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Effective three-body interactions in the f72 shell
II.B: I.D.I I
Nuclear Physics A270 (1976) 388 --398; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
E F F E C T I V E T H R E E - B O D Y I N T E R A C T I O N S IN T H E f~ S H E L L F. A N D R E O Z Z I and G. SARTORIS lstituto di Fisica Teorica dell' Universitd and lstttuto Nazionale di Fisica Nucleare, Mostra d' Oltremare. pad. 19, Napoli, Italy Received 13 February 1976 (Revised 24 May 1976) Abstract: The contributions o f effective three-body forces to the interaction energy of three particles
in the 0f.~ shell are calculated using reaction matrix elements derived from the Hamada and Johnston potential. The results obtained by second-order perturbation theory are found to be much smaller than those obtained from phenomenological fits. An effective interaction is then derived for the 0f~_shell performing a shell-model calculation for three particles in the whole 0f-lp major shell and through the use of biorthogonal states; effective three-body contributions turn out to be large, as expected from the results of phenomenological fits. 1. Introduction
Shell-model calculations are usually performed under the assumption that the influence of configurations not included in the model space can be accounted for by using an effective interaction. A method for calculating effective interactions, which consists in the derivation of a reaction matrix G f r o m a free nucleon-nucleon potential and in a subsequent renormalization of G in second-order perturbation theory, was proposed several years ago by K u o and Brown '). When more than two valence particles are considered, their approach gives rise to effective three-body forces even though the original interaction was of a two-body character. This feature, however, is not peculiar to the method. Effective many-body forces should be considered, at least in principle, even when the effective interaction is derived from the experimental data directly 2, 3). Many-body effects are usually assumed to be small and therefore ignored, but the validity of this assumption depends strongly on the choice of the model-space configurations. The very popular f~_ model, in which the configuration space is truncated within the 0f-lp major shell - the energy difference between the 0f~_ and lp~ singleparticle levels is estimated to be only a couple of MeV - appears to be a good candidate for many-body effects to play a relevant role. Three-body contributions to the parameters of the Talmi formula 4), which gives the ground-state energy in t h e j " configuration, were evaluated in second-order perturbation theory by Osnes 5) for the calcium isotopes using the Kuo-Brown bare interaction 1, 6), and were found to be small. The same interaction was used by Bertsch 7) in a note on the relationship between the ground-state energy of closed-shell nuclei and the two-body interaction energies deduced from the spectrum of a nucleus with two valence particles in the shell. Bertsch pointed out that some two-body correlations 388
f~ SHELL
389
which are present in the nucleus with two valence particles cannot be effective in the closed-shell nucleus. By subtracting out the contributions of such correlations to the two-particle energies, he obtained sizable corrections to the binding energies of 4SCa and 56Ni' which he interpreted as an indirect estimate of appreciable three-body effects in the 0ft shell. A phenomenological shell-model fit to energy levels of f~ nuclei was performed by Quesne 2) taking explicitely into account the contribution of effective many-body forces. Quesne extracts the two-body part of the effective interaction from the binding energies and the spectra of 4o-42Ca and uses these phenomenological values to calculate the energy levels of 43Ca with standard shell-model techniques. The differences between the calculated and observed energy levels of 4aCa are then identified with the matrix elements of an effective three-body interaction. The same procedure is applied to the other calcium isotopes to extract four-body forces and so on up to eight-body forces. A similar calculation is performed for the N = 28 isotones. The many-body forces so obtained turn out to be very large. A different philosophy was adopted by Eisenstein and Kirson a) who applied a best-fit procedure to a great number of energy levels of f~ nuclei to determine the matrix elements of an effective interaction containing just two- and three-body terms. Three-body effects are found to be quite large, especially when the three particles interact in a T = ½ state. More recently, three-body contributions to the effective interaction in the 0f~ subshell were calculated by Yariv a) in second-order perturbation theory using reaction matrix elements derived from Reid potentials 9). The results are much smaller than those obtained from phenomenological fits. In Yariv's work an attempt is made to reduce this discrepancy by calculating the influence of the excluded lp~ level on the effective T = ~ interaction in the (0f~) a model space through a non-perturbative procedure. No substantial improvement is obtained. To summarize, the results ofrefs. 2, a, 7), although not directly comparable, indicate that many-body correlations play a relevant role in a pure f~ model. The results of refs. 5, s) indicate that calculations based on second-order perturbation theory are unable to reproduce such large correlations. On the other hand, Yariv's attempt s) to calculate three-particle correlations to all orders in G is limited to the T = ½ case and neglects the contribution of the 0f~_and lp~. levels, which, as suggested by the second-order results, may be quite significant. In the present paper, we calculate the contribution of effective three-body forces to the interaction energy of three particles in the 0f~ subshell using reaction matrix elements derived by Kuo and Brown i, 6) from the Hamada and Johnston potential 1o). We first recalculate these contributions in second-order perturbation theory to test the dependence of Yariv's results s) on the choice of the reaction matrix elements. We then perform a non-perturbative calculation for the T = ½, ~z matrix elements of the effective interaction taking into account the influence of the whole f-p major shell.
390
F. ANDREOZZI AND G. SARTORIS 2. Two-body interaction
The G-matrix elements used in this work are those derived from the HamadaJohnston potential t o) by Kuo and Brown t, 6) with the inert *°Ca core as reference state and the value ho~ = 10.5 MeV for the oscillator parameter. An effective twobody interaction for the 0f-lp shell was calculated, following the Kuo-Brown procedure 1, 6), by adding the diagrams shown in fig: 1. [Arguments for neglecting secondorder contributions corresponding to the excitation of two particles from the core have been given in ref. 6).] It is worth mentioning that our results for the G3p-~ diagram differ slightly from those obtained by Kuo and Brown 6) using the same bare matrix elements and the same average energy denominators 2ho9. This is because they used a redundant set of three-particle, one-hole intermediate states, while in our calculation this approximation was removed by using uncoupled (m-scheme) intermediate states.
