Exact formula for the statistical distribution function of the fluctuating cross section is presented. Comparison with the results obtained from the approximate formula shows that such a formula can be used for calculation of the cumulative distribution function in the range of its values from 0.01 to 0.99

Jarczyk, L.

Lang, J.

Müller, R.

Strzałkowski, A.

Witała, Henryk

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. B14 (1983) ACTA PHYSICA POLONICA No 7ON THE DISTRIBUTION OF THE FLUCTUATING REACTION CROSS SECTIONBy H. Witała, L. Jarczyk, A. StrzałkowskiInstitute of Physics, Jagellonian University, Cracow*  *** Address: Instytut Fizyki UJ, Reymonta 4, 30-059 Kraków, Poland.** Address: Laboratorium fur Kernphysik, Eidg. Techn. Hochschule, 8093 Zurich, Switzerland.and J. Lang, R. MullerLaboratorium fur Kernphysik, Eidg. Techn. Hochschule, Zurich*  *(Received September 23, 1982)Exact formula for the statistical distribution function of the fluctuating cross section is presented. Comparison with the results obtained from the approximate formula shows that such a formula can be used for calculation of the cumulative distribution function in the rangę of its values from 0.01 to 0.99.PACS numbers: 25.70.FgRecently a great increase of interest in molecular resonances in heavy ion collision has been observed [1-3], Usually such resonances are expected in the region of nuciear excitation where a strong fluctuating statistical component of the reaction cross section is also present. The aim of the analysis of the experimental data in a search for resonances consists in the first linę in checking to what extent the observed structures in the excitation curves are exρlanable in terms of the statistical model. In the next step of analysis the possible resonances should be fished out from the fluctuating background. These tasks have given rise to a revival of interest in the methods of statistical analysis and their cricital examination [4, 5].Rudimental for the construction of many test functions used in a such analysis (e.g. deviation function, cross correlation function) and for establishing their distributions is the knowledge of the distribution function of the cross section which follows from the statistical model of the reaction. An approximate formula is usually applied for this purpose [6], A problem arises to what extent such an approximate formula is consistent with the exact one. The aim of the present work was to derive the accurate statistical distribution function (555)556of the cross section and to compare its predictions with those calculated with the approxi- mate one.The differential cross section for reaction to a specific finał state is expressed as a sum of the partial cross sections corresponding to the different substates β = (μ, μ', M, M,) of nuclei participating in the reactionNβσ = Σ °β> (Dβ = iwhere Nβ, the maximum number of independent partial cross sections is related to the number of magnetic substates in the entrance and exit channels of the reaction. In the presence of the direct reaction contribution the averaged cross section could be split into two incoherent parts corresponding to the two different reaction mechanisms:<σ> = ∑ <σz,> = ∑(σd + <σzj>c'',) = ad + <a>CN. β β(2)According to the statistical reaction model [7] the probability distribution function for the partial cross section normalized by its average value yβ = o√<o⅛>is given by the formula:fyβa(yβ') ~^ exP L (>⅛÷J'β)∕(l J⅛)]∙fo[2( v y^y∣3∕(l v,β)]∕(l yβ), (3)where = σiβ∣{σβy denotes the relative direct reaction contribution to the <σ^> and Jr0(∕z) is the cylindrical Bessel function of order zero of the imaginary argument iz.Introducing a notationpβ = <√⅞>cn∕<X>cn,where is a selected value of the average partial cross section we defineand557The exact distribution function of y could be obtained as usual as the Fourier transform of the characteristic function:f(y) = (2τr) 1 ∫ e∖p(-ity)Φy(t)dt.— oo(5)In two particular cases the evaluation of the integral in formula (5) leads to the closed form analytical expressions:1. When all the partial cross sections <a/(>CN have the equal average values and thusFig. 