Fusion Engineering and Design 37 (1997) 89-93
Fusion Engineering and Design
ELSEVIER
Inclusion of two-body effects in calculating nuclear level densities S . M . G r i m e s *, T . N . M a s s e y Ohio University, Athens, OH 45701. USA
Abstract The nuclear level density is normally addressed through the use of non-interactive Fermion models. These are easy to understand, involve minimal input and are efficient for making calculations. There are some questions about the use of these models off of the stability line and in regions of extreme deformation. New developments in the theory of moment methods allow better calculations involving a two-body interaction. These techniques and some results will be presented. © 1997 Elsevier Science S.A. Keywords: Two-body effects; Nuclear level densities; Fermion models
1. Introduction The non-interacting model for nuclear level densities has been used extensively for many years. This model in its most simplified form relates the nuclear level density for a particular nucleus to a formula with only two parameters. In general, the model works rather well. There are some situations, however, where such a simple parameterization would be questionable. A model which ignores interaction would not predict any collective states. Details of the difference between n - p and n n or p - p interactions may average out in a particular fashion for nuclei on the stability line but might require a more sophisticated treatment for nuclei off of the stability line. Finally, since the two-body interaction * Corresponding author. Tel.: + 1 614 5931979; fax: + t 614 5931436.
clearly depends on relative orbital angular momentum of the two particles, it is plausible that the inclusion of two-body force effects on nuclear level densities might influence the energy distribution of levels differently as a function of spin and parity. This last possibility has substantial consequences, since the parity ratio and the spin cutoff parameters are needed to interpret neutron resonance data in terms of level densities.
2. Method Early studies of the level density for a group of nucleons interacting with a two-body force were performed by French and co-workers [1-5]. These calculations were based on algebraic relations which allowed the calculation of the first and second moment of the Hamiltonian from straightforward sums. Comparison of these results with
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S.M. Grimes, T.N. Massey /Fusion Engineering and Design 37 (1997) 89-93
those from shell-model calculations in small spaces utilizing the same Hamiltonian showed that the distribution of levels with energy was nearly Gaussian but that deviations from the Gaussian form did occur. To improve on these calculations, it was necessary to calculate the third and fourth moments of the Hamiltonian. Such calculations are virtually impossible using the algebraic technique because of the complexity of the sums. It has been shown [6,7] that these moments can be calculated using a Monte Carlo technique. The principle of unitarity implies that moments of the Hamiltonian are invariant with respect to the representation. They may, therefore, be evaluated in a non-diagonal basis. Thus, by simply selecting Slater determinants at random and evaluating ( H ) , (H2), ( H 3) and (H4), these moments can be calculated as the average over many such trials. Some of the bases for which calculations have been made have sizes as large as 10~2; even a small fractional sample of size about 106 provides reasonable convergence for these moments. Such calculations can be done in about 72-96 h on a modern work station. A major complication remained, however. The obvious procedure for deriving a level density from the moments was to use an Hermite polynomial expansion. This orthogonal polynomial expansion is based on a complete set and would be expected to converge with a small number of terms if the expansion is of a function which is close to Gaussian. Further, given the moments, the coefficients for the expansion are available in closed form. A fundamental drawback is also present if an orthogonal polynomial series is used. All polynomials beyond H 0 have negative excursions, required in order to make them orthogonal to H 0. If convergence is not complete at fourth order, then the remaining discrepancies between the actual function and the low-order expansion are expressible in terms of an Hermite expansion of order five and higher. Given that all of these functions have zero average, it is essentially guaranteed that if convergence is not complete, the fitting function will have negative excursions. In general, these occur in the tail region, which is an area of particular physics interest.
