EXPERIMENTAL VERIFICATION OF THE STATISTICAL THEORY OF NUCLEI

Recently a number of papers have been published\(^{1-9}\) devoted to the experimental verification of the statistical theory of nuclei. As is known, the statistical treatment of nuclear properties assumes that, in the interaction of a nucleus with an incident particle, an intermediate nucleus is formed, whose subsequent decay does not depend on the manner of its formation\(^{10}\). In this case the effective cross section \(\sigma(ab)\) of the reaction \(A+a \to C^* \to B+b\) can be represented in the form

\[ \sigma(ab)=\sigma_a(\varepsilon)\eta_b(\varepsilon), \tag{1} \]

where \(\sigma_a(\varepsilon)\) is the cross section for absorption by the nucleus \(A\) of a particle \(a\) having kinetic energy \(\varepsilon\), with formation of the compound nucleus \(C^*\), and \(\eta_b(\varepsilon)\) is the probability of decay of the nucleus \(C^*\) into the final state \(B+b\). A direct verification of this important proposition of the theory was carried out in work\(^{1}\), performed on two accelerators: a 60-inch cyclotron, producing a beam of \(\alpha\)-particles with energy \(E=40\) MeV, and a linear accelerator, in which protons with \(\varepsilon=32\) MeV are obtained. The author investigated the reactions: \(\mathrm{Ni}^{60}(\alpha,n)\mathrm{Zn}^{63}\), \(\mathrm{Ni}^{60}(\alpha,pn)\mathrm{Cu}^{62}\), \(\mathrm{Ni}^{60}(\alpha,2n)\mathrm{Zn}^{62}\), and \(\mathrm{Cu}^{63}(p,n)\mathrm{Zn}^{63}\), \(\mathrm{Cu}^{63}(p,2n)\mathrm{Zn}^{62}\), \(\mathrm{Cu}^{63}(p,pn)\mathrm{Cu}^{62}\), in which the intermediate nucleus \(\mathrm{Zn}^{64}\) is formed. In the case of irradiation with \(\alpha\)-particles, nickel enriched in the isotope \(\mathrm{Ni}^{60}\) was taken; in the case of irradiation with protons, pure copper in the form of the usual mixture of isotopes was used. The excitation curves were obtained by the “foil stack” method; their \(\beta\)-activity was measured with the aid of a thin-walled counter. The maxima of the curves of the dependence of the effective cross section on the energy of protons and \(\alpha\)-particles (Fig. 1) are shifted relative to one another by \(7\pm1\) MeV, which is explained by the difference of the masses \(\mathrm{Cu}^{63}+\mathrm{H}^1\) and \(\mathrm{Ni}^{60}+\mathrm{He}^4\), equal, according to mass-spectrographic measurements, to \(5.74\pm0.5\) MeV. The ratios \(\sigma(\alpha n):\sigma(\alpha,pn):\sigma(\alpha,2n)\) are, within the errors, equal to \(\sigma(p,n):\sigma(p,pn):\sigma(p,2n)\), which proves the correctness of (1). The paper also compares the experimental effective cross sections for absorption of \(p\) and

α-particles (found by summation over σ of all observed reactions) with the theoretical cross sections calculated by Weisskopf on the basis of the statistical model (Fig. 2). The bend in the experimental curve is, apparently, due to the unobserved reaction of elastic (p,p) scattering.

As is seen from Fig. 1, σ(α,pn) and σ(p,pn) are approximately 4 times greater than σ(τ,2n) and σ(α,pn). The author explains this fact by the smaller (by 3 MeV) binding energy of the proton in the nucleus, and also by the circumstance that in the first case an odd-odd residual nucleus is formed (for the isotope Cu^63

Fig. 1. Experimental cross sections for the (pn), (p,2n), (p,pn) reactions on Cu^63 and for the (α,n), (τ,2n), and (α,pn) reactions on Ni^60 as functions of the energy of protons and α-particles, respectively.

Fig. 2. Comparison of the sum of the cross sections of the (p,n), (p,2n), and (p,pn) reactions on Cu^63 as a function of proton energy with the theoretically calculated cross sections for absorption of a proton by the Cu^63 nucleus.

$N$ and $Z$ are odd numbers), the level density of which should be greater than that of the even-even nucleus Zn^62. The latter is confirmed in the work of Hughes with 320 MeV γ-rays^11 and in work^2 on the investigation of (n,2n) and (np) reactions. Here the source of neutrons with energy 4–17 MeV was a thick beryllium target irradiated with 15 MeV deuterons. As a result of measuring the activity of the residual nuclei, σ(n,p) was obtained for 25 elements from O^16 to La^139 and σ(n,2n) for 19 elements. The ratio

$$ \frac{\sigma_{\text{even-even}}}{\sigma_{\text{odd-odd}}} $$

turned out to be equal to $3 \pm 1$, in agreement with the preceding work^1 and theoretical calculations.

