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Many-body terms in the folded diagram expansion
Nuclear Physics @ North-Holland
A442 (1985) 68-78 Publishing Company
MANY-BODY TERMS IN THE FOLDED DIAGRAM EXPANSION H. MOTHER, Institut ffir Theoretische
A. POLLS*,
P.K. RATH
and
Physik, Auf der Morgenstelle
AMAND
FAESSLER
14, D-7400 Tiibingen, West Germany
Received 27 December 1984 (Revised 8 March 1985) Abstract: The folded diagram expansion for the effective hamiltonian to be used in shell-model calculations for a system with A valence nucleons yields folded diagrams connecting up to A nucleons. The importance of such effective many-nucleon forces is investigated for the example of 4 valence nucleons in the 1sOd shell (“‘Ne). The folded diagram terms turn out to reduce the binding energy for the low-lying states. The effects of the folded diagrams involving three nucleons are of the same importance as those of the two-body terms. The four-nucleon forces originating from folded diagrams are considerably weaker. This suggests that three-body operators have to be considered in a microscopic calculation of an effective energy-independent shell-model hamiltonian, whereas terms involving four nucieons may be ignored.
1. Introduction The investigation of many-nucleon forces has always been a very attractive subject in nuclear physics. Such many-body forces, especially of course the three-nucleon forces, might be important for the understanding of the bulk properties of nuclear calculation of the excitation ground-states im6) as well as for a detailed microscopic spectra 7,8). The hope is that one can establish the effects of a “real” three-nucleon force originating, e.g. from meson rescattering or inner excitations of the nucleons in a nuclear many-body system. In all nuclear structure calculations, however, there is the problem that the effects of such real many-nucleon forces interfere with many-body terms of the conventional nuclear many-body theory which assumes two-nucleon forces only. For the calculation of ground-state properties one must be aware that effects of three-nucleon correlations, which can be taken into account by solving a Bethe-Faddeev equation, are of similar importance to those from three-nucleon forces 3). A three-body operator in the hamiltonian to be used in nuclear shell-model calculations may originate from a real three-nucleon force, but may also be due to the renormalisation of the hamiltonian, which is necessary to account for the degrees of freedom outside the shell-model space under consideration. Therefore one has to perform a careful * Permanent
address:
Departament
de Fisica Teorica, 68
Universitat
de Barcelona,
Barcelona-28,
Spain.
69
H. Miither et al. / Many-body terns
investigation structure
of such effective
three-nucleon
effects to real three-nucleon
forces before
one may relate nuclear
interactions.
It has been observed that the lowest-order terms in a conventional perturbation expansion for the effective three-nucleon interaction (terms of second order in the Brueckner G-matrix) give a very weak contribution if a realistic shell-model space is considered which contains all configurations within one major shell “). In this case, the effects of the three-nucleon force on systems with three valence nucleons in the 1sOd shell were found to be negligible. The weakly repulsive effects of this effective three-nucleon force become visible only if four or more valence nucleons are considered ‘). For a small model space containing, e.g., the configurations within one subshell only, the effective three-nucleon terms seem to be much larger, since in this case a renormalisation due to intermediate Ohw is required ‘OT1r). The shell-model hamiltonian determined in a perturbation expansion to account for the nucleonic degrees of freedom outside the model space depends on the energy of the state to be calculated. This energy dependence makes the hamiltonian to be used for the calculation of a system with three valence nucleons different from the one to be used for two nucleons. Therefore this energy dependence can also be understood as an effective three-body, or in general many-body, interaction. The explicit energy dependence of the hamiltonian can now be eliminated by introducing folded diagrams I*). Applying the folded diagram expansion to obtain an energyindependent shell-model hamiltonian for a system of three valence nucleons, one obtains connected folded diagrams which involve three nucleons 13). This supports the interpretation of the energy-dependent hamiltonian in terms of effective manybody forces. In fact, for the example of three valence nucleons in the 1sOd shell 13), these three-body folded diagrams turned out to be much more important than the terms of second order in the G-matrix discussed, e.g., in ref. “). Therefore the folded diagram expansion yields a non-negligible effective threebody force. The same three-body force should also be contained in energy-independent empirical shell-model hamiltonians which are determined by a fit to experimental data. It might be possible, however, to simulate the effects of the many-body forces by a proper adjustment of the matrix elements. of the two-body operator or by introducing an effective NN interaction, which depends on the mass number 14). Now the next question is of course: what happens to the folded diagram expansion in the case of four or more valence nucleons? Is it sufficient to consider folded diagrams involving up to three nucleons or must we take into account the effects of effective four-body forces as well? If n-body forces turn out to be important in general for the case of n valence nucleons, one should probably give up the idea of using the folded diagram expansion to determine the effective hamiltonian to be used for shell-model calculations for all nuclei within a certain shell. It is the aim of the present paper to investigate this question. For that purpose we repeat some features of the folded diagram expansion in sect. 2. Results for calculations
70
H. Miiiher et al. / Many-body terms
considering four valence nucleons are presented section contains our main conclusions.
and discussed
in sect. 3. The last
2. Folded diagram expansion To review some features of the folded diagram expansion for an effective shellmodel hamiltonian we split the hamiltonian H for the A-nucleon system into a one-body operator Ho and the rest H,, which may contain one- and two-nucleon operators, H=Ho+H,.