Examination of the difficulty in calculating moments of the Hamiltonian higher than four established that, although the fifth and sixth moments could be calculated using the Monte Carlo method, extending the moment calculations as high as tenth order was probably not possible. The reason for this is that the Slater determinant basis is not an eigenbasis. Thus, each successive application of the Hamiltonian produces a vector which is considerably longer. In typical calculations, one application of the Hamiltonian produces a multiplication of between 200 and 1500, depending on the number of particles included in the calculation. While the multiplication factor decreases slightly with each application, it is clear that a small number of applications of the Hamiltonian will produce a vector with l06 Slater determinants. The calculation eventually becomes too slow, even if computer memory limitations can be circumvented. This problem has led to a number of modifications to the original procedure. The first was to change the sequence of moment averaging and expansion. One possible method of performing the expansion involves calculating each moment a large number of times ( ~ 106); once an average value has been obtained, one then calculates the Hermite expansion. This was the procedure first used and is called the average moment expansion (AME). It was subsequently realized that an alternative technique would be to make the Hermite expansion for each Slater determinant and add the individual contributions to make the total density. This is called the individual Slater determinant expansion (ISDE). It might seem that the same information is being input to the expansion and that the two results would be the same. In fact, they differ and the second yields better results. A mathematical analysis of the difference has been performed [8,9] which leads to an unambiguous preference for the second method. A nonmathematical way to see this is to consider what the difference between the two methods would be in the eigenbasis (basis states are eigenstates). In this situation, method one would produce an Hermite expansion based on four moments which would, in general, not be completely converged. On the other hand, method two would give very
S.M. Grimes, T.N. Massey / Fusion Engineering and Design 37 (1997) 89 93
good convergence with only one moment, since each basis state has no width about its centroid. A realistic calculation will not be in the eigenbasis, but until the extreme limit is reached in which each basis state contains equal amounts of each eigenvector, the second method will be superior. Even when this limit is reached, the two methods are equally good, so the use of the second procedure is never worse and is generally better than the first. Unfortunately, while this produced improvement, it did not remove the negative excursions entirely. A second modification was to attempt an expansion which was not based on orthogonal polynomials. An expansion in terms of delta functions can be constructed which reproduces the lowest 2 N - 1 moments of the distribution. Thus, with moments through H 3, a two delta function expansion can be calculated and with moments as high as H 5, a three delta function expansion can be made. Neither of these would be remotely realistic with the AME, but when used with the ISDE, about l0 6 delta functions are produced. Thus, despite the reliance on delta functions, a quasi continuous distribution could be produced. Clearly, no negative excursions could occur with this method. Comparisons of the results of this delta function expansion show that indeed the result is essentially continuous but does show more structure than is in the actual distribution. Moreover, as could be anticipated from the discrete nature of the expansion, there is a tendency to underestimate strength in the tails of the distribution. Although the delta function distribution clearly eliminates the negative excursions, it suffers from a similar problem as the Hermite expansion in that the distribution is poorest where there is the most physical interest. More recently, a further alternative has been tried [10]. Instead of using an orthogonal polynomial expansion or a non-continuous distribution, an expansion based on an exponential with polynomial argument (EPA) has been tried. As an example, assume the first five moments (H °, H, H 2, H 3 and H 4) are available for a Slater determinant. We expand the strength in the form exp (a~ + a2E + a3E 2 + a4E3 --}-asE4). This form is
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obviously continuous and positive definite. It has the further advantage that it can be made asymptotically Gaussian for distributions which are close to Gaussian. Comparison with the exact distribution for some cases where the exact results are known indicates that the EPA produces results which are better than an Hermite expansion or a delta function expansion. One final problem remained. Unlike the other two expansions, no closed form solution for the coefficients aK in terms of the moments exists. Although a search procedure which led reliably to best fit parameters was found, it was extremely slow and not practical for use in a calculation where ~ l06 expansions had to be made. A possible solution would be to store the parameters in a look-up table so that a search would not be necessary. With five moments, the search would need to be over five parameters, and it seems hopeless to define a five-dimensional grid in fine enough steps to allow the direct use of stored parameters. A solution to this problem was found by reducing the dimensions of the problem. All distributions are first normalized to one and the energy scale shifted so that ( H s ) = 0. Finally, the energy scale is changed by replacing H with K = Hs/ ( H 4 ) 1/4. This automatically makes the fourth moment one. In this new system, only the second and third moments vary. A two-dimensional grid of the appropriate size is easily defined and the solutions simply stored.
