Shell-model study of 56Ni, 58Ni, 54Fe and 56Fe

f-ix-j

Nuclear Physics A191 (1972) 577-595; Not to

@ North-Holland Publishing Co., Amsterdam

be reproduced by photoprint or microfilm without written permission from the publisher

SHELL-MODEL

STUDY OF 56Ni, ‘*Ni, “Fe AND 56Fe

G. OBERLECHNER Dkpartement de Physique Nucltfaire Thdorique,

and J. RICHERT

Centre de Recherches

Nucldaires, Strasbourg,

France

Received 30 December 1971 (Revised 21 March 1972) Abstract: Shell-model calculations of 56Ni, 58Ni, 54Fe and 56Fe are presented. States are assumed to belong to configurations involving no excitation and two-particle excitations from the 14 shell. A limited number of core-excited states corresponding to four-particle excitations are also taken into account in the cases of 56Ni and 58Ni. The effective two-body interaction in the (If, 2p) shell is deduced from a non-local potential which reproduces the nucleon-nucleon scattering data. Calculated energy spectra and transition rates are not in good agreement with experiment. This is certainly related to the truncation of the basis. Some improvements are obtained when single-particle energy spacings are allowed to vary.

1. Introduction

Theoretical investigations of the nickel isotopes have, until recently, been based on the assumption of an inert 56Ni core ’ - “). The studies of Auerbach “) and Cohen et al. “) reproduced successfully the low-lying energy spectra. However, electromagnetic transition rates could not be satisfactorily explained. In particular, the neutron effective charge used in the calculation of E2 transition probabilities was unrealistically large. This was an indication of the presence of core-excited states in the wave functions. More recently two-particle and four-particle excitations from the lft shell have been introduced 6- ‘) in shell-model calculations of 56Ni, 58Ni and 54Fe. Wong and Davies “) reproduced quite well the energy spectrum of 56Ni and found that 4p-4h states could play an essential role in the description of the low-lying levels of this nucleus. They used the Kuo-Brown matrix elements lo). The 2p*, 14 and 2p* singleparticle energy spacings were determined from the 5‘Ni spectrum. They allowed, however, a variation of the lfa-2p3 single-particle splitting. In order to get agreement with experiment this splitting was fixed at a rather small value (1.5 MeV). The same procedure has also been adopted by Pittel “) who succeeded, in this way, in the description of the low-lying J = O+ levels of 54Fe. On the other hand, the lf%-2p+ single-particle energy difference was much larger (about 5 MeV) in the work of Goode and Zamick ‘) and Jaffrin ‘). This gap was obtained through binding-energy considerations. We present here shell-model calculations of 56Ni, 58Ni, 54Fe and 56Fe with configurations including two-particle excitations from the 14 shell. A restricted number of four-hole states are also taken into account for 56Ni and 58Ni. The two-body 577

578

G. OBERLE~HNER

AND J. RICHERT

effective interaction is deduced from a realistic non-local potential lo). The corresponding matrix elements in the (if, 2p) shell are, as it will be shown, comparable with the Kuo-Brown matrix elements. The single-particle energies are related to the single-particle energies in 41Ca [ref. “)I. By adding the interaction due to the filled If; shell, this procedure leads to a lf,-2p, single-particle splitting of about 3.4 MeV. In sect. 2 we give some details of the calculation of the two-body effective interaction and the effective charges in the (If, 2p) shell. The basic shell-model states are described in sect. 3. Energy spectra and E2 transition rates are presented and discussed in sect. 4. 2. Effective ~teractio~

and effective charges

2.1. REACTION MATRIX ELEMENTS

Our starting point is a non-local, separable potential which reproduces nucleonnucleon scattering data up to 350 MeV. This potential has been given elsewhere “); it contains strong repulsive terms and is therefore renormalized within the framework of Brueckner theory. Brueckner’s G-matrix satisfies the equation

(2.1)

G=v-vgG, e

with -.-= Q e

c a, b

unoccupied states

lab)
(2.2)

where .s@and et, are respectively the energies of the single-particle states la) and lb) and W is the starting energy. The operator Q excludes occupied states and also the individual valence states which are needed in the definition of the model space “, 13). There are two main difficulties in the computation of the G-matrix elements appropriate to finite nuclei. The first is due to the fact that so far no suitable definition of the shell-model potential has been found. Recent progress has been made in this direction by Kirson r4) and Brandow ’ “) for closed-shell nuclei but their conclusions are not yet of much practical use. The second difficulty is due to the geometry of finite systems, a clean resolution of eq. (2.1) being impossible in finite nuclei. Several methods have been used in order to overcome these di~c~ties 16-21). In our work, reaction matrix elements are taken as appropriate averages of the values in nuclear matter. The procedure looks much like the local-density approximation described by Wong “). Treating the propagator Q/e as in nuclear matter, we are led to the following expression: -Q z e

s

dkdKlkK)

‘g

(kKI, 3

(2.3)

where (rr rz[kK) = (2z)-3e”k”e’e’g’R, r = rl-r2

,

R = j(r, +rJ.

