Dependence of the Photocapture Cross Section on the Nuclear Charge

A study of the yields of photonuclear reactions*) carried out for a large number of elements makes it possible to establish the dependence of the photon-absorption cross section by the nucleus, \(\sigma(\gamma)\), on the atomic number \(Z\) of the target. Below we present the contents of two papers\(^{1,2}\) devoted to this question. The authors of the first paper, using experimental data on \((\gamma n)\)-reactions induced by the \(\gamma\)-rays of Li \((p\gamma)^3\), calculate \(\sigma(\gamma)\) for: 1) light nuclei with \(Z\) between 8 and 20; 2) nuclei of medium weight, Cu and Ni; and 3) monoisotopic heavy nuclei J, Ta, Bi, and U. For the calculation it is necessary to know the binding energies of the emitted particles, the level densities of the excited nuclei, and the cross section for penetration through the potential barrier. In the systems considered, excitation may be followed by the emission of one or two particles (a proton or a neutron) in various sequences. The binding-energy data are taken from review material, from \((n\gamma)\)- and \((\gamma n)\)-experiments, beta spectra, the semiempirical formula, or from the relation

\[ L_n + L_{np} = L_p + L_{pn}, \]

where \(L_n, L_p\) are the binding energies of the first neutron and proton, and \(L_{np}, L_{pn}\) are the binding energies of the particle (proton or neutron) emitted after evaporation of the first particle.

Fig. 1. Penetration cross section, in \(10^{-24}\,\mathrm{cm}^2\), for protons at \(Z = 8, 12, 16, 20\).
\(x = \dfrac{\varepsilon}{B}\) is the ratio of the proton energy to the height of the Coulomb potential barrier.

The level density is determined from the semiempirical formula \(\omega(\varepsilon)=a\exp(b\sqrt{\varepsilon})\), where \(a\) and \(b\) are parameters obtained from experimental data; \(\varepsilon\) is the energy. For the region \(15 < A < 70\), the value of \(b\) found

) See UFN*, XLII, 166 (1950).

in the work, is \(b^2 = 0.14(A-12)\ \mathrm{MeV}^{-1}\). Practically all the data of Fig. 1 refer to initial nuclei with odd \(A\); the level density for even nuclei should be larger.

The barrier penetrability is calculated according to\(^4\) for small \(Z\) and \(R = 1.4 A^{1/3}10^{-13}\ \mathrm{cm}\) (Fig. 1). The penetrability for neutrons at \(Z > 16\) is taken from the data of\(^5\), and for smaller \(Z\) from the formula \(\sigma_n = 4\pi\lambda/k_0\), where \(\lambda\) is the neutron wavelength and \(k_0 \sim 1\cdot 10^{13}\ \mathrm{cm}^{-1}\) is the neutron wave number in the nucleus. The cross section for penetration of \(\alpha\)-particles through the potential barrier is presented in Table I.

Table I

\(R_0 = 1.3\,A^{1/3}\cdot 10^{-13}\ \mathrm{cm}\), \(\varepsilon\) is the energy of the \(\alpha\)-particle, \(B\) is the height of the potential barrier.

Integrating over particle energies, the authors obtain the dependences of the total probability of proton and neutron emission on the excitation energy for \(Z = 8,\ 12,\ 16,\ 20\), and 30, and of \(\alpha\)-particles for \(Z = 8,\ W = 10\ \mathrm{MeV}\) and \(Z = 12,\ W = 7\ \mathrm{MeV}\). In this way the number of neutrons evaporating from nuclei upon their excitation by Li \(\gamma\)-rays was calculated (Table II).

Table II

Number of evaporating neutrons \((N)\)

As is seen from the table, a) light nuclei evaporate predominantly protons, except for isotopes with \(N > Z\), which, on the contrary, almost do not evaporate protons; b) for heavy nuclei the emission of two or more particles is probable; c) for light nuclei there is an odd-even correlation (in \(Z\)) in the number of evaporated neutrons.

Knowing from experiment \(\sigma(\gamma n)\), one can calculate the cross section for photon capture by the nucleus,

\[ \sigma_\gamma=\frac{\sigma(\gamma n)}{N}, \]

which is well described by the expression\(^3\)

\[ \sigma_\gamma = 2.4\, Z \cdot 10^{-27}\ \text{cm}^2 \]

(Fig. 2).

The exceptions are the nuclei O, Na, and Ca. In the second work the data of the latest photonuclear experiments are compared with the theoretical results obtained earlier\(^6\). The authors note two substantial discrepancies. First, the experimentally observed neutron yield is not a smooth function of \(Z\), as follows from the theory. This is apparently explained by competition from the \((\gamma p)\) reaction. In fact, the total yield of \(n\) and \(p\) for elements from \(\mathrm{Mg}_{12}\) to \(\mathrm{Zn}_{30}\) increases smoothly with \(Z\). The second discrepancy consists in the fact that the experimental yield of photoneutrons varies as \(Z^2\), and not as

\[ \frac{NZ}{A} \]

(or \(Z^{1,2}\)). The latter is caused by the considerable probability of emission of two neutrons, increasing with \(Z\), owing to the decrease in the neutron binding energy.

Fig. 2. Cross section for photoabsorption in \(10^{-24}\ \text{cm}^2\) as a function of the logarithm of the atomic number \(Z\).

In the work a relation is derived between the integral cross section

\[ \int \sigma(\omega)\,d\omega \]

and the photoneutron yield observed in experiment \(Y\):

\[ \int_{0}^{\infty}\sigma(\omega)d\omega = \frac{E_n Y}{1500}\cdot 10^{-24}\ \text{MeV cm}^2. \]

The neutron binding energy \(E_n\) was calculated according to the statistical model for the elements \(\mathrm{Cu}_{29}\), \(\mathrm{J}_{53}\), \(\mathrm{Ta}_{73}\), \(\mathrm{Bi}_{83}\), and \(\mathrm{U}_{92}\).

For comparison with experiment, works\(^7,8\) were taken, carried out with synchrotrons at \(\omega = 330\) and \(320\ \text{MeV}\), using the method of rhodium foils and \(\mathrm{BF}_3\) counters. For the selected elements the experimental ratio of neutron yields in both cases agrees within the limits from 1.44 to 1.17. Averaging the data of both works, the authors find that the straight line

\[ \int_{0}^{\infty}\sigma\,d\omega = 0.14\,\frac{NZ}{A} \]

agrees well with experiment (Fig. 3). The exception is uranium, which gives a 35% deviation on account of photofission. In this case

Fig. 3. Integral photoneutron cross section. The indicated points correspond to experimental data for the elements Cu\(_{29}\), J\(_{53}\), Ta\(_{73}\), Bi\(_{83}\), and U\(_{92}\). \(N = A - Z\) is the number of neutrons in the nucleus.

the ratio of the probabilities of the latter process and the \((\gamma n)\) reaction is \(y = 0.23\).

Thus the conclusions of both papers considered reduce to the fact that, in the region of resonance absorption of photons by nuclei (\(h\nu \sim 20\) MeV), the cross section of such a process is proportional to the nuclear charge.

B. R.

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