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Investigation of a matrix technique to produce an effective hamiltonian as applied to the p-shell
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 594 (1995) 57-69
Investigation of a matrix technique to produce an effective hamiltonian as applied to the p-shell B.R. Caldwell, M.R. Meder Department of Physics and Astronomy, GeorgiaState University,Atlanta, GA 30303, USA
Received 25 May 1994; revised 5 July 1995
Abstract Several aspects of a technique for calculating effective interactions, which avoids the perturbation expansion while eliminating the effects of unlinked graphs, are investigated. In the course of this investigation the effective p-shell interactions are calculated for the Reid Soft Core, the Paris, and the coordinate space Bonn interactions. The convergence properties of two approaches to summing the folded diagram series, that of Krenciglowa and Kuo and that of Suzuki and Lee are compared. The convergence properties of the latter, in the cases studied, are found to be superior. We find that removing the influence of unlinked graphs has little effect on excitation energies and wave functions and may be characterized by a mass dependent shift in the binding energy. In addition we note that these calculated excitation energies agree quite well with experiment.
1. Introduction Shell-model calculations are traditionally performed with an effective interaction which contains only one- and two-body terms. However, an effective interaction derived from a two-body nucleon-nucleon interaction in a truncated model space acquires a many-body character. The justification generally given for including only one- and two-body terms is that the contributions of effective three-body and higher terms, the many-body terms, are assumed to be negligible. The success of calculations designed to extract the matrix elements of the two-body effective interaction from the experimental energies of the p-shell nuclei, e.g. Ref. [1], suggests that the many-body components of the effective interaction are indeed small. However, these types of calculations have difficulty reproducing some of the experimental energy levels, particularly for A = 6 - 8 . There are two equally plausible explanations for this behavior. One is that some of the levels of the p-shell nuclei should have wave functions with large amplitudes for configurations outside the p-shell model space. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00328-2
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B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
Therefore the energies of these states cannot be calculated from an effective two-body interaction appropriate to the p-shell. Alternatively, the behavior could signal the existence of a significant three-body component in the effective interaction originating from either the NN interaction or a possible NNN interaction. In a recent attempt to produce an empirical effective interaction appropriate to the p-shell [2,3] van Hees et al. have, in addition to the usual parameters appearing in such a calculation (the single-particle energies and two-body matrix elements) introduced a two-parameter schematic three-body force into the effective interaction, which we will denote as V ~3). Their approach produced a significant improvement in the fit to the experimental energy levels, reducing the rms deviation between the empirical and fitted energies of the normal parity levels by nearly a third. However, they were unable to estimate how much of the three-body component of the effective interaction was due to the existence of a possible nuclear three-body (NNN) force and how much was simply the NN interaction manifesting itself as a three-body effective interaction due to the truncated model space. In the present study we wish to examine some aspects of these problems more closely within the context of modern realistic interactions. To this end we calculate effective p-shell interactions derived from modern realistic NN interactions for nuclei of mass 4 through 7. These calculations include all excitations through 2 h to so that when the mass-7 interaction is projected into the p-sheU it acquires a many-body (three-body) character. In one standard method, the effective interactions would be calculated perturbatively through a diagrammatic approach. However, matrix methods are more attractive in that diagonalization of the hamiltonian matrix is equivalent to infinite order perturbation theory. Due to the necessary truncation of the basis [4], a difficulty arises when implementing the matrix approach. The effects of unlinked diagrams, which may be removed in the perturbative picture, will be included in the resulting effective interaction. Fortunately there is a procedure available, previously used by Skouras and Varvitisiotis [5] in p-shell calculations, which will allow us to use matrix methods to find the interaction while at the same time removing the effects of the unlinked diagrams. In their calculations, Skouras and Varvitisiotis found the energy-independent interaction using the Q-box technique [6] of Krenciglowa and Kuo (KK). In some cases they found that KK was not able to converge the matrix elements of the effective interaction completely. In this work we investigate the convergence properties of an alternative method for introducing energy independence, an iterative scheme [7] due to Suzuki and Lee (SL). To summarize, we calculate the zero- through three-body components of the p-sheU effective interaction from realistic interactions. In this context we compare the convergence properties of SL and KK in producing these energy-independent interactions. Further we examine the importance of the contributions of the unlinked diagrams, both to energies and wave functions. In addition we hope to provide a more accurate measure of the importance of the three-body components of the effective interaction derived from
B.R. Caldwel~ M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
59
realistic NN interactions than has been previously possible [8-10]. Finally, motivated by the success of van Hees et al., we will test the flexibility V <3) affords in fitting these matrix elements.
2. Model calculations for masses 4 through 7
2.1. The model hamiltonian In this subsection we briefly discuss the model hamiltonian and the form of the residual interaction. A more detailed discussion may be found in Ref. [11]. The hamiltonian of a system consisting of A nucleons of mass m may be written in the following translationally invariant form: A 2. )
where Gij is the two-body interaction between nucleons i and j, and Pij is relative momentum of the pair, Pij = (Pi-P~)A/2, that is the momentum conjugate to the relative coordinate Pii = (ri - rj)/~/2. The first term in Eq. (1) yields the kinetic energy of the system relative to the center of mass and the second, the system's potential energy. It is convenient to rewrite Eq. (1) as follows:
H
=
~ ( -2- h o i j i
+Gi/-
--AUOidj
+ /3[H0(c.m. ) -- 3htol ,
(2)
where ho~j =p2./2m + Uo~j and Uoi/ = mto2p2./2, a harmonic-oscillator hamiltonian and a harmonic-oscillator potential written in the relative coordinates of particles i and j. The term proportional to fl gives the excitation energy of the center of mass of the system and /3 is a dimensionless parameter which may be adjusted to lift states for which the center of mass of the system is in an excited state out of the energy region of interest. The hamiltonian may now be written in the standard shell-model form A
A
H = ~.,Ho(i ) - n0(c.m. ) + E V/i + fl [n0(c-m.) - 3hw] • i
(3)
i
Ho(i) is a harmonic-oscillator hamiltonian operating on the coordinates of particle i, and V~i, the residual interaction, is given by the expression 2
Vii = Gij - -~Uoi j.
(4)
2.2. The two-body effective interaction It is well known that it is impossible to solve the Schr6dinger equation with the nuclear interaction exactly. This is primarily due to an inability to deal with the
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B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
short-ranged repulsion exhibited by all realistic NN interactions. The truncated model space simply does not allow the wave function the necessary freedom to respond to the repulsion. The effects of the short-ranged repulsion may be overcome by introducing the reaction matrix or, as it is generally called when dealing with bound-state problems, the G-matrix. The two-particle G-matrix is defined as follows: q G(to) = v + v G(to). (5) to - Ho(2p) In Eq. (5), v is the realistic two-body interaction, H0(2 p) = H0(1) + H0(2) , and to is a parameter of the calculation, the starting energy. In the calculations described below the operator q is a projection operator which projects onto the space of two-particle harmonic-oscillator states which have energies greater than 4 h to. The G-matrix elements appearing in the calculations described below have been calculated using a variation [12] on the reference spectrum method of Barrett et al. [13].