(2.4)

s6Ni, 58Ni, 54Fe, s6Fe

579

The function e(k, K) is an angle average of the true values of Q:

(2.5) and therefore does not depend on the angle of the relative and c.m. momenta. Finally, the energy denominator e(k, K) is e(k, K) = (l/m*)(k2f~K2)+2A-IV,

(2.6)

where the effective mass m* and the constant A are used in order to include the onebody potential in the intermediate states. Because it is not possible to introduce explicitly the c.m. coordinate R into eq. (2.3), we use the mixed representation (k, RI and get (s).,,.

= IdkdRdR’lkR)

[ I(%

E

eiK*(R-R’)] (kR’I,

(2.7)

where(Q/e),.,. stands for the right-hand side of eq. (2.3). A simplified and more manageable expression for (Q/e),.,. is obtained if the variable K appearing in Q(k, K) and e(k, K) is replaced by some effective value K& Such an approximation is justified if Q(k, K)/e(k, K) is a slowly varying function of k Then:

(24 At this stage it is possible to take into account the density p(R) of the nucleus by means of a varying Fermi momentum k,(R). Of course, the constants m*, A and lu,,, should also depend on R. More details of the local-density approximation and numerical results are given by Wong *I). Instead of following the procedure outlined above, we keep the exact form (2.3) of the propagator Q/e in nuclear matter. Reaction matrix elements in momentum space, G(k’, k, m) are then easily calculated for a separable potential. If

(2.9) then G(k’, k, K) = x w,(k’)8&

K),

(2.10)

where the functions CYj(k, K) are obtained by solving a system of linear equations. The calculation of shell-model reaction matrix elements proceeds now in the usual manner. We work inj_j coupling, and lZ6 JM TMT) represents a normalized and antisymmetrized state where 5 and 6 couple to the total angular momentum (JM) and total isospin (TM=). The single-particle states are those of the harmonic oscillator so that a Moshinsky ‘*) transformation into relative and c.m. states can be applied.

G. OBERLECHNER

580

AND

J. RICHERT

We have: lZ&JMTM,)

\

= I j,

X (d,

_ib

Jj

NL, @Ia&, nblb, l)(-l)J+““+j[l-(-l)‘+s+=]U(LIJS;

x InlSj, NL; JMTM,).

+) (2.11)

The X-coefficients transform from j-j to L-S coupling. The bracket (nl, NL, II In,,Za,nblb, A) is the Brody-Moshinsky transformation bracket “) and the U-coefficient U(LlJS; Aj) is related to the Racah coefficient by U(LlJS;

Aj) = d(U+

1)(2j+ l)W(LlJS;

Aj).

(2.12)

The kets Id) and INL) represent the states of relative and c.m. motion. In the {k, K} representation and following the Brody-Moshinsky conventions, (klnlm)


= b”( - l)“R,r(b,/~k)Y;“(~), = b3( - 1)NR,(bK/J2)Y,M(&

(2.13)

where b = (h/Mm)* is the oscillator constant and (2.14) with Erdelyi’s definition of the Laguerre polynomials Lr* [ref. ‘“)I. Let us now specify the values we take for the Fermi momentum k, and the starting energy W. First of all, results obtained with various values of kF and W have shown that the reduced matrix elements
(2.15)

Another possibility would be to take the value of k, corresponding to the statedependent average density (P)~~, but Wong 2’) has shown that this would exaggerate the importance of the Q-operator, especially in the nuclear surface region where the density is low. We noted also that the G-matrix elements do not critically depend on W. Therefore, we refer to the experimental single-particle energies for the determination of W. For two interacting valence nucleons in the (2p, lf) shell, W z - 10 MeV. It comes out of the work of Bethe 24) that the single-particle potential in intermediate states should be close to zero. For this reason, we choose M* = 1 and A = 0 in eq. (2.6).

