Parameters of the best approximation for distribution of the reduced neutron widths. Specificity of full-scale method of analysis
aa r X i v : . [ nu c l - e x ] M a y PARAMETERS OF THE BEST APPROXIMATION FORDISTRIBUTION OF THE REDUCED NEUTRON WIDTHS.SPECIFICITY OF FULL-SCALE METHOD OF ANALYSISA.M. Sukhovoj, V.A. Khitrov
Joint Institute for Nuclear Research, Dubna, Russia
Abstract
The method is described and tested for analysis of statistical parameters ofreduced neutron widths distributions accounting for possibility of coexistence ofsuperposition of some functions with non-zero mean values of neutron amplitudeand its arbitrary dispersion. The possibility to obtain reliable values of distributionparameters at variation of number of resonances involved in analysis and change ofregistration threshold of resonances with the lowest widths is studied.
Experimentally measured reduced widths Γ n (Γ n ) of neutron resonances – strong-ly fluctuating values. This circumstance very much complicates determination of theirmean values (the averaged spacing D and strength function S = < Γ n > /D ) from realexperimental data distorted by different systematical uncertainties. The generally accept-ed notion of shape of their distribution was suggested in 1956 [1] and was not up to nowtested in full scale.This test is non-trivial procedure because only the part of the measured distributionis observed in experiment but it’s independent parameter X can be determined only forthe total spectrum of possible values of widths. Id est, approximation of experimentaldata is performed at presence of unknown error parameter X = Γ n / < Γ n > . Real valueand error < Γ n > , of cause, cannot be determined experimentally. And value of the δX depends on accepted model notions.The ordinary test of distribution Γ n consists in determination of effective value ofnumber of degrees of freedom ν of χ -distribution for given set of widths (with fixed orbitalmomentum l ) and rather subjective choice of neutron energy interval where distortionsof D and S are minimal. The deeper test must answer the questions, in what degree arerealized the conditions of applicability of χ -distribution to real data. Id est:(a) whether mathematical expectation of amplitude A = q Γ n is equal to zero,(b) its dispersion – to mean value < Γ n > and(c) whether the function providing description of experimental data with maximumpossible precision is the unique?t should be also taken into account that practical investigation of nucleus propertiesincludes obligatory stage – creation of mathematical model of process under study. Bythis, any model is created on limited basis of data having non-estimated systematicalerrors which are inevitably projected on the following investigation of nucleus properties.Therefore, predictive ability of a model and its quality are strongly correlated values.It follows from this the necessity to test hypothesis [1] whether real distribution A canbe composition of several Gauss distributions with different mean values and dispersions. Analysis of status of the problem from the point of view of both theoretical ideasand totality of experimental data allows one to expect for maximal discrepancy betweenexperimental data and hypothesis [1] in region of maximal widths. In practice, it canbe caused by influence [2] of large components of wave functions of nuclear states withmaximal number of quasi-particles and phonons owing to their weak fragmentation [3] inthe excitation energy region E ex ≈ B n .Distribution Γ n can be approximated in both its “differential” and “integral” formsin function of width or square root from this resonance parameter. Experimental datacontain fixed quantity of information. Therefore, the volume of available information doesnot depend on form of data presentation and its choice is determined only by mathematicalproblems of obtaining of the sought values and visualization of results. In principle, it ispossible to ahalyse both distribution itself or large enough set of its momentums.Specific problem of distribution analysis of changeable values at presence of thresholdof their registration - the lack of information on portion of distribution of neutron widthis really observed in experiment. As a consequence, there appears the problem of unit ofmeasurement for random value it does not depend on form of distribution presentation.The most suitable form for presentation of the data for the problem under solution iscumulative sum of experimental values of X = Γ n / < Γ n > , increasing when increases X .This sum includes all the observed experimentally and included in the used compilation(for example, in [4] or library ENDF/B-VII [5]) values of widths.The selective average < Γ n > for experimental cumulative sum was determined fromthis set without accounting for missed resonances and their unresolved multiplets. Itsinevitable displacement with respect to unknown value is compensated at approximationby deflection of approximated σ value from the most probable value (in particularly, from σ = 1). This uncertainty does not influence χ - the shape of relative difference betweenexperimental and approximated distributions does not depend on units determining thewidth Γ n .Approximation region in all calculations was limited by the interval from zero to twicemaximal experimental values X max . Cumulative sum was normalized in point X max tonumber of experimentally determined widths. The region (0 − X max ) included in allcases not less than 1000 points, in which was minimized the difference of experimentalumulative sum and its approximating function. Dispersion of cumulative sum at thisnormalization changes from zero in extreme points to maximal value in region X ∼ − χ was calculatedas a sum of squares of difference of experimental and approximated values of cumulativesums. Naturally, all statistical errors in region of the lowest widths in this case exert thelowest influence on determined parameters of distributions. For convenience of comparisonof different data the value χ was divided by number of freedom degrees of approximation.All the obtained experimentally values of widths were included in practical analy-sis except obvious errors of experiment (misprints in compilation, strong discrepancy indifferent data sets). Practically, the latter can be revealed only in region of maximalvalues of Γ n , therefore corresponding correction decreases degree of discrepancy betweenexperiment and [1].This form of presentation of experimental data permits one to involve simply enoughin approximation, in principle, any factor distorting width distributions. Besides, thisallows determination of probable resonance parameters for any nucleus at presence of sys-tematical errors of Γ n , if only influence of such systematical error can be take into accountin any (numerical or analytical) form of functional dependence with free parameters. Experimental level density in region of neutron resonances of nuclei from mass region40 ≤ A ≤
