Parameters of the best approximation of reduced neutron widths distribution. Actinides
aa r X i v : . [ nu c l - e x ] M a y PARAMETERS OF THE BEST APPROXIMATION FORDISTRIBUTION OF THE REDUCED NEUTRON WIDTHS . ACTINIDESA.M. Sukhovoj, V.A. Khitrov
Joint Institute for Nuclear Research, Dubna, Russia
Abstract
The data of ENDF/B-VII library on Γ n (Γ n ) for nuclei Pa,
Th, , , , , U, Np, , , , Pu, , Am and
Cm (including p-resonances of
Th, U, Pu) in form of cumulative sums in function onΓ n / < Γ n > were approximated by variable number K of partial items (1 ≤ K ≤ K in their totalsum. The problems of their determination from distributions of different numberof squares of normally distributed random values with variable threshold of loss ofsome part of the lowest Γ n values were studied.It was obtained for some part of neutron resonances that their mean amplitudescan considerably differ from zero value, and dispersions – from < Γ n > . And it isworth while to perform any quantitative analysis of distributions Γ n by means ofcomparison of different model notions with obligatory estimation of random disper-sion of the desired parameters. The experimental data on reduced neutron widths Γ n (Γ n ) of s- or p-resonances po-tentially contain diversiform information. In particular, of interest are the data on theirreal density ρ and on few-quasi-particle components of wave function of high-lying level[1]. Restoring of this information from the data of any experiment on the neutron timeof flight method cannot be made without the use of hypothesis on form of the frequencydistribution of neutron widths. Porter and Thomas first assumed [2], on the grounds oflarge fluctuations of Γ n , that it corresponds to χ -distribution with one degree of freedom.In the other words – to distribution of squares of normally distributed random values ξ with mathematics expectation M ( ξ ) = 0 and dispersion D ( ξ ) = 1. These conditions arerealized to higher or lower extent in case when wave function of neutron resonance con-tains many small items of different sign which determine the value Γ n . Within frameworksof notions of quasi-particle-phonon nuclear model, this means that fragmentation of anynuclear states like m quasi-particles ⊗ n phonons over nuclear levels in region of neutronresonances (for truth of [2]) must be very strong.Theoretical analysis of fragmentation process of nuclear states of different type over thelevels with different excitation energy [3] showed that this condition can be not realizedin case of large enough values m and n . This conclusion follows and from results ofapproximation below B n [4, 5] of methodically more precise experimental data for ρ [6, 7]n density ρ n of n -quasi-particle levels by Strutinsky model [8]. It can be performed atwide enough variation of assumptions on shape of correlation function of nucleon pair inheated nucleus and on coefficients of its vibration enhancement. Nevertheless, results ofapproximation already gave some notion on structure of high-lying levels of any nuclei.The results [4, 5] unambiguously show that at the simplest hypotheses of correlationfunctions δ n of nucleon Cooper pairs in heated nucleus, the number n increases by 2 quasi-particles with excitation energy interval being some less than ∆ E ex ≈ δ . Id est, structureof wave function of highly-excited levels (and, it is not excluded, of neutron resonances)can cyclically change. These data are enough for qualitative explanation of change inform of radiative strength functions as increases mass of nucleus A [9]. Id est, there aretheoretical and experimental grounds for detailed and methodically independent analysisof the data on Γ n . The primary goal of this analysis – discovery of possible deviationsof neutron width distribution from the Porter-Thomas distribution and estimation ofreliability extent for their observation in experiment.Reanalysis of the data on gamma-transition intensities from reaction ( n, γ ) [10, 11]revealed strong influence of nuclear structure on distribution of parameters of radiativewidths of primary gamma-transitions from neutron resonances. This is an additionalargument for independent full-scale analysis of the data on neutron widths with lessquantity of the used model ideas. Direct determination of structure of arbitrary nuclear levels above some MeV usuallyis inaccessible for all the known experiments. Therefore, any information on this accountcan be derived only from indirect data (as it was shown in [1]). First of all – from analysisof the results of model approximation of experimental distributions of Γ n (2 g Γ n – forresonances with different spins). Numerous problems of analysis of experimental dataof this type are described, for example, in [12]. Analysis of modern data on parametersof neutron resonances of U in neutron energy region up to 20 keV is presented, forexample, in [13],