(1)
It is useful of course to choose H,, in such a way that H, is made small and can be treated as a perturbation. This could be achieved by using the Hat-tree-Fock or Brueckner-Hartree-Fock approximation to define Ho [ref. “)I. The A-nucleon eigenfunctions of Ho are then used to define the model-space or P-space. For a shell-model calculation the basis of the P-space typically consists of those Slater determinants for which the shells of a core nucleus (e.g. 160) are completely occupied and the remaining valence nucleons occupy states within the major shell above the core (e.g. the 1sOd shell). It is then the aim to derive an effective hamiltonian He* such that PHe,P!l’i = E,py,
(2)
is fulfilled for the low-lying states of all nuclei one would like to investigate within the given shell-model approach. The energies Ei and wave functions !Pi are the exact eigenvalues and states of the complete hamiltonian H, and P is the projection operator projecting be written as
onto the model
space. This effective
hamiltonian
linked
.
can formally
(3)
The projection operator Q is used to restrict all intermediate states to configurations outside the model space. The subscript “linked” on the right-hand side of eq. (3) is used to indicate that only those terms should be considered in the expansion which are represented by a linked diagram. Examples of linked diagrams are given in figs. la and lb. They are characterized by the fact that all H, interaction lines (wiggly lines) are connected to at least one valence nucleon (upward-going lines). Core excitations which are not connected to the valence nucleons (see fig. lc) have to be considered as unlinked diagrams. It should be noticed that the energy denominator of eq. (3) contains the exact energy Ei of the state under consideration. Therefore the effective hamiltonian is different for each state, which of course also implies that it is different for different numbers of valence nucleons. As a consequence this energy dependence can be understood as a source of effective many-body forces.
H. Miither et al. / Many-body terms
da1 a)
b)
71
c)
Fig. 1. A linked diagram (a), a disconnected diagram (b), and an unlinked diagram (c) for a system of four valence nucleons.
There are two arguments which speak against the use of an effective energydependent hamiltonian. First, it is not so convenient for treating an energy-dependent hamiltonian. The evaluation of the eigenvalues is more complicated and the resulting eigenstates are not orthogonal. Secondly, the success of the energy-independent empirical hamiltonians demonstrates that it should be possible to determine an effective hamiltonian which does not depend on the energy. Formally this can be achieved by introducing the folded diagram expansion. Loosely speaking, one expands the energy denominator of eq. (3) to eliminate the energy difference between the energy of the complete hamiltonian H and the energy E0 of the unperturbed hamiltonian Z-J0from this denominator. This yields the folded diagram expansion of &i, HeE= H,,+H,+
H, & I
Note, that in this formulation Ei any more. The eigenvalues
He,+ folded 0
0
diagrams
. I
(4)
linked
He@ does not explicitly depend on the exact energy of the hamiltonian given in eq. (4) are identical to
solutions of the hamiltonian of eq. (3). In general, however, the energy-dependent formulation yields more solutions than the energy-independent one. A more detailed discussion of the folded diagram expansion can be found, e.g., in refs. ‘**i6). In a practical application of the folded diagram expansion one normally proceeds as follows. One first selects the terms to. be calculated in the energy-dependent expansion of eq. (3). Translated in the language of diagrams this means that one is selecting a certain set of diagrams which must be linked (see above) and irreducible (i.e. all intermediate states should be outside the model space). As typical examples for such diagrams we display in fig. 2 those diagrams which shall also be considered in our numerical investigations. Now the lowest-order approximation of the folded diagram expansion is obtained by calculating these diagrams using the unperturbed Rayleigh-Schrodinger energy denominators. These terms are identified as the Q-box terms. F,:= Q. (5) For a system with A valence nucleons these Q-box terms must be understood as an operator connecting A-nucleon wave functions. The first-order correction (Fl)
H. M&her et al. / Many-body terns
12
a) Fig. 2. Two-body
b)
d)
c)
diagrams which are used to define the Q-box. The intermediate diagram (b) are restricted to two particles in the 1pOf shell.
2-particle
states in
now contains all terms with one fold. In terms of diagrams this means that two Q-box terms must be connected leading to an intermediate model space state, and the lines of the intermediate state must be folded. For three valence nucleons such combinations are displayed in fig. 3. If the Q-box contains two-body interaction terms only and no single-particle terms, all the folded diagrams for three nucleons are connected. The diagram displayed in fig. 3a represents a two-body term with
-
al Fig. 3. Two-body
b)
(a) and three-body
terms (b, c) in the folded valence nucleons.
c) diagram
expansion
for a system of three
the third valence nucleon being a spectator, while diagrams 3b and 3c are real three-nucleon terms. In the case of four valence nucleons one finds diagrams for F1 which are disconnected (see fig. 4a). These diagrams, which are of course also contained in the energy-dependent formulation are identical in magnitude to Q-box terms displayed in fig. 4b. Due to the additional minus sign in the folded diagram
H---+Hl-.=O b)
0)
Fig. 4. The disconnected
folded
diagram
(a) is cancelled
by the disconnected
Q-box
term (b).
H. Miither et al. / Many-body terms
the two diagrams the disconnected
cancel
each other.
diagrams
The calculation
that one should
in the Q-box terms and also in the folded
of the folded
the unperturbed energies calculated very easily
This demonstrates
73
diagrams
is in general
of the model-space F,:=
states
quite involved. are degenerate,
leave out
diagrams. If, however, F, can
be
(6)
Q,Q,
if we define
The product Q,Q in eq. (7) is a product of operators functions. Also higher-order corrections in the folded calculated in a similar way 13).
acting on A-nucleon wave diagram expansion can be
It turned out, however, that a grouping of the folded diagram terms in this conventional method leads to convergence problems already for systems with three valence nucleons r3). The convergence is much better if the folded diagrams are regrouped following a proposal by Lee and Suzuki “). In this case the effective hamiltonian is defined as &= The members
(8)
n-02
R, are given
of the sequence Ro=
Ho+ lim R,. as
Q,
RI = Cl- QA-‘9,
(9) Since the convergence
of the Lee-Suzuki
iteration
for the conventional method, the Lee-Suzuki folded diagrams in the calculations discussed
scheme
seems to be better than
method has been used to sum the in the next section.
