Parameters of the best approximation for distribution of the reduced neutron widths. The most probable density of neutron resonances in 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. THE MOSTPROBABLE DENSITY OF NEUTRON RESONANCES IN ACTINIDESA.M. Sukhovoj, V.A. Khitrov
Joint Institute for Nuclear Research, Dubna, Russia, 141980
Abstract
In the frameworks of hypothesis of practical constancy of the neutron resonancenumber in small fixed intervals ∆ E of neutron energy, their most probable valuewas determined for nucleus mass region 231 ≤ A ≤
243 from approximation of thereduced neutron widths by superposition of two or four independent distributions.This was done under assumption that a set of the measured neutron amplitudes cancorrespond to one or to superposition of some normal distributions with non-zeroaverage and dispersion differing from < Γ n > .The main result of the analysis: the mean D and S values can be determined onlywith unknown systematical uncertainty whose magnitude is determined by unknownprecision of the Porter-Thomas hypothesis correspondence to concrete experimentalsets of resonances and unknown experimental mean < Γ n > widths. The density of neutron resonances ρ λ = D − λ is one of the main results of analysis ofthe data of all experiments performed using the neutron time-of-flight method. It is thebasis point for any experiments where nucleus level density is derived from the spectraof gamma-quanta or evaporation nucleons. High precision in determination of D − λ isstipulated by excellent resolution of corresponding method, but it can be realized only bycareful accounting or correction of all systematical errors of experiment.The most serious and not removable from them is “omission” of resonances whosereduced neutron width Γ n (Γ n ...) is less than the sensitivity threshold of experiment. Inprinciple, determination of the most probable value of D − λ in this situation is possibleonly by means of the most precise approximation of distribution of Γ n in all region oftheir values and extrapolation of the obtained function into the region below threshold. Ofcourse, precision of this procedure is determined by degree of correspondence of theoreticalnotions about distribution Γ n to experiment.According to theoretical analysis [1], the value Γ n in nuclei of intermediate and largemass is determined by few-quasi-particle components of wave function whose square con-tribution in its normalization is estimated by value of about 10 − -10 − . Their small andchaotic values for different resonances are determined by strong fragmentation [2] of low-lying one- and two-quasi-particle states of a nucleus. There is the first necessary conditionfor description of fluctuations of Γ n by the Porter-Thomas distribution [3].nother condition is that the mathematics expectation of mean value of amplitude A = q Γ n must be equal to zero, and its dispersion – to mean < Γ n > . Both conditions: M ( A ) = 0 ,D ( A ) = < Γ n > (1)are not tested in modern analysis of the experimental Γ n values [4]. I.e., applicabilityof the Porter-Thomas distribution is postulated but is not proved. The experimentalwidth distribution is not tested also for possibility of existence of superposition of severaldistributions with different values M ( A ) and D ( A ). Approximation [5, 6] of level densityderived from the two-step cascade intensities shows that the structure of any nucleuschanges cyclically as increasing excitation energy. This fact is determined so far as atpresent there is the only methodically model-free method for determination of ρ – [7].This occurs, at least, due to excitation of nucleus states with increasing number of quasi-particles and, probably, due to variation of number and type of phonons. Fragmentationof these complicating nucleus states inevitably changes coefficients of wave functions ofneutron resonances (as it follows from basis notions of quasi-particle-phonon model ofnucleus). As a result, it is possible violation of the Porter-Thomas distribution in existingtoday interpretation (1). The method for analysis of the data on Γ n accounting for these factors is described in[8], concrete results of the best fitting of the experimental data for actinides are given in[9]. Cumulative sum of Γ n in suggested there analysis is approximated by one or severaldistributions of the variables: X = (( A − b ) /σ ) (2)with the initial values of fitted parameters b = M ( A ) = 0 ,σ = D ( A ) = < Γ n > . (3)Parameters of the best approximation of distribution of the experimental values of Γ n in actinides for variants of their one ( K = 1) or, maximum, four ( K = 4) distributions withdifferent M and D are compared in [9] between themselves or with approximation of thedistorted by given registration threshold pure model random values. This analysis bringsto the conclusion that at present it is inadmissibly to exclude a possibility of existenceof superposition of several differing by parameters b and σ width distributions in everynucleus. Although unambiguous conclusion about its presence cannot be made on thebasis of the modern experimental data on the resonance neutron widths. Therefore, themean spacing between resonances in actinides is determined below in different ( K = 2 and K = 4) variants. The suggested in [9] possibility to estimate the most probable number ofomitted resonances in any experiment calls no doubts if only the functional dependencef their part ∆ S th from the total number S was set on the grounds of some data forconcrete intervals of resonance energies. Then the parameters of unknown distributionsare determined by equation: χ = ( S − ( ψ ( A, b, σ ) − ∆ ψ th )) (4)Here ψ ( A, b, σ ) = R X ∗ Γ( X ) dX for gamma-function Γ with variable X . The value∆ ψ th is determined only by difference N t − N exp for the varied expected resonance N t number in interval δE and the obtained experimentally N exp . The number of these inter-vals practically was varied from 5 to 20 in dependence on quantity of experimental valuesof widths. Moreover, negative values N t − N exp in all cases were changed by zero.The calculated and experimental cumulative sums in this equation have differing