Nuclei Off the Line of Stability - ACS Publications - American Chemical

15 The Importance of Level Structure in Nuclear Reaction Cross-Section Calculations M. A. Gardner and D. G. Gardner

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Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550 We continue to observe the necessity of describing nuclei in the first few MeV above their ground states with discrete levels in order to obtain accurate cross-section and isomer-ratio calculations. There appears to be no adequate way in which one can use a level-density expression to represent or substitute for the level structure. The level information should not only consist of those levels that are obtained from experiment, but must be supplemented with the level structure theoretically known to be present by model calculations. For the calculation of isomer ratios, gamma-ray branchings among the discrete levels are also required. Discrete-level sets are not only important in describing nuclei at low excitation energies, but are also valuable in estimating several calculational parameters, particularly for nuclei off the line of stability. Introduction In the course of carrying out nuclear reaction cross-section calculations of the statistical-model Hauser-Feshbach type, we continue to observe the necessity of describing nuclei in the first few MeV above their ground states with discrete levels in order to obtain accurate results[GAR85a]. There is no adequate way in which one can use a level-density expression to represent or substitute for the level structure. Furthermore, all levels must be included; the level information obtained from experiment must be supplemented with that known to be present theoretically. This is true both for spherical nuclei, where typically tens of levels are found in the first 1 to 4 MeV of nuclear excitation, and for deformed nuclei, where perhaps a thousand levels are present in thefirst1.5 MeV. We have found that discrete-level sets are not only important in the description of nuclei at low excitation energies but are also very valuable in the prediction of a number of calculational parameters, particularly for nuclei off the line of stability. Discrete-level information can be used to obtain improvements on global parameterizations of level densities; to predict unknown values of D , the average s-wave neutron resonance spacing; to compute correct radiation widths; and to deduce absolute El and Ml gamma-ray strength functions. If correct gamma-ray branchings among the discrete levels are provided, reliable isomer ratios can be computed. In addition, complete discrete-level sets allow one to analyze primary gamma-ray spectra. ob

Discrete-Level Descriptions of Nuclei at Low Excitation Energies The accurate calculation of a reaction cross section near the threshold requires that the product nucleus be described with a sufficient number of discrete levels. Figure 1 illustrates this with the calculation of the Tm(n,3n) excitation function for incident neutron energies in the first few MeV above the reaction threshold of 14.96 MeV[GAR84a]. We obtained good agreement with the experimental data[BAY75,NET72,NET76,VEE77] when we used 63 modeled discrete levels for Tm in its first 1.4 MeV of nuclear excitation[HOF84]. However, when we described Tm down to its ground state with the Gilbert-Cameron level-density expression and the Cook-modified parameters[C0067], we computed an excitation function that was about 60% lower in the first 3 MeV above the threshold. Figure 2 shows another example of the need for discrete-level descriptions of nuclei. Two computations of the Y(n,y) excitation function[GAR84b] were made. In the first, the Y and *°Y nuclei were described above the ground state with an additional 24 levels provided by E. A. Henry[HEN77]; in the second, the additonal levels were replaced with the Gilbert-Cameron level-density formulae and the Cook-modified parameters. Since the first level above the ground state in Y lies at 0.9 MeV, no inelastic 169

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0097^6156/86/0324-0100$06.00/0 © 1986 American Chemical Society

Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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101

The Importance of Level Structure

Figure 1. Calculated (11,2η) and (n,3n) excitation functions for * T m [GAR84aJ, w i t h discrete-level descriptions of T m that i n c l u d e d 63 levels (solid curves) and o n l y the ground-state level (broken curves), compared w i t h experimental data from [BAY75,NET72,NET76,VEE77]. 1 6