2.3. The p-shell energy-independenteffective hamiltonian A standard working hypothesis would be that the diagonalization of Eq. (3) in the largest possible space provides us with the best estimate of the energies and wave functions of the p-shell nuclei. However, two issues arise when these results of such a calculation are projected into the p-shell model space. The two-body interaction appearing in Eq. (3) becomes a many-body, energy-dependent effective interaction. That is, each eigenstate appears to arise from a different effective hamiltonian. Further, from the point of view of perturbation theory, it can be shown that if the space in which Eq. (3) is diagonalized is not infinite [14] one must be careful to exclude the spurious contributions to the energy due to unlinked diagrams. Fortunately methods have been devised [14,15] which deal with both of these difficulties, at least within a model space which allows 0 + 2 h to excitations. We will briefly describe these methods here. A more detailed discussion may be found in Ref. [16] and the references contained therein. We introduce an operator P which projects onto the 0 hto space, which is of dimension /x, and a operator Q which projects onto the 2 h to space of dimension M. As was shown in Ref. [16], it is convenient to perform the required calculations in a basis where PHP and QHQ are separately diagonal with eigenvalues w,~ and W/ respectively:
PnPly~)=w~ly~), analYi)=wilY~),
a = 1, 2 , . . . , / x ,
(6)
i = 1 , 2 . . . . . M.
(7)
The basic building blocks to be used in constructing the effective interaction are the matrix elements of the operator R n, M
(y=lR, ly a)
=
( - 1 ) " ~ (y=lVIY~)(wa i=1
w,)
-(,+1)
Iv I
).
(8)
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
61
In a situation where the Q-space is restricted to 2 h to excitations each R n consists of linked-plus-folded valence diagrams and linked-core diagrams. In order to remove the effects of unlinked diagrams on the energy of the core, one must also calculate the matrix elements of the operator EI, where E 1 = PVP. The total energy of 4He may now be written in terms of these quantities,
EC = H~ + E~ + R~ .
(9)
The superscript c refers to the ground state of the closed core, in this case 4He. The single-particle energies, two-body matrix elements, and the three-body matrix elements of the p-shell effective interaction are also formed from combinations of the operators R , and E v In order to write the somewhat complex expressions for these operators most efficiently, Skouras and Varvitsiotis expanded the p-shell basis vectors, ](n 1 nz)"JTN), in terms of an alternative set of vectors which readily allow one to evaluate an m-body operator. The quantities n 1 and n 2 are the occupation numbers of the P3/z and P1/2 states, where n I + n 2 = n, and n is the number of p-shell valence particles. This expansion may be written in terms of the products of coefficients of fractional parentage and states of m and m' particles, which are separately antisymmetric, ](nln2)"JTU)
= ~,
~,
~,
[(mlmz)mj1T1Nl,(rttln~z)m'j2TzNz;JT)
rnlmz J1T1N1 JzTzN2
× [(m~mz)mjtT1Nl,(~ltrgz)m'jzTzN2;
JTI}(nlnz)"JTN ] .
(10)
The symbol [ [}] is a fractional parentage coefficient, m'i = n i - mi, and N is a quantum number which distinguishes among states having identical occupation numbers, spin and isospin. We now define the matrix elements of the n-body operators, which are denoted as e 1 and rk, as follows:
(( nlnz) nj T u I w l( n'ln'2) njTN ') = (( nln 2) "JTN I W I(n'ln'2) "JTN')
-- E rn= 1
E
E E ((mlm2)
J1TINIIwI(m3m4)
J1T1N~)
mlm 2 J1T1 NIN2
'n3m4 J2r2 ~V~
× [ ( m l m 2 ) mj1T1NI( m'a m'2 ) r"'J2T2N2; J r l } ( n,n:)"JrU] × [(m3m4)rnJ1T1g~(tt~3tr[4)m'J2T2N2;
-
8,,,,,w c,
JTI}(n'In'2)"JTN'] (11)
where W stands for the quantities Ea and R k defined above and w for the quantities e 1 and r k respectively.
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B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
2.4. Summing folded diagrams The effective interaction can be obtained by adopting the Q-box techniques of Krenciglowa and Kuo [6]. The quantity r o is considered to be the Q-box while r k (k > 0) is taken to be the kth derivative of the Q-box. The effective interaction is then found from the series Veff = E F i , i
(12)
where
F o = e l + r o,
Fx=rlro,
F 2 = r z r 2 + r ~ r o,
F3 = r3r3 + r2rlr2 + r2rorlro + rlr2r2 + r3ro"
(13)
These expressions, which represent the contributions of the additional folded diagrams, and those of higher order, which we omit due to their complexity, may be found using the rules contained in Ref. [6]. An alternative approach is to implement the method of Suzuki and Lee [7]. We will refer to this method as " S L " while the previous method will be denoted as " K K " . In the SL approach the quantity r~ is still treated as the nth derivative of the Q-box but now the effective interaction is calculated iteratively as follows: We define a matrix Z~ = V~f 1 - E 1,
(14)
where n = 1,2 . . . . and Ve°f= r 0 .
(15)
The iteration process is then defined as follows: Z. =A~xZ1,
(16)
where A n --
n-1 1 - r 1 - E rk k=2
n-1 I-I
Zm.
(17)
m=n-k+l
If the matrix Z~ converges, the effective interaction is Vdf = Z. + e~,
(18)
and so nef f = H o + Veff.
(19)
2.5. Parameterization of the three-body interaction We follow the suggestion of Ref. [2] and introduce a simple schematic parameterized form for the three-body interaction V°) = E ( a ( o - t "~rz, °'1" °'3}(~'1 "z2, zl" ~'3} cycl
+C[O-1 "0"2' O"1" O"3]["/'1 "72, '/'1" ~/'3])"
(20)
B.R. CaMwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
63
The summation runs over the three cyclic permutations of particles 1, 2 and 3. The braces and brackets represent anticommutators and commutators respectively. The structure of this force is similar to that which would be obtained from the exchange of two pions. However, the tensor contribution to the force and the radial dependence have been suppressed. Since this simplified version of the three-body interaction was able to successfully reproduce the empirical data [2] it might also exhibit similarities to the three-body interaction calculated in the manner described above, at least to the extent in which our calculations mirror reality. There is no a priori reason to assume that the form of V ~3) is appropriate for the effective three-body force originating from a two-body force. However, if V ~3) has sufficient flexibility it should be possible to condense the information obtained from the calculations described above into the values of the two strength parameters, A and C. Further, this parameterization allows a direct comparison with the results of Ref. [2].