56Ni, 58Ni, 54Fe, 56Fe 2.2. CORE RENORMALIZATION

581

OF THE EFFECTIVE INTERACTION

In our study of 56Ni, ‘*Ni, 54Fe and 56Fe we consider all nucleons in the (If, 2p) shell as valence nucleons. The inert core is then 40Ca, although we only take fourparticle excitations from the If4 shell into account. It appears in the work of Kuo and Brown lo )t hat the most important second-order correction to the effective interaction b

d

a

b

b

Fig. 1. Core polarization diagrams.

comes from core polarization processes. We note this correction by AG. The antisymmetrized matrix element (ablAG]cd-dc) is then the sum of the contributions of the four diagrams shown in fig. 1. Explicitly <44cd--dc)

= ,ch

,[

<4Glc~), t (~blGlhd>,+(hblGl~d>.;
terms with the exchange c t, d)

where
=
and, applying the diagram rules of the linked valence expansion 13): e1 = &h--Ep+-E*-Eb, e2 = &h-&p+&,-&,.

1 ,

(2.16)

G. OBERLECHNER

582

AND

J. RICHERT

These expressions of the energy denominators Iead generally to a non-hermitian operator AG. To overcome this difficulty we replace el and e2 by e = ~~-8~ which is the right expression in the case of exact degeneracy. The following single-particle and TABLE 1 Numerical a

b

values c

(in MeV) d

of some matrix elements (abJT~G~cd/T> J

T

G

2 3 4 5 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 3 4 3 4 1 2 3 4 1 2 3 4 1 2 1 2 2 3 2 3

1 1 1 I 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0

-0.806 -0.211 -0.307 -0.143 -0.666 - 1.098 -0.238 -2.355 -0.385 -0.192 -0.397 -0.322 -0,298 - 1.454 -3.821 -2.250 - 1.265 - 1.494 -0.629 -2.651 -0.221 -0.547 -1.412 -0.962 -0.232 -0.098 -0.253 -0.552 - 1.946 -1.147 -0.581 -1.119 --0.159 -0.950 -2.260 -2.535 -0.341 -0.125 -0.515 - 1.660

-

GCB

-0.631 -0.213 -0.216 -0.119 -0.266 -0.681 -0.105 -2.075 -0.258 -0.073 -0.218 -0.235 -0.236 -1.147 -3.866 -2.837 - 1.087 -1.919 -0.393 -2.432 -0.152 -0.408 -1,446 -0.745 -0.234 +0.004 -0.205 -0,384 -2.222 -1.239 -0.597 -0.881 -0.137 -0.817 -2.413 -2.272 -0.265 -0.123 -0.163 -1.391

Values given in columns GKB and dGK, are those of the corresponding Kuo and Brown I*). 2:2p*; 3: lf*; l:lfg; 4:2&.

and

(abJTIdGlcdJT)

AG -0.230 +0.1s5 +0.196 +0.277 -0.140 to.159 -0.025 -0.128 +0.017 -0.039 +0.437 1-0.283 10.661 +0.017 +0.01 I +0.162 +0.022 1-0.033 +0.314 to.019 +0.229 10.142 -0.029 +0.003 +0.166 +0.256 +0.393 +0.067 +o.oso +0.102 +0.137 +0.053 10.315 1-0.102 -0.052 10.213 to.1 16 +0.290 -0.021 +0.136

dGKB -0.230 +0.186 -i-O,169 j-O.264 -0.027 -to*077 -0.059 -0.090 -0.029 -0.023 +0.240 +0.266 +0.392 +0.253 +0.245 +0.106 $0.102 1-0.03 3 +0.28 1 +0.215 -1-0.181 j-o.134 -0.038 --0.001 1-0.202 +0.211 4-0.325 50.141 +o.oXS +0.003 i-O.138 -t-o.159 -i-O.289 i-O.129 +0.122 to.375 10.130 $0.328 +o.os2 +0.157

elements calculated by

583

single-hole orbitals have been taken into account: P:

If;,

ZP+, lf+,

2p,>

lg+,

h:

lp+, Ip+, Id,, 2s,, Id,.

2d,,

kg,

3~9 2d,;

Like Kuo and Brown lo), we have used an average energy denominator of 2hw (sp-% = 21 MeV) sincep and h must have the same parity. The energies of the singleparticle and hole states p and h enter also into the calculation of the G-matrix elements which are necessary to evaluate the right-hand side of (2.16). Thus, the starting energies are W = E,+E* for (ah]G]cp> and W = +,-I-Q for
CHARGES

Like the two-body interaction, the electromagnetic transition operators must also be renormalized in order to take account of the presence of the core nucleons. In the following we calculate B(E2) values. The E2 transition operator is (2.17) where r3 = + 1 for a proton and zf = - 1 for a neutron. This transition operator has to be replaced by an effective operator Sz,,, if the transition rates are calculated with the eigenfunctions of the effective Ha~ltonian instead of the true energy eigenfunctions. We have considered only the one-body part of C&r which can be written (2.18) Since we do not yet know how to rearrange the linked cluster expansion of F in order to obtain a good convergence 28), we have limited our calculation of 9” up to the first-order contribution. The matrix elements (alFlb>, where a and b are indices of valence states, are then given by the sum of the contributions of the diagrams shown in fig. 2. We show our results in table 2.