3. Results and discussion In order to investigate the importance of many-body forces in the folded diagram expansion we consider the example of four valence nucleons occupying a singleparticle states in the 1sOd shell. The single-particle energies (the eigenvalues of the hamiltonian Ho) are chosen to be degenerate for all states within one major shell of the oscillator model ~~=h~(~~-22)-5 In this equation
N (i = 0, 1,2, . . .) is th e main
(10)
MeV. quantum
number
of the oscillator
74
model choice
H. M&her et al. / Many-body terms
and the energy hw has been chosen to be 14 MeV. With this degenerate for the single-particle energies the “derivative” method (see eqs. (6)-(g))
can be used to calculate the folded diagrams. The residual energy-dependent interaction between the valence nucleons is defined by the Q-box terms displayed in fig. 2. The wiggly lines in these diagrams should be identified with matrix elements of the Brueckner G-matrix. For our calculations this G-matrix was obtained by solving the Bethe-Goldstone equation using the Reid soft-core ‘*) potential. The Pauli operator in the Bethe-Goldstone equation has been chosen to suppress scattering into intermediate states with one interacting nucleon in the OS or Op shell or both nucleons in the 1sOd or 1pOf shell. Therefore we can add explicitly the Q-box terms of fig. 2b, where the nucleons are scattered into intermediate states with two particles in the 1pOf shell. The summation over intermediate states in diagrams 2c and 2d was restricted to states with 2hw excitation energy. The matrixelements of G were calculated in an oscillator basis with an oscillator constant (b = 1.72 fm) which corresponds to hw = 14 MeV and the dependence of G on the starting energy was taken into account. For a more realistic calculation one should also consider single-particle terms for the Q-box. These single-particle terms, however, give rise to many disconnected diagrams which should be eliminated from the expansion. For the case of two valence nucleons this can be achieved by using the method discussed, e.g., in of the refs. 12,i3). But already for the case of three valence nucleons the elimination disconnected diagrams seems to be very difficult. In order to avoid these difficulties we have ignored any single-particle terms in the Q-box. For the two-body terms as displayed e.g. in fig. 2 all folded diagrams are connected for the cases of two and three valence nucleons and even in the case of four valence nucleons the disconnected diagrams are suppressed by a statistical factor of i as compared to the connected diagrams with one fold. Some results of shell-model calculations for energies of states in “Ne with isospin T= 0 are displayed in fig. 5. The spectrum displayed in the leftmost column is obtained effective
by ignoring hamiltonian
the degenerate
all folded diagrams and using the approximation R. in the of eq. (8). This means that the hamiltonian consists out of
single-particle
energies
(-5 MeV) and the two-body
Q-box terms of
fig. 2 calculated at the unperturbed energy of -20 MeV for the four-nucleon system, which corresponds to a starting energy of -10 MeV for the two-body diagrams. If one determines the starting energy of the Q-box for each individual state in a self-consistent way by requesting that this energy corresponds to the final energy of this state, the energy-dependent solution is obtained, which is shown in the fourth column of fig. 5. Looking at the ground state of “Ne we find for example that the binding energy of -52.5 MeV, which is obtained by calculating the Q-box at -20 MeV, is reduced to -46.8 MeV if the energy dependence of the Q-box is treated self-consistently. A similar reduction of the binding energy can also be observed for the other states displayed in fig. 5.
H. Miither et al. / Many-body
75
terms
A=20,T=O
-37.01 -3&O-39.0-
4
2
2 -6
-z
0
-42.0 -43.0
4
1
-4o.o-41.0-
1 4
1
-!
62 -2”
5: 2 -44.0-
0 0
; -45.0F g -46.0-
4
4
2
2
4
w -47.0-
0
0 4
-4&o-
2
-49,0-5o.o-
2
0
-51.0-52.0-53.0-
a No Folds
+ P-body
+ 3-body
E-depend
Fig. 5. Energy spectra resulting from shell-model calculations for four valence nucleons in the 1sOd shell with isospin T=O. The result in the first column (no folds) is obtained when the Q-box terms of fig. 2 are calculated at the unperturbed starting energy. If the two-body folded diagrams are added, the spectrum in the second column (+2-body) is obtained. The results which are obtained after inclusion of the three-body folded diagrams, are displayed in the third column, while the results for the energy-dependent hamiltonian are given in the last column.
This feature can be understood simply: The Q-box terms of second order in G (figs. 2b-d, the argument can also be extended to the bare-G term of fig. 2a) give in general attractive contributions to the energies of the low-lying states. This attraction is reduced if the absolute value of the energy denominators in these terms is enhanced by using more negative starting energies. Therefore the Q-box calculated at -50 MeV is less attractive than calculated at -20 MeV, which would be the starting energy for four non-interacting nucleons. This reduction of the binding energy is reproduced in the folded diagram expansion. As a first approximation we restrict the folded diagrams to two-body terms. This can be done by evaluating the folded diagram expansion for a system of two valence nucleons. The effective hamiltonian defined by eq. (13) is then an operator consisting of a one- and a two-body operator, which can also be used for the calculation of 20Ne. Results of such a calculation are displayed in the second column of fig. 5. Looking again at the energy of the ground state, we observe that two-body terms of the folded diagrams only account for a reduction of the binding energy of 2.0 MeV as compared to the complete reduction of 5.7 MeV.