G
G3p.,h
Fig. 1. The G-matrix and core-polarization diagrams.
G3p
G4p.lh
Fig. 2. Second-orderdiagrams representing three-body correlations. 3. Second-order calculation of three-body correlations
The three-body correlations to second order in G were obtained by adding the diagrams shown in fig. 2. When the model space is restricted to the 0ft subshell, graph G4p-ta corresponds to the excitation of a 0p particle to the 0ft level and graph G3p corresponds to the excitation of a 0ft particle to a higher negative-parity level. The configuration (f~)3 has 22 independent states; 16 states with T = ½ and J = $2,½,{ [2], -7z[3], { [2], ~ - [2], &2 ~ [2], ~ , ~ , ~ (the numbers in square brackets give the multiplicity of the states), and six states with T = ½ and J = ½, {, {, 9, ~-,
f:z SHELL
391
]NTJ) -= ~, C~,rss,[[a+a + ]r'Z'a+ ]rJI0),
(1)
1 s We adopted the basis vectors T'J" +
where antjm~ denotes the creation operator of a particle in the (nljrnT) single-particle state (for simplicity, we shall omit the quantum numbers of the 0f~ states). The coefficients r,NrJ •-,r'z' were calculated by applying the Schmidt orthonormalization procedure to the states [[a+a + ]r'S'a+ ]rsl0),
(2)
ordered with increasing T'J' (J' changing faster than T'). The quantum number N is obviously unnecessary to classify the states with multiplicity 1, which can be obtained by normalizing any of the states (2) having the required T, J values. For the T --- ½, J states having multiplicity greater than 1, N keeps track of the order in which the basis vectors are generated (N = 1 means the first T, J state generated by the Schmidt procedure and so on). In the adopted basis, graphs G3p and G4p- lh are given by
(N1TJIv3IN2TJ) = + ~, ~r,s,rs r,N2rs T'J' ~'T"J'" T'J" T"J"
x • ( - 1)i+i+r+JU(TrjJTz; J"J')U(~-IzT½; T"T') nij × Gr,s,(nly) ~
1
(3)
Gr"s"(nlj)'
where the U are normalized Racah coefficients and Grs(nlj) is the antisymmetrized but not normalized bare reaction matrix element
(O[([a+ a~j]ra)+G[a+ a + ]rSl0).
(4)
The minus sign and the plus sign correspond to graph G3p and G4p- lh, respectively; the quantum numbers (nlj) are those of the internal particle or hole line. Graph G4p-lh was calculated using an average energy denominator A E = -2hto. Graph G3p contains contributions corresponding to the excitation of a particle within the f-p shell; these were calculated with energy denominators given by minus the sum of the single-particle energies in the intermediate state, for which the values ef~ = 0,
ep~ = 2.00,
~p½ = 4.25,
~f~ = 6.25 MeV
(5)
were used. In the calculation of graph Gap, contributions from intermediate states with 2hto excitation energy should also be considered. We ignored them, since the G-matrix elements required for such calculation were not available. However, as we shall see below, this restriction is likely to be not important and cannot alter the conclusions of this section. The results obtained for the three-body correlations Gap and G4p-lh are given in
392
F. A N D R E O Z Z I A N D G. SARTORIS TABLE 1
Second-order contributions o f three-body correlations to the interaction energy of three particles in the 0f~ shell (in MeV) T J
Gap (a)
½ { * 7 -~
* * *
*
--0.252 --0.310 --0.236 --0.079 0.112 0.051 : --0.074 --0.177 --0.068 --0.046 0.023 0.113 0.095 0.085
G4p-lh
T
J
(b) --0.193 --0.153 --0.249 --0.086 0.120 0,044
--0.064 :--0.174 --0.098 0.020 --0~015 0.147 0.141 --0.006
Gap (a)
0.027 0.028 0.004 --0.001 --0.003 --0.002 0.003 0.090 --0.002 0.016 0.003 0.001 0.006 --0.017
½
¥ *
0.078 --0.158 --0.074
o.144 *
0.061 0.016
-0.109 -~ ajt
0.228 --0.031
j j
O.OlO 0.002
{-
0.039 0.033 0.027 0.007
G.p-th (b) 0,072 -0.110 -0.007 0.082 0,122 --0.004 --0.051 0.188 --0.036 0.035 0,017 0.035 0.022 0.026 0.000
-0,003 0.014 0.012 -0.023 0.009 --0.011 0.019 --0.021 0 , 0.003 0.001 0.000 --0.003 0.001 --0.002
The ordering o f the matrix elements is explained in the text; off-diagonal matrix elements are starred. Labels (a) and (b) refer to different choices o f the energy denominators for diagram G3p.