584

G. OBERLECHNER

AND J. RKHERT

The values that are given there are those of the effective charges defined by (2.19)

and eP = e(rs = -t l), e, = e(tg = -. 1). These values are typical of effective charges calculated up to the first order with realistic forces 28t2“). In this connection, we would like to mention that this type of cakulation has also been done with phenomenological forces by Arima and Horie 30).

h 0

h /\

b Fig. 2. First-order diagrams contributing

to the calculation of effective charges.

TABLE2 Effective neutron and proton charges for E2 transitions in the fp shell d

b

en

en

‘4 ‘4

‘4 2p*

0.54 0.47 0.74 0.39 0.40 0.42 0.53 0.40

1.18 1.19 1.37 1.14 1.13 1.16 1.19 1.12

’ fi

’4

2p* 2p* 2pP

2p+ ‘4 2pi If3 2p*

’ f* ’ f%

3. ConQurations and states Basic wave functions are written in the following way:

W, , Jr4 9JM)= c&&>~“&I1”“~ where JP is the total angular momentum of the proton group, J, is that of the neutron group and JP coupled to .T,,gives J. The wave functions of the proton group and those of the neutron group are completely antisymmetrized.

585 3.1. CONFIGURATIONS

The following configurations are considered in the description of 56Ni:

where the letters p and n stand respectively for protons and neutrons, the subscript 1 stands for the If% shell and the subscripts 2 and 3 stand for any one of the 2p+, If+ and 2p+ shells. The co~gurations with 1 proton hole and 1 neutron hole in the 14 shell and those with 4 proton holes or 4 neutron holes are not considered because they give much less binding energy than that of configurations of the last three types mentioned above (2 proton holes, 2 neutron holes, 2 proton and 2 neutron holes). We have also neglected 3h-3p configurations. It has been shown ‘, ‘l) that the excitations of an odd number of particles provide small contributions in these nuclei. Finally for 2 proton-hole and 2 neutron-hole configurations, we consider only those with 2 protons in the same level and 2 neutrons in the same level. In accordance with the preceding discussion, the following configurations are taken into account in the case of ‘aNi: j,“, .C, A, A,,

$, _ii2 j,“, X2 ,

.$ j,:! .& .in”,.ii,

$, Iit 2, iit AL.

For s4Fe, they are jf, .C,,

j,4 jp2 jp3 AZ,,

j,“& 2, A, A, 3

and for 56Fe: jfl jL A, A,,

j,4 jp”* 2, jiL9

jfi 2, At2 ii3 -

In these two last cases also, the 4h configurations are those with an even number of protons and an even number of neutrons in the same level. 3.2. STATES

The states corresponding to the configurations given above are characterized by the resulting angular momentum of each group of particles. Thus, for the last three types of configurations considered in the study of 56Ni, the states are described as follows:

C~(j%,

C(&)Jt, y (j,, Qf&

(3.1)

C(X,)Jtn3 CL A3J2nlJ~

(3.2)

l W&,;

N.tX npt&P&&

(3.3)

The intermediate couplings are indicated by braces and brackets. When necessary, we use seniority z?as an additional quantum number. Thus, for 58Ni, the 2h4p and 4h-6p states are described as: (3.4)

CKjL)J1,, {~~*}~~~~~~;(i3fJJ,

if jpzf jPf;

U~j;tl)u~pJlp9

C(j&)J2p9

(j&)J3pIlJ23plJp;

(j3JnlJ,

(3Sb)

(3.7) (3.8) (3.9) and for 56Fe: (3.10) (3.11) (3.12a) (3.12b) The calculation, with the basic states defined above, of the matrix elements of a one-body or two-body operator is done without difficulty with the help of fractional parentage coefficients 32). 4. Results and discussion The single-particle energies we choose are those obtained from the 41Ca spectrum [ref. I ‘)I: If; : 0 MeV,

2p, : 2.07 MeV,

lfr : 5.50 MeV,

2p, : 4.13 MeV.

(4.0

This choice is justified by the fact that the model space extends to the fr- shell. But the four nuclei we study here have 14, 16 or 18 particles above the closed 4*Ca core, so that the mean field in which the particles move is strongly modified by their presence. There would be no problem if one could get the single-particle energies with regard to a 56Ni core by adding the Hartree-Fock contribution of the lf% shell to the singleparticle energies extracted from the spectrum of 41Ca. This does not exactly happen with the G+ AG force; we obtain in this case for 57Ni If* : 0 MeV,

2p, : 3.38 MeV,

lfS : 3.66 Me’?,

2p, : 4.88 MeV.