1 6 7

scattering can occur below this energy. Yet, the calculation that used only level-density expressions pre dicted that the (η,η') cross section was already 400 mb at 0.9 MeV! This overestimation of the inelastic cross section leads to a significant lowering of the (n,y) cross section. Another effect of substituting a level-density description for discrete levels can also be seen in this figure. Note that the calculation done using level densities immediately above the ground state is still lower than that done using 25 levels for ^ty in the low-keV energy range. This cannot be due to incorrectly computed inelastic scattering competition. It comes from the fact that the description of with only level-density formulae leads to partial radiation widths that are calculated to all be about the same. In Table 1 we see that with 25 levels in ^ Y , the p-wave width and the f-wave width are calculated to be twice as large as that for s-wave neutrons (in agreement with experiment[BOL77]), and the larger capture cross section is accounted for. The role of a good discrete-level description of the target nucleus i n the calculation of a neutron capture excitation function is shown again in Fig. 3, where our initial computation of the Bi(n,y) cross section vs neutron energy is plotted. In this calculation, the B i is described by 36 discrete levels up to 3.5 M e V in nuclear excitation; these levels were provided by R. A . Meyer[MEY85]. Note that for incident energies between 0.5 and 3 MeV, the capture cross section actually increases, in agreement with 209

2 0 9


102

NUCLEI OFF T H E LINE OF STABILITY

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Figure 2. (η,χ) a n d (η,η') excitation functions for Y , calculated with discrete-level descrip tions for Y a n d *Ύ that i n c l u d e d 25 levels (solid curves), a n d o n l y the ground-state level (dashed curves).

1

I I I

2

F i g u r e 3. O u r i n i t i a l c a l c u l a t i o n o f t h e B i ( n , / ) excitation function compared w i t h measurements from [VOI83].

209

Table 1. Calculated a n d measured partial ra diation widths for **Y. Neutron partial wave s Ρ d f g

Calculated width (meV) 25 levels 1 level in^Y 183 342 186 251 179

183 170 152 132 113

Measured width (meV) [BOL77J 115 307

— — —

measurements[VOI83], because of the lack of inelastic scattering competition. This is to be expected, since the first three levels above the ground state in B i are widely spaced at 0.896, 1.608, and 2.443 MeV, respectively. A level-density description of B i above its ground state would surely overestimate the inelastic scattering competition in this region, leading to an incorrect energy dependence for the capture cross section. 2 0 9

2 0 9

Discrete Levels and L e v e l - D e n s i t y Parameterizations Most modern statistical-model codes allow the matching of the low-energy portion of a level-density expression to an available discrete-level set. Since we use the Gilbert-Cameron level-density formalism in our version of the S T A P R E code[UHL70], we accomplish this by adjusting the values of the energy constant, E , in the constant-temperature portion and of the pairing energy, δ, in the Fermi-gas portion in such a way that the two functions and first derivatives remain continuous. We find this procedure to be quite valid for spherical nuclei, provided that the level set is sufficiently complete. From such a set, information can be extracted about the spin cut-off parameter as well as about other parameters, enabling one to obtain good agreement with known D values or to predict such quantities for nuclei off the line of stability. In this way, the level structure can be used to improve on global parameterizations of the level density. 0

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GARDNER AND GARDNER

For example, in calculations that we made involving a series of strontium and yttrium isotopes in the mass range of 86 to 90, we used the discrete-level schemes provided by Henry[HEN77,HEN85]; each set was composed of 25 to 60 levels. We found that almost every isotope required a different value of k, the constant in the mean-square spin projection expression (m ) = kA , which is related to the energydependent spin cut-off parameter σ © by the formula σ ^) = (6/ic^m ){a\E - δ]) . Although it is common practice to use average values of k = 0.146[GIL65] or 0.24[REF80], we obtained a range of k values, from 0.146 to 0.29, when we attempted to extract them from the discrete-level sets. Table 2 shows our findings. Such fluctuations are in agreement with the observations of Reffo[REF80]. Table 3 summarizes the two ways in which o can be deduced from discrete-level sets. The first is the maximum likelihood estimator expression, where there are Ν levels of various J spins. This method can be more sensitive to small differences in o unless the levels exhibit a nonrepresentative distribution of mainly high or low spins. The second expression allows one to extract σ from the spin distribution and is useful unless several spins are missing. Figures 4(a) and (c) show the results of the estimator method for Y, described with 60 discrete levels, and for Y , described with 25 levels. Difficulties are encountered in both cases since each nucleus has low-lying, high-spin states that introduce disproportionate weightings because of the (/ + 1/2) term. Note that for Y in Fig. 4(a), if one had information on the first 15 levels only, up to an excitation of 3.1 MeV, the choice of k = 0.24 would be made. In Fig. 4(b), although the spin distribution method indicates that k = 0.146 is reasonable for Y , the estimator method indicates that k should have a somewhat larger value. Finally, in Fig. 4(d), the spin distribution in Y clearly indicates that the global k value of 0.146 is not suitable and that a A: of about 0.29 is preferred. The following are examples of what happens when one value of k is chosen over another. The use of a k value of 0.24 instead of 0.146, with no other changes, leads to about a 30% difference in the Fermi-gas level density. In the calculation of the Υ(η,γγ°Υ cross section, the choice of k = 0.17 instead 0.146 for *°Y, based on the discrete levels, led to a D change of 15% and a lowering of the capture cross section by about 10% in the E range of a few hundred keV. 2