table 1. The values of G3D corresponding to the choice (5) for the energy denominators are listed under the label (a). We first observe that three-body corrections to matrix elements with T = ~}turn out to be negligibly small, in agreement with the results of refs. 5, 8), andin any case much smaller than the corrections to T = ½ matrix elements, in qualitative agreement with the results of phenomenological fits 3). The contributions from G4p-th are much smaller than those from G3p and are practically irrelevant. This result is explained mainly by the larger energy denominators of graph G4p_ th and suggests that we did not lose much by neglecting intermediate states with 2hco excitation energy in the calculation of graph Gap. To compare our results with those obtained by Yariv a), graph G3p was recalculated using an average energy denominator - d E = ½hco = 5.25 MeV; the values so obtained are listed in table 1 under the label (b). In spite of minor differences, the two sets of values (a) and (b) can both be considered in overall agreement with Yariv's results a) and have the common feature of being much smaller than the values obtained from phenomenologieal fits 3), which for T = ) and low J are as large as 2 MeV. This feature does not seem to depend critically on the choice of the bare reaction matrix elements nor on the choice of reasonable singleparticle energies. The discrepancy between second-order and phenomenological results might be attributed to the fact that in the calculation of three-body correlations to second order in G the strong renormalization of the two-body interaction through core exci-
f~ SHELL
393
tations is completely ignored. It is therefore tempting to repeat the calculation using the renormalized interaction of sect. 2 instead of G as a vertex. This procedure is open to objection since it implies, for instance, violations of the Pauli principle of the type described in fig. 3. In fact, of the third-order contributions (a) and (b), which should cancel each other for j = ~ and m = m', only (a) is taken into account. However, there are many diagrams of type (a) which do not violate the Pauli principle, whereas the results of table 1 suggest that the exchange contributions (b) should not be too relevant. On these grounds, we recalculated graph G3p using the two-body renormalized interaction as a vertex; on the whole the contributions of Gap turn out to be at most doubled.
a
b
Fig. 3. Third-order contributions which cancel each other forj = ~[and m = m'. It is therefore possible to conclude that, whether we use G or the renormalized interaction G +G3p-lh, the perturbative approach discussed in this section is inadequate to reproduce the large three-body correlations obtained by phenomenological fits.
4. Non perturbative calculation of three-body correlations The procedure adopted can be summarized as follows. First, we performed standard shell-model calculations for two and three particles in the 0f-lp major shell. Then, the results of the (f.p)a calculation were considered as the starting point for the derivation of an effective interaction for the f.~ subshell through the use of biorthogonal states. Finally, the results of the (f_p)2 calculation were used to extract the two-body contributions from the obtained three-particle effective interaction. 4.1. DIAGONALIZATION IN THE f-p SHELL We solved the shell-model eigenvalue problem H [ ~ , ) = (Ho + W)IV~) = EJ~/',),
(6)
for the (f.p)2 and (f_p)a configurations. As usual, Ho denotes the one-body shellmodel Hamiltonian that we assume to be diagonal in the adopted harmonic oscillator basis (hto = 10.5 MeV). The single-particle energies (5) were used. Two calculations were performed with W = G and W = G+Gap-tw The corresponding energy spectra obtained for 4aSc are shown in fig. 4, where they are compared with experiment.
394
F. ANDREOZZI AND G. SARTORIS M~v
4.5
,/7-/-/7T/77 ~'2",T=~ 4.(
3.5 7. T=3~ J ~2~" /5 .
½
('y.-)
3.O 11/2 ~5
'F,
%-,%-
V5 s/5
%-
°/5
2.5.
.0.
"/5 '5
1.5.
/i'% ½ k5
I o0.
"/2 sh
I,a
¾
% 0.5.
O.
G
G+G3p.I h
E X P
Fig. 4. Spectra of 4SSc calculated using the interactions G and G~-Ga~.Ih. The experimental levels shown on the right side of the figure arc those whose wave functions are expected to contain large (f.p)S components and are taken from ref. 11). 4.2. EFFECTIVE INTERACTION IN THE f~ SHELL; BIORTHOGONAL STATES Let us consider the eigenvector I ~ ) corresponding to the eigenvalue E~ o f eq. (6.) F o r the present, [ ~ v ) m a y be either a two-particle state or a three-particle state. Let us denote by D the space defined by the (0f-lp)" configurations and by d the space defined by the (0f~)" configuration. We n o w expand 1 ~ ) in the adopted h a r m o n i c
f~ S H E L L
395
oscillator basis Iv}
(7)
I~,,) = ~ a~,,,Iv) + ~,, a~,,,Iv}. yea
The effective interaction Wetf in the d-space is defined by [see for example ref. 12)] H, ul~b,) = (Ho + W©ff)l~at} = E,l~b,),
(8)
]~b,) = X a,,[v)
(9)
where vc=d
is the projection of I~v~) in d. In general, the Itk~) are not mutually orthogonal. However, one can find a second set of states Iq~}, such that
(10)
The two sets of states Iq~} and I/~} are said to be biorthogonal 12). By expanding I~p), Iep) = E ap,lv),
(ll)
eq. (lO) can be written in the form ~, ~ , a ~ , = E a~*~pv = 6~,a.
(12)
In the present work the a (and the a) are real; thus eq. (12) becomes X a~, aa, -- 6,.p.
(13)
red
The amplitudes a can be calculated from the a by matrix inversion. The matrix elements of H, ff in the space d a r e then given by [see for example ref. ,a)] (14)
(vlH, fdv'} = ~a~,~,a~,,,E,,, fit
and, with the choice (5) for the single-particle energies, coincide with the matrix elements of W©ff. When the model space is one dimensional, as in the case of (f_p)2 configurations and (f_p)3 T = ½ configurations, eq. (14) reduces simply to (15)
(vlnarlv} = E.
Thus, the matrix elements of the effective interactions W,t2) and W,tf~) in the (f~)2 and (ft) 3 configuration spaces, respectively, were calculated by inserting in eq. (14) the results of the diagonalizations described in subsect. 4.1. The contribution of
(3) t ~ (2) JT
,_IT
d T
Fig. 5. Schematic representation o f three-body correlations. The interactions ,.w(a)cftand Wt~t~ are defined in the text.