Th - [14].The basis for all the performed earlier analyses is an assumption that the hypothesis[2] describes tested set of the data on Γ n with a precision exceeding accuracy of theexperiment. By this it was assumed that in analysis is in some form realized correctaccounting (or exclusion) of experimental distortions of the data under consideration(omission of weak levels, unresolved multiplets, admixture of resonances with other orbitalmomentum l and so on). Or observed discrepancies of experiment with distribution [2] arecompletely explicable by enumerated factors. In practice, it is tested up to now only thehypothesis of deviation of experimental distribution of Γ n from the expected theoreticalone owing only to deviation of parameter ν of theoretical distribution from unit.Whereas, the form of distribution of Γ n strongly depends on degree of execution ofcondition of equality to zero of mean amplitude A n (Γ n = A n ) of the tested set of res-onances. It is absolutely impossible also to exclude a possibility that the experimentaldata are superposition from K distributions even for their set with precisely determinedpins and orbital momenta of resonance neutron. Qualitatively, the possibility M ( A n ) = 0directly follows from [3], K >
The fitted function in full-scale analysis is the sum of K distributions P ( X ) of squaresof normally distributed random values with independent variables X k each. The de-sired parameters in compared variants are the most probable value b k of amplitude A = q Γ n / < Γ n > , dispersion σ k and total contribution C k of function number k forthe variable X k = (( A k − b k ) ) /σ k (1)in the total experimental sum of widths. Statistically significant result b k = 0 allowsone to state that the neutron resonance is not completely chaotic system, σ k < K >
1) is not caused by random fluctuations of X , can givenew information on nuclear structure in region B n . First of all – information on possibleexistence of neutron resonances with different structure of their wave functions and onregions of Γ n values, where radiative strength function (the total gamma-spectrum) haveessentially different form (see, for example [9]).Cyclic change in structure of neutron resonances at different neutron energies (directlyfollowing from successive break up of Cooper nucleon pairs [4, 5]) can stipulate non-monotonous character of change in density of nuclear excited levels and above neutronbinding energy. This means additional systematical error of the data on ρ in the mostimportant for this nuclear-physics parameter point.As it was obtained by modeling [16], the values of b k , σ k and C k with small statisticalerror for accumulated by now sets of neutron widths cannot be get not only for K > K = 1. Most probably, this circumstance has principle character and appearsitself, first of all, by extraction of level density and emission probability of the nuclearreaction product from the spectra (cross sections) of nuclear reactions. By analysis of experimental data in low energy nuclear physics (at least in some itssections) is really used the postulate on principle possibility of unambiguous determina-tion of desired nuclear parameters. For example, level density in fixed interval ∆ E foriven nuclear excitation energy and emission probability of some nuclear reaction productat their de-excitation. Or excitation – at decay of higher-lying levels. However, the expe-rience of determination of ρ and radiative strength functions k from intensities of two-stepgamma-cascades [6, 7] together with analysis of possibilities of existing methods of anal-ogous experiments [15] shows that their unambiguous determination is impossible. It istrue, at least, for the present and for region of high level density. Practically, it followsfrom this circumstance that the ρ and k values can be determined only with inevitablesystematical error or there can be found only final interval of values of these parameterswhich contains desired parameter. And its asymptotical width is not equal to zero.The task under consideration obviously belongs to the same class. Id est:(a) the parameter ν of χ -distribution can be unambiguously determined (with pre-cision up to experimental uncertainty and statistical fluctuations), but there cannot betested all the necessary conditions of applicability of hypothesis [2], or(b) there can be found only asymptotically non-zero interval of values of parameters ofexpression (1). Below is realized only the second possibility. The more detailed descriptionthe analysis method of and results of its test are presented in [16]. Comparison of experimental cumulative sums with approximating curves for 15 setsof s - and three sets of p -resonances in variants:the distributions K = 1 and superposition of four possible distributions ( K = 4) ispresented in figures 1 and 2.The ratios of χ in function of nuclear mass for two variants of approximation areshown in Fig. 3. (Approximation in all the cases was performed, as a minimum, over X =1000 values.It follows from these figures that approximation of the experimental data by superpo-sition of four distributions improved precision for the greater part of experimental data.And maximally – for s -resonances of A -odd nuclei and all sets of available p -resonances.The simplest possible explanation is obvious: the 2 g Γ n values for different spins ofresonances differ by parameters of distributions of their neutron amplitudes. It is enoughfor their appearance in the obtained experimental data as a superposition of two dis-tributions with different σ and/or b values. But, the tendency of change of ratio ofapproximation parameter χ for different nuclei allows and possibility of change in shapeof width distribution when nucleus mass changes.Large dispersion of random values X brings to large fluctuations of cumulative sumsof both experimental data and model distributions [16]. And, respectively, to essentialvariations of the best values of parameters (1). That is why, the conclusions about possibledeviations of b and σ parameters from expected values 0 and 1, respectively, can have, asit was mentioned above, only probabilistic character. In Fig. 4 are compared frequencydistributions of these parameters for modeling sets with N=150, 500 and 2000 random Pa Th,l=0 Th,l=1 C u m u l a ti v e s u m o f G / < G > U U U U G /< G > Np U, l=0