3. Results and discussion 3.1. N N interactions for which effective interactions are derived The effective interactions for masses 4, 5, 6 and 7 were calculated in the manner described above. Three realistic interactions were used in these calculations: the Reid Soft Core (RSC) [17], Paris [18], and Bonn (OBEPR) [19]. The RSC was chosen because it is widely used as a benchmark, while the other two were chosen because they are relatively modem. Further, these interactions span (and exceed) the generally accepted range for the strength of the tensor component of the interaction as characterized by the D-state admixture in the deuteron wave function calculated from each of them. Finally, the group included both energy-dependent and energy-independent interactions. (We note that, in this context, the term "energy dependent" indicates a dependence on the relative kinetic energy of the pair of interacting nucleons rather than a dependence on the energy eigenvalue as is the case with the effective interaction). The effective interaction was also calculated for the Sussex interaction [20]. Our purpose in doing so was to check the results obtained with our programs against those presented in Ref. [5]. 3.2. Convergence In this section we investigate the relative rate of convergence for the two methods of obtaining the effective interaction, KK and SL, discussed in Section 2.4. The KK results presented in the current section include terms through F 5 although the expressions for F 4 and F 5 were not included in Eq. (13) due to the unwieldy number of terms in each, 14 in F 4 and 43 in F 5. In contrast to KK, the number of iterations which may be comfortably performed in SL is limited only by numerical considerations. The level of
64
B.R. CaldweU, M.R. Meder ~Nuclear Physics A 594 (1995) 57-69 500
>~
400
'" Z
300
¢ J,
SL KK
m I<3 -
-
200
100
!
1
2
3
4
5
6
ITERATION
Fig. 1. Comparison of the convergence rate of SL and KK for the diagonal and T = 0 calculated with the Sussex interaction.
(P3/2)2 matrix element with J
= 1
approximation represented by k folds is comparable to k iterations of SL since each require at most the kth derivative of the quantity r. For most of the examples presented in this paper a maximum k of five was sufficient to produce reasonable convergence. However, in all cases SL converges, at least marginally, more quickly than KK and in some cases a maximum value of k = 5 is not sufficient to produce convergence. Hence, in these instances SL is preferable. An example of this behavior is displayed in Fig. 1.
3.3. The effective p-sheU interaction If one considers at most three valence particles, the p-shell effective interaction has at most four terms, the zero-, one-, two-, and three-body parts, as indicated in Section 2. We will consider, in turn, the results for each of these terms below. First we examine the binding energy of 4He, that is the case where no valence particles are present. This quantity is characterized by EC: the core energy. The value of E ¢ depends strongly on the starting energy, to, used in calculating the two-body G-matrix (see Eq. (5)). A starting energy of h to = 16.2 MeV was chosen. Interestingly this choice gave a core energy approximately equal to the available mass-4 binding-energy calculations for the three NN potentials considered here. Hence the values obtained for E c represent an underbinding of the core with respect to experiment by 4 to 5 MeV.
Table 1 Core and single-particle energies (MeV) Nucleus 4He SHe
State
RSC
Paris
OBEPR
3/2 1/2
- 22.004 4.004 7.008
- 21.592 4.075 7.164
- 26.056 3.721 7.021
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69 6
---br--
CK
Paris
O
4
65
2
o =J
0
e¢D
=_
-2
eo
-4 -6 J
-8
0i
i~
2i
2i
w ~
~
•"
0i
1i
2i
0i
~
g
g
g
T:I
"
T=O
Fig. 2. Comparison of the TBME calculated from the Paris interaction and the empirical TBME of Cohen and Kurath (CK). The matrix elements are identified by 2j] 2j2 2j3 2j4. Each calculated off-diagonal matrix element is represented by two values, the matrix element indicated and its hermitian conjugate. Thus the small extent to which the calculated off-diagonal matrix elements are non-hermitian is displayed.
The results for both E ¢ and the single-particle energies (the one-body contribution to the effective interaction) are presented in Table 1. The single-particle energies are found to be essentially independent of the realistic two-body interaction from which they are calculated. Further, the P1/2-P3/2 splitting compares favorably with experimental estimates, although the energies themselves do not. The effective two-body matrix elements for all realistic interactions were found to be similar to each other and to experimentally determined matrix elements. In Fig. 2 the two-body matrix elements of Veff for the Pads potential are compared with the empirical matrix elements of Cohen and Kurath. In order to display the degree of non-hermitian behavior we present both the matrix elements and their hermitian conjugates. It can be seen from Fig. 2 that the asymmetry in the off-diagonal matrix elements is generally quite small, at most of the order of 100 or 200 keV as measured from the mean. However, the non-hermitian character of the matrix elements was preserved in our calculations. The three-body matrix elements of the effective interaction were extracted from the mass-7 calculations. The converged matrix elements range in magnitude from tens of keV to the order of an MeV. Unfortunately it was found that neither KK nor SL produced convergence in the J ~ = 1 / 2 - , T = 1 / 2 matrix elements for the Paris and Bonn interactions. We interpreted this to mean that the three-body part of these interactions was too small to obtain accurate results when subtracting, and thus in calculations where these matrix elements were needed they were taken to be zero. However, we note that in the Reid calculation for mass 7 these matrix elements shift the energy of the first excited state enough to make it the ground state. There are strong indications that the Bonn interaction gives a similar result, although complete convergence was not obtained.
66
B.R. Caldwel~ M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
Q
=E
2,0
P Q
_
_
0,1
C m
3,0
......
m
Diag.
Linked
1,0
Exp.
Fig. 3. Comparison of the energies, labeled J, T, obtained for mass 6 from the Paris potential by diagonalizing the bamiltonian and through the linked cluster expansion. The experimental excitation energies are shown for purposes of reference.
Unfortunately no direct comparison with experiment is possible for the three-body matrix elements (3BME). Those indirect comparisons which are possible will be discussed below.
3.4. Effects of removing unlinked diagrams on energy spectra Our purpose in this section is to assess the importance of including only the contribution of linked diagrams in the effective interaction. We will consider the effect on both the energy spectra and the wave functions. The spectra of masses 6 and 7 obtained from the Paris potential both by including and excluding unlinked diagrams are presented in Figs. 3 and 4. In both cases we find that the primary effect of removing the contribution of the unlinked diagrams is to lower the energy of each state by approximately 1.5 MeV. A similar shift in energy was found for each of the three realistic interactions considered. These shifts are presented in Table 2. We note that for each mass number, ff the core shift is removed the resulting shift is remarkably independent of the choice of realistic interaction. In order to allow comparison of the wave functions, the wave function obtained through diagonalization of the hamiltonian within the entire 2 h co space was projected into the 0 h ~o space and renormalized. It was found that the overlap between the wave functions obtained in this manner and those obtained excluding unlinked diagrams differed from unity by less than 1%. In some cases the 0 h~o basis contains only one state; of course in such a case this result is not significant. However, other bases contain as many as five states and yet the overlap in these cases remains nearly unity. This result is clearly consistent with the observation that the difference between the two calculations may be characterized by a simple shift in energy.
B.R. Caldwell, M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
67
11
10 9 •
m
iiiiiii
8
......... ""
5,1
5,1
6 5
o
I: ILl
7,1
""
4 3
1,1
3,1 Diag.
TBME
3BME
Exp.
Fig. 4. C o m p a r i s o n o f the energies, labeled 2 J , 2 T , obtained for m a s s 7 f r o m the Paris potential b y d i a g o n a l i z i n g the hamiltonian a n d through the linked cluster expansion. The experimental excitation energies are s h o w n for purposes o f reference.
It is also interesting to note that the renormalization described above showed that, within a few percent, the p-shell contribution to the wave function was 85% in both masses 6 and 7. Hence 2 6 to excitations contributed only 15% of the wave function.
3.5. Comparison between calculated spectra and experiment As previously stated the starting energy chosen for the G-matrix calculation yields a two-body interaction which severely underbinds mass 4. Thus we expect that masses 5, 6, and 7 will also be underbound. We have therefore aligned the experimental ground state with our calculated ground state in Figs. 2 and 3. The excitation energies obtained for mass 7, with only the TBME, and for mass 6 agree reasonably well with experiment. Although the individual 3BME may be large, the inclusion of the 3BME in the mass-7 calculation has little effect on either the binding energy or the excitation energies of the low-lying states. In fact one finds that, in a least-squares sense, the inclusion of the 3BME actually degrades the agreement with experiment slightly. This result suggests that a larger space may be necessary to calculate the 3BME accurately. Results have been presented only for the Paris potential. However, RSC and Bonn yield similar results in that the order of the states and the excitation energies are reasonably well reproduced while the binding energies are not.
Table 2 A v e r a g e shift in energy as a result o f r e m o v i n g the effects o f unlinked d i a g r a m s ( M e V ) A
RSC
Pads
OBEPR
4 5 6 7
-
-
-
0.64 0.92 1.31 1.77
0.72 0.99 1.39 1.67
1.34 1.50 2.03 2.45
68
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
3.6. The schematic three-body interaction A variety of calculations were performed to test the ability of the schematic three-body interaction, V ~3) (see Eq. (20)), to emulate the behavior of the effective three-body matrix elements and also to improve the fit of the calculated mass-7 energies, both in the sense of excitation energies and binding energy. A least-squares fit was made to the 3BME with the strengths A and C of V <3) as parameters, although, as previously noted, there is no a priori reason to think that V <3) has the appropriate form to approximate the 3BME. Thus it was not surprising that for each interaction the uncertainty in the parameters was found to be larger than the value of the parameter itself. These results were consistent with values of zero for both A and C. Further a non-linear least-squares fit to the absolute energies of mass 7 relative to that of 4He was performed. Again we found the results consistent with values of zero for A and C.
4. Summary and conclusions In this work we have calculated effective interactions for the p-shell from three realistic potentials. It has been often observed that all realistic interactions yield similar results in nuclear structure calculations. Our calculations confirm this view in that the similarities among the resulting interactions are striking. In the course of calculating the effective interactions one of our aims was to test the relative rate of convergence of the two available methods, KK and SL, to produce the energy-independent interaction. We found the rate of convergence was similar for the two methods studied in this work. However, SL is more easily extended to a large number of iterations when needed and showed a slight advantage in the rate of convergence in some cases. Concerning the effective interaction itself, the results obtained depend strongly on the starting energy used in the G-matrix calculation. In this calculation the starting energy was chosen to produce a 4He binding energy for each interaction roughly equal to published values. Hence the core was severely underbound in all cases. Further, in all cases the single-particle energy splitting was found to be similar to the experimental value while the energies themselves were of the order of 3 MeV too high. An interesting aspect of the methods employed in this work is that the effective interaction obtained is non-hermitian. However, we found the asymmetry to be small. While we preserved the non-hermitian quality in our energy calculations, simply symmetrizing the matrix elements would produce essentially identical results. The effective TBMEs were similar to the empirical TBMEs of Cohen and Kurath. Thus the effective interaction was found to give a reasonable approximation to excitation energies for masses 5, 6, and 7. The effects on these spectra of including the 3BMEs were at best negligible. We suspect that a space including more than 2 h to excitations is required to produce more accurate 3BMEs.
B.R. Caldwel~ M.R. Meder / Nuclear Physics A 594 (1995) 57-69
69
M a n y authors have emphasized the importance of excluding u n l i n k e d diagrams from nuclear structure calculations. It was found that the i n c l u s i o n of the effects of u n l i n k e d diagrams on energy spectra and w a v e functions m a y be simply approximated b y a m a s s - d e p e n d e n t shift in the energy. Further we note that for a given mass n u m b e r the energy shifts relative to the core are essentially i n d e p e n d e n t of the choice of realistic interaction. Finally, we observe that the schematic interaction V (3) w a s not sufficiently flexible to fit the 3 B M E or to improve the fit to the energies of the states of mass 7 relative to 4He. W e assume that differences in the symmetries of wave functions found in this work (which are largely determined by the T B M E ) and those of v a n Hees et al. [2] are responsible for this latter result.
References [1] [2] [3] [4]
S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 A.G.M. van Hees, J.G.L. Booten and P.W.M. Glaudemans, Phys. Rev. Lett. 62 (1989) 2245 A.G.M. van Hees, J.G.L Booten and P.W.M. Glaudemans, NucL Phys. A 507 (1990) 55 H.A. Mavromatis, Nucl. Phys. A 206 (1973) 477; P. Goode, Nucl. Phys. A 341 (1975) 311 [5] L.D. Skouras and J.C. Varvitsiotis, Nucl. Phys. A 513 (1990) 264 [6] E.M. Krenciglowa and T.T.S. Kuo, Nucl. Phys. A 235 (1974) 171; T.T.S. Kuo, Lecture notes in physics, Vol. 144, eds. T.T.S. Kuo and S.S.M. Wong (Springer, Berlin, 1981) [7] K. Suzuki and S.Y. Lee, Prog. Theor, Phys. 64 (1980) 2091; S.Y. Lee and K. Suzuld, Phys. Lett. B 91 (1980) 173. [8] P. Goldhammer, J.R. Hill and J. Nachamkin, Nucl. Phys. A 106 (1968) 62 [9] J.L. Norton and P. Goldhammer, Nucl. Phys. A 165 (1971) 33 [10] H. Dirim, J.P. Elliot and J.A. Evans, Nucl, Phys. A 244 (1975) 301 [11] B.R. Caldwell and M.R. Meder, Nucl. Phys. A 567 (1994) 477 [12] M.R. Meder, S.W. Walker and B.R. Caldwell, Nucl. Phys. A 556 (1993) 228 [13] B.R. Barrett, R.G.L. Hewitt and R.J. McCarthy, Phys. Rev. C 3 (1971) 1137 [14] P.J. Ellis, Phys. Lett. B 56 (1976) 232 [15] H.A. Mavromatis and L.D. Skouras, Nucl. Phys. A 326 (1979) 83 [16] L.D. Skouras and H.M. Miither, Nucl. Phys. A 515 (1990) 93 [17] R. Reid, Ann. of Phys. 50 (1968) 411 [18] M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. C6t6, P. Pir~s and R. de Tourreil, Phys. Rev. C 21 (1980) 861 [19] R. Machleidt, K. Holinde and Ch. Elster, Phys. Reports 149 (1987) 1 [20] J.P. Elliott, A.D. Jackson, H.A. Mavromatis, E.A. Sanderson and B. Singh, Nucl. Phys. A 121 (1968) 241
Nuclear Physics A 594 (1995) 57-69
Investigation of a matrix technique to produce an effective hamiltonian as applied to the p-shell B.R. Caldwell, M.R. Meder Department of Physics and Astronomy, GeorgiaState University,Atlanta, GA 30303, USA
Received 25 May 1994; revised 5 July 1995
Abstract Several aspects of a technique for calculating effective interactions, which avoids the perturbation expansion while eliminating the effects of unlinked graphs, are investigated. In the course of this investigation the effective p-shell interactions are calculated for the Reid Soft Core, the Paris, and the coordinate space Bonn interactions. The convergence properties of two approaches to summing the folded diagram series, that of Krenciglowa and Kuo and that of Suzuki and Lee are compared. The convergence properties of the latter, in the cases studied, are found to be superior. We find that removing the influence of unlinked graphs has little effect on excitation energies and wave functions and may be characterized by a mass dependent shift in the binding energy. In addition we note that these calculated excitation energies agree quite well with experiment.