1. Comparison of the exact and approximate distributions of the fluctuating reaction cross section. The abscisa are the values of the cumulative distribution function while the ordinates give the fractiles of the approximate distribution (curve 7) or the ratio of fractiles for the exact and approximate distributions (curve 2)As the variable y is a linear combination of the statistically independent random variables yβ its characteristic function is equal(4)where558pβ = 1 for all β', then Netf = Nβ and/(>’) = ∕tve"exp [-(y + ydX]Λff~ ,¼ff-1(2L4 √y⅛(L4 x⅛>"^1]∙2. Without contribution from the direct reaction processes yd = 0 and/(v) ∑Pββ____β k^p(-y∑pi∣Pk)iII (l/A-i/Pt) ’l≠k(6)(7)The cumulative distribution function has than the form:(8)Fig. 2. The same as in Fig. 1In generał case however it is necessary to determine the density and cumulative distri­bution functions by means of the numerical integration of (5).It should be pointed out however that the exact expressions for the distribution function developed here are not suitable for the practical calculations not only due to the rather complicated form but also due to the lack of the a priori information concerning 559the values of parameters pβ and yj appearing in the formula. Thus for the practical application an approximate formula was develoρed [6] and is in common use:*"ω = (⅞P""',p(^s-≡χ jfiVeff-l E2f‰ Vyj,d∕(i-j-,d)][≈‰√Pd∕(i-Jd)l^f^1In cases when the contributions from the direct reaction processes as well as the average cross sections for the compound nucleus reactions are equal for all magnetic substates (i.e. yβ — ydβ. and pβ = 1) the approximate formula leads to the results identical with those obtained from the exact expression with Λ,cff = Nβ- This could be considered as some justification of the approximate procedurę. In that case for the analysis of experimentalFig. 3. The same as in Fig. 1data only the knowledge of the compound nucleus contribution (<<7>cn, Λζff) and the average cross section, which could be obtained by appropriate averaging of experimental data, is demanded. However in a generał case the agreement between the exact and approxi- mate calculations should be tested by a direct comparison of the distribution calculated with both procedures.(9)560 In the present paper such calculations are performed for a considerable rangę of the values of Neff and jd. The results are presented in Figs. 1, 2, 3. In the diagrams the abscisa are the values of the cumulative distribution function while the ordinates give the fractiles of the approximate distribution (curve 1), or the ratio of fractiles for the exact and approxi- mate distributions (curve 2).The largest discrepancies in the values of fractiles appear for smali values of cumulative distribution and for smali values of Nc[t and yd. E.g. the disagreement between the fractiles for the exact and approximate formulas reaches ca 15% for Fx 0.01, Nc{[ ≈ 2 and yd x 0. For F ≈ 0.99 the discrepancies are not larger than ca 5 % with the fractiles for the exact distribution always larger than those for the approximate one. With increasing Ne(( and yd these discrepancies tend rather fast to zero.Thus the application of the approximate formula is justified for calculation of the fractiles in the rangę of values of the cumulative distribution function from 0.01 to 0.99.REFERENCES[1] Nuclear Molecular Phenomena, ed. by N. Cindro, North-Holland, Amsterdam 1978.[2] Proceedings of the lnternational Conference on Resonance Behavior of Heavy łon Systems, Aegean Sea-1980, ed. by G. Vourvopoulos, Nuclear Research Center “Democritos”, Athens 1981.[3] Resonances in Heaυy łon Reactions, Lecture Notes in Physics 156, ed. by K. A. Eberhard, Springer- -Verlag, Berlin, Heidelberg, New York 1982.[4] L. C. Dennis, S. T. Thornton, K. R. Cordell, Phys. Rev. C19, 777 (1979).[51 J. Lang, M. Hugi, R. Muller, J. Sromicki, E. Ungricht, H. Witała, L. Jarczyk, A. Strzał­kowski, Phys. Lett. 104B, 369 (1981).[6] R. A. Dayras, R. G, Stokstad, Z. E. Switkowski, R. M. Wieland, Nuci. Phys. A265, 153 (1976).[7] T. Ericson, in Statistical Theories of Spectra'. Fluctuations, ed. by Charles E. Porter, Academic Press, New York and London 1965, p. 559.

On the distribution of the fluctuating reaction cross section

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On the distribution of the fluctuating reaction cross section

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