Fig. 1. Histogram - cumulative sum of Γ n / < Γ n > for their values, less than givenmagnitude. Thick solid curve – the best approximation by four distributions, dottedcurve – by one distribution for nuclei with mass 231 ≤ A ≤ < K ≤ U, l=1 Pu, l=0 Pu, l=1 C u m u l a ti v e s u m o f G / < G > Pu Pu Pu Am G /< G > Am Cm Fig. 2. The same, as in Fig. 1, for 238 ≤ A ≤
32 236 240 2440,00,20,40,60,81,0 c ( K = ) / c ( K = ) A l=0 l=1 Fig. 3. The ratio of the lowest χ values for the sets from K approximating distributionsfor actinides under consideration. X values. Modeling was performed for the variant of absence of omission of small X values and for omission corresponding to exclusion of L = 30% of their lowest magnitudes(linearly changing with number of random value). -8 -6 -4 -2 0 2 40510152025 -5 0 505100 2 4051015 0 1 2 3 4051015 b s E v e n t s Fig. 4. The comparison of the frequency distributions of appearance of given values of b (upper) and σ (lower) row accordingly for K = 1. Left column – in modeling are includedall the possible random values; right column – there are excluded L = 30% of the lowestrandom values in each tested set. Thick solid curve – experimental data set, thin solid,dashed and dotted curves – the data for N =2000, 500 and 150 random values in modeledsets.The widths of corresponding distributions decrease when N increases. They are min-imal for L = 0 and in all practical cases – less than the maximal width of b k and σ k experimental frequency distribution. Figures 1 and 2 permit one to conclude that de-viation of experimental width distribution from the Porter-Thomas distribution appearsitself mainly at X = (Γ n / < Γ n > ) > −
5. Discrepancy between experimental data andhypothesis [2] at less values of X can be related, first of all, to omission of weak resonances E v e n t s b s Fig. 5. The same, as in Fig. 4, for the case K = 4.or to other systematical errors of experiment. But, it is not excluded and possibility ofreal deviations of parameters b and σ from the values corresponding to hypothesis [2].Although deficiency of experimental Γ n values for analyzed here actinides did not allowone to get unambiguous conclusions on real parameters of expression (1), the informationfrom the data of estimation of mean spacing between their resonances is much moreunexpected [17].
1. The analysis performed shows that the probability of correspondence of distribution2 g Γ n to the unique functional dependence ( K = 1) in nuclei of different mass is less thanto the set of visibly different functions ( K = 4). Therefore, any quantitative test ofhypothesis [2] should be performed by comparison of two or more different model notionsin maximal set of nuclei.2. The results of performed analysis, probably, do not contradict to hypothesis [2]on equality of mean value of amplitude to zero for the main part of the determined Γ n values. More precise statement on this account can be made only after significant decreaseof experimental threshold of resonance registration.3. The more unambiguous conclusions on equality of dispersion of experimental dis-tribution of Γ to the fitted value cannot be made on basis of present analysis.4. Parameters of approximation of the experimental data permit a presence of, atleast, superposition of two sets of resonances for K > ν of the Porter-Thomas distribution from ν = 1 forhe majority of experimental sets of neutron widths can be interpreted only after modelestimation of its random fluctuations.6. Unambiguous conclusions on problems considered here require very significantincrease of sets of resonances with experimentally determined Γ n (Γ n ) values at theiraccordingly decreased distortions. References [1] V.G. Soloviev, Sov. Phys. Part. Nuc. (1972) 390.[2] C.F. Porter and R.G. Thomas, Phys. Rev. (1956) 483.[3] L.A. Malov, V.G. Solov’ev, Yad. Phys., (1977) 729 .[4] A.M. Sukhovoj and V.A. Khitrov, Preprint No. E3-2005-196, JINR (Dubna, 2005).[5] A.M. Sukhovoj, V.A. Khitrov, Physics of Paricl. and Nuclei, (2006) 899.[6] E.V. Vasilieva, A.M. Sukhovoj, V.A. Khitrov, Phys. At. Nucl. (2001) 153,(nucl-ex/0110017)[7] A.M. Sukhovoj, V.A. Khitrov, Phys. Particl. and Nuclei, (2005) 359.[8] V.M. Strutinsky, In: Proc. of Int. Conf. Nucl. Phys. , (Paris, 1958), p. 617.[9] A.M. Sukhovoj, W.I. Furman, V.A. Khitrov, Physics of atomic nuclei, (2008) 982.[10] A. M. Sukhovoj, Physics of atomic nuclei, (2008) 1907.[11] V. A. Khitrov and A. M. Sukhovoj, In: Proceedings of the XVI International Semi-nar on Interaction of Neutrons with Nuclei, Dubna, June 2008 , E3-2009-33 (Dubna,2008), p. 192: p. 162; p. 230.[12] F.H. Frohner, In
Proceedings of the IAEA advisory group meeting on basic and appliedproblems of nuclear level densities, Brookhaven, april 11-15, 1983 , p.219.[13] H. Derrien et al., ORNL/TM-2005/241[14] H. Derrien, L.C. Leal, N.M. Larson, Nucl. Sci. Eng., 160 (2008) 149.[15] A. M. Sukhovoj, V. A. Khitrov, W. I. Furman, Physics of atomic nuclei, (2009)1759.[16] A.M. Sukhovoj, V.A. Khitrov, In: Proceedings of the XVIII International Seminaron Interaction of Neutrons with Nuclei, Dubna, May 2010 , E3-2011-26, Dubna, 2011,p. 199.[17] A.M. Sukhovoj, V.A. Khitrov,