H. Miither et al. / Many-body terms
16
As a next step we perform
the folded diagram
expansion
for a system of 3 valence
nucleons. The hamiltonian resulting from this procedure the two-body folded diagrams obtained from the expansion real three-body
terms.
If this hamiltonian
contains in addition to in the two-nucleon case,
is used for a calculation
of *‘Ne results
are obtained as displayed in the third column. In this case one obtains a reduction of the binding energy of the ground state by 5.3 MeV, which is in a good agreement with the energy-dependent result. The agreement is even better for most of the excited states. Therefore one may say that the inclusion of three-body terms in the folded diagram expansion is indispensable. This has already been discussed in ref. r3). The four-body folded diagrams, however, which explain the differences between the third and fourth column of fig. 5, seem not to be that important. A=20,T=l
-30.0 -31.0 1 -32.07
-3 3.0
6 -34.0: 3
-35.0-
6
-5
&i L -36.0wc” -37.0-
I
7
6
6 -f
-65
_I
6 -f
i 0 1 ,z
-3&O-
4 A&
-39.0-
-;
2 0
-ul.o-
Fig. 6. Energy
2
-41.o-
No Folds
spectra
for four
+ 2-body valence
nucleons explanation
+ 3-body
E-depend
in the 1sOd shell with isospin see fig. 5.
T= 1. For further
The same features can also be observed for the states of four valence nucleons with isospin T = 1, which are presented in fig. 6. These states are less bound than the lowest states with T = 0 and therefore the energy difference between the states calculated without folds and the energy-dependent results are typically only 2 MeV. Also in this case the three-body folded diagrams are as important as the two-body terms and both yield repulsion. The situation is slightly different in the case of states with T =2 as displayed in fig. 7. In this case the three-body folded diagrams are in general attractive. For the
H. Miither et al. / Many-body
terms
77
A=20,T=2 -24.0
5
5
6
6
-25.0 1
5
5 -26.0 -;
-1
-27.0 2
6 -:
4
4
-2&O-
-‘;
-; -z % F -29.0w
&;
0
2
2
2
2
2 2
0
0 0 0 - 33.0 Fig. 7. Energy
spectra
for four
+ 3-body
+ 2-body
No Folds valence
nucleons explanation
E-depend
in the 1sOd shell with isospin see fig. 5.
T= 2. For further
ground state this attraction even spoils the agreement of the folded diagram expansion with the energy-dependent solution obtained on the two-body level. Here one should keep in mind that, in this case, the corrections are smaller (less than 1 MeV) than for the states discussed above. For most of the excited states, however, the three-body terms again improve the agreement. This is especially true for states like the lowest 6+, which are more bound in the calculation without folds.
in the energy-dependent
calculation
than
4. Conclusions The folded diagram expansion of the effective hamiltonian to be used in shellmodel calculations contains two-body, three-body and up to N-body terms, if N is the number of valence nucleons under consideration. It is the aim of the present work to discuss the relative importance of these effective many-body forces. For that purpose we investigate the case of four valence nucleons in the 1sOd shell. For the single-particle energies the degenerate energies of the harmonic oscillator model, which allows the use of the derivative method in calculating the folded diagrams. For the expansion of the effective hamiltonian only two-body Q-box terms of second order in the Brueckener G-matrix were considered. One-body terms in the Q-box were neglected to minimize the effects of disconnected diagrams. Using these
H. Miither et al. / Many-body terns
78
approximations good agreement
one cannot expect to obtain shell-model results which are in very with the experimental data. However, the model is realistic enough
to allow conclusions about the importance of the many-body forces mentioned above. It is observed that the three-body folded diagrams are of the same importance as the two-body folded diagrams. The effects of the four-body smaller. Altogether the folded diagrams give rise to changes
terms are considerably in energy up to almost
6 MeV. The folded diagrams are in general repulsive for low-lying states. Only for states with aligned isospin (T = 2; 2oO) attractive contributions are obtained. The effective three-body forces obtained in the folded diagram expansion seem to be much more important than the “real” three-body forces which are discussed, e.g., in refs. 7,8). These three nucleon forces, both real and effective, might be represented by an effective two-body force which depends on the mass number of the nucleon under consideration. We would like to thank the “Bundesministerium fiir Forschung (BMFT) for partial financial support of this work.
