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Fission Neutron Spectrum of U\(^{235}\)
The journal has already reported\(^{1}\) on a study\(^{2}\) devoted to the fission-neutron spectrum of U\(^{235}\) in the range \(0.4 \div 7\) MeV. Below three further works\(^{3,4,5}\) are reviewed, extending the investigated energy region of U\(^{235}\) fission neutrons from 0.05 MeV to 17 MeV.
In work\(^{3}\) the spectrum of (prompt) neutrons in the fission of U\(^{235}\) was investigated in the range \(0.05 \div 0.7\) MeV. The lower limit is determined by the minimum track length of the recoil proton that can be measured with sufficient accuracy, and the upper limit by the maximum track length of the recoil proton that could be accommodated in the Wilson chamber. The source of thermal neutrons was a heavy-water boiler used as a moderator. The neutron beam emerged from the thermal column and, after a collimator of cadmium and lead, struck uranium foil enriched in the isotope U\(^{235}\). In the path of the beam there was a cadmium shutter, so that neutrons struck the uranium foil only when the shutter was open. The neutron detector was a Wilson chamber 12 inches in diameter and 9 inches deep. The chamber was filled with a mixture of hydrogen and water vapor to a pressure of \(\frac{1}{3}\) atmosphere. Under these conditions recoil protons with an energy of 50 keV had tracks 7 mm long. The chamber was protected from \(\gamma\)-rays by a thick layer of lead.
Fig. 1.
A total of 2800 stereoscopic photographs were obtained, on which there were 25,000 recoil-proton tracks. Of these, 437 tracks began in a certain region selected in advance in the chamber volume and had an angle with the direction of the neutrons from the uranium foil of not more than \(15^\circ\). The protons corresponding to these tracks received almost all the energy of the neutrons that collided with them. Of the 437 tracks, 237 ended in the gas of the chamber and were the objects of study. Fig. 1 reproduces the neutron spectrum obtained in this work. Along the abscissa axis is plotted the fission-neutron energy in keV, and along the ordinate axis the relative number of neutrons. The curve corresponds to Watt’s semiempirical formula\(^{5}\): \(N(E_n)=e^{-E_n}\cdot \operatorname{sh}\sqrt{2E_n}\). It is seen that
the semiempirical formula describes the experimental results sufficiently well.
In paper 4 the spectrum of fission neutrons of U\(^{235}\) (prompt and delayed) was investigated when uranium foil was irradiated by a beam of neutrons from a thermal column of a heavy-water pile. The investigated region of the spectrum was \(0.4 \div 7\) MeV. The distribution of recoil protons by ranges was observed,
Fig. 2.
and then the spectrum of neutrons that knocked these protons out of a paraffin layer was calculated. The ranges of the recoil protons were found with an apparatus consisting of a paraffin layer, an absorber of various thicknesses, and four counters arranged as a telescope. Of the four counters, three were connected in coincidence, and the last in anticoincidence with the three. Only those protons were recorded which passed through the absorber layer and three counters but could not penetrate into the fourth. The neutron spectrum has a broad maximum near \(0.75\) MeV. For neutron energies from \(2\) MeV to \(7\) MeV the spectrum has an exponential falloff with “attenuation energy” equal to \(1.55\) MeV. The results of papers 4 and 5 are shown in Fig. 2. Along the abscissa is plotted the energy of the fission neutrons, and along the ordinate—the intensity in arbitrary units.
In paper 5 the investigated energy region lies within \(3.3 \div 17\) MeV. In this case the source of thermal neutrons was a homogeneous heavy-water pile. The distribution by ranges of recoil protons from a paraffin layer was investigated with the aid of absorbers and a telescope of three counters connected in a coincidence circuit. All recoil protons with energy greater than a minimum determined by the thickness of the absorber, the paraffin, and the walls of the telescope were recorded. The differential distribution ...
The latter was obtained from the difference of the integral distributions. The results of the work are shown in Fig. 2. The data of work\(^4\) are reproduced in the same figure. The curve corresponds to Watt’s semiempirical formula:
\[ N(E_n)=4.75\cdot 10^6\cdot e^{-E_n}\cdot \operatorname{sh}\sqrt{2E_n}. \]
It follows from the figure that this formula describes the experimental data sufficiently well. The same formula describes the energy region \(0.075\div 0.6\) MeV (see Fig. 1).
Work\(^5\) reports how the formula was derived. After experimental data had been obtained for the spectrum of fission neutrons of \(U^{235}\), an attempt was made to calculate it theoretically. Three assumptions were made: 1) emission of neutrons by the fragment is isotropic in the center-of-mass systems; 2) the neutron distribution in the center-of-mass system is proportional to
\[ E'_n\cdot e^{-E'_n/Q}, \]
where \(E'_n\) is the neutron energy in the center-of-mass system, and \(Q\) corresponds to the nuclear “temperature” of the fragment; 3) the velocity of the fragment at the emission of the neutron is maximal. This assumption is based on the liquid-drop model.
The assumption that there is only one average fragment mass did not lead to agreement with experiment. Two fragments with two average masses were assumed. The calculation became more complicated, but the formula did not correspond to the experimental data.
However, by considering the neutron distribution to be Maxwellian, i.e., proportional to \(E_n'^{1/2}\cdot e^{-E'_n/Q}\), rather than \(E'_n e^{-E'_n/Q}\), the formula was obtained
\[ N(E_n)=\operatorname{const}\cdot e^{-E_n/Q}\cdot \operatorname{sh}\left[2Q^{-1}\cdot (E_n\cdot E_{\text{frag}}\cdot m_n/M_{\text{frag}})^{1/2}\right], \]
where \(E_{\text{frag}}\) and \(M_{\text{frag}}\) are the energy and mass of the fragment, and \(E_n\) and \(m_n\) are the energy and mass of the neutron. This formula describes the experimental data well under the condition that the fragments have equal masses. The constants \(Q\) and \((E_{\text{frag}}\cdot m_n/M_{\text{frag}})\) were, for simplicity, taken equal to:
\[ Q=1.00\ \text{MeV},\quad E_{\text{frag}}\cdot m_n/M_{\text{frag}}=0.5\ \text{MeV}. \]
Then the following formula was obtained:
\[ N(E_n)=\operatorname{const}\cdot e^{-E_n}\cdot \operatorname{sh}\sqrt{2E_n}, \]
where \(E_n\) is in MeV. This formula is discussed in all the preceding works.
Assuming that the fragments are different, the authors obtained a more complicated formula, but in this case too they did not obtain better agreement with the experimental data. The experimental spectrum can be described by a better formula if one assumes that the fragments have different \(Q\). Complicated formulas are obtained, which are practically inconvenient. Watt believes that the three assumptions indicated above are reasonable, but it must be taken into account that there are fragments with different masses, velocities, and excitations. However, the formula will be complicated and practically unsuitable for calculation.
L. K.
REFERENCES
- UFN, vol. XLVIII, 585 (1952).
- N. Nereson, Phys. Rev. 85, 600 (1952).
- T. W. Bonner et al., Phys. Rev. 87, 1032 (1952).
- D. L. Hill, Phys. Rev. 87, 1034 (1952).
- B. E. Watt, Phys. Rev. 87, 1037 (1952).