valuesof variables: function S was obtained under assumption that the unknown mean value ofneutron width corresponds to P N exp , but the mean neutron width for the calculated valueis determined by sum P N t . Therefore, calculation of χ is carried out after correspondingchange in variable X for difference ψ − ∆ ψ th .The serious enough problem is setting of dispersion of cumulative sum for arbitraryvalue X . Methodically this problem has simple solution: there are generated large sets ofcumulative sums of squares of normally distributed numbers with given b and σ values foreach “partial” function number K and for them by means of usual relations of mathemat-ical statistics is determined function σ = f ( X ) for each magnitude of variable X . But,in practice, this procedure requires unacceptable expenditures of computer time. Thatis why, possible change of the χ value for different densities of neutron resonances forrealistic values of dispersions of cumulative sums was performed only for Th, , Uand
Pu (only in approach of validity of the Porter-Thomas distribution).The difference of principle between the results of this approximation and the datagiven below was not revealed.Function (4) has not real minimum and in this variant of analysis of distributions ofreduced neutron widths. Comparison between the calculated and experimental cumulativesums shows that some small difference of χ for tested N t values is mainly caused by strongfluctuations of cumulative sums in region of the largest X values.Naturally, function ∆ ψ can take into account and other factors distorting experimentaldistribution of widths. This accounting can be performed in frameworks of both somemodel approaches and concrete experimental data. Of course, function ∆ ψ cannot be setunambiguously for the majority of factors which distort the neutron widths distributions.The desired D = P δE/ P N t value corresponds to minimum of χ for varied values D . Fluctuations of different strength in the found function χ = f ( D ) are connected withambiguity of the best fit in the region of the large X values or change in parameters forelements of the tested superposition at K >
1. In particular, at change of D in case K = 2, for example, the smaller values of χ can be really realized not for two, but in fact– three distributions: sum of widths distributions for both spin values of resonances andadditional distribution of widths corresponding to the largest values of Γ n and parameter b >> N exp is shown in Fig. 1. The parameter of N e xp E n , eV Fig. 1. Experimental number of resonances in interval δE = 113 eV for U.analysis (4) for this nucleus was tested for interval 110 ≤ N t ≤ U corresponding to dif-ferent expected density of neutron resonances for D = 0 . D = 0 . D ≥ . − . D ≥ − b and σ at noticeably smaller values D cannot give small χ by use of superposition ofboth two and four different distributions. However, the values χ for K = 2, respectively,increase with respect to K = 4. Sometimes – very essentially.Besides, it should be taken into account that the practical search of parameters b and σ , which guarantees minimum of χ in the used method of approximation cannot securethe best approximation of the experimental data in arbitrary variant of calculation. Onlythe repeated variation of initial values and ways of random processes can provide thesufficient for practical applications precision of determination of the lowest possible χ value.The obtained by us distributions χ = f ( D ) for different variants of approximation ofthe experimental data for nuclei from the mass region 231 ≤ A ≤
243 are given in figures3-4.Estimated values of widths were taken from library ENDF/B-VII [10]. In order tocompensate “omitted” resonances in
Th and
U, the authors of the neutron resonanceevaluation included for these nuclei in the library data the resonances whose randomwidths are less than registration threshold. Naturally, presented here analysis of such C u m u l a ti v e s u m o f G / < G > G /< G > D>0.2 eV
D=0.04 eV
Fig. 2. Typical forms of the best approximations of cumulative sums for the experimentaldata on the reduced neutron widths. As an example, there are presented the data for K = 4 U in region of strong increase of χ and region of its practically constant value. ,0 0,2 0,4 0,6 0,80204060 0 5 10 151015202530 0 1 2 3 4 54060800,0 0,2 0,4 0,6 0,81020304050 0 2 4 6 8 10051015 0,0 0,2 0,4 0,602004006000 5 10 1502468 0,0 0,2 0,4 0,6 0,8050100 0 5 10 15 20 2520304050 Pa Th, l=0
Th, l=1 c U U U U D, eV Np U, l=0
Fig. 3. The χ value for the tested D parameter for the nuclei with mass 231 ≤ A ≤ D values from [11] or [10].mixture may gives somewhat distorted information on density of neutron resonances andis added below, most probably, for demonstration of potential of the suggested method.Results of fitting of the D value, as it is seen from the data presented in figures 3-4 foreach nucleus, depend on model notions. In practice, one can conclude that:(a) the analysis gives wide spectrum of possible D values corresponding to eitherpractically constant χ value or – weakly fluctuating function of this parameter;(b) weak local minima of χ are caused by bad stipulation of approximation processfor variant K >
U, l=1
Pu, l=0
Pu, l=1 c Pu Pu Pu Am D, eV Am Cm Fig. 4. The same, as in Fig. 3, for nuclei with mass 238 ≤ A ≤ The most important result obtained in frameworks of described analysis of the exper-imental data on values of Γ n or Γ n – the mean Γ n and ρ values are at present determinedwith on principle unknown systematical error. Really this result is expected: parametersof any process under study cannot be found even from mathematically strong extrapola-tion (or interpolation) of corresponding data (in given case – for the studied regions ofnucleus excitation energy). The unexpected point was the found here possibility that themean value of widths can be much less than registration threshold of experiment .In original paper [3] is stated without any proof that: “As a consequence of experimen-tal limitations, levels with small widths may escape detection, and also there may be onlyfew of them...”. Authors bring as an example for X = 0 .