(4.2)

If+ : 4.78 MeV,

2p+ : 5.08 MeV.

(4.3)

The experimental values are 14 : 0 MeV,

2pt : 4.00 MeV,

56Ni, 58Ni, 54Fe, 56Fe

587

In our calculations we use the set (4.2). This is, in our sense, the most consistent choice. In the study of ‘*Ni with Oh-2p states, we take the values given by the set (4.3); 56Ni is then considered as a closed core. The question of single-particle energies is the central problem of shell-model calculations with core-excited states. Several authors *I 33) have been obliged to reduce the single-particle splittings in order to get agreement between the experimental and theoretical level schemes in the study of doubly even (If, 2p) shell nuclei. This procedure is then interpreted as a simulation of the absence of states which were not taken into account in the diagonalization. The difficulty in getting correct spectra is also present in our work. However we do not change the single-particle splittings because we feel that it is not an adequate method. On the contrary, if we keep the consistent set (4.2) we are more able to judge the importance of rejected states in the description of the four nuclei. The effect of single-particle splitting reduction on spectra and wave functions will only be discussed for comparison. 4.3. SPECTRA

OF 56Ni

This nucleus has been experimentally investigated by different authors 34-37). Theoretical investigations have been presented by Wong and Davis “) and Jaffrin ‘) who found good agreement with experiment. Bruge et al. report in their paper a calculation of McGrory who includes up to 2h-2p excitations in his diagonalization. In our study the whole set of (3.1) and (3.2) states, i.e. 2h-2p states, is taken into account. This is not the case for (3.3) states which are very numerous. The states we use are those of lowest seniority contribution the most to potential-energy matrix elements. These (3.3) states are characterized by: J, = 0,

J,, = 0,

J, = J, J,,

= J,,

or or

J, = J, J,,

= J,,

J, = 0,

Jzp = 0,

and similar couplings for the neutron group. A few (3.3) states with JP = J,, and total seniority u = 4 are also included, but they do not modify appreciably the results. This is certainly due to the fact that if one introduces these states, it must be in a coherent scheme like the aligned scheme ‘“). The first diagonalization shown in spectrum (1) of fig. 3 includes only the 2h-2p states and one Oh-Op state for O+; their total number is 21 for O+, 60 for 2+ and 68 for 4+. The ground state is mostly a Oh-Op state (74 %) and the first excited 0: at 6.12 MeV is purely 2p-2h. The big gap between the ground state and the first excited state has also been observed by McGrory and Wong and Davies. The addition of (3.3) states (28 Of, 32 2+ and 32 4+ states) lowers 0: to 4.5 MeV but it is still impossible to bring the first 2+ and 4+ states to a correct position (spectrum (2) fig. 3). The percentages of the different configurations in the ground and low excited states is shown in table 3; 0: remains mostly a Oh-Opstate, 2: a 2h-2p state, and 4: contains 54 % 4h-4p admixtures. The situation is drastically changed if the single-particle

G. OBERLECHNER

588

AND J. RICHERT

energy splittings used by Wong and Davies are used (spectrum (3), fig. 3). The spectrum as well as the wave function components of 0:) 2: and 4: are now predominantly 4h-4p states (table 3). This is well understood because 4h-4p potential contributions overwhelm all other contributions due to the narrow single-particle splittings. It is possible to remedy the situation found in spectrum (2) and get a spectrum which is in relatively good agreement with experiment. This is possible with a reduco+

a34 -----__, al2 ---

2+

x9----

-

4+

693-2+

6.66 6.641-

534-o+ 612-

664. &oB---0+ 600 y-to+

o+

6!59-

636-4+ 590-_(4+)

0+

533-(2+)

/:: /2+ YO-

4+

533-o+

532-P

495-o+

4.77i-/0+ 4.52 r4?==-'

449-o+

I+ II*

403-(2+) NO-4+

2.69-

2+

2.50-

4+

m.H-w+

H

196

~

-

o+ Exp.

-o+

o+

Fig. 3. Spectra of 56Ni; explanations

Percentages

of configuration

2 61 6

o+

(3)

are given in the text.

TABLE 3 admixtures in low-lying states of 56Ni

Oh-Op

01 + 02 + 21 + 41 +

0

-

(2)

(1)

2h-2p 3 5 43

Case 2 refers to eigenstates corresponding to spectrum (3).