und Technologie”
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
S.A. Coon, M.D. Scadron and B.R. Barrett, Nucl. Phys. A242 (1975) 467 T. Kouki, C.M.W. Smulter and A.M. Green, Nucl. Phys. A290 (1977) 38 W.H. Dickhoff, A. Faessler and H. Miither, Nucl. Phys. A389 (1982) 492 C. Hajduk, P.U. Sauer and W. Strueve, Nucl. Phys. A405 (1983) 581 S.A. Coon, J.G. Zabolitzky and D.W.E. Blatt, Z. Phys. A281 (1977) 137 A. Faessler, H. Miither, K. Shimizu and W. Wadia, Nucl. Phys. A333 (1980) 428 S.A. Coon, R.J. McCarthy and C.P. Malta, J. of Phys. G4 (1978) 183 A. Polls, H. Miither, A. Faessler, T.T.S. Kuo and E. Osnes, Nucl. Phys. A401 (1983) 124 P.K. Rath, H. Miither, A. Polls, A. Faessler, B. Metsch and D. Zwarts, Nucl. Phys. A427 (1984) 511 A. Poves and A. Zucker, Phys. Reports 70 (1981) 235 B.R. Barrett, private communication T.T.S. Kuo, in Lecture Notes in Physics, ed. T.T.S. Kuo and S.S.M. Wong, vol. 144 (Springer, Berlin, 1981) p. 248 H. Miither, A. Polls and T.T.S. Kuo, Nucl. Phys. A435 (1985) 548 B.H. Wildenthal, Bull. Am. Phys. Sot. 27 (1982) 725 J. Shurpin, H. Miither, T.T.S. Kuo and A. Faessler, Nucl. Phys. A293 (1977) 61 B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771 S.Y. Lee and K. Suzuki, Phys. Lett. 91B (1980) 173 R.V. Reid, Ann. of Phys. 82 (1974) 411
A442 (1985) 68-78 Publishing Company
MANY-BODY TERMS IN THE FOLDED DIAGRAM EXPANSION H. MOTHER, Institut ffir Theoretische
A. POLLS*,
P.K. RATH
and
Physik, Auf der Morgenstelle
AMAND
FAESSLER
14, D-7400 Tiibingen, West Germany
Received 27 December 1984 (Revised 8 March 1985) Abstract: The folded diagram expansion for the effective hamiltonian to be used in shell-model calculations for a system with A valence nucleons yields folded diagrams connecting up to A nucleons. The importance of such effective many-nucleon forces is investigated for the example of 4 valence nucleons in the 1sOd shell (“‘Ne). The folded diagram terms turn out to reduce the binding energy for the low-lying states. The effects of the folded diagrams involving three nucleons are of the same importance as those of the two-body terms. The four-nucleon forces originating from folded diagrams are considerably weaker. This suggests that three-body operators have to be considered in a microscopic calculation of an effective energy-independent shell-model hamiltonian, whereas terms involving four nucieons may be ignored.
1. Introduction The investigation of many-nucleon forces has always been a very attractive subject in nuclear physics. Such many-body forces, especially of course the three-nucleon forces, might be important for the understanding of the bulk properties of nuclear calculation of the excitation ground-states im6) as well as for a detailed microscopic spectra 7,8). The hope is that one can establish the effects of a “real” three-nucleon force originating, e.g. from meson rescattering or inner excitations of the nucleons in a nuclear many-body system. In all nuclear structure calculations, however, there is the problem that the effects of such real many-nucleon forces interfere with many-body terms of the conventional nuclear many-body theory which assumes two-nucleon forces only. For the calculation of ground-state properties one must be aware that effects of three-nucleon correlations, which can be taken into account by solving a Bethe-Faddeev equation, are of similar importance to those from three-nucleon forces 3). A three-body operator in the hamiltonian to be used in nuclear shell-model calculations may originate from a real three-nucleon force, but may also be due to the renormalisation of the hamiltonian, which is necessary to account for the degrees of freedom outside the shell-model space under consideration. Therefore one has to perform a careful * Permanent
address:
Departament
de Fisica Teorica, 68
Universitat
de Barcelona,
Barcelona-28,
Spain.
69
H. Miither et al. / Many-body terns
investigation structure
of such effective
three-nucleon
effects to real three-nucleon
forces before
one may relate nuclear
interactions.
It has been observed that the lowest-order terms in a conventional perturbation expansion for the effective three-nucleon interaction (terms of second order in the Brueckner G-matrix) give a very weak contribution if a realistic shell-model space is considered which contains all configurations within one major shell “). In this case, the effects of the three-nucleon force on systems with three valence nucleons in the 1sOd shell were found to be negligible. The weakly repulsive effects of this effective three-nucleon force become visible only if four or more valence nucleons are considered ‘). For a small model space containing, e.g., the configurations within one subshell only, the effective three-nucleon terms seem to be much larger, since in this case a renormalisation due to intermediate Ohw is required ‘OT1r). The shell-model hamiltonian determined in a perturbation expansion to account for the nucleonic degrees of freedom outside the model space depends on the energy of the state to be calculated. This energy dependence makes the hamiltonian to be used for the calculation of a system with three valence nucleons different from the one to be used for two nucleons. Therefore this energy dependence can also be understood as an effective three-body, or in general many-body, interaction. The explicit energy dependence of the hamiltonian can now be eliminated by introducing folded diagrams I*). Applying the folded diagram expansion to obtain an energyindependent shell-model hamiltonian for a system of three valence nucleons, one obtains connected folded diagrams which involve three nucleons 13). This supports the interpretation of the energy-dependent hamiltonian in terms of effective manybody forces. In fact, for the example of three valence nucleons in the 1sOd shell 13), these three-body folded diagrams turned out to be much more important than the terms of second order in the G-matrix discussed, e.g., in ref. “). Therefore the folded diagram expansion yields a non-negligible effective threebody force. The same three-body force should also be contained in energy-independent empirical shell-model hamiltonians which are determined by a fit to experimental data. It might be possible, however, to simulate the effects of the many-body forces by a proper adjustment of the matrix elements. of the two-body operator or by introducing an effective NN interaction, which depends on the mass number 14). Now the next question is of course: what happens to the folded diagram expansion in the case of four or more valence nucleons? Is it sufficient to consider folded diagrams involving up to three nucleons or must we take into account the effects of effective four-body forces as well? If n-body forces turn out to be important in general for the case of n valence nucleons, one should probably give up the idea of using the folded diagram expansion to determine the effective hamiltonian to be used for shell-model calculations for all nuclei within a certain shell. It is the aim of the present paper to investigate this question. For that purpose we repeat some features of the folded diagram expansion in sect. 2. Results for calculations
70
H. Miiiher et al. / Many-body terms
considering four valence nucleons are presented section contains our main conclusions.
and discussed
in sect. 3. The last
2. Folded diagram expansion To review some features of the folded diagram expansion for an effective shellmodel hamiltonian we split the hamiltonian H for the A-nucleon system into a one-body operator Ho and the rest H,, which may contain one- and two-nucleon operators, H=Ho+H,.