The authors found the dependence of σ(np) as a function of $Z$, whose general course agrees with theoretical curves calculated for different values-

... \(r_0\) and the nuclear temperature \(T\) (Fig. 3). The dependence of \(T\) on the atomic weight \(A\) of the element upon excitation of nuclei by protons with \(\varepsilon = 16\) MeV was obtained from a study of the energy spectrum of neutrons emitted from various targets in a cyclotron (Be, Al, Fe, Rh, and Tl were investigated)\(^3\). The neutron energy was found from the tracks of recoil protons in photonuclear emulsions.

Fig. 3. Cross section of the \((n,p)\) reaction as a function of \(Z\). Solid lines are theoretical curves:
\(A\): \(r_0 = 1.3 \cdot 10^{-13}\) cm and \(T\) from the work of V. Weisskopf, Lecture Series in Nuclear Physics (U. S. Government Printing Office, Washington D. C., 1947);
\(B\): \(r_0 = 1.5 \cdot 10^{-13}\) cm;
\(C\): \(r_0 = 1.3 \cdot 10^{-13}\) cm, \(T' = 0.75T\);
\(D\): \(r_0 = 1.3 \cdot 10^{-13}\) cm, \(T'' = 1.25T\);
\(E\): \(r_0 = 1.3 \cdot 10^{-13}\) cm, \(T\) from the work of V. Weisskopf and J. M. Blatt (unpublished), according to the data of Ref. 10.

In the figure: vertical axis — “cross section \((n,p)\) in \(10^{-27}\ \mathrm{cm}^2\)”; horizontal axis — “atomic number, \(Z\).” Legend:
\(\bullet\) Observed (even, even), \(\div 4\);
\(\circ\) Observed (even, odd or odd, even);
\(\times\) Corrected for the threshold;
\(\;>\) Less than.

In Fig. 4 the dependence of the coefficient

\[ a=\frac{4(\varepsilon_{\max}-\varepsilon)}{T^2} \]

\([(\varepsilon_{\max}-\varepsilon)\) is the excitation energy of the nucleus] on \(A\) is presented in comparison with Weisskopf’s theoretical curve.

Knowing the spectrum of particles emitted from the intermediate nucleus, one can find the density of levels of the residual nucleus \(\omega_R\) from the relation\(^ {12}\):

\[ \omega_R(\varepsilon_{\max}-\varepsilon)=C\,\frac{n}{\varepsilon}, \tag{2} \]

where \(n\) is the number of neutrons in the energy interval. In all cases \(\omega_R\) can be represented as

\[ \omega_R = C e^{\varepsilon/T}, \tag{3} \]

but the values of \(T\) obtained are smaller than in the statistical theory. A similar law of increase of \(\omega_R\) \((\varepsilon_{\max}-\varepsilon)\) has been obtained in a number of works. In individual cases, as, for example, in the study of the inelastic scattering of protons by \(\mathrm{Al}^{27}\) at \(E_p = 30\) MeV, a number of fast particles is observed\(^{4}\) that greatly exceeds the statistical value. It is assumed that this phenomenon (as also in the photodisintegration of Ag and Rh*) is connected with a direct interaction, i.e., without formation of an intermediate nucleus, of the incident particle with the nucleon of the nucleus.

Fig. 4. \(a = 4E/T^2\) as a function of atomic weight.

B. R.

References

1. S. N. Ghoshal, Phys. Rev. 80, 939 (1950).
2. B. L. Cohen, Phys. Rev. 81, 184 (1951).
3. P. C. Gugelot, Phys. Rev. 81, 51 (1951).
4. Levintal, Martinelli and Silverman, Phys. Rev. 78, 199 (1950).
5. R. A. Peck, Phys. Rev. 76, 1279 (1949).
6. H. Wäffler, Helv. Phys. Acta 23, 239 (1950).
7. Pollard, Sailor and Wyly, Phys. Rev. 75, 727 (1949).
8. K. J. Le Couteur, Proc. Phys. Soc. A63, 259 (1950).
9. L. G. Grosskreutz, Phys. Rev. 76, 482 (1949).
10. V. F. Weisskopf and D. H. Ewing, Phys. Rev. 57, 472 (1940).
11. K. Strauch, Phys. Rev. 81, 973 (1937).
12. V. F. Weisskopf, Phys. Rev. 52, 995 (1937).

* See UFN 44, no. 2, 437 (1951).