200 obtained in Dubna (within the model-free method for analysis of thetwo-step cascade intensities) is described [6, 7, 8] by sum of three (or more) partial leveldensities with different number of quasi-particles and phonons. Practically, it was accept-ed on calculation problems that in limit case the experimentally observed resonances canbelong (as a maximum) to four different distributions of Γ n for even-even target-nuclei.This is true and for A odd nuclei at equality of < g Γ n > for resonances with differentspins J . In the other case the results of approximation contain and information on spindependence of neutron strength functions.Physically, according to the parameters of different approximation variants of thetotal set of level density obtained for ≈
40 nuclei in Dubna, it is also worth while tolimit maximal value of K by K = 4. In this case, the system of corresponding nonlinearequations will be most probably always degenerated. Therefore, instead of determinationof the unique value of any parameter, it is necessary and possibly to determine the widthof limited interval of their values corresponding to χ minimum.A smallness of set of experimental values of the widths and exponential functionaldependence of probability for their observation at different Γ n very strongly complicateprocess of determination of the parameters for approximating function. Therefore, it isworth while to perform this operation so that the algorithm of search for minimum of χ would permit stable approximation of the experimental data at presence of two and moredistributions with practically coinciding parameters. Comparison of the data obtainedfor K > B n . First of all – information on possibleexistence of neutron resonances with different structures of their wave functions (as it wassuggested in [9]).Practical degeneration of the realized process together with exponential change of theanalyzed dependencies complicate (but do not exclude) the use of the Gauss methodfor solution of systems of nonlinear equations in form of existing library programs. Theproblem of the use of this method is very complicated by events of appearance (as the mostprobable value) of near to zero values σ and corresponding to them steps in cumulativesums. Id est – to some sets of non-random width values. There is easier realized theMonte-Carlo method for solution of systems of degenerated nonlinear equations. Namely– random set of elements correction vector of parameters of fitted function with arbitraryvariation of their initial values.The fitted function is sum K of the distributions P ( X ) of normally distributed randomvalues with independent variables X k each. The required parameters in compared variantsare the most probable value b k of the amplitude A = q Γ n / < Γ n > , its dispersion σ k andtotal contribution C k of function number k for variable X k = (( A k − b k ) ) /σ k (1)in the total experimental cumulative sum of widths.The number of distribution and sign of amplitude A k for given resonance are unknown.Further was used its positive value because (1) is invariant with respect to simultaneouschange of signs of A k and b k . But, it was everywhere supposed that in the considereddistribution number K can exist the only value of b k . I.e., any distribution of widths K has only one maximally possible value of amplitude. There is the main (and absolutelynecessary) hypothesis of the performed by us analysis of distributions of the resonancereduced widths. Concrete value of function P ( X ) for variable (1) in the described analysiswas obtained by compression and shifting of the generally known Euler gamma-function.The obtained in this way value corresponds to the magnitude of this mention functionfor the variable X = ( A × σ + b ) . At present the basis for this algorithm for settingof parameters of approximating function is excellent degree of description of all knownexperimental distributions of the widths. In addition it should be noted that the modulesof the b k and σ k values are strongly correlated variables, at least, for large enough b k values. The test of analysis method was performed by approximation of different sets of ran-dom X values. By this, the mean values of normally distributed random values, theirdispersion, number of variables in sets and distortions of different types appearing in theexperiment are easily varied.The random value X = ξ , corresponding to the χ - distribution with one degree offreedom and unit dispersion and corresponding average was generated from the normally C u m u l a ti v e s u m o f X s=0.85 s=1.15 s=1.1s=0.9 s=1.05s=0.95 X Fig. 1. Thin lines – the example of cumulative sums for some tens of sets from 150, 500and 2000 random X values (upper row) Thick lines – the minimal and maximal valueswith corresponding parameters σ . Cumulative sums for the same sets after exclusion of30% of the lowest X values (lower row).distributed random values ξ . The later were set using Neumann algorithm as a productof two random numbers: δ = sin (2 πγ ) and δ = − ln ( γ ), where γ – the uniformlydistributed in interval [0,1] random value. Modeling of experimental distortions of widthsin the case under consideration reduces to corresponding arithmetic operations with ξ andapproximation of cumulative sums of the distorted values X d with necessary repetitionnumber of this process. For example, below modeling of influence of the