(1)
It is useful of course to choose H,, in such a way that H, is made small and can be treated as a perturbation. This could be achieved by using the Hat-tree-Fock or Brueckner-Hartree-Fock approximation to define Ho [ref. “)I. The A-nucleon eigenfunctions of Ho are then used to define the model-space or P-space. For a shell-model calculation the basis of the P-space typically consists of those Slater determinants for which the shells of a core nucleus (e.g. 160) are completely occupied and the remaining valence nucleons occupy states within the major shell above the core (e.g. the 1sOd shell). It is then the aim to derive an effective hamiltonian He* such that PHe,P!l’i = E,py,
(2)
is fulfilled for the low-lying states of all nuclei one would like to investigate within the given shell-model approach. The energies Ei and wave functions !Pi are the exact eigenvalues and states of the complete hamiltonian H, and P is the projection operator projecting be written as
onto the model
space. This effective
hamiltonian
linked
.
can formally
(3)
The projection operator Q is used to restrict all intermediate states to configurations outside the model space. The subscript “linked” on the right-hand side of eq. (3) is used to indicate that only those terms should be considered in the expansion which are represented by a linked diagram. Examples of linked diagrams are given in figs. la and lb. They are characterized by the fact that all H, interaction lines (wiggly lines) are connected to at least one valence nucleon (upward-going lines). Core excitations which are not connected to the valence nucleons (see fig. lc) have to be considered as unlinked diagrams. It should be noticed that the energy denominator of eq. (3) contains the exact energy Ei of the state under consideration. Therefore the effective hamiltonian is different for each state, which of course also implies that it is different for different numbers of valence nucleons. As a consequence this energy dependence can be understood as a source of effective many-body forces.
H. Miither et al. / Many-body terms
da1 a)
b)
71
c)
Fig. 1. A linked diagram (a), a disconnected diagram (b), and an unlinked diagram (c) for a system of four valence nucleons.
There are two arguments which speak against the use of an effective energydependent hamiltonian. First, it is not so convenient for treating an energy-dependent hamiltonian. The evaluation of the eigenvalues is more complicated and the resulting eigenstates are not orthogonal. Secondly, the success of the energy-independent empirical hamiltonians demonstrates that it should be possible to determine an effective hamiltonian which does not depend on the energy. Formally this can be achieved by introducing the folded diagram expansion. Loosely speaking, one expands the energy denominator of eq. (3) to eliminate the energy difference between the energy of the complete hamiltonian H and the energy E0 of the unperturbed hamiltonian Z-J0from this denominator. This yields the folded diagram expansion of &i, HeE= H,,+H,+
H, & I
Note, that in this formulation Ei any more. The eigenvalues
He,+ folded 0
0
diagrams
. I
(4)
linked
He@ does not explicitly depend on the exact energy of the hamiltonian given in eq. (4) are identical to
solutions of the hamiltonian of eq. (3). In general, however, the energy-dependent formulation yields more solutions than the energy-independent one. A more detailed discussion of the folded diagram expansion can be found, e.g., in refs. ‘**i6). In a practical application of the folded diagram expansion one normally proceeds as follows. One first selects the terms to. be calculated in the energy-dependent expansion of eq. (3). Translated in the language of diagrams this means that one is selecting a certain set of diagrams which must be linked (see above) and irreducible (i.e. all intermediate states should be outside the model space). As typical examples for such diagrams we display in fig. 2 those diagrams which shall also be considered in our numerical investigations. Now the lowest-order approximation of the folded diagram expansion is obtained by calculating these diagrams using the unperturbed Rayleigh-Schrodinger energy denominators. These terms are identified as the Q-box terms. F,:= Q. (5) For a system with A valence nucleons these Q-box terms must be understood as an operator connecting A-nucleon wave functions. The first-order correction (Fl)
H. M&her et al. / Many-body terns
12
a) Fig. 2. Two-body
b)
d)
c)
diagrams which are used to define the Q-box. The intermediate diagram (b) are restricted to two particles in the 1pOf shell.
2-particle
states in
now contains all terms with one fold. In terms of diagrams this means that two Q-box terms must be connected leading to an intermediate model space state, and the lines of the intermediate state must be folded. For three valence nucleons such combinations are displayed in fig. 3. If the Q-box contains two-body interaction terms only and no single-particle terms, all the folded diagrams for three nucleons are connected. The diagram displayed in fig. 3a represents a two-body term with
-
al Fig. 3. Two-body
b)
(a) and three-body
terms (b, c) in the folded valence nucleons.
c) diagram
expansion
for a system of three
the third valence nucleon being a spectator, while diagrams 3b and 3c are real three-nucleon terms. In the case of four valence nucleons one finds diagrams for F1 which are disconnected (see fig. 4a). These diagrams, which are of course also contained in the energy-dependent formulation are identical in magnitude to Q-box terms displayed in fig. 4b. Due to the additional minus sign in the folded diagram
H---+Hl-.=O b)
0)
Fig. 4. The disconnected
folded
diagram
(a) is cancelled
by the disconnected
Q-box
term (b).