396
F. ANDREOZZI A N D G. SARTORIS
three-body correlations to W~faf) was then obtained, as shown schematically in fig, 5, by subtracting from the matrix elements of W ~ ) the corresponding three-particle matrix elements calculated with standard shell-model techniques using the effective two-body interaction W~fzf). 4.3. RESULTS
The matrix elements of W~f2f) and w(a) ©ff w=re calculated using the lowest eigenvalues and corresponding eigenstates obtained from the f-p diagonalizations. The eigenvalues E= to be reproduced by W=ffin eq. (8) can be, at least in principle, arbitrarily chosen. Our choice was determined by the fact that the lowest theoretical levels reproduce reasonably well the experimental levels which are usually included in pure f~ phenomenological fits. The contributions of three-body correlations to W~3) are given in tables 2 and 3. The ordering of the T = ½ states is the same as in sect. 3. rr
TABLE 2 Three-body contributions to the matrix elements o f the effective interaction wo~,, err between (f~)3 states with multiplicity I (in MeV) T
J
This work
Ref. 3)
Ref. ' )
.~
~
--0.002 --0.190
--0.328
--0.026
0.026 0.022
0.008
--0,010
0.064 0.156
0.112
0.009
0.057 0.230
0.271
0.021
0.046 0.136
0.012
0.004
0.069 0.102
-- 0.062
0.004
--1.511 --2.175
--2.341
-- 1.252 1.839
--2.350
--0.092 --0.246
--0.335
J~
0.348 0.471
0.920
~.g
--0.049 --0.101
--0.963
½
~½
-~ ~-
--
The results correspond to the choice W = G (first r o w ) a n d W = G+G3p_ ~ (second row).
f~ S H E L L
397
TXBLt 3
T h r e e - b o d y c o n t r i b u t i o n s to the m a t r i x elements o f the effective interaction W,trat~ between (f~)3 T = ½ states with multiplicity > 1 (in M e V )
1 2
1 --0.662 -- 1:!26
2 --0.181 --0.231
--0.071 -- 0.288
0.140 0.201
y-~ ]
1 2
1 0.469 2.783
2 --0.342 -- 3.054
3 --0.584 0.015
2
0.016 0.799
--0.194 -- 3.399
--0.447 0.405
3
0.003 2.345
--0.398 -- 3.307
0.071 1.136
i
.t--- ¥
1 0.467 0.370
2 0.212 0.169
0.045 0.053
0.330 0.546
1 2
J = :~
1 0.452 1.273
2 --0.605 --1.509
0.107 0.863
--0.365 -- 1.222
1 2
1 0.224 0.214
2 0.073 0.115
0.035 --0.016
0.311 0.497
The ordering o f t h e basis states is explained in the text. T h e results c o r r e s p o n d to t h e choice W = G (first row) a n d W = G+G3p-~u (second row).
The values obtained are considerably larger than those calculated in second-order perturbation theory, especially for T = ½and low J. The T = ½contributions are also generally larger than those corresponding to T = ½. The interaction W = G + G3p- i b yields energy levels in better agreement with experiment and larger three-body correlations. This makes clear that core excitations play a relevant role in the description of 43Ca and 43Sc and cannot be ignored in a microscopic calculation of three-body correlations. Possible objections to the use of the renormalized interaction in the calculation of such correlations have already been discussed in connection with the perturbative approach of sect. 3. They could be raised also here. However, a more accurate treatment of core excitations 13), implying diagonalizations in a three-and fourparticle, one-hole configuration space was beyond the scope of this work. Table 2 shows that phenomenological three-body forces 3) can be reasonably accounted for by the present calculation, although some discrepancies still remain for the highest angular momenta. However, it should be emphasized that a detailed comparison between our results and those of ref. 3) would be misleading. As a matter of fact, Eisenstein and Kirson's values, obtained by fitting the levels of many f~ nuclei, certainly include averaged contributions from n-body (n > 3) forces. The non-hermiticity of the T --- ½effective interaction, especially marked for J = 5, can be attributed to the large configuration mixing which takes place in the (f-p) shell. In fact, the lowest two-particle and T = ½ three-particle states are found to have a large projection in the f~ model space, whereas for the T = ½ states the f~ strength is highly fragmented and there are cases in which the lowest eigenstates
398
F. ANDREOZZI AND G. SARTORIS
have only a small fraction of it. This result can also help to explain why perturbation theory is inadequate to reproduce the large three-body correlations predicted for T = ½ by phenomenological fits. As a matter of fact, small overlaps between f-p wave functions and model wave functions imply that the corresponding expansion which defines the effective interaction for the ft subshell is likely to diverge, Therefore, at least for some T = ½ states, the choice of the f ; model space appears to be inadequate for three-body correlations to be treated in perturbation theory and a non-perturbative approach is the only way to follow. In conclusion, we have shown by a microscopic calculation that a two-body force, acting in the (f.p)a configuration space, can give rise to an effective interaction in the (ft)a model space with fairly large three-body correlations as demanded by phenomenological fits, provided that the effective interaction is derived by a nonperturbative procedure. We thank T. T. S. Kuo for providing us with the G-matrix elements and N. Lo Iudice for valuable discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13)
T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 C. Quesne, Phys. Lett. 31B (1970) 7 I. Eisenstein and M. W. Kirson, Phys. Lett. 47B (1973) 315 I. Talmi, Roy. Mod. Phys. 34 0962) 704 E. Osnes, Phys. Lett. 26B (1967) 274 T. T. S. Kuo and G. E. Brown, Nucl. Phys. AU4 0968) 241 G. F. Bertsch, Phys. Rev. Lett. 21 (1968) 1694 Y. Yariv, Nucl. Phys. A225 (1974) 382 R. V. Reid, Jr., Ph.D. thesis, Cornell University, Ithaca, 1968; and Ann. of Phys. 50 (1968) 41 l T. Hamada and I. D. Johnston, Nucl. Phys. 34 0962) 382 W. P. Alford, N. Schulz and A. Jamshide, Nucl. Phys. A174 (1971) 148 B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771 N. Lo Iudice, D. J. Rowe and S. S. M. Wong, Nucl. Phys. A219 (1974) 171
Nuclear Physics A270 (1976) 388 --398; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
E F F E C T I V E T H R E E - B O D Y I N T E R A C T I O N S IN T H E f~ S H E L L F. A N D R E O Z Z I and G. SARTORIS lstituto di Fisica Teorica dell' Universitd and lstttuto Nazionale di Fisica Nucleare, Mostra d' Oltremare. pad. 19, Napoli, Italy Received 13 February 1976 (Revised 24 May 1976) Abstract: The contributions o f effective three-body forces to the interaction energy of three particles