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It has been our experience that one frequently can use discrete-level information to predict unknown values of D . For example, we have recently begun cross-section calculations for a suite of bismuth isotopes with target masses ranging from 203 to 212 and are using preliminary discrete-level sets provided by Meyer[MEY85]. These schemes presently consist of 30 to 40 levels each. In a number of cases, the parity distributions are poor, with most levels having the same parity; Meyer is now carrying out further evaluations for these sets. Since D is only known experimentally for B i , we used Meyer's preliminary level schemes for the various bismuth isotopes to adjust our level densities and thereby obtain initial estimates of all the D values. These are summarized in Table 4. A s one sees, the values decrease for the lighter-mass isotopes, spanning more than two orders of magnitude, as one moves away from the closed neutron shell. For the compound nucleus B i , the calculated value of 5.21 keV is in reasonable agreement o b

2 1 0

o b

o b

2 1 0

Table 2. Values of k, the constant in the meansquare spin projection expression, that were ex tracted from the discrete-level sets for isotopes of yttrium and strontium.

Table 3. The two methods of deducing the energy-dependent spin cut-off parameter, ^(E) = (6/* κ*Λ )(α[Ε - S\) , from discretelevel sets.

Isotope

(1) Use of the maximum likelihood estimator expression:

90γ 89γ 88γ 87γ 86γ

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Isotope

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Figure 9.

C o m p a r i s o n of the calculated and measured [HOF85b] reduced transition intensities for

primary gamma rays f o l l o w i n g 2-keV neutron capture b y

1 7 5

Lu.

obtained from the estimated number of radiating states. The bars indicate experimental values and their error limits, with some showing spin and parity assignments. A l l measurements and calculations are normalized to the single indicated transition. Originally, the comparison showed that our calculated M l / E l ratio was low by a factor of 3. In order to preserve the total radiation width value that had given agreement with experiment, and also match the M l / E l ratio from the A R C measurements, we had to increase our M l strength function by a factor of 3 and decrease our E l strength in the region around 5 to 6 MeV. Figure 10 compares the newly derived strength functions (the solid curves) with those predicted by our original systematics (the dashed curves). We revised the E l strength function downward around 5 MeV, while still maintaining continuity with the photonuclear region, by changing the value of the one free parameter in our systematics, E , from 5 to x

11 MeV, as indicated by the arrows in the figure. This parameter, E , is the energy at which our energyx

dependent Breit-Wigner line shape[GAR84b] has the same value as that for the Lorentzian line shape (the dotted curve), both defined with the same giant dipole resonance parameters. A s seen in Fig. 11, when we applied the new E l systematics in the mass-90 region, we obtained excellent agreement with experimental measurements for Y[AXE70] and Zr[SZE79]. Our original sys 89

90

tematics had predicted E l strengths that were about 30% higher. We are continuing this work in other mass regions to verify and further improve our systematics.


15.