1. Introduction Shell-model calculations are traditionally performed with an effective interaction which contains only one- and two-body terms. However, an effective interaction derived from a two-body nucleon-nucleon interaction in a truncated model space acquires a many-body character. The justification generally given for including only one- and two-body terms is that the contributions of effective three-body and higher terms, the many-body terms, are assumed to be negligible. The success of calculations designed to extract the matrix elements of the two-body effective interaction from the experimental energies of the p-shell nuclei, e.g. Ref. [1], suggests that the many-body components of the effective interaction are indeed small. However, these types of calculations have difficulty reproducing some of the experimental energy levels, particularly for A = 6 - 8 . There are two equally plausible explanations for this behavior. One is that some of the levels of the p-shell nuclei should have wave functions with large amplitudes for configurations outside the p-shell model space. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95)00328-2
58
B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
Therefore the energies of these states cannot be calculated from an effective two-body interaction appropriate to the p-shell. Alternatively, the behavior could signal the existence of a significant three-body component in the effective interaction originating from either the NN interaction or a possible NNN interaction. In a recent attempt to produce an empirical effective interaction appropriate to the p-shell [2,3] van Hees et al. have, in addition to the usual parameters appearing in such a calculation (the single-particle energies and two-body matrix elements) introduced a two-parameter schematic three-body force into the effective interaction, which we will denote as V ~3). Their approach produced a significant improvement in the fit to the experimental energy levels, reducing the rms deviation between the empirical and fitted energies of the normal parity levels by nearly a third. However, they were unable to estimate how much of the three-body component of the effective interaction was due to the existence of a possible nuclear three-body (NNN) force and how much was simply the NN interaction manifesting itself as a three-body effective interaction due to the truncated model space. In the present study we wish to examine some aspects of these problems more closely within the context of modern realistic interactions. To this end we calculate effective p-shell interactions derived from modern realistic NN interactions for nuclei of mass 4 through 7. These calculations include all excitations through 2 h to so that when the mass-7 interaction is projected into the p-sheU it acquires a many-body (three-body) character. In one standard method, the effective interactions would be calculated perturbatively through a diagrammatic approach. However, matrix methods are more attractive in that diagonalization of the hamiltonian matrix is equivalent to infinite order perturbation theory. Due to the necessary truncation of the basis [4], a difficulty arises when implementing the matrix approach. The effects of unlinked diagrams, which may be removed in the perturbative picture, will be included in the resulting effective interaction. Fortunately there is a procedure available, previously used by Skouras and Varvitisiotis [5] in p-shell calculations, which will allow us to use matrix methods to find the interaction while at the same time removing the effects of the unlinked diagrams. In their calculations, Skouras and Varvitisiotis found the energy-independent interaction using the Q-box technique [6] of Krenciglowa and Kuo (KK). In some cases they found that KK was not able to converge the matrix elements of the effective interaction completely. In this work we investigate the convergence properties of an alternative method for introducing energy independence, an iterative scheme [7] due to Suzuki and Lee (SL). To summarize, we calculate the zero- through three-body components of the p-sheU effective interaction from realistic interactions. In this context we compare the convergence properties of SL and KK in producing these energy-independent interactions. Further we examine the importance of the contributions of the unlinked diagrams, both to energies and wave functions. In addition we hope to provide a more accurate measure of the importance of the three-body components of the effective interaction derived from
B.R. Caldwel~ M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
59
realistic NN interactions than has been previously possible [8-10]. Finally, motivated by the success of van Hees et al., we will test the flexibility V <3) affords in fitting these matrix elements.
2. Model calculations for masses 4 through 7
2.1. The model hamiltonian In this subsection we briefly discuss the model hamiltonian and the form of the residual interaction. A more detailed discussion may be found in Ref. [11]. The hamiltonian of a system consisting of A nucleons of mass m may be written in the following translationally invariant form: A 2. )
where Gij is the two-body interaction between nucleons i and j, and Pij is relative momentum of the pair, Pij = (Pi-P~)A/2, that is the momentum conjugate to the relative coordinate Pii = (ri - rj)/~/2. The first term in Eq. (1) yields the kinetic energy of the system relative to the center of mass and the second, the system's potential energy. It is convenient to rewrite Eq. (1) as follows:
H
=
~ ( -2- h o i j i
+Gi/-
--AUOidj
+ /3[H0(c.m. ) -- 3htol ,
(2)
where ho~j =p2./2m + Uo~j and Uoi/ = mto2p2./2, a harmonic-oscillator hamiltonian and a harmonic-oscillator potential written in the relative coordinates of particles i and j. The term proportional to fl gives the excitation energy of the center of mass of the system and /3 is a dimensionless parameter which may be adjusted to lift states for which the center of mass of the system is in an excited state out of the energy region of interest. The hamiltonian may now be written in the standard shell-model form A
A
H = ~.,Ho(i ) - n0(c.m. ) + E V/i + fl [n0(c-m.) - 3hw] • i
(3)
i
Ho(i) is a harmonic-oscillator hamiltonian operating on the coordinates of particle i, and V~i, the residual interaction, is given by the expression 2
Vii = Gij - -~Uoi j.
(4)
2.2. The two-body effective interaction It is well known that it is impossible to solve the Schr6dinger equation with the nuclear interaction exactly. This is primarily due to an inability to deal with the
60
B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
short-ranged repulsion exhibited by all realistic NN interactions. The truncated model space simply does not allow the wave function the necessary freedom to respond to the repulsion. The effects of the short-ranged repulsion may be overcome by introducing the reaction matrix or, as it is generally called when dealing with bound-state problems, the G-matrix. The two-particle G-matrix is defined as follows: q G(to) = v + v G(to). (5) to - Ho(2p) In Eq. (5), v is the realistic two-body interaction, H0(2 p) = H0(1) + H0(2) , and to is a parameter of the calculation, the starting energy. In the calculations described below the operator q is a projection operator which projects onto the space of two-particle harmonic-oscillator states which have energies greater than 4 h to. The G-matrix elements appearing in the calculations described below have been calculated using a variation [12] on the reference spectrum method of Barrett et al. [13].