in the 0f.~ shell are calculated using reaction matrix elements derived from the Hamada and Johnston potential. The results obtained by second-order perturbation theory are found to be much smaller than those obtained from phenomenological fits. An effective interaction is then derived for the 0f~_shell performing a shell-model calculation for three particles in the whole 0f-lp major shell and through the use of biorthogonal states; effective three-body contributions turn out to be large, as expected from the results of phenomenological fits. 1. Introduction
Shell-model calculations are usually performed under the assumption that the influence of configurations not included in the model space can be accounted for by using an effective interaction. A method for calculating effective interactions, which consists in the derivation of a reaction matrix G f r o m a free nucleon-nucleon potential and in a subsequent renormalization of G in second-order perturbation theory, was proposed several years ago by K u o and Brown '). When more than two valence particles are considered, their approach gives rise to effective three-body forces even though the original interaction was of a two-body character. This feature, however, is not peculiar to the method. Effective many-body forces should be considered, at least in principle, even when the effective interaction is derived from the experimental data directly 2, 3). Many-body effects are usually assumed to be small and therefore ignored, but the validity of this assumption depends strongly on the choice of the model-space configurations. The very popular f~_ model, in which the configuration space is truncated within the 0f-lp major shell - the energy difference between the 0f~_ and lp~ singleparticle levels is estimated to be only a couple of MeV - appears to be a good candidate for many-body effects to play a relevant role. Three-body contributions to the parameters of the Talmi formula 4), which gives the ground-state energy in t h e j " configuration, were evaluated in second-order perturbation theory by Osnes 5) for the calcium isotopes using the Kuo-Brown bare interaction 1, 6), and were found to be small. The same interaction was used by Bertsch 7) in a note on the relationship between the ground-state energy of closed-shell nuclei and the two-body interaction energies deduced from the spectrum of a nucleus with two valence particles in the shell. Bertsch pointed out that some two-body correlations 388
f~ SHELL
389
which are present in the nucleus with two valence particles cannot be effective in the closed-shell nucleus. By subtracting out the contributions of such correlations to the two-particle energies, he obtained sizable corrections to the binding energies of 4SCa and 56Ni' which he interpreted as an indirect estimate of appreciable three-body effects in the 0ft shell. A phenomenological shell-model fit to energy levels of f~ nuclei was performed by Quesne 2) taking explicitely into account the contribution of effective many-body forces. Quesne extracts the two-body part of the effective interaction from the binding energies and the spectra of 4o-42Ca and uses these phenomenological values to calculate the energy levels of 43Ca with standard shell-model techniques. The differences between the calculated and observed energy levels of 4aCa are then identified with the matrix elements of an effective three-body interaction. The same procedure is applied to the other calcium isotopes to extract four-body forces and so on up to eight-body forces. A similar calculation is performed for the N = 28 isotones. The many-body forces so obtained turn out to be very large. A different philosophy was adopted by Eisenstein and Kirson a) who applied a best-fit procedure to a great number of energy levels of f~ nuclei to determine the matrix elements of an effective interaction containing just two- and three-body terms. Three-body effects are found to be quite large, especially when the three particles interact in a T = ½ state. More recently, three-body contributions to the effective interaction in the 0f~ subshell were calculated by Yariv a) in second-order perturbation theory using reaction matrix elements derived from Reid potentials 9). The results are much smaller than those obtained from phenomenological fits. In Yariv's work an attempt is made to reduce this discrepancy by calculating the influence of the excluded lp~ level on the effective T = ~ interaction in the (0f~) a model space through a non-perturbative procedure. No substantial improvement is obtained. To summarize, the results ofrefs. 2, a, 7), although not directly comparable, indicate that many-body correlations play a relevant role in a pure f~ model. The results of refs. 5, s) indicate that calculations based on second-order perturbation theory are unable to reproduce such large correlations. On the other hand, Yariv's attempt s) to calculate three-particle correlations to all orders in G is limited to the T = ½ case and neglects the contribution of the 0f~_and lp~. levels, which, as suggested by the second-order results, may be quite significant. In the present paper, we calculate the contribution of effective three-body forces to the interaction energy of three particles in the 0f~ subshell using reaction matrix elements derived by Kuo and Brown i, 6) from the Hamada and Johnston potential 1o). We first recalculate these contributions in second-order perturbation theory to test the dependence of Yariv's results s) on the choice of the reaction matrix elements. We then perform a non-perturbative calculation for the T = ½, ~z matrix elements of the effective interaction taking into account the influence of the whole f-p major shell.
390
F. ANDREOZZI AND G. SARTORIS 2. Two-body interaction
The G-matrix elements used in this work are those derived from the HamadaJohnston potential t o) by Kuo and Brown t, 6) with the inert *°Ca core as reference state and the value ho~ = 10.5 MeV for the oscillator parameter. An effective twobody interaction for the 0f-lp shell was calculated, following the Kuo-Brown procedure 1, 6), by adding the diagrams shown in fig: 1. [Arguments for neglecting secondorder contributions corresponding to the excitation of two particles from the core have been given in ref. 6).] It is worth mentioning that our results for the G3p-~ diagram differ slightly from those obtained by Kuo and Brown 6) using the same bare matrix elements and the same average energy denominators 2ho9. This is because they used a redundant set of three-particle, one-hole intermediate states, while in our calculation this approximation was removed by using uncoupled (m-scheme) intermediate states.
G
G3p.,h
Fig. 1. The G-matrix and core-polarization diagrams.
G3p
G4p.lh
Fig. 2. Second-orderdiagrams representing three-body correlations. 3. Second-order calculation of three-body correlations
The three-body correlations to second order in G were obtained by adding the diagrams shown in fig. 2. When the model space is restricted to the 0ft subshell, graph G4p-ta corresponds to the excitation of a 0p particle to the 0ft level and graph G3p corresponds to the excitation of a 0ft particle to a higher negative-parity level. The configuration (f~)3 has 22 independent states; 16 states with T = ½ and J = $2,½,{ [2], -7z[3], { [2], ~ - [2], &2 ~ [2], ~ , ~ , ~ (the numbers in square brackets give the multiplicity of the states), and six states with T = ½ and J = ½, {, {, 9, ~-,
f:z SHELL
391
]NTJ) -= ~, C~,rss,[[a+a + ]r'Z'a+ ]rJI0),
(1)
1 s We adopted the basis vectors T'J" +
where antjm~ denotes the creation operator of a particle in the (nljrnT) single-particle state (for simplicity, we shall omit the quantum numbers of the 0f~ states). The coefficients r,NrJ •-,r'z' were calculated by applying the Schmidt orthonormalization procedure to the states [[a+a + ]r'S'a+ ]rsl0),
(2)
ordered with increasing T'J' (J' changing faster than T'). The quantum number N is obviously unnecessary to classify the states with multiplicity 1, which can be obtained by normalizing any of the states (2) having the required T, J values. For the T --- ½, J states having multiplicity greater than 1, N keeps track of the order in which the basis vectors are generated (N = 1 means the first T, J state generated by the Schmidt procedure and so on). In the adopted basis, graphs G3p and G4p- lh are given by
(N1TJIv3IN2TJ) = + ~, ~r,s,rs r,N2rs T'J' ~'T"J'" T'J" T"J"
x • ( - 1)i+i+r+JU(TrjJTz; J"J')U(~-IzT½; T"T') nij × Gr,s,(nly) ~
1
(3)
Gr"s"(nlj)'
where the U are normalized Racah coefficients and Grs(nlj) is the antisymmetrized but not normalized bare reaction matrix element
(O[([a+ a~j]ra)+G[a+ a + ]rSl0).
(4)
The minus sign and the plus sign correspond to graph G3p and G4p- lh, respectively; the quantum numbers (nlj) are those of the internal particle or hole line. Graph G4p-lh was calculated using an average energy denominator A E = -2hto. Graph G3p contains contributions corresponding to the excitation of a particle within the f-p shell; these were calculated with energy denominators given by minus the sum of the single-particle energies in the intermediate state, for which the values ef~ = 0,
ep~ = 2.00,
~p½ = 4.25,
~f~ = 6.25 MeV
(5)
were used. In the calculation of graph Gap, contributions from intermediate states with 2hto excitation energy should also be considered. We ignored them, since the G-matrix elements required for such calculation were not available. However, as we shall see below, this restriction is likely to be not important and cannot alter the conclusions of this section. The results obtained for the three-body correlations Gap and G4p-lh are given in
392
F. A N D R E O Z Z I A N D G. SARTORIS TABLE 1
Second-order contributions o f three-body correlations to the interaction energy of three particles in the 0f~ shell (in MeV) T J
Gap (a)
½ { * 7 -~
* * *
*
--0.252 --0.310 --0.236 --0.079 0.112 0.051 : --0.074 --0.177 --0.068 --0.046 0.023 0.113 0.095 0.085
G4p-lh
T
J
(b) --0.193 --0.153 --0.249 --0.086 0.120 0,044
--0.064 :--0.174 --0.098 0.020 --0~015 0.147 0.141 --0.006
Gap (a)
0.027 0.028 0.004 --0.001 --0.003 --0.002 0.003 0.090 --0.002 0.016 0.003 0.001 0.006 --0.017
½
¥ *
0.078 --0.158 --0.074
o.144 *
0.061 0.016
-0.109 -~ ajt
0.228 --0.031
j j
O.OlO 0.002
{-
0.039 0.033 0.027 0.007
G.p-th (b) 0,072 -0.110 -0.007 0.082 0,122 --0.004 --0.051 0.188 --0.036 0.035 0,017 0.035 0.022 0.026 0.000
-0,003 0.014 0.012 -0.023 0.009 --0.011 0.019 --0.021 0 , 0.003 0.001 0.000 --0.003 0.001 --0.002
The ordering o f the matrix elements is explained in the text; off-diagonal matrix elements are starred. Labels (a) and (b) refer to different choices o f the energy denominators for diagram G3p.
table 1. The values of G3D corresponding to the choice (5) for the energy denominators are listed under the label (a). We first observe that three-body corrections to matrix elements with T = ~}turn out to be negligibly small, in agreement with the results of refs. 5, 8), andin any case much smaller than the corrections to T = ½ matrix elements, in qualitative agreement with the results of phenomenological fits 3). The contributions from G4p-th are much smaller than those from G3p and are practically irrelevant. This result is explained mainly by the larger energy denominators of graph G4p_ th and suggests that we did not lose much by neglecting intermediate states with 2hco excitation energy in the calculation of graph Gap. To compare our results with those obtained by Yariv a), graph G3p was recalculated using an average energy denominator - d E = ½hco = 5.25 MeV; the values so obtained are listed in table 1 under the label (b). In spite of minor differences, the two sets of values (a) and (b) can both be considered in overall agreement with Yariv's results a) and have the common feature of being much smaller than the values obtained from phenomenologieal fits 3), which for T = ) and low J are as large as 2 MeV. This feature does not seem to depend critically on the choice of the bare reaction matrix elements nor on the choice of reasonable singleparticle energies. The discrepancy between second-order and phenomenological results might be attributed to the fact that in the calculation of three-body correlations to second order in G the strong renormalization of the two-body interaction through core exci-
f~ SHELL
393
tations is completely ignored. It is therefore tempting to repeat the calculation using the renormalized interaction of sect. 2 instead of G as a vertex. This procedure is open to objection since it implies, for instance, violations of the Pauli principle of the type described in fig. 3. In fact, of the third-order contributions (a) and (b), which should cancel each other for j = ~ and m = m', only (a) is taken into account. However, there are many diagrams of type (a) which do not violate the Pauli principle, whereas the results of table 1 suggest that the exchange contributions (b) should not be too relevant. On these grounds, we recalculated graph G3p using the two-body renormalized interaction as a vertex; on the whole the contributions of Gap turn out to be at most doubled.
a
b
Fig. 3. Third-order contributions which cancel each other forj = ~[and m = m'. It is therefore possible to conclude that, whether we use G or the renormalized interaction G +G3p-lh, the perturbative approach discussed in this section is inadequate to reproduce the large three-body correlations obtained by phenomenological fits.
4. Non perturbative calculation of three-body correlations The procedure adopted can be summarized as follows. First, we performed standard shell-model calculations for two and three particles in the 0f-lp major shell. Then, the results of the (f.p)a calculation were considered as the starting point for the derivation of an effective interaction for the f.~ subshell through the use of biorthogonal states. Finally, the results of the (f_p)2 calculation were used to extract the two-body contributions from the obtained three-particle effective interaction. 4.1. DIAGONALIZATION IN THE f-p SHELL We solved the shell-model eigenvalue problem H [ ~ , ) = (Ho + W)IV~) = EJ~/',),
(6)
for the (f.p)2 and (f_p)a configurations. As usual, Ho denotes the one-body shellmodel Hamiltonian that we assume to be diagonal in the adopted harmonic oscillator basis (hto = 10.5 MeV). The single-particle energies (5) were used. Two calculations were performed with W = G and W = G+Gap-tw The corresponding energy spectra obtained for 4aSc are shown in fig. 4, where they are compared with experiment.
394
F. ANDREOZZI AND G. SARTORIS M~v
4.5
,/7-/-/7T/77 ~'2",T=~ 4.(
3.5 7. T=3~ J ~2~" /5 .
½
('y.-)
3.O 11/2 ~5
'F,
%-,%-
V5 s/5
%-
°/5
2.5.
.0.
"/5 '5
1.5.
/i'% ½ k5
I o0.
"/2 sh
I,a
¾
% 0.5.
O.
G
G+G3p.I h
E X P
Fig. 4. Spectra of 4SSc calculated using the interactions G and G~-Ga~.Ih. The experimental levels shown on the right side of the figure arc those whose wave functions are expected to contain large (f.p)S components and are taken from ref. 11). 4.2. EFFECTIVE INTERACTION IN THE f~ SHELL; BIORTHOGONAL STATES Let us consider the eigenvector I ~ ) corresponding to the eigenvalue E~ o f eq. (6.) F o r the present, [ ~ v ) m a y be either a two-particle state or a three-particle state. Let us denote by D the space defined by the (0f-lp)" configurations and by d the space defined by the (0f~)" configuration. We n o w expand 1 ~ ) in the adopted h a r m o n i c
f~ S H E L L
395
oscillator basis Iv}
(7)
I~,,) = ~ a~,,,Iv) + ~,, a~,,,Iv}. yea
The effective interaction Wetf in the d-space is defined by [see for example ref. 12)] H, ul~b,) = (Ho + W©ff)l~at} = E,l~b,),
(8)
]~b,) = X a,,[v)
(9)
where vc=d
is the projection of I~v~) in d. In general, the Itk~) are not mutually orthogonal. However, one can find a second set of states Iq~}, such that
(10)
The two sets of states Iq~} and I/~} are said to be biorthogonal 12). By expanding I~p), Iep) = E ap,lv),
(ll)
eq. (lO) can be written in the form ~, ~ , a ~ , = E a~*~pv = 6~,a.
(12)
In the present work the a (and the a) are real; thus eq. (12) becomes X a~, aa, -- 6,.p.
(13)
red
The amplitudes a can be calculated from the a by matrix inversion. The matrix elements of H, ff in the space d a r e then given by [see for example ref. ,a)] (14)
(vlH, fdv'} = ~a~,~,a~,,,E,,, fit
and, with the choice (5) for the single-particle energies, coincide with the matrix elements of W©ff. When the model space is one dimensional, as in the case of (f_p)2 configurations and (f_p)3 T = ½ configurations, eq. (14) reduces simply to (15)
(vlnarlv} = E.
Thus, the matrix elements of the effective interactions W,t2) and W,tf~) in the (f~)2 and (ft) 3 configuration spaces, respectively, were calculated by inserting in eq. (14) the results of the diagonalizations described in subsect. 4.1. The contribution of
(3) t ~ (2) JT
,_IT
d T
Fig. 5. Schematic representation o f three-body correlations. The interactions ,.w(a)cftand Wt~t~ are defined in the text.