observationthreshold of resonance was done using linear function of number X with the parametersproviding in sum exclusion from the tested set L = 30% of the lowest random X values.The spectrum of possible values of cumulative sums for any practically achievable valuesof number of observed resonances can be obtained by interpolation of the data presentedin Fig. 1.Large dispersion of random values X brings to large fluctuations of cumulative sumsof both experimental data and model distributions. And, respectively, to essential varia-tions of the best values of the parameters (1). Therefore, the conclusion about possibledeviations of the parameters b and σ from the expected values 0 and 1, respectively, canhave only probabilistic character.Frequency distributions of these parameters were obtained from modeling sets for theN=150, 500 and 2000 random values X . Modeling was performed for variant of the non-distorted values X and omission which corresponds to exclusion of L = 30% of their lowestvalues (linearly changing with number of random value). The results of approximation ofthese distributions (corresponding to practical maximum of χ ) are given in Fig. 2. C u m u l a t i v e s u m o f X X Fig. 2. The examples of approximation of cumulative sums from the sets presented inFig.1 curves for case of maximal χ values. The upper row – for zero threshold, thelower row – with exclusion of 30% of the least random X values. Dotted curves – partialdistributions for K = 4, points – their sum, solid curves – approximation for K = 1.The widths of corresponding distributions decrease as N increases and at small X depend on value L . One can conclude from the data presented in figures 1 and 2 that adeviation of the experimental distribution of widths from the Porter-Thomas distributionappears itself mainly at X = (Γ n / < Γ n > ) > − X values canbe related, first of all, with omission of weak resonances or other systematical errors ofthe experiment. But, it is not excluded and possibility of real deviation of parameters b and σ from values corresponding to hypothesis [1].Probabilistic conclusions on this account can be made only from comparison betweenfrequency distribution of the parameters (1) for different model distributions and experi-mental data. For the case b = 0 and σ = 1 they are shown in figures 3 and 4. Any errors of experimental values of the tested set inevitably increase dispersion ofthe obtained best values of b and σ . But, in principle they can be taken into accountby determination of the most probable parameters even for distorted distribution Γ n . E v e n t s s Fig. 3. Comparison of frequency distributions of given values of b (upper) and σ (lower)rows, respectively for K = 1. Left column – all possible random values are included inmodeling, right column – L = 30% of the lowest random values are excluded from eachtested set. -8 -6 -4 -2 0 2 40510152025 -5 0 505100 2 4051015 0 1 2 3 4051015 s L=30% E v e n t s
500 L=30%
Fig. 4. The same, as in Fig. 3, for K = 4. C u m u l a t i v e s u m o f X N=500 N=350 N=500, L=30%
Fig. 5. Comparison of averaged cumulative sums for equal values of N with thresholds L = 0 and L = 30%.For example, the problem of resonance omission can be solved easily enough at presenceof reliably established dependence of threshold of its registration on neutron energy. Apossibility to realize of this computational process follows from the data presented in Fig.5. As it is seen from comparison of mean values of cumulative sums, their form for thesame number of resonances (in given case N exp =350) depends on presence/lack of omittedresonances.Direct estimation of the most probable number of omitted resonances in any exper-iment does not call troubles and can be simple if only functional dependence of theirportion δψ th from the total number S is set on the ground of some data or hypotheses forconcrete intervals of resonance energies. Then χ = ( S − ψ ( A, b, σ ) − δψ th ) (2)Here ψ ( A, b, σ ) = R X ∗ P ( X ) dX for any fitted distribution P in function of ratio X .The value δψ th depends only on difference of N t − N exp for varied from variant to variantexpected number of resonances N t in interval δE and determined in the experiment N exp .A number of these intervals was varied in interval 5-20 in dependence on bulk of theexperimental width values. Moreover, negative values N t − N exp in any intervals werechanged by zero. The desired value D = P δE/ P N t corresponds to minimum of χ .Naturally, function δψ can take into account and other factors distorting experimentalwidth distribution. This accounting can be performed in frameworks of both some modelapproaches and concrete experimental data.Modeling of the process of determination of the most probable D value was performedby approximation of cumulative sums of sets of the random X values for χ distributionswith some different N t values. Approximately 30% of their lowest values were excludedfrom every set (the threshold – linearly increasing with number of random value).Direct use of equation (2) for determination of the most probable N t with high reliabil-ty, most probably, is not worth-while without solution of, as a minimum, two problems:(a) The set of the precise enough (relative or absolute) dispersion of cumulative sumfor every value of X and(b) The guarantied determination of location of absolute minimum of χ correspondingto the desired N t value.Although these problems are not irresistible of principle but their solution is not foundup to now. The described and tested method of the reduced neutron widths distributions analysisallows one to get new of principle information on properties of neutron resonances. Inparticular to suppose that the values of the distribution parameters of their neutronamplitudes can correspond to the set of several distributions with their different meanvalues and dispersions.
References [1] C.F. Porter and R.G. Thomas, Phys. Rev. , 483 (1956)[2] V.G. Soloviev, Sov. Phys. Part. Nuc. (1972) 390.[3] L.A. Malov, V.G. Solov’ev, Yad. Phys., (2006) 899.[8] A. M. Sukhovoj, V. A. Khitrov, W. I. Furman, Physics of Atomic Nuclei, (2009)1759.[9] A.M. Sukhovoj, W.I. Furman, V.A. Khitrov, Physics of Atomic Nuclei,71(6)