2+

2 28 37 76 46

4h-4p 3 15 28 8 6

2 5 48 24 54

3 80 29 92 94

to spectrum (2). Case 3 refers to eigenstates corresponding

56Ni, 58Ni, 54Fe, s6Fe

589

tion of the f$-f+ splitting by an amount of 2.5 MeV. One gets then the following sequence of states: 2: at 2.37 MeV, 2+ at 3.95 MeV, 4: at 4.15 MeV and 0: at 4.26 MeV. The ground state contains then 20 % Oh-Op,40 % 2h-2p and 40 % 4h-4p components; the 2: and 4: states contain 40 % 2h-2p and 60 % 4h-4p components. The wave functions contain in this case similar admixtures as in the work of Wong and Davies. Finally it is interesting to notice that the introduction of lh-lp states does not change the main features of the spectra in this last case. The corresponding components in the wave functions are small. This is in contradiction to the result of Jaffrin “) who describes 2: and 4: as lh-lp states. 4 .2 . SPECTRA OF 58Ni

Different types of reactions have been used for the investigation of this nucleus [refs. 3g-41)], Recently heavy-ion reactions have been performed by Faivre et nl. 42) in order to study nickel isotopes by a-transfer on iron targets. A large number of states has been excited from 4 to 10 MeV in the spectrum. The presence of these excitations has been interpreted by Jaffrin 43) as being due to 2h-4p states. Multiparticle excitations should also exist in the low-lying states of “PIi. This assertion is related to the existence of such excitations in 56Ni and, on the other hand, to some specific properties of this nucleus such as E2 transition rates. Spectrum (1) of fig. 4 shows the results of a diagonalization with the (4.3) set of single-particle energies in the Oh-2p space. The agreement with experiment is satisfactory for the states between 0 and 3.5 MeV. It is of course impossible to explain the

d26-

r

392-P

tm-

3-r

3.77-

2* G+

‘II

cw-

0’

291-

-

1+

o*

3-r OI

-a+

-0,

Exp.

(2)

a+

?.,11-

1*

2-w

2.L

-0’

32?-

(31

Fig. 4. Spectra of 5*Ni.

2?9-

9*

-a+

Mf

r

r.n-

-a+

(5)

590

G. OBERLECHNER

AND

J. RICHERT

large B(E2; 2: + 0:) value without introducing large effective charges. Using the values given in table 2, B(E2; 2: -+ 0:) = 7.3e2 9fm4 which is far from 136 e* - fm4 which is the experimental value. The addition of 2h-4p states gives us spectrum

(2) of fig. 4. The (3.4) states which

are used are those with lowest seniority and J, = 0, J,, = J or J,, = J, J,, = 0. All intermediate momenta for the separate neutron group and proton group are - with perhaps one exception - equal to zero. We are left with 15 2h-4p states in the diagonalization of the Of, 43 in the case of the 2+ and 41 in the case of the 4+ state. Spectrum (2) shows that the agreement between experiment and theory becomes worse; 4: goes up to 4.2 MeV. The 2h-4p admixtures in the low-lying states are shown in table 4. TABLE 4

Percentages ~

2 70 79 72 80 82

The numbers

admixtures

in low-lying

Oh-2p

-.

01+ 02 + 21 + 22 + 41+

of configuration

3 54 14 38 20 0

in the columns

states

of 58Ni

2h-4p 4 48 34 44 11 15

5 10 17 7 6 0

2 30 21 28 20 18

refer to the eigenstates

3 46 86 62 80 100

4h-6p 4 42 46 44 61 57

5 47 9 55 39 38

in the corresponding

spectra.

4 10 20 12 28 28

5 43 74 38 55 62

The states have predominant Oh-2p components but the 2h-4p ones are still important. Spectrum (4) shows the effect of the presence of 4h-6p states which are taken with the same criteria as the 2h-4p ones. The total number of states is respectively 33, 52 and 48 for O+, 2+ and 4+. The spectrum is similar to spectrum (2). Table 4 shows that there are between 10 and 30 ‘A 4h-6p admixtures in the wave functions. Similarly to the study of ’ 6Ni, spectra (3) and (5) show the results of a diagonalization with the single-particle energies used by Wong and Davies. The agreement with experiment is better, although the first excited 4+ level is not at the right position in spectrum

(3). The Oh-2p components

of the wave functions

are considerably

reduced

(table 4); the reason for this is similar to the one given in the discussion of 56Ni. A certain number of E2 transitions between low-lying levels corresponding to spectra (2) and (4) have been calculated using effective charges of table 2. There is some improvement in the results when one compares the results to those obtained in the pure Oh-2p case. However the B(E2; 2: -+ 0:) does not exceed 30 e* 1fm” in case (4). 4.3.