2.3. The p-shell energy-independenteffective hamiltonian A standard working hypothesis would be that the diagonalization of Eq. (3) in the largest possible space provides us with the best estimate of the energies and wave functions of the p-shell nuclei. However, two issues arise when these results of such a calculation are projected into the p-shell model space. The two-body interaction appearing in Eq. (3) becomes a many-body, energy-dependent effective interaction. That is, each eigenstate appears to arise from a different effective hamiltonian. Further, from the point of view of perturbation theory, it can be shown that if the space in which Eq. (3) is diagonalized is not infinite [14] one must be careful to exclude the spurious contributions to the energy due to unlinked diagrams. Fortunately methods have been devised [14,15] which deal with both of these difficulties, at least within a model space which allows 0 + 2 h to excitations. We will briefly describe these methods here. A more detailed discussion may be found in Ref. [16] and the references contained therein. We introduce an operator P which projects onto the 0 hto space, which is of dimension /x, and a operator Q which projects onto the 2 h to space of dimension M. As was shown in Ref. [16], it is convenient to perform the required calculations in a basis where PHP and QHQ are separately diagonal with eigenvalues w,~ and W/ respectively:
PnPly~)=w~ly~), analYi)=wilY~),
a = 1, 2 , . . . , / x ,
(6)
i = 1 , 2 . . . . . M.
(7)
The basic building blocks to be used in constructing the effective interaction are the matrix elements of the operator R n, M
(y=lR, ly a)
=
( - 1 ) " ~ (y=lVIY~)(wa i=1
w,)
-(,+1)
Iv I
).
(8)
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
61
In a situation where the Q-space is restricted to 2 h to excitations each R n consists of linked-plus-folded valence diagrams and linked-core diagrams. In order to remove the effects of unlinked diagrams on the energy of the core, one must also calculate the matrix elements of the operator EI, where E 1 = PVP. The total energy of 4He may now be written in terms of these quantities,
EC = H~ + E~ + R~ .
(9)
The superscript c refers to the ground state of the closed core, in this case 4He. The single-particle energies, two-body matrix elements, and the three-body matrix elements of the p-shell effective interaction are also formed from combinations of the operators R , and E v In order to write the somewhat complex expressions for these operators most efficiently, Skouras and Varvitsiotis expanded the p-shell basis vectors, ](n 1 nz)"JTN), in terms of an alternative set of vectors which readily allow one to evaluate an m-body operator. The quantities n 1 and n 2 are the occupation numbers of the P3/z and P1/2 states, where n I + n 2 = n, and n is the number of p-shell valence particles. This expansion may be written in terms of the products of coefficients of fractional parentage and states of m and m' particles, which are separately antisymmetric, ](nln2)"JTU)
= ~,
~,
~,
[(mlmz)mj1T1Nl,(rttln~z)m'j2TzNz;JT)
rnlmz J1T1N1 JzTzN2
× [(m~mz)mjtT1Nl,(~ltrgz)m'jzTzN2;
JTI}(nlnz)"JTN ] .
(10)
The symbol [ [}] is a fractional parentage coefficient, m'i = n i - mi, and N is a quantum number which distinguishes among states having identical occupation numbers, spin and isospin. We now define the matrix elements of the n-body operators, which are denoted as e 1 and rk, as follows:
(( nlnz) nj T u I w l( n'ln'2) njTN ') = (( nln 2) "JTN I W I(n'ln'2) "JTN')
-- E rn= 1
E
E E ((mlm2)
J1TINIIwI(m3m4)
J1T1N~)
mlm 2 J1T1 NIN2
'n3m4 J2r2 ~V~
× [ ( m l m 2 ) mj1T1NI( m'a m'2 ) r"'J2T2N2; J r l } ( n,n:)"JrU] × [(m3m4)rnJ1T1g~(tt~3tr[4)m'J2T2N2;
-
8,,,,,w c,
JTI}(n'In'2)"JTN'] (11)
where W stands for the quantities Ea and R k defined above and w for the quantities e 1 and r k respectively.
62
B.R. Caldwell, M.R. Meder / Nuclear Physics A 594 (1995) 57-69
2.4. Summing folded diagrams The effective interaction can be obtained by adopting the Q-box techniques of Krenciglowa and Kuo [6]. The quantity r o is considered to be the Q-box while r k (k > 0) is taken to be the kth derivative of the Q-box. The effective interaction is then found from the series Veff = E F i , i
(12)
where
F o = e l + r o,
Fx=rlro,
F 2 = r z r 2 + r ~ r o,
F3 = r3r3 + r2rlr2 + r2rorlro + rlr2r2 + r3ro"
(13)
These expressions, which represent the contributions of the additional folded diagrams, and those of higher order, which we omit due to their complexity, may be found using the rules contained in Ref. [6]. An alternative approach is to implement the method of Suzuki and Lee [7]. We will refer to this method as " S L " while the previous method will be denoted as " K K " . In the SL approach the quantity r~ is still treated as the nth derivative of the Q-box but now the effective interaction is calculated iteratively as follows: We define a matrix Z~ = V~f 1 - E 1,
(14)
where n = 1,2 . . . . and Ve°f= r 0 .
(15)
The iteration process is then defined as follows: Z. =A~xZ1,
(16)
where A n --
n-1 1 - r 1 - E rk k=2
n-1 I-I
Zm.
(17)
m=n-k+l
If the matrix Z~ converges, the effective interaction is Vdf = Z. + e~,
(18)
and so nef f = H o + Veff.
(19)
2.5. Parameterization of the three-body interaction We follow the suggestion of Ref. [2] and introduce a simple schematic parameterized form for the three-body interaction V°) = E ( a ( o - t "~rz, °'1" °'3}(~'1 "z2, zl" ~'3} cycl
+C[O-1 "0"2' O"1" O"3]["/'1 "72, '/'1" ~/'3])"
(20)
B.R. CaMwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
63
The summation runs over the three cyclic permutations of particles 1, 2 and 3. The braces and brackets represent anticommutators and commutators respectively. The structure of this force is similar to that which would be obtained from the exchange of two pions. However, the tensor contribution to the force and the radial dependence have been suppressed. Since this simplified version of the three-body interaction was able to successfully reproduce the empirical data [2] it might also exhibit similarities to the three-body interaction calculated in the manner described above, at least to the extent in which our calculations mirror reality. There is no a priori reason to assume that the form of V ~3) is appropriate for the effective three-body force originating from a two-body force. However, if V ~3) has sufficient flexibility it should be possible to condense the information obtained from the calculations described above into the values of the two strength parameters, A and C. Further, this parameterization allows a direct comparison with the results of Ref. [2].