01 the estimation of deviation in9% between the average over measured widths and the expected one’s average over thetotal distribution. These statements are quite true in case of small part of widths whichare less than the threshold value. And they are absolutely mistaken – in case when themain part of neutron widths lie below registration threshold of experiment. Belonging ofhe tested set to one of these extreme (as and intermediate) cases is determined by valueof < Γ n > . In turn, it can be obtained only on the basis of necessary amount of additionalexperimental information.Accordingly, all the published estimates of density of neutron resonances contain un-known systematical error. In the best case it is enough (for practical aims, for example)small; in the worse – changes the values of Γ n and ρ by many times. The errors of pa-rameters under consideration anticorrelate with each other. Accordingly, at calculationof, for example, averaged neutron-interaction cross-sections, their uncertainties can benegligibly small even for large δ Γ n and δρ . However, for understanding of occurring innucleus processes of interaction and transition between Bose and Fermi systems and de-termining them properties of nuclear matter, the achieved precision for determination oflevel density can be insufficient.Presentation of experimental data in form of cumulative sums of Γ n chosen for anal-ysis has the lowest dependence on error of determination of < Γ n > . Therefore, theresult obtained here could not be determined earlier in the simplest analysis methods ofdistributions of < Γ n > .As a consequence, any method for determination of mean parameters of neu-tron widths distributions can give only some their probabilistic values . The main result of the neutron widths distribution analysis: the D and S values can bedetermined only with unknown systematical uncertainty whose magnitude is determinednow by unknown precision of the Porter-Thomas distribution correspondence to concreteexperimental mean < Γ n > widths.1. The suggested approximation of the total set of all the existing data on widthsof neutron resonances does not allow one to find unambiguously determined absoluteminimum of χ for the unique value of D .2. The use for this aim of superposition of several distributions with the differentaverage and dispersion allows one to obtain the lowest value of χ , first of all, for theexperimental data with number of widths exceeding ∼ n to the simple functional dependence in nuclei of different mass is less than thatfor the set of noticeably differing functions. Therefore, any quantitative determinationof parameters of their distribution should be performed by comparison of two or moredifferent model notions in maximum set of nuclei.4. The obtaining of the more unambiguous conclusions with respect to the problemconsidered here requires very significant increase of sets of resonances with the experi-mentally determined values Γ n (Γ n ) at their correspondingly decreased distortions.5. The increase of precision for determination of the mean parameters of the neutronidth distributions requires, most probably, considerable making more precise of modelnotions [3]. First of all of degree of influence of structure of the nuclear excited levels onlevel density and probability of emission of the nuclear reaction products in wide diapasonof their energy. In particular – in region of neutron resonances.6. Selection of neutron resonances by orbital momentum must be performed in com-mon – by minimum sum value of χ for obtained distributions with l = 0 and l = 1, forexample. References [1] V.G. Soloviev, Sov. Phys. Part. Nuc. (1972) 390.[2] L.A. Malov, V.G. Solov’ev, Yad. Phys., (1977) 729.[3] C.F. Porter, R.G. Thomas, Phys. Rev. (1956) 483.[4] H. Derrien, L.C. Leal, N.M. Larson, Nucl. Sci. Eng., 160, 149 (2008).[5] A.M. Sukhovoj, V.A. Khitrov, Preprint No. E3-2005-196, JINR (Dubna, 2005).[6] A. M. Sukhovoj, V. A. Khitrov, Physics of Paricl. and Nuclei, (2006) 899.[7] A.M. Sukhovoj, V.A. Khitrov, Phys. Particl. and Nuclei, (2005) 359.[8] 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.[9] A.M. Sukhovoj, V.A. Khitrov, In: