Full Text
On the Theory of the Beryllium Nucleus
V. Chavchanidze
The limited applicability of the self-consistent-field method to problems of nuclear physics is well known. According to Bohr’s theory, atomic nuclei are systems of nuclear particles interacting so strongly with one another that it is impossible to regard any one particle as isolated, moving in the averaged field of the remaining nuclear particles.
Since Bohr’s theory has a statistical character, its application to light and the lightest nuclei is not legitimate. In this connection one may expect that, for the lightest nuclei, the self-consistent-field method can acquire heuristic power. As is known, this method has proved itself well in application to the deuteron nucleus. Bethe and Peierls¹ showed that the main characteristics of the deuteron nucleus can be found despite ignorance of the exact form of the nuclear potential. The possibility of solving such a problem is due to the circumstance that the dimensions of the wave packet corresponding to the deuteron are considerably larger than the radius of action of the nuclear forces.
The nucleus \(\mathrm{Be}^{9}_{4}\) contains 4 protons and 5 neutrons. Consequently, generally speaking, it cannot be assigned either to the lightest nuclei or, still less, to the heavy nuclei for which Bohr’s theory is valid. Nevertheless, there are certain specific circumstances which, in calculating decay phenomena, make it possible to assign this nucleus to the type of the lightest nuclei.
Indeed, from the experiments of Szilard and Chalmers², L. I. Rusinov and A. N. Sagaidak³, and others, it follows that one unpaired neutron in the \(\mathrm{Be}^{9}_{4}\) nucleus is bound much more weakly than the other particles of this nucleus, since the photodisintegration reaction of \(\mathrm{Be}^{9}_{4}\) by \(\gamma\)-rays, proceeding according to the scheme
\[ \mathrm{Be}^{9}_{4} + h\nu \rightarrow \mathrm{Be}^{8}_{4} + n, \]
has as its threshold a \(\gamma\)-quantum energy equal to \(1.63\ \text{Mev}\).
It follows from this that the binding energy of the odd neutron in the nucleus \(\mathrm{Be}_4^9\) is equal to \(\varepsilon_0 = 1.63\) MeV. This value is considerably smaller than the average binding energy of the remaining particles, which is close to 7 MeV. (The total binding energy of the nucleus \(\mathrm{Be}_4^9\) is 57.8 MeV.)
Consequently, 8 particles (4 protons and 4 neutrons) form in the nucleus \(\mathrm{Be}_4^9\) a tightly packed nuclear core, weakly bound to the odd neutron. In this connection there arises the tempting idea of considering the nucleus \(\mathrm{Be}_4^9\) as a system of two bodies: a nuclear core \(\mathrm{Be}_4^8\) and an odd neutron, assuming that the latter is in the field of the residual nucleus.
Generally speaking, the following possibilities are conceivable for a theoretical approach to the problem of the nucleus \(\mathrm{Be}_4^9\).
I. The nucleus \(\mathrm{Be}_4^9\) may be considered as a system consisting of 9 particles (5 neutrons and 4 protons).
II. One may suppose that the nucleus \(\mathrm{Be}_4^9\) consists of three basic nuclear bodies: two \(\alpha\)-particles and one neutron (the model: \(2\alpha + n\)).
III. Finally, as indicated above, one may consider that the nucleus \(\mathrm{Be}_4^9\) consists of two basic nuclear bodies: one neutron and a nuclear core identified with the nucleus \(\mathrm{Be}_4^8\).
Rejecting the first possibility as at present unpromising, it is necessary to choose between the two remaining ones.
It is known that the nucleus \(\mathrm{Be}_4^8\) produced as a result of the nuclear reaction
\[ \mathrm{Be}_4^9 + h\nu = \mathrm{Be}_4^8 + n \]
is unstable and decays into two \(\alpha\)-particles.
On this basis it might be supposed that the representation of the nucleus \(\mathrm{Be}_4^9\) as a system consisting of two \(\alpha\)-particles and one neutron is the most acceptable.
Taking into account, however, that the energy released in the decay of the nucleus \(\mathrm{Be}_4^8\) into two \(\alpha\)-particles is 0.116 MeV, whereas the minimum energy required to liberate a neutron from the nucleus \(\mathrm{Be}_4^9\) is 1.63 MeV, one must conclude that the lifetime of the nucleus \(\mathrm{Be}_4^8\) with respect to decay into two \(\alpha\)-particles is much longer than the time spent in tearing the neutron from the nucleus \(\mathrm{Be}_4^9\).
This circumstance compels us to give preference to the last possibility characterized above, since during the process of emission of the neutron it is permissible to regard the nucleus \(\mathrm{Be}_4^8\) as a still stable and unchanged nuclear body. Obviously, such an assumption is legitimate so long as the excitation energy of the nuclear system \(\mathrm{Be}_4^9\) is considerably smaller than the energy with which the particles of the residual nucleus \(\mathrm{Be}_4^8\) are bound to one another.
V. CHAVCHANIDZE
The idea of such an approach to the problem of the nucleus \( \mathrm{Be}_4^9 \) was first expressed in 1936 by Mamasakhlisov \(^{4,5,6,7,8,9,10*}\), who from this point of view considered all questions connected with the interaction of an odd neutron with the residual nucleus \( \mathrm{Be}_4^8 \) (photoelectric disintegration, photomagnetic disintegration, electronic disintegration, capture of neutrons by the nucleus \( \mathrm{Be}_4^8 \), etc.).
Let us turn to the consideration of individual effects connected with the interaction of an odd neutron with the residual nucleus.
PHOTOELECTRIC DISINTEGRATION
By 1935, the experiments of Chadwick and Goldhaber \(^{12}\) on the photodisintegration of \( \mathrm{Be}_4^9 \) were known. These authors found spherical symmetry of the emitted neutrons in the case when the energy of the \(\gamma\)-quantum is close to the disintegration threshold, i.e. to \(1.63\) MeV. It followed from this that the neutrons emitted from the nucleus \( \mathrm{Be}_4^9 \) are in an \(S\)-state.
On the other hand, according to the selection rules for dipole radiation (the quadrupole is small), one must conclude that the bound state of the neutron in the nucleus \( \mathrm{Be}_4^9 \) is a \(P\)-state (in contrast to the deuteron, where the bound state, apart from a small admixture of the \(D\)-state, is an \(S\)-state).
Thus, in the process of photoelectric disintegration of the beryllium nucleus, the bound neutron, which is in a \(P\)-state, passes into a free \(S\)-state. It is clear that if one starts from the \(P\)-state as the initial state, then the final state may be either an \(S\)- or a \(D\)-state. The presence of a \(D\)-state must show itself in a deviation from spherical symmetry in the distribution of the intensity of neutrons in the free state.
As indicated above, the initial experiments did not lead to deviations from spherical symmetry. However, these experiments were carried out with \(\gamma\)-quanta whose energy only slightly exceeded the binding energy of the neutron. Experiments on the splitting of the beryllium nucleus carried out in recent years lead to a deviation from spherical symmetry for the neutrons released from the nucleus \( \mathrm{Be}_4^9 \) in the case of large values of the \(\gamma\)-quantum energy. As will be shown below, this deviation is explained by the fact that the free neutrons are characterized not only by an \(S\)- but also by a \(D\)-state.
The transition of the system \( \mathrm{Be}_4^8 + n \) from one state to another is caused by the interaction of the \(\gamma\)-quantum with the dipole moment of the nucleus \( \mathrm{Be}_4^9 \),
*) Guth and Mullin indicate \(^{11}\) that this idea was first expressed by Guth in 1937. Such an assertion cannot but cause surprise, since the authors are well acquainted (they cite it) with the work of Mamasakhlisov \(^{4}\), published in 1936, in which this idea was not only expressed, but on its basis a calculation of the photodisintegration of the beryllium nucleus was also carried out.
arising from the noncoincidence of the center of mass with the center of charge of the system under consideration.
The electric dipole moment of the system is equal to:
\[ D=-\frac{e Z M_0}{M_0+M}\,z=-\frac{e Z M_0}{M_0+M}\,r\cos\vartheta, \tag{1} \]
where \(M_0\) is the mass of the neutron, \(M\) is the mass of \(\mathrm{Be}_4^8\), \(Z\) is the number of protons in the \(\mathrm{Be}_4^9\) nucleus, and \(z=r\cos\vartheta\) is the projection of the dipole-moment vector onto the direction of the \(z\)-axis.
Mamasakhlisov\(^4\) calculated the total cross section of the photoelectric transition \(P \to S\), under the assumption that the radius of the potential well (a rectangular well of depth \(V_0\)) \(r_0\) is smaller than the dimensions of the wave packet corresponding to the bound neutron,
\[ d=\frac{\hbar}{\sqrt{2\mu\varepsilon_0}}=3.8\cdot 10^{-13}\ \text{cm}, \]
where \(\varepsilon_0\) is the binding energy of the neutron and \(\mu\) is the effective mass of the neutron relative to the beryllium nucleus.
The total cross section turned out to be equal to
\[ \sigma_{P\to S}= \frac{48\pi}{3} \left(\frac{e^2}{\hbar c}\right) \left(\frac{ZM}{M_0+M}\right)^2 \frac{r_0d}{2+\beta_1}\,F(\gamma), \tag{2} \]
where
\[ F(\gamma)= \frac{(\gamma-1)^{\frac12}}{\gamma^3} (m\gamma+n)^2, \]
with
\[ m=\beta_1^2-\beta_1-1,\qquad n=2+2\beta_1, \]
\[ \beta_1=\frac{r_0}{\alpha},\qquad \gamma=\frac{\varepsilon}{\varepsilon_0},\qquad \alpha=\frac{1}{d}=\frac{\sqrt{2\mu\varepsilon_0}}{\hbar}, \qquad \text{and}\qquad \varepsilon=h\nu. \]
It is easy to see that the cross section has a maximum near \(\gamma \simeq 1\), after which it decreases slowly. A comparison of formula (2) with experimental data was carried out by a number of authors.
The experiments of Rusinov and Sagaidak\(^3\) (1936) indicated a more or less satisfactory agreement of formula (2) with experimental data for two values of the \(\gamma\)-quantum energy: 1.1 and 1.6 MeV.
Tandberg\(^ {13}\), in his dissertation, published in 1937, studied the absorption of a \(\gamma\)-quantum in beryllium, occurring as a result of the nuclear photoeffect. He proceeded from two possible assumptions: that the bound state of the neutron in beryllium may be either an \(S\)- or a \(P\)-state. In this connection he compared the experimental data obtained by him with the formula of Bethe and Peierls, on the one hand, and with Mamasakhlisov’s formula, on the other. As a result of the comparison Tandberg came to the conclusion that formula (2), which assumes that the bound state in the beryllium nucleus is a \(P\)-state, agrees better with the experimental data.
V. CHAVCHANIDZE
Korsunskii, Nikolaevskaya, and Bak[^14] pointed to a certain discrepancy between formula (2) and experimental data; in their experiments they used the continuous spectrum of γ-rays emitted by a pulse tube as a result of the bremsstrahlung of electrons. According to their data, not confirmed, however, by subsequent investigations, formula (2) leads to a stronger dependence of the effective cross section on the potential difference applied to the pulse tube, at γ-quantum energies close to the threshold value, than follows from their experiments. Formula (2) was derived under the assumption that the radius \(r_0\) of the potential well of the interaction between the neutron and the nuclear residue \(\mathrm{Be}_4^9\) is so small that integration over all space can be replaced by integration over the region outside the potential well. However, the circumstance that the bound state of the neutron in the nucleus \(\mathrm{Be}_4^9\) is a \(P\)-state (and not an \(S\)-state) makes it necessary to consider such neglect of the region inside the potential well not entirely legitimate. Therefore in 1940 V. Mamasakhlisov[^5],[^7] considered the problem of the photodisintegration of beryllium in a more general form, taking into account the region \(r<r_0\).
The formula obtained by him for the cross section of photodisintegration of beryllium has the form
\[ \sigma_{P\to S} = \frac{4\pi}{9}\frac{e^2}{\hbar c} \left(\frac{Z\mu}{M}\right)^2 \frac{1}{\alpha^2} \frac{(\gamma-1)^{\frac12}}{\gamma^3} F^2(\gamma), \tag{3} \]
where
\[ F(\gamma) = - \left(\frac{\beta}{\alpha}\right)^{\frac32} a\sin \beta r_0 \cos \theta \left\{ 2+(1+\alpha r_0)\gamma + \frac{\alpha}{\beta}(2+\alpha r_0)\gamma \frac{\operatorname{tg} s\beta r_0}{s} +\Phi(s) \right\}. \tag{4} \]
In this formula \(\Phi(s)\) denotes the expression
\[ \Phi(s) = \frac{(s^2-1)^{\frac32}}{2s\sin s\beta r_0\cdot \cos s\beta r_0} \left\{ \frac{\sin(s+1)\beta r_0}{s+1} - \frac{\sin(s-1)\beta r_0}{s-1} + \right. \]
\[ \left. + \frac{1}{(s+1)^2} \left[ \sin(s+1)\beta r_0 - (s+1)\beta r_0\cdot \cos(s+1)\beta r_0 \right] + \right. \]
\[ \left. + \frac{1}{(s-1)^2} \left[ \sin(s-1)\beta r_0 - (s-1)\beta r_0\cdot \cos(s-1)\beta r_0 \right] \right\}, \tag{5} \]
with
\[ \alpha=\sqrt{\frac{2\mu\varepsilon_0}{\hbar^2}}, \qquad \beta=\sqrt{\frac{2\mu(V_0-\varepsilon_0)}{\hbar^2}}, \qquad \gamma=\frac{h\nu}{\varepsilon_0}, \]
\[ s=\left(1+\left(\frac{\alpha}{\beta}\right)^2\gamma\right)^{\frac12}, \qquad \text{and}\qquad \operatorname{tg}\theta=\frac{k}{l}\operatorname{tg}lr_0, \]
where
\[ k=(\gamma-1)^{\frac12}\alpha=\sqrt{\frac{2\mu E}{\hbar^2}},\qquad l=\sqrt{\frac{2\mu(V_0+E)}{\hbar^2}} \]
and
\[ E=h\nu-\varepsilon_0. \]
The coefficient \(a\), entering into (4), is determined from the normalization condition, namely:
\[ \frac{a^3}{2}\left\{\beta r_0+\left[\left(2+\alpha r_0\right)\frac{\beta^4}{\alpha^4} +\left(1+\alpha r_0\right)\frac{\beta^2}{\alpha^2}-1\right] \frac{\sin^2\beta r_0}{\beta r_0}\right\}=1. \]
Upon substitution into (3) of the numerical values \(\left(\alpha=0.266\cdot 10^{-13}\ \mathrm{cm}^{-1},\ \frac{\mu}{M}=\frac19\right)\), one obtains:
\[ \sigma=2.84\cdot 10^{-28}\frac{(\gamma-1)^{\frac12}}{\gamma^3}F^2(\gamma). \tag{6} \]
The function \(F(\gamma)\) contains the depth \(V_0\) and the width \(r_0\) of the potential well, which, generally speaking, are unknown. If one proceeds from the assumption that only a single \(P\)-level, corresponding to the bound state of the neutron, is accommodated in the potential well, then the requirement of continuity of the wave function and of its derivative at the point \(r=r_0\) leads to a condition that relates the width and depth of the potential well to the binding energy of the neutron in the beryllium nucleus.
Having specified the value of the radius of the well and using the experimental value of the binding energy, one can determine the depth of the well \(V_0\). Thus, for example, for \(r_0=5\cdot 10^{-13}\ \mathrm{cm}\) one obtains \(V_0=11.25\ \mathrm{MeV}\).
As is seen from formula (6), the cross section for photodisintegration, at a certain value of the energy of the incident quanta, reaches a maximum, after which it decreases slowly. It is easy to show that the position of the maximum depends only weakly on the value of the radius of the potential well. This indicates that the structure of the nuclear residue \(\mathrm{Be}^{9}_{4}\), from the standpoint of the position of the maximum of the effective cross section, is of no essential significance.
Satisfactory agreement of formula (6) with the experimental data is restricted to the region of small \(\gamma\)-quantum energies close to the threshold for disintegration of the beryllium nucleus.
The discrepancy of formula (6) with the experimental data in the case of larger \(\gamma\)-quantum energies was discovered by Wattenberg and his collaborators\(^{15}\). These authors found that the curve of the dependence of the effective cross section for photodisintegration of beryllium on the \(\gamma\)-quantum energy possesses, in addition to a maximum in the interval \(1.7\text{--}1.8\ \mathrm{MeV}\), also a minimum in the region from 2 to \(2.5\ \mathrm{MeV}\) (see the figure on p. 113).
The presence of such a minimum on the effective cross-section curve was not provided for by formula (3).
The theory of beryllium disintegration set forth in papers ⁵˒⁷ was supplemented and developed in 1949 by Guth and Mullin¹¹*). These authors developed the theory of photodisintegration of beryllium in two directions. First, they took into account the dependence of the interaction energy \((\mathrm{Be}^{8}_{4}, n)\) on the orientation of the neutron spin and, second, considered also the case in which the neutron emitted from the beryllium nucleus, according to the selection rules for an electric transition, corresponds not only to an \(S\)- but also to a \(D\)-state.
Taking into account the dependence of the interaction energy on the direction of the spin leads to the result that the position of the neutron energy level inside the potential well proves to depend on whether the neutron spin is parallel or antiparallel to its orbital angular momentum in the initial state.
The position of the energy level, in turn, determines the depth of the potential well for a given width. For example, the assumption that the ground level lying at a depth of \(1.63\ \text{MeV}\) corresponds to a neutron spin parallel to the orbital angular momentum gives, for the depth of the well, \(V_{1,\ 3/2}=12.16\ \text{MeV}\) (well radius \(r=5\cdot10^{-13}\ \text{cm}\)). It is necessary to note further that the depth of the potential well, for a given value of the radius, depends strongly on the value of the neutron orbital angular momentum. If in the initial state with \(l=1\), for the well depth, as indicated above, we have \(V_{1,\ 3/2}=12.16\ \text{MeV}\), then in the final state with \(l=0\) it proves to be equal to \(V_{0,\ 1/2}=3\ \text{MeV}\).
The latter value is determined by the presence inside the potential well of a virtual \(S\)-level at a depth \(E\approx100\ \text{keV}\). Guth and Mullin note that other authors engaged in the study of the photodisintegration of beryllium assigned the final virtual \(S\)-state of the neutron to the same potential well to which they assigned the initial \(P\)-state. In reality, however, the dimensions of the potential well depend essentially on whether the neutron is in an \(S\) or \(P\) state. Taking account of the virtual \(S\)-level inside the potential well leads to the result that the effective cross section for the transition \(P\to S\), as Guth and Mullin indicate, possesses near the disintegration threshold a sharp maximum, which is in good agreement with the experimental data.
*) Borsellino¹⁸, in his paper published in 1948, gives a theory of the photodisintegration of beryllium that almost completely coincides with Mamasakhlisov’s theory⁵˒⁷, developed by the latter in 1940 and 1947. The difference consists in the fact that the coefficient in Borsellino’s formula for the effective cross section is exaggerated by a factor of 3. The error occurred because the mentioned author did not average the transition probability over the initial values of the magnetic quantum number.
ON THE THEORY OF THE BERYLLIUM NUCLEUS
The figure shows curves of theoretically calculated photodisintegration cross sections as functions of the energy of the γ-quantum. The experimental points lie in a sufficiently satisfactory way on the theoretical curve of the total photodisintegration cross section (solid curve), representing the sum of the effective cross sections of the transition \(P \to S\) (curve \(I\)) and the transition \(P \to D\) (curve \(II\)).
Owing to the use of improved neutron counters, it became possible to measure the yield of neutrons from beryllium when it is irradiated by high-energy quanta. Experiments of this kind were carried out in 1948 by Watenberg’s group \({}^{15}\). The idea of the experiment is quite simple. A source of γ-rays in the form of artificially radioactive elements (for example, \(\mathrm{Na}^{24}\), \(\mathrm{Mn}^{56}\), \(\mathrm{Ga}^{72}\), \(\mathrm{In}^{116}\), \(\mathrm{La}^{140}\)) was placed in a hollow graphite cylinder. This cylinder was inserted into another cylinder made of beryllium. The yield of neutrons from the beryllium cylinder was measured by special neutron counters (counters filled with \(\mathrm{B}^{10}\mathrm{F}_3\) and surrounded by paraffin).
[Figure on page: vertical axis—“Photodisintegration cross sections \(\sigma \times 10^{28}\ \mathrm{cm}^{2}\)”; horizontal axis—“Energy of γ-quanta in MeV.” Curves are labeled \(I\) and \(II\), with experimental points marked by crosses.]
The most reliable results were obtained when using radioactive \(\mathrm{Na}^{24}\), which, as is known, gives a single line with an energy of \(2.76\ \mathrm{MeV}\). For this energy the cross section proved to be equal to \(7 \cdot 10^{-28}\ \mathrm{cm}^{2}\).
The ratio of the photodisintegration cross sections of beryllium and of the deuteron was measured with sufficient accuracy (in the latter case the graphite cylinder with the source is inserted into a vessel with heavy water).
The ratio
\[ \frac{\sigma_{\mathrm{Be}}(\gamma,n)}{\sigma_{\mathrm{D}}(\gamma,n)} \]
proved to be equal to \(0.43\) for photons with energy \(2.76\ \mathrm{MeV}\), and \(0.3\) for photons with energy \(2.50\ \mathrm{MeV}\) (source \(\mathrm{La}^{140}\)).
Using the existing data on the photodisintegration of the deuteron,
\[ \sigma_{\mathrm{D}}(2.50\ \mathrm{MeV}) = 0.67 \cdot \sigma_{\mathrm{D}}(2.76\ \mathrm{MeV}), \]
it is easy to find that
\[ \sigma_{\mathrm{Be}}(2.50\ \mathrm{MeV}) = 0.47 \cdot \sigma_{\mathrm{Be}}(2.76\ \mathrm{MeV}). \]
Consequently, \(\sigma_{\mathrm{Be}}(2.50\ \mathrm{MeV}) = 3.3\cdot 10^{-28}\ \mathrm{cm}^2\). On the other hand, for photon energies of \(1.67\ \mathrm{MeV}\) the cross section is \(\sigma_{\mathrm{Be}}(1.67\ \mathrm{MeV}) = 9.7\cdot 10^{-28}\ \mathrm{cm}^2\) (source \( \mathrm{Sb}^{124}\)).
Thus, at \(\gamma\)-quantum energies of \(1.67\ \mathrm{MeV}\), \(2.5\ \mathrm{MeV}\), and \(2.76\ \mathrm{MeV}\), the cross section for the photodisintegration of beryllium takes, respectively, the values \(9.7\cdot 10^{-28}\ \mathrm{cm}^2\), \(3.3\cdot 10^{-28}\ \mathrm{cm}^2\), and \(7.0\cdot 10^{-28}\ \mathrm{cm}^2\), which obviously indicates that the effective cross section assumes a minimum value between \(\gamma\)-quantum energies of \(1.67\) and \(2.67\ \mathrm{MeV}\).
The presence of a minimum in the cross section is explained by the different character of the dependence of the photodisintegration cross sections \(\sigma_{P\to S}\) and \(\sigma_{P\to D}\) on energy. As can be seen from the figure, \(\sigma_{P\to S}\) decreases rather rapidly with energy after the maximum point, while \(\sigma_{P\to D}\) increases slowly with energy, which in combination leads to a curve with a minimum.
More direct proof of the existence of \(P\to D\) transitions can be found by studying the angular distribution of the emitted neutrons. The absence of spherical symmetry in the angular distribution of neutrons was first pointed out by T. Goloborodko\({}^{26}\) in 1941.
Further confirmation was obtained in 1949 by B. Hamermesh, A. Wattenberg, and M. Hamermesh\({}^{17}\).
The absence of spherical symmetry of the emitted neutrons suggests the idea of possible transitions of the neutron from a bound \(P\)-state to a free \(D\)-state (in addition to transitions to an \(S\)-state). The corresponding theoretical treatment of the question was given by Guth and Mullin\({}^{11}\) in 1949. Their calculations are in satisfactory agreement with the experimental data both with respect to the presence of a minimum on the curve of the dependence of the effective photodisintegration cross section (see figure) on the energy of the incident \(\gamma\)-quanta, and with respect to the angular distribution of neutrons (under the assumption of a dependence of the energy of interaction of the neutron with the nuclear residue \(\mathrm{Be}^{8}_{4}\) on the neutron spin).
For the differential effective cross section of the photodisintegration of beryllium they obtained:
\[ d\sigma=(a+b\sin^2\theta)\,d\Omega, \tag{7} \]
where \(\theta\) is the angle between the direction of the ejected neutron and the direction of the incident quantum,
\[ \frac{a}{b}=\frac{25\sigma_{P\to S}}{12\sigma_{P\to D}}+\frac{17}{12}, \]
\(\sigma_{P\to S}\) and \(\sigma_{P\to D}\) are the total effective cross sections corresponding to the transitions \(P\to S\) and \(P\to D\), respectively.
According to the data of B. Hamermesh, M. Hamermesh, and A. Wattenberg^17, the ratio \(\frac{a}{b}\) is equal to 1.5 at a \(\gamma\)-quantum energy of \(2.76\,\text{MeV}\) (\(\gamma\)-radiation of \(\mathrm{