H. Miither et al. / Many-body terms
the two diagrams the disconnected
cancel
each other.
diagrams
The calculation
that one should
in the Q-box terms and also in the folded
of the folded
the unperturbed energies calculated very easily
This demonstrates
73
diagrams
is in general
of the model-space F,:=
states
quite involved. are degenerate,
leave out
diagrams. If, however, F, can
be
(6)
Q,Q,
if we define
The product Q,Q in eq. (7) is a product of operators functions. Also higher-order corrections in the folded calculated in a similar way 13).
acting on A-nucleon wave diagram expansion can be
It turned out, however, that a grouping of the folded diagram terms in this conventional method leads to convergence problems already for systems with three valence nucleons r3). The convergence is much better if the folded diagrams are regrouped following a proposal by Lee and Suzuki “). In this case the effective hamiltonian is defined as &= The members
(8)
n-02
R, are given
of the sequence Ro=
Ho+ lim R,. as
Q,
RI = Cl- QA-‘9,
(9) Since the convergence
of the Lee-Suzuki
iteration
for the conventional method, the Lee-Suzuki folded diagrams in the calculations discussed
scheme
seems to be better than
method has been used to sum the in the next section.
3. Results and discussion In order to investigate the importance of many-body forces in the folded diagram expansion we consider the example of four valence nucleons occupying a singleparticle states in the 1sOd shell. The single-particle energies (the eigenvalues of the hamiltonian Ho) are chosen to be degenerate for all states within one major shell of the oscillator model ~~=h~(~~-22)-5 In this equation
N (i = 0, 1,2, . . .) is th e main
(10)
MeV. quantum
number
of the oscillator
74
model choice
H. M&her et al. / Many-body terms
and the energy hw has been chosen to be 14 MeV. With this degenerate for the single-particle energies the “derivative” method (see eqs. (6)-(g))
can be used to calculate the folded diagrams. The residual energy-dependent interaction between the valence nucleons is defined by the Q-box terms displayed in fig. 2. The wiggly lines in these diagrams should be identified with matrix elements of the Brueckner G-matrix. For our calculations this G-matrix was obtained by solving the Bethe-Goldstone equation using the Reid soft-core ‘*) potential. The Pauli operator in the Bethe-Goldstone equation has been chosen to suppress scattering into intermediate states with one interacting nucleon in the OS or Op shell or both nucleons in the 1sOd or 1pOf shell. Therefore we can add explicitly the Q-box terms of fig. 2b, where the nucleons are scattered into intermediate states with two particles in the 1pOf shell. The summation over intermediate states in diagrams 2c and 2d was restricted to states with 2hw excitation energy. The matrixelements of G were calculated in an oscillator basis with an oscillator constant (b = 1.72 fm) which corresponds to hw = 14 MeV and the dependence of G on the starting energy was taken into account. For a more realistic calculation one should also consider single-particle terms for the Q-box. These single-particle terms, however, give rise to many disconnected diagrams which should be eliminated from the expansion. For the case of two valence nucleons this can be achieved by using the method discussed, e.g., in of the refs. 12,i3). But already for the case of three valence nucleons the elimination disconnected diagrams seems to be very difficult. In order to avoid these difficulties we have ignored any single-particle terms in the Q-box. For the two-body terms as displayed e.g. in fig. 2 all folded diagrams are connected for the cases of two and three valence nucleons and even in the case of four valence nucleons the disconnected diagrams are suppressed by a statistical factor of i as compared to the connected diagrams with one fold. Some results of shell-model calculations for energies of states in “Ne with isospin T= 0 are displayed in fig. 5. The spectrum displayed in the leftmost column is obtained effective
by ignoring hamiltonian
the degenerate
all folded diagrams and using the approximation R. in the of eq. (8). This means that the hamiltonian consists out of
single-particle
energies
(-5 MeV) and the two-body
Q-box terms of
fig. 2 calculated at the unperturbed energy of -20 MeV for the four-nucleon system, which corresponds to a starting energy of -10 MeV for the two-body diagrams. If one determines the starting energy of the Q-box for each individual state in a self-consistent way by requesting that this energy corresponds to the final energy of this state, the energy-dependent solution is obtained, which is shown in the fourth column of fig. 5. Looking at the ground state of “Ne we find for example that the binding energy of -52.5 MeV, which is obtained by calculating the Q-box at -20 MeV, is reduced to -46.8 MeV if the energy dependence of the Q-box is treated self-consistently. A similar reduction of the binding energy can also be observed for the other states displayed in fig. 5.
H. Miither et al. / Many-body
75
terms
A=20,T=O
-37.01 -3&O-39.0-
4
2
2 -6
-z
0
-42.0 -43.0
4
1
-4o.o-41.0-
1 4
1
-!
62 -2”
5: 2 -44.0-
0 0
; -45.0F g -46.0-
4
4
2
2
4
w -47.0-
0
0 4
-4&o-
2
-49,0-5o.o-
2
0
-51.0-52.0-53.0-
a No Folds
+ P-body
+ 3-body
E-depend
Fig. 5. Energy spectra resulting from shell-model calculations for four valence nucleons in the 1sOd shell with isospin T=O. The result in the first column (no folds) is obtained when the Q-box terms of fig. 2 are calculated at the unperturbed starting energy. If the two-body folded diagrams are added, the spectrum in the second column (+2-body) is obtained. The results which are obtained after inclusion of the three-body folded diagrams, are displayed in the third column, while the results for the energy-dependent hamiltonian are given in the last column.