3. Results and discussion 3.1. N N interactions for which effective interactions are derived The effective interactions for masses 4, 5, 6 and 7 were calculated in the manner described above. Three realistic interactions were used in these calculations: the Reid Soft Core (RSC) [17], Paris [18], and Bonn (OBEPR) [19]. The RSC was chosen because it is widely used as a benchmark, while the other two were chosen because they are relatively modem. Further, these interactions span (and exceed) the generally accepted range for the strength of the tensor component of the interaction as characterized by the D-state admixture in the deuteron wave function calculated from each of them. Finally, the group included both energy-dependent and energy-independent interactions. (We note that, in this context, the term "energy dependent" indicates a dependence on the relative kinetic energy of the pair of interacting nucleons rather than a dependence on the energy eigenvalue as is the case with the effective interaction). The effective interaction was also calculated for the Sussex interaction [20]. Our purpose in doing so was to check the results obtained with our programs against those presented in Ref. [5]. 3.2. Convergence In this section we investigate the relative rate of convergence for the two methods of obtaining the effective interaction, KK and SL, discussed in Section 2.4. The KK results presented in the current section include terms through F 5 although the expressions for F 4 and F 5 were not included in Eq. (13) due to the unwieldy number of terms in each, 14 in F 4 and 43 in F 5. In contrast to KK, the number of iterations which may be comfortably performed in SL is limited only by numerical considerations. The level of
64
B.R. CaldweU, M.R. Meder ~Nuclear Physics A 594 (1995) 57-69 500
>~
400
'" Z
300
¢ J,
SL KK
m I<3 -
-
200
100
!
1
2
3
4
5
6
ITERATION
Fig. 1. Comparison of the convergence rate of SL and KK for the diagonal and T = 0 calculated with the Sussex interaction.
(P3/2)2 matrix element with J
= 1
approximation represented by k folds is comparable to k iterations of SL since each require at most the kth derivative of the quantity r. For most of the examples presented in this paper a maximum k of five was sufficient to produce reasonable convergence. However, in all cases SL converges, at least marginally, more quickly than KK and in some cases a maximum value of k = 5 is not sufficient to produce convergence. Hence, in these instances SL is preferable. An example of this behavior is displayed in Fig. 1.
3.3. The effective p-sheU interaction If one considers at most three valence particles, the p-shell effective interaction has at most four terms, the zero-, one-, two-, and three-body parts, as indicated in Section 2. We will consider, in turn, the results for each of these terms below. First we examine the binding energy of 4He, that is the case where no valence particles are present. This quantity is characterized by EC: the core energy. The value of E ¢ depends strongly on the starting energy, to, used in calculating the two-body G-matrix (see Eq. (5)). A starting energy of h to = 16.2 MeV was chosen. Interestingly this choice gave a core energy approximately equal to the available mass-4 binding-energy calculations for the three NN potentials considered here. Hence the values obtained for E c represent an underbinding of the core with respect to experiment by 4 to 5 MeV.
Table 1 Core and single-particle energies (MeV) Nucleus 4He SHe
State
RSC
Paris
OBEPR
3/2 1/2
- 22.004 4.004 7.008
- 21.592 4.075 7.164
- 26.056 3.721 7.021
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69 6
---br--
CK
Paris
O
4
65
2
o =J
0
e¢D
=_
-2
eo
-4 -6 J
-8
0i
i~
2i
2i
w ~
~
•"
0i
1i
2i
0i
~
g
g
g
T:I
"
T=O
Fig. 2. Comparison of the TBME calculated from the Paris interaction and the empirical TBME of Cohen and Kurath (CK). The matrix elements are identified by 2j] 2j2 2j3 2j4. Each calculated off-diagonal matrix element is represented by two values, the matrix element indicated and its hermitian conjugate. Thus the small extent to which the calculated off-diagonal matrix elements are non-hermitian is displayed.
The results for both E ¢ and the single-particle energies (the one-body contribution to the effective interaction) are presented in Table 1. The single-particle energies are found to be essentially independent of the realistic two-body interaction from which they are calculated. Further, the P1/2-P3/2 splitting compares favorably with experimental estimates, although the energies themselves do not. The effective two-body matrix elements for all realistic interactions were found to be similar to each other and to experimentally determined matrix elements. In Fig. 2 the two-body matrix elements of Veff for the Pads potential are compared with the empirical matrix elements of Cohen and Kurath. In order to display the degree of non-hermitian behavior we present both the matrix elements and their hermitian conjugates. It can be seen from Fig. 2 that the asymmetry in the off-diagonal matrix elements is generally quite small, at most of the order of 100 or 200 keV as measured from the mean. However, the non-hermitian character of the matrix elements was preserved in our calculations. The three-body matrix elements of the effective interaction were extracted from the mass-7 calculations. The converged matrix elements range in magnitude from tens of keV to the order of an MeV. Unfortunately it was found that neither KK nor SL produced convergence in the J ~ = 1 / 2 - , T = 1 / 2 matrix elements for the Paris and Bonn interactions. We interpreted this to mean that the three-body part of these interactions was too small to obtain accurate results when subtracting, and thus in calculations where these matrix elements were needed they were taken to be zero. However, we note that in the Reid calculation for mass 7 these matrix elements shift the energy of the first excited state enough to make it the ground state. There are strong indications that the Bonn interaction gives a similar result, although complete convergence was not obtained.
66
B.R. Caldwel~ M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
Q
=E
2,0
P Q
_
_
0,1
C m
3,0
......
m
Diag.
Linked
1,0
Exp.
Fig. 3. Comparison of the energies, labeled J, T, obtained for mass 6 from the Paris potential by diagonalizing the bamiltonian and through the linked cluster expansion. The experimental excitation energies are shown for purposes of reference.
Unfortunately no direct comparison with experiment is possible for the three-body matrix elements (3BME). Those indirect comparisons which are possible will be discussed below.
3.4. Effects of removing unlinked diagrams on energy spectra Our purpose in this section is to assess the importance of including only the contribution of linked diagrams in the effective interaction. We will consider the effect on both the energy spectra and the wave functions. The spectra of masses 6 and 7 obtained from the Paris potential both by including and excluding unlinked diagrams are presented in Figs. 3 and 4. In both cases we find that the primary effect of removing the contribution of the unlinked diagrams is to lower the energy of each state by approximately 1.5 MeV. A similar shift in energy was found for each of the three realistic interactions considered. These shifts are presented in Table 2. We note that for each mass number, ff the core shift is removed the resulting shift is remarkably independent of the choice of realistic interaction. In order to allow comparison of the wave functions, the wave function obtained through diagonalization of the hamiltonian within the entire 2 h co space was projected into the 0 h ~o space and renormalized. It was found that the overlap between the wave functions obtained in this manner and those obtained excluding unlinked diagrams differed from unity by less than 1%. In some cases the 0 h~o basis contains only one state; of course in such a case this result is not significant. However, other bases contain as many as five states and yet the overlap in these cases remains nearly unity. This result is clearly consistent with the observation that the difference between the two calculations may be characterized by a simple shift in energy.
B.R. Caldwell, M.R. Meder ~Nuclear Physics A 594 (1995) 57-69
67
11
10 9 •
m
iiiiiii
8
......... ""
5,1
5,1
6 5
o
I: ILl
7,1
""
4 3
1,1
3,1 Diag.
TBME
3BME
Exp.
Fig. 4. C o m p a r i s o n o f the energies, labeled 2 J , 2 T , obtained for m a s s 7 f r o m the Paris potential b y d i a g o n a l i z i n g the hamiltonian a n d through the linked cluster expansion. The experimental excitation energies are s h o w n for purposes o f reference.