396
F. ANDREOZZI A N D G. SARTORIS
three-body correlations to W~faf) was then obtained, as shown schematically in fig, 5, by subtracting from the matrix elements of W ~ ) the corresponding three-particle matrix elements calculated with standard shell-model techniques using the effective two-body interaction W~fzf). 4.3. RESULTS
The matrix elements of W~f2f) and w(a) ©ff w=re calculated using the lowest eigenvalues and corresponding eigenstates obtained from the f-p diagonalizations. The eigenvalues E= to be reproduced by W=ffin eq. (8) can be, at least in principle, arbitrarily chosen. Our choice was determined by the fact that the lowest theoretical levels reproduce reasonably well the experimental levels which are usually included in pure f~ phenomenological fits. The contributions of three-body correlations to W~3) are given in tables 2 and 3. The ordering of the T = ½ states is the same as in sect. 3. rr
TABLE 2 Three-body contributions to the matrix elements o f the effective interaction wo~,, err between (f~)3 states with multiplicity I (in MeV) T
J
This work
Ref. 3)
Ref. ' )
.~
~
--0.002 --0.190
--0.328
--0.026
0.026 0.022
0.008
--0,010
0.064 0.156
0.112
0.009
0.057 0.230
0.271
0.021
0.046 0.136
0.012
0.004
0.069 0.102
-- 0.062
0.004
--1.511 --2.175
--2.341
-- 1.252 1.839
--2.350
--0.092 --0.246
--0.335
J~
0.348 0.471
0.920
~.g
--0.049 --0.101
--0.963
½
~½
-~ ~-
--
The results correspond to the choice W = G (first r o w ) a n d W = G+G3p_ ~ (second row).
f~ S H E L L
397
TXBLt 3
T h r e e - b o d y c o n t r i b u t i o n s to the m a t r i x elements o f the effective interaction W,trat~ between (f~)3 T = ½ states with multiplicity > 1 (in M e V )
1 2
1 --0.662 -- 1:!26
2 --0.181 --0.231
--0.071 -- 0.288
0.140 0.201
y-~ ]
1 2
1 0.469 2.783
2 --0.342 -- 3.054
3 --0.584 0.015
2
0.016 0.799
--0.194 -- 3.399
--0.447 0.405
3
0.003 2.345
--0.398 -- 3.307
0.071 1.136
i
.t--- ¥
1 0.467 0.370
2 0.212 0.169
0.045 0.053
0.330 0.546
1 2
J = :~
1 0.452 1.273
2 --0.605 --1.509
0.107 0.863
--0.365 -- 1.222
1 2
1 0.224 0.214
2 0.073 0.115
0.035 --0.016
0.311 0.497
The ordering o f t h e basis states is explained in the text. T h e results c o r r e s p o n d to t h e choice W = G (first row) a n d W = G+G3p-~u (second row).
The values obtained are considerably larger than those calculated in second-order perturbation theory, especially for T = ½and low J. The T = ½contributions are also generally larger than those corresponding to T = ½. The interaction W = G + G3p- i b yields energy levels in better agreement with experiment and larger three-body correlations. This makes clear that core excitations play a relevant role in the description of 43Ca and 43Sc and cannot be ignored in a microscopic calculation of three-body correlations. Possible objections to the use of the renormalized interaction in the calculation of such correlations have already been discussed in connection with the perturbative approach of sect. 3. They could be raised also here. However, a more accurate treatment of core excitations 13), implying diagonalizations in a three-and fourparticle, one-hole configuration space was beyond the scope of this work. Table 2 shows that phenomenological three-body forces 3) can be reasonably accounted for by the present calculation, although some discrepancies still remain for the highest angular momenta. However, it should be emphasized that a detailed comparison between our results and those of ref. 3) would be misleading. As a matter of fact, Eisenstein and Kirson's values, obtained by fitting the levels of many f~ nuclei, certainly include averaged contributions from n-body (n > 3) forces. The non-hermiticity of the T --- ½effective interaction, especially marked for J = 5, can be attributed to the large configuration mixing which takes place in the (f-p) shell. In fact, the lowest two-particle and T = ½ three-particle states are found to have a large projection in the f~ model space, whereas for the T = ½ states the f~ strength is highly fragmented and there are cases in which the lowest eigenstates
398
F. ANDREOZZI AND G. SARTORIS
have only a small fraction of it. This result can also help to explain why perturbation theory is inadequate to reproduce the large three-body correlations predicted for T = ½ by phenomenological fits. As a matter of fact, small overlaps between f-p wave functions and model wave functions imply that the corresponding expansion which defines the effective interaction for the ft subshell is likely to diverge, Therefore, at least for some T = ½ states, the choice of the f ; model space appears to be inadequate for three-body correlations to be treated in perturbation theory and a non-perturbative approach is the only way to follow. In conclusion, we have shown by a microscopic calculation that a two-body force, acting in the (f.p)a configuration space, can give rise to an effective interaction in the (ft)a model space with fairly large three-body correlations as demanded by phenomenological fits, provided that the effective interaction is derived by a nonperturbative procedure. We thank T. T. S. Kuo for providing us with the G-matrix elements and N. Lo Iudice for valuable discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13)
T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 C. Quesne, Phys. Lett. 31B (1970) 7 I. Eisenstein and M. W. Kirson, Phys. Lett. 47B (1973) 315 I. Talmi, Roy. Mod. Phys. 34 0962) 704 E. Osnes, Phys. Lett. 26B (1967) 274 T. T. S. Kuo and G. E. Brown, Nucl. Phys. AU4 0968) 241 G. F. Bertsch, Phys. Rev. Lett. 21 (1968) 1694 Y. Yariv, Nucl. Phys. A225 (1974) 382 R. V. Reid, Jr., Ph.D. thesis, Cornell University, Ithaca, 1968; and Ann. of Phys. 50 (1968) 41 l T. Hamada and I. D. Johnston, Nucl. Phys. 34 0962) 382 W. P. Alford, N. Schulz and A. Jamshide, Nucl. Phys. A174 (1971) 148 B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771 N. Lo Iudice, D. J. Rowe and S. S. M. Wong, Nucl. Phys. A219 (1974) 171