)ô 2P3n]} )

IAR IN f-p SHELL

329

presence of three valence-shell neutrons . The pairing strength G = 0.4 and theparticle-vibration coupling strength ~z = 1 .8 were employed. The resulting energyspectrum is given in fig . 7A . Fig. 7B shows the result of a calculation for the phononenergy ~coz = 1 .45 MeV (E(2i, saNi)) and the particle-vibration coupling strength~z = 1 .6 MeV. In calculating the energy spectra, a truncation ofthe diagonal energywas performed so as to take into account a maximal dimension of S 155.

Fig. 7 also shows the calculated spectroscopic factors for each level . In table 6,besides the spectroscopic factors corresponding to ~z = 1 .8, we present also thecorresponding spectroscopic factors for slightly largerand smaller couplingstrengths~z = 1 .6 and ~z = 2.0, respectively, thus revealing their sensitivity to the interactionstrength . In table 7, the most important configurations for the JR = ~i, z , ~i , ~,,~i and ~i levels are given. In order to calculate spectroscopic factors S,~(J;~, theseNi ground state wave function has to be calculated within a two-particle core-coupling calculation, with the resultIssNi; Oi ~ = 0.571(2p~)z , 0 + ~+0.431(lft)z, 0+~+0.351(2p})z, 0+~

-0.261(lg~)z, 0 +) -0.271(2p~2p~)2 +, 12 ; 0+ ~+0.241(lft)z2 +, 12+ ; 0+~-0.211(2p~)z2+,12 + ; 0+~.

No specific configuration dominates this wave function resulting, by means of eq .(2 .10), in a large number of contributions for these spectroscopic factors. For theS,~( J;~ values with J; _ ~i, z ; ~i, z ~ ~ i, z ~ ~ i, z, rather stable values result if smallchanges in the Hamiltonian parameters and truncation effects are allowed . In thecase of the J; _ ~} , . z levels, however, a large sensitivity remains in the range ofparameters as indicated in table 8. Also, in enlarging the model space, higher-lyingbasis states make the amplitudes ofthe I(2p~) z 0 +, 2pti, (lf~)z0+, 2p#~ andl(lgt)z 0+,2p~~ configurations increase continuously, thereby also increasing (decreasing) thespectroscopic factor in the J;` _ ~i~z~ states, respectively .

In the three-particle-vibrational-core calculation, the lowest calculated positive-parity state is ~i , with I(2p~) z 0+, lgt~, I(lf t)z0+, lg .~) and I(2p~) z 0+, lgt) as thelargest components. This state, therefore, has a large spectroscopic factor (0.68) . Thecalculated ~i state lies 2 MeVhigher, and the calculated spectroscopic factor is 0.04.This state might correspond to the experimental state (~)+ lying 1 .64 MeV above~; , with the spectroscopic factor 0.07. Of the remaining calculated states, the ~;state (S = 0.13), which lies 1 .16MeV above the ~i state, has the largest spectroscopicfactor . The experimental ~i state at 1 .44 MeVabove the ~i state has a spectroscopicfactor of 0.20.

In comparing now both the pure shell-model calculations a°-az) with the particle-core coupling calculations, we remark that the ASDI gives the best overall agreementwith experiment . Comparing both sets of wave functions (table 7), wave functionscontaining the purest configurations occur with the ASDI . In the particle-corecoupling calculations, mixing with additional collective configurations will reduce

330

K. HEYDE et al .

in general the amplitudes of the contributing pure three-particle configurations .From looking at the wave functions describing the lowest J~ _ ~-, ~- and ~+states, a good approximation can be made by writing these wave functions in adirect product representation of the sBNi J` = 0+ ground state with the corre-sponding single-particle configuration, and thus

IJi) .= IseNi(Ui) ® .1; Jx)b1 .r (for Ji = ~i, ~i, ~i)'

A further comparison of the results of the shell model and the cluster-vibrationmodel reveals a qualitative similarity . The lowest triplet (~i , ~~ , ~ 1) is followed bya group of four states (~ ~ , ~~, ~ ~ , ~ i). In both calculations the ordering of theseseven states is the same, but compared with experiment the J; _ ~s and ~s statesare interchanged . In both calculations a group of four close-lying states (~i , ~3 ,

~ z , ~s ) follows, though with different orderings. Above these states, more low-spinstates appear in the three-particle core than in the shell-model calculation, but thereis obvious correspondence between higher spin states (J~ ? ~-) in both calculations(~2+ ~3+ ~4+ ~3)'The largest spectroscopic factors for negative-parity states are qualitatively similar

in both calculations and in experiment (~~ , ~ 1, ~~ , ~z). The ratios S(~ ~)/S(~z )and S(~~ )/S(~2 ) are much larger than S(~~)/S(~z ). However, the magnitude of thespectroscopic factors calculated in the three-particle-vibration model are systemati-cally too small. It may be that the collectivity introduced via sizeable phonon admix-tures would cause an overall increase in the calculated magnitude due to two-stepprocesses, which are not included in the present calculation.

4. Conclusioo

By means of(z,dp) reactions on aBCa, s°Ti, szCr, s4Feand sBNi,wecould selectivelyexcite high-lying high-angular momentum IAR. The subsequent proton decay wascarried out by the sequential (z, dp') reaction giving information on the neutronsingle-particle andneutron single-particle+core excited (Jt = 2i , 3~ , 4; ) configu-rations. We have tried to correlate these data by meansofunified-model calculationswith the neutron occupying the lf~, 2pß, 2pß, lgt and2dt orbits coupled with collec-tive excitations of the underlying core nucleus. Level schemes, spectroscopic factorsandparentage coefficients on theJ; = 2i , 3~, 4i levels were obtained and comparedextensively with experimental data . The change in single-particle energy (relativeto the 2p.t orbit) in going from 49Ca towards ssFe can be explained by means of theresidual proton-neutron interaction acting when the proton 1 f~ shell becomes filled .Comparison with other calculations (unified model and shell model) is made. Ageneral problem remains in the description of the selective splitting of the 2p}neutron strength in s3Cr and ssFe into two levels separated by 3.0 < dE, < 3 .5MeV. These experimental facts cannot actually be understood either in the unified-model or in the shell-model calculations .

IAR IN f-p SHELL

33 1

For S9Ni, both three-particle shell-model calculations and three-particle core-coupling calculations (only quadrupole phonons) were performed, allowing a studyof the influence of core excitations . It should be noted that S9Ni is sufficiently closeto the doubly closed shells to be considered as a good case for shell-model calcu-lations and as not too good a case for the three-particle-vibration coupling . There-fore, the qualitative similarity between the results of both models is rather surprisingin the case ofthis nucleus. This should also be considered in the light of the differencein residual forces : In the three-particlwibration model, a bare pairing force plusan effective Q-Q force generated by phonon exchange are responsible for the inter-action between the three neutrons . Shell-model calculations are characterized by amodified or adjusted surface delta interaction . Below Ex x 2 MeV, good agreementresults with experiment although the pure shell-model calculation, taking as effectiveforce the ASDI, gives the better results.

The authors are indebted to Prof. Dr. A. J. Deruytter for his interest during thecourse of this work . They are most grateful to Dr . P. W. M. Glaudemans for kindlysupplying the three-particle shell-model wave functions for S9Ni . One of the authors(K.H.) has benefitted from the hospitality obtained at the IPN, Lyon from Prof. R.Chery, and to J. Sau for illuminating discussions .

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