RESULTS

FOR

54Fe

The spectroscopic properties of this nucleus have been extensively studied by several authors 44-46) and the level spectrum at low energy is now well established. Recently Pittel “) performed a shell-model calculation for this nucleus and succeeded rather well in describing the level scheme.

56Ni, ssNi, 54Fe , 56Fe

591

If we include first the 2h-Op configurations corresponding shell, the levels are given by the two-body matrix elements (f&W = llG+AGlf&JT The energies are core polarization are better if one two-body matrix

= 1)

for

to two holes in the &

J = 0,2,4

and 6.

then respectively 0, 1.10, 1.83 and 2.14 MeV. This shows that the contribution AG is not adapted to this unique configuration. Results calculates AG with respect to a 48Ca core; with the corresponding elements of Kuo and Brown lo) the excitation energies are respec6384,

<;I

k%6:62=

3,836

4+

3,297 3163 2,961\ 2,948 2,564.

4*

- 4+ - 4+

5,85

o+

515

0+

_?+2+ --0* / o+ - 4+ ’ 2*

1,407

2,20

4+

1,59

2’

3,62

2+

2s86

O+

2.29. 2,25-

- 4+ --r

2+ 135

0+

EXP

2*

0+

0+

(Ii Fig. 5. Spectra of 54Fe.

(2)

G. OBERLECHNER

592

AND

J. RICHERT

tively 1.31,2.10 and 2.35 MeV for J = 0,2,4,6. The E2 transition rate B(E2; 2: + 0:) from the first 2+ level to the ground state is not in agreement with experiment. Using the effective proton charge calculated up to first order one gets 35.5 ez - fm4. The experimental value, 106 e2 * fm4, shows that the effective proton charge in the f; shell should be equal to 2.2. The set of 4h-2p states which is added to the 2h-Op states is the following: (3.8) states with alp = 0, 2 and (3.9) states with Jp = J, J, = 0 or J, = 0, J, = J. The results are shown in spectrum (1) of fig. 5. Only the first O+, 2+ and 4+ levels are well reproduced. The percentages of 4h-2p admixtures are 32 % for O:, 33 % for 2: and 28 % for 4:. The B(E2; 2: + 0:) transition rate is not much improved. Its value is 38.5 e2 * fm4. A special point is the dramatically high position of the first excited O+ level situated at 5.15 MeV. This value is very similar to the schematic estimation made by Zamick 47), who gives a prescription for reducing this gap to the experimental value by reducing the attractiveness of the f+ pa T = 1 matrix elements. Following the choice made by Pittel *), we lower the p+ f%and p+ single-particle energies by 2 MeV. In this way we get spectrum (2) of fig. 5. The triplet of states at 2.53 MeV is now better reproduced; 0: comes down to 2.86 MeV. The wave functions contain important 4h-2p components, which are respectively 51 %, 77 %, 52 % and 97 % for O:, 2:, 4: and 0:. These results contradict the arguments presented by Zamick who increases the ft -pa splitting in order to get the correct position of the 0: level. The addition of 4h-2p states does not improve the transition rates; B(E2; 2: + 0:) remains practically unchanged. This indicates that the 4h-2p components produce no collective effect in the wave functions. 4.4.

SPECTRA

AND

B(E2)

OF 56Fe

A considerable amount of work has been done on 56Fe. In refs. 48,49) are presented some of the most recent results, including level spectra as well as B(E2) transition rates. From a theoretical point of view, attempts were made years ago to describe this nucleus in the framework of the vibrational model 50), but agreement with experiment could hardly be achieved. As far as we know, shell-model calculations on this nucleus have been restricted up to now to 2h-2p configurations 51’52). McGrory got reasonable agreement with experiment. Spectrum (1) in fig. 6 shows the results obtained with 2h-2p configurations only, i.e. the whole set of (3.10) states. The excitation energies are generally too small. With single-particle energies taken with reference to a 56Ni core (set 4.3), levels above the first excited 2: states go up, 4: and 2: are at the right position but higher states are shifted upwards and the first excited 0: state lies too high by an amount of 1 MeV (spectrum (2), fig. 6). In this case the B(E2) values are B(E2; 2: -+ 0:) = 54.5 e2 * fm4 and B(E2; 2: + 0:) = 0.5 e2 * fm4; the experimental values are respectively 144 and 24 e2 * fm4.