This feature can be understood simply: The Q-box terms of second order in G (figs. 2b-d, the argument can also be extended to the bare-G term of fig. 2a) give in general attractive contributions to the energies of the low-lying states. This attraction is reduced if the absolute value of the energy denominators in these terms is enhanced by using more negative starting energies. Therefore the Q-box calculated at -50 MeV is less attractive than calculated at -20 MeV, which would be the starting energy for four non-interacting nucleons. This reduction of the binding energy is reproduced in the folded diagram expansion. As a first approximation we restrict the folded diagrams to two-body terms. This can be done by evaluating the folded diagram expansion for a system of two valence nucleons. The effective hamiltonian defined by eq. (13) is then an operator consisting of a one- and a two-body operator, which can also be used for the calculation of 20Ne. Results of such a calculation are displayed in the second column of fig. 5. Looking again at the energy of the ground state, we observe that two-body terms of the folded diagrams only account for a reduction of the binding energy of 2.0 MeV as compared to the complete reduction of 5.7 MeV.
H. Miither et al. / Many-body terms
16
As a next step we perform
the folded diagram
expansion
for a system of 3 valence
nucleons. The hamiltonian resulting from this procedure the two-body folded diagrams obtained from the expansion real three-body
terms.
If this hamiltonian
contains in addition to in the two-nucleon case,
is used for a calculation
of *‘Ne results
are obtained as displayed in the third column. In this case one obtains a reduction of the binding energy of the ground state by 5.3 MeV, which is in a good agreement with the energy-dependent result. The agreement is even better for most of the excited states. Therefore one may say that the inclusion of three-body terms in the folded diagram expansion is indispensable. This has already been discussed in ref. r3). The four-body folded diagrams, however, which explain the differences between the third and fourth column of fig. 5, seem not to be that important. A=20,T=l
-30.0 -31.0 1 -32.07
-3 3.0
6 -34.0: 3
-35.0-
6
-5
&i L -36.0wc” -37.0-
I
7
6
6 -f
-65
_I
6 -f
i 0 1 ,z
-3&O-
4 A&
-39.0-
-;
2 0
-ul.o-
Fig. 6. Energy
2
-41.o-
No Folds
spectra
for four
+ 2-body valence
nucleons explanation
+ 3-body
E-depend
in the 1sOd shell with isospin see fig. 5.
T= 1. For further
The same features can also be observed for the states of four valence nucleons with isospin T = 1, which are presented in fig. 6. These states are less bound than the lowest states with T = 0 and therefore the energy difference between the states calculated without folds and the energy-dependent results are typically only 2 MeV. Also in this case the three-body folded diagrams are as important as the two-body terms and both yield repulsion. The situation is slightly different in the case of states with T =2 as displayed in fig. 7. In this case the three-body folded diagrams are in general attractive. For the
H. Miither et al. / Many-body
terms
77
A=20,T=2 -24.0
5
5
6
6
-25.0 1
5
5 -26.0 -;
-1
-27.0 2
6 -:
4
4
-2&O-
-‘;
-; -z % F -29.0w
&;
0
2
2
2
2
2 2
0
0 0 0 - 33.0 Fig. 7. Energy
spectra
for four
+ 3-body
+ 2-body
No Folds valence
nucleons explanation
E-depend
in the 1sOd shell with isospin see fig. 5.
T= 2. For further
ground state this attraction even spoils the agreement of the folded diagram expansion with the energy-dependent solution obtained on the two-body level. Here one should keep in mind that, in this case, the corrections are smaller (less than 1 MeV) than for the states discussed above. For most of the excited states, however, the three-body terms again improve the agreement. This is especially true for states like the lowest 6+, which are more bound in the calculation without folds.
in the energy-dependent
calculation
than
4. Conclusions The folded diagram expansion of the effective hamiltonian to be used in shellmodel calculations contains two-body, three-body and up to N-body terms, if N is the number of valence nucleons under consideration. It is the aim of the present work to discuss the relative importance of these effective many-body forces. For that purpose we investigate the case of four valence nucleons in the 1sOd shell. For the single-particle energies the degenerate energies of the harmonic oscillator model, which allows the use of the derivative method in calculating the folded diagrams. For the expansion of the effective hamiltonian only two-body Q-box terms of second order in the Brueckener G-matrix were considered. One-body terms in the Q-box were neglected to minimize the effects of disconnected diagrams. Using these
H. Miither et al. / Many-body terns
78
approximations good agreement
one cannot expect to obtain shell-model results which are in very with the experimental data. However, the model is realistic enough
to allow conclusions about the importance of the many-body forces mentioned above. It is observed that the three-body folded diagrams are of the same importance as the two-body folded diagrams. The effects of the four-body smaller. Altogether the folded diagrams give rise to changes
terms are considerably in energy up to almost
6 MeV. The folded diagrams are in general repulsive for low-lying states. Only for states with aligned isospin (T = 2; 2oO) attractive contributions are obtained. The effective three-body forces obtained in the folded diagram expansion seem to be much more important than the “real” three-body forces which are discussed, e.g., in refs. 7,8). These three nucleon forces, both real and effective, might be represented by an effective two-body force which depends on the mass number of the nucleon under consideration. We would like to thank the “Bundesministerium fiir Forschung (BMFT) for partial financial support of this work.
und Technologie”
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
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