3. Results
To evaluate the quality of fit achievable with this expansion procedure, a shell model diagonalization was carried out for the space d5/2, s~/2, d3/2, f7/2 with two neutrons and two protons allowed to occupy any orbits in the space. This represents 2°Ne calculated with a core of ~60. This calculation produces a total of 2040 positive parity states and 1960 negative parity states. Since every single eigenvalue was calculated, we have 'exact' values for the level density, the spin cutoff parameter and the parity ratio. These values are compared with those deduced using the moment method
S.M. Grimes, T.N. Massey /Fusion Engineering and Design 37 (1997) 89 93
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code in Figs. 1-3. Results for each of these parameters were calculated using the EPA, the delta function method and the Hermite polynomial expansion. Note that the delta function method though positive definite, clearly produces
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too many oscillations in the level density, the parity ratio and the spin cutoff factor. The Lanczos also produces a distribution which systematically underestimates the values of the level density in the tail region. The Hermite polynomial expansion also has extra structure in the spin cutoff parameter and the parity ratio, but the number and magnitude of the oscillation is less than for the delta function expansion; a least-squares analysis gives a definite preference for the EPA for all three parameters. It appears the use of the EPA expansion has removed the problems of negative excursion and poor convergence in the tail regions associated with use of orthogonal polynomial expansions. Because the two-body calculations require considerably more input and because they are slower than Fermi gas calculations, finding optimum parameter sets will be changing. We are making Hartree-Fock calculations for nuclei off of the stability line to try to see if the single particle energy systematics can be better understood in this region. Preliminary results show a dependence of the level density parameter a on N - Z as well as A, with proton-rich nuclei having higher and neutron-rich nuclei having lower level densities than nuclei with the same A on the stability line.
S.M. Grimes, T.N. Massey /Fusion Engineering and Design 37 (1997) 89-93
References [l] J.B. French, Combinatorial methods for spectroscopic averaging and spectral moments, Phys. Lett. 23 (4) (1965) 248 250. [2] F.S. Chang, J.B. French, K.F. Ratcliff, Centroid energies and widths of nuclear configurations, Phys. Lett. 23 (4) (1965) 251 254. [3] J.B. French, K.F. Ratcliff, Spectral distributions in nuclei, Phys. Rev. C3 (l) (1971) 94 117. [4] K.F. Ratcliff, Applications of spectral distributions in nuclear spectroscopy, Phys. Rev. C3 (1) (1971) 117-429. [5] F.S. Chang, A. Zuker, Validity of spectral distribution methods in nuclei, Nucl. Phys. A 198 (3) (1972) 417 429.
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[6] S.M. Grimes, S.D. Bloom, R.F. Hausman Jr., B.J. Dalton, Method for calculating operator traces, Phys. Rev. C19 (6) (1979) 2378 2386. [7] S.M. Grimes, S.D. Bloom, H.K. Vonach, R.F. Hausman Jr., Spectral distribution calculations of the level density and spin cutoff parameters of 28Si, Phys. Rev. C27 (6) (1983) 2893 2901. [8] B. Strohmaier, S.M. Grimes, H. Satyanarayana, Spectral distribution calculations of the level density of 24Mg, Phys. Rev. C36 (4) (1987) 1604 1610. [9] B. Strohmaier, S.M. Grimes, Spectral distribution calculations of the level density of 2°Ne, Zeit. Physik A 329 (2) (1988) 431 440. [10] S.M. Grimes, T.N. Massey, New expansion technique for spectral distribution calculations, Phys. Rev. C51 (2) (1995) 606 610.