56Ni, 5SNi, s4Fe, s6Fe

593

The addition of 4h-4p states of lowest seniority leads to spectrum (3) of fig. 6. The selected (3.11) and (3.12) states are chosen with the same criteria as in the case of 56Ni and ‘*Ni. The total number of states is 28 for O+, 77 for 2+ and 84 for 4+. The agreement with experiment gets poorer, 2: goes up to 1.34 MeV and the level ordering o+

SW460-

2+

464-F

1$7-2+ 4.41-4+

4.lo-

4+

382 -----_=;: 3.77360-O+

160-o+ 3.44-

2+

350-

2+

3.16-

II+

3,37-2+ 312296 --s--2+ --‘0+ 294' 2.66-

2+

2.672.7678

4+ 2+

255-

4+

3.4?-

1c

324-

4+

o.os-

2+

-o+ Exp.

2.lo-

2+

1.75-o+ 1.61-

4+

0.77-

2+

-

3122-

2+

2.@4-4+ 2&i-2+ 2.73-1, 2.43-

232-2' 2m-4+

3.64P4+

2.04-

4’

0.73-

2+

o+

-o+

-0+

(21

(11

0+

(3)

Fig. 6. Spectra of 5sFe.

is disturbed. The 4h-4p contributions lie between 15 and 25 % for O:, 2: and 4:. The transition amplitudes are essentially unchanged; B(E2; 2: --f 0:) = 55 e2 - fm4. A better spectral description of the first excited states is possible if the f+-f, single-particle splitting is reduced by 2.5 MeV as in “Ni. Using this procedure enhances the percentages of 4h-4p admixtures up to 35 x-40 % in O:, 2: and 4: but does not much improve the E2 transition rates. 5. collclusion We have tried in this paper to describe the low-lying states of 56Ni, “*Ni, 54Fe and s6Fe with a shell-model space including two-particle excitations from the If% shell,

G. OBERLECHNER

594

AND

J. RICHERT

and also four-particle excitations in the case of ’ 6Ni and 5*Ni. The effective two-body matrix elements are similar to those of Kuo and Brown ’ “) calculated with respect to a 40Ca core. The resulting spectra are not in very good agreement with experiment. Our choice of single-particle energies, although it is not unique, is consistent with the effective two-body interaction we use. The splittings of these energies with respect to a 56Ni core are not exactly the experimental ones but are still not too different from them. It is always possible to improve the spectroscopic results by changing these splittings or some two-body matrix elements. In our sense, however, this is not a good procedure because it conceals the role played by configurations and states which are rejected in the description. E2 transition rates are improved by the presence of core-excited states although they are not well described. Their values stay generally in the ratio 1 : 3 with the experimental ones. Here also there is certainly a link between this failure and the truncation. The situation is particularly dramatic for 58Ni where it is impossible to get either a correct spectrum or the correct transition amplitude. Last of all, as an illustration of our assertion concerning rejected configurations and states, we come back to the 4h-4p states in 56Ni. Only states with an even number of particles in a given shell entered the diagonalization. On the other hand, we performed calculations which indicate that there is a great difference between spectra of 60Zn (2 protons and 2 neutrons in the p+ f+ and p+ shells) calculated with configurations restricted to an even number of particles on a given shell, and spectra obtained from the complete calculation. This fact is an indication that the restriction we put on the choice of configurations should be removed. The authors

wish to thank

Dr. D. Banerjee

for interesting

discussions.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

R. Arvieu, E. Salusti and M. Vbneroni, Phys. Lett. 8 (1964) 334 M. K. Pal, Y. K. Gambhir and Ram Raj, Phys. Rev. 155 (1967) 1144 Ram Raj and Y. K. Gambhir, Phys. Rev. 163 (1967) 1004 N. Auerbach, Nucl. Phys. 76 (1966) 321; Phys. Lett. 24B (1967) 260; Phys. Rev. 163 (1967) 1203 S. Cohen, R. D. Lawson, M. H. Macfarlane, S. P. Pandya and M Soga, Phys. Rev. 160 (1967) 903 S. S. M. Wong and W. G. Davies, Phys. Lett. 28B (1968) 77 P. Goode and L. Zamick, Phys. Rev. Lett. 22 (1969) 958 S. Pittel, Phys. Lett. 33B (1970) 158 A. Jaffrin, Phys. Lett. 32B (1970) 158 T. T. S. Kuo and G. E. Brown, Nucl. Phys. All4 (1968) 241 R. A. Ricci, Proc. Int. School of Physics Enrico Fermi, Varenna, 1967, ed. M. Jean (Academic Press, New York, 1969) p. 80 Th. Hammann, G. Oberlechner, G. Trapp and J. Yoccoz, J. de Phys. 28 (1967) 756 B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771; G. Oberlechner, F. Owono-N’Guema and J. Richert, Nuovo Cim. 68B (1970) 23 M. W. Kirson, Nucl. Phys. All5 (1968) 49 B. H. Brandow, Ann. of Phys. 57 (1970) 214

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