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The well-known problem of significant disagreements between partial photoneutron reaction cross sections obtained using the method of neutron multiplicity-sorting at Livermore (USA) and Saclay (France) was investigated in detail. As a rule for 19 nuclei from 51V to 238U (g,1n) reaction cross sections are larger at Saclay, but (g,2n) cross sections vice versa larger at Livermore. The averaged ratio Saclay/Livermore of integrated cross sections sSint/sLint = 1.08 for (g,1n) reaction but 0.83 for (g,2n) reaction. Using the objective physical criteria for data reliability it was found that the main reason of significant systematic uncertainties is unreliable sorting of many neutrons between 1n, 2n and, 3n channels because of definite shortcoming of the method of neutron multiplicity-sorting. Three very interesting cases were found in the systematics mentioned above: nuclei 127I (sSint/sLint for (g,1n) and (g,2n) reactions are relatively equal to 1.34 and 1.08), 181Ta (1.25 and 0.89), and 208Pb (1.22 and 0.77). For all three nuclei the neutron yield reaction cross sections (g,xn) = (g,1n)+2(g,2n)+3(g,3n)+… obtained at Saclay and Livermore are significantly different at photon energies before the threshold B2n of (g,2n) reaction, where one has no multiplicity sorting problems. The similar situation with (g,xn) reaction cross sections takes place additionally for 75As, but in this case sSint/sLint(1n) = sSint/sLint(2n) »=1.22. Partial photoneutron reactions for 75As, 127I, 181Ta, and 208Pb were investigated in detail using the experimental-theoretical method for evaluation of partial photoneutron reaction cross sections. It was shown that the main reason of such type significant systematic uncertainties of data is that at Livermore many neutrons from the reaction (g,1n) were lost.