It is also interesting to note that the renormalization described above showed that, within a few percent, the p-shell contribution to the wave function was 85% in both masses 6 and 7. Hence 2 6 to excitations contributed only 15% of the wave function.
3.5. Comparison between calculated spectra and experiment As previously stated the starting energy chosen for the G-matrix calculation yields a two-body interaction which severely underbinds mass 4. Thus we expect that masses 5, 6, and 7 will also be underbound. We have therefore aligned the experimental ground state with our calculated ground state in Figs. 2 and 3. The excitation energies obtained for mass 7, with only the TBME, and for mass 6 agree reasonably well with experiment. Although the individual 3BME may be large, the inclusion of the 3BME in the mass-7 calculation has little effect on either the binding energy or the excitation energies of the low-lying states. In fact one finds that, in a least-squares sense, the inclusion of the 3BME actually degrades the agreement with experiment slightly. This result suggests that a larger space may be necessary to calculate the 3BME accurately. Results have been presented only for the Paris potential. However, RSC and Bonn yield similar results in that the order of the states and the excitation energies are reasonably well reproduced while the binding energies are not.
Table 2 A v e r a g e shift in energy as a result o f r e m o v i n g the effects o f unlinked d i a g r a m s ( M e V ) A
RSC
Pads
OBEPR
4 5 6 7
-
-
-
0.64 0.92 1.31 1.77
0.72 0.99 1.39 1.67
1.34 1.50 2.03 2.45
68
B.R. Caldwell, M.R. Meder /Nuclear Physics A 594 (1995) 57-69
3.6. The schematic three-body interaction A variety of calculations were performed to test the ability of the schematic three-body interaction, V ~3) (see Eq. (20)), to emulate the behavior of the effective three-body matrix elements and also to improve the fit of the calculated mass-7 energies, both in the sense of excitation energies and binding energy. A least-squares fit was made to the 3BME with the strengths A and C of V <3) as parameters, although, as previously noted, there is no a priori reason to think that V <3) has the appropriate form to approximate the 3BME. Thus it was not surprising that for each interaction the uncertainty in the parameters was found to be larger than the value of the parameter itself. These results were consistent with values of zero for both A and C. Further a non-linear least-squares fit to the absolute energies of mass 7 relative to that of 4He was performed. Again we found the results consistent with values of zero for A and C.
4. Summary and conclusions In this work we have calculated effective interactions for the p-shell from three realistic potentials. It has been often observed that all realistic interactions yield similar results in nuclear structure calculations. Our calculations confirm this view in that the similarities among the resulting interactions are striking. In the course of calculating the effective interactions one of our aims was to test the relative rate of convergence of the two available methods, KK and SL, to produce the energy-independent interaction. We found the rate of convergence was similar for the two methods studied in this work. However, SL is more easily extended to a large number of iterations when needed and showed a slight advantage in the rate of convergence in some cases. Concerning the effective interaction itself, the results obtained depend strongly on the starting energy used in the G-matrix calculation. In this calculation the starting energy was chosen to produce a 4He binding energy for each interaction roughly equal to published values. Hence the core was severely underbound in all cases. Further, in all cases the single-particle energy splitting was found to be similar to the experimental value while the energies themselves were of the order of 3 MeV too high. An interesting aspect of the methods employed in this work is that the effective interaction obtained is non-hermitian. However, we found the asymmetry to be small. While we preserved the non-hermitian quality in our energy calculations, simply symmetrizing the matrix elements would produce essentially identical results. The effective TBMEs were similar to the empirical TBMEs of Cohen and Kurath. Thus the effective interaction was found to give a reasonable approximation to excitation energies for masses 5, 6, and 7. The effects on these spectra of including the 3BMEs were at best negligible. We suspect that a space including more than 2 h to excitations is required to produce more accurate 3BMEs.
B.R. Caldwel~ M.R. Meder / Nuclear Physics A 594 (1995) 57-69
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M a n y authors have emphasized the importance of excluding u n l i n k e d diagrams from nuclear structure calculations. It was found that the i n c l u s i o n of the effects of u n l i n k e d diagrams on energy spectra and w a v e functions m a y be simply approximated b y a m a s s - d e p e n d e n t shift in the energy. Further we note that for a given mass n u m b e r the energy shifts relative to the core are essentially i n d e p e n d e n t of the choice of realistic interaction. Finally, we observe that the schematic interaction V (3) w a s not sufficiently flexible to fit the 3 B M E or to improve the fit to the energies of the states of mass 7 relative to 4He. W e assume that differences in the symmetries of wave functions found in this work (which are largely determined by the T B M E ) and those of v a n Hees et al. [2] are responsible for this latter result.
References [1] [2] [3] [4]
S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 A.G.M. van Hees, J.G.L. Booten and P.W.M. Glaudemans, Phys. Rev. Lett. 62 (1989) 2245 A.G.M. van Hees, J.G.L Booten and P.W.M. Glaudemans, NucL Phys. A 507 (1990) 55 H.A. Mavromatis, Nucl. Phys. A 206 (1973) 477; P. Goode, Nucl. Phys. A 341 (1975) 311 [5] L.D. Skouras and J.C. Varvitsiotis, Nucl. Phys. A 513 (1990) 264 [6] E.M. Krenciglowa and T.T.S. Kuo, Nucl. Phys. A 235 (1974) 171; T.T.S. Kuo, Lecture notes in physics, Vol. 144, eds. T.T.S. Kuo and S.S.M. Wong (Springer, Berlin, 1981) [7] K. Suzuki and S.Y. Lee, Prog. Theor, Phys. 64 (1980) 2091; S.Y. Lee and K. Suzuld, Phys. Lett. B 91 (1980) 173. [8] P. Goldhammer, J.R. Hill and J. Nachamkin, Nucl. Phys. A 106 (1968) 62 [9] J.L. Norton and P. Goldhammer, Nucl. Phys. A 165 (1971) 33 [10] H. Dirim, J.P. Elliot and J.A. Evans, Nucl, Phys. A 244 (1975) 301 [11] B.R. Caldwell and M.R. Meder, Nucl. Phys. A 567 (1994) 477 [12] M.R. Meder, S.W. Walker and B.R. Caldwell, Nucl. Phys. A 556 (1993) 228 [13] B.R. Barrett, R.G.L. Hewitt and R.J. McCarthy, Phys. Rev. C 3 (1971) 1137 [14] P.J. Ellis, Phys. Lett. B 56 (1976) 232 [15] H.A. Mavromatis and L.D. Skouras, Nucl. Phys. A 326 (1979) 83 [16] L.D. Skouras and H.M. Miither, Nucl. Phys. A 515 (1990) 93 [17] R. Reid, Ann. of Phys. 50 (1968) 411 [18] M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. C6t6, P. Pir~s and R. de Tourreil, Phys. Rev. C 21 (1980) 861 [19] R. Machleidt, K. Holinde and Ch. Elster, Phys. Reports 149 (1987) 1 [20] J.P. Elliott, A.D. Jackson, H.A. Mavromatis, E.A. Sanderson and B. Singh, Nucl. Phys. A 121 (1968) 241