GARDNER AND GARDNER

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Figure 10. Recently derived absolute dipole strength functions for L u (solid curves), com pared w i t h those predicted b y our original sys tematics (dashed curves). T h e arrows indicate the value assigned to the one free parameter, E . In our recent derivation, E = 11 M e V , and i n our original systematics, E = 5 M e V . 1 7 6

x

x

x

Figure 11. E l strength functions for Y a n d " Z r , calculated using our revised systematics, compared w i t h measurements from [AXE70, SZE79], 8 9

Use of Discrete Levels i n A n a l y s i s of Primary G a m m a - R a y Spectra Another interesting utilization of Hoff's modeled discrete levels for L u is presented in Fig. 12, where we show calculated primary dipole transitions to the L u modeled levels with energies in the range of 0.40 to 0.82 M e V following 2-keV neutron capture by L u . Also shown are the positions of the accessible levels and their spins and parities. The spectrum was constructed by smearing the calculated transition intensities with a unit Gaussian line shape with a 6-keV resolution. Note these features of the plot: (a) fcr gamma rays from 5.5 to 5.9 M e V we see 5 doublets, 1 triplet, and 1 quadruplet, all unresolved, more than expected on statistical grounds[HOF85b]; (b) the 2 " + 5 " doublet just below 0.44 M e V excita tion appears to have the same shape and intensity as the singlet 4" peak at 0.466 MeV, while the triplet, quadruplet, and doublet peaks at 0.66, 0.75, and 0.77 MeV, respectively, also show a close resemblance in shape and intensity; (c) the true singlet E l peaks, due to transitions to 4" levels at 0.82, 0.60, and 0.47 MeV, show a pronounced increase in intensity because of the energy dependence of the E l strength function itself, in addition to the expected E energy dependence. This kind of plot should be quite useful in interpreting the experimental data, particularly where unresolved multiple peaks occur. 1 7 6

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Figure 12. A portion of the calculated primary gamma-ray spectrum for the Lu(n,/) reaction. 175

Conclusion It is evident from the above examples that the use of complete sets of discrete levels is an essential key in making accurate statistical-model Hauser-Feshbach calculations. Since these level sets must include both the observed and modeled information for completeness, additional personnel and expertise are required to produce them. It should also be pointed out that the use of large discrete-level sets in calcula tions usually requires significant increases in computer time. We see that many problems still need to be solved in order to obtain accurate results in HauserFeshbach calculations. Some examples are the energy dependence of rotational enhancement of levels in deformed nuclei, the energy and mass dependence of M l gamma-ray transitions, the importance of E2 transitions, and better estimates of fission barriers. Work in each of these areas will benefit greatly from a better understanding of the discrete levels, particularly in nuclei away from stability.

Acknowledgments The authors are grateful to M . N . Namboodiri of the Lawrence Livermore National Laboratory for presenting this paper at the Symposium. This work was performed under the auspices of the U.S. Depart ment of Energy by Lawrence Livermore National Laboratory under contract N o . W-7405-ENG-48.


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The Importance of Level Structure References

[AXE70] [BAR83]

P. Axel et al., Phys. Rev. C 2, 689 (1970). D. W. Barr, S. A. Beatty, M. M . Fowler, J. S. Gilmore, R. J. Prestwood, Ε. N . Treher, and J. B. Wilhelmy, "(p,xn) Measurements on Strontium Isotopes," Nuclear Chemistry Division Annual Report FY 82, Los Alamos National Laboratory, Los Alamos, NM, LA-9797-PR (1983). [BAY75] B. P. Bayhurst et al., Phys. Rev. C 12, 451 (1975). [BEE80] H. Beer, F. Kappeler, and K. Wisshak, "The Neutron Capture Cross Sections of Natural Yb, Yb, Lu, and W in the Energy Range from 5 to 200 keV for the Lu-Chronometer," Proc. Int. Conf. Nuclear Cross Sections for Technology, Knoxville, TN, 1979, National Bureau of Stan dards, Gaithersburg, MD, NBS Special Publication 594 (1980), p. 340. [BOL77] J. W. Boldeman et al., Nucl. Sci. Eng. 64, 744 (1977). [COO67] J. L. Cook, H. Ferguson, and A. R. Musgrove, Nuclear Level Densities in Intermediate and Heavy Nuclei, Australian Atomic Energy Commission, AAEC/TM-392 (1967). [GAR84a] M. A. Gardner, " Tm(n,3n) Tm Cross-Section Calculations Near Threshold," Nuclear Chem istry Division FY 84 Annual Report, Lawrence Livermore National Laboratory, Livermore, CA, UCAR-10062-84/1 (1984). [GAR84b] D. G. Gardner, "Methods for Calculating Neutron Capture Cross Sections and Gamma-Ray Energy Spectra," Neutron Physics and Nuclear Data in Science and Technology, Volume 3, Neu tron Radiative Capture (Pergamon Press, New York, 1984), Chapter III. [GAR85a] M. A. Gardner, Calculational Tools for the Evaluation of Nuclear Cross-Section and Spectra Data, Lawrence Livermore National Laboratory, Livermore, CA, UCRL-91947 (1985); to be published in Proc. Int. Conf. Nuclear Data for Basic and Applied Science, Santa Fe, NM, May 13-17, 1985. [GAR85b] D. G. Gardner, M. A. Gardner, and R. W. Hoff, "Absolute Dipole Gamma-Ray Strength Func tions for Lu," Proc. Conf. Capture Gamma-Ray Spectroscopy and Related Topics, Knoxville, TN, American Institute of Physics, New York, NY, AIP Conf. Proc. #125 (1985), p. 513. [GIL65] A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43, 1446 (1965). [HAN83] G. Hansen and A. S. Jensen, "Energy Dependence of the Rotational Enhancement Factor in the Level Density," IAEA Advisory Group Meeting on Basic and Applied Problems of Nuclear Level Densities, Brookhaven National Laboratory, Upton, NY, BNL-NCS-51694 (1983), p. 161. [HEN77] E. A. Henry, Lawrence Livermore National Laboratory, Livermore, CA, private communication (1977). [HEN85] E. A. Henry, Lawrence Livermore National Laboratory, Livermore, CA, private communication (1985). [HOF84] R. W. Hoff and W. R. Willis, "Level-Structure Modeling of Odd-Mass Deformed Nuclei," Nuclear Chemistry Division FY 84 Annual Report, Lawrence Livermore National Laboratory, Livermore, CA, UCAR 10062-84/1 (1984). [HOF85a] R. W. Hoff, J. Kern, R. Piepenbring, and J. B. Boisson, "Modeling Level Structures of Odd-Odd Deformed Nuclei," Proc. Conf. on Capture Gamma-Ray Spectroscopy and Related Topics, Knox ville, TN, American Institute of Physics, New York, NY, AIP Conf. Proc. #125 (1985), p. 274. [HOF85b] R. W. Hoff et al., Nucl. Phys. A437, 285 (1985). [KAT78] S. Kataria et al., Phys. Rev. C 18, 549 (1978). [MAC78] R. L. Macklin, D. M. Drake, and J. J. Malanify, Fast Neutron Capture Cross Sections of Tm, Ir, Ir, and Lu for 3 < E < 2000 keV, Los Alamos National Laboratory, Los Alamos, NM, LA7479-MS (1978). [MEY85] R. A. Meyer, Lawrence Livermore National Laboratory, Livermore, CA, private communication (1985). [MUG84] S. F. Mughabghab, Neutron Cross Sections, Vol. 1, Part Β (Academic Press, New York, 1984). [MYE75] W. Myers et al., J. Inorg. Nucl. Chem. 37, 637 (1975). [NET72] D. R. Nethaway, Nucl. Phys. A190, 635 (1972); also, unpublished (n,3n) cross-section measure ments, Lawrence Livermore National Laboratory, Livermore, CA (1972). [NET76] D. R. Nethaway, unpublished (n,2n) cross-section measurements, Lawrence Livermore Na tional Laboratory, Livermore, CA (1976). [REF80] G. Reffo, "Phenomenological Approach to Nuclear Level Densities," Theory and Applications of Moment Methods in Many-Fermion Systems (Plenum Press, New York, 1980). 170

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G. Szeflinska et al., Nucl. Phys. A323, 253 (1979). M. Uhl, Acta Phys. Austriaca 31, 245 (1970). L. R. Veeser et al., Phys. Rev. C 16, 1792 (1977). J. Voignier, S. Joly, and G. Grenier, "Mesure De La Section Efficace De Capture Radiative Du Lanthane, Du Bismuth, Du Cuivre Naturel et De Ses Isotopes Pour Des Neutrons D'Energie Comprise Entre 0,5 et 3 MeV," Proc. Int. Conf. Nuclear Data for Science and Technology, A Belgium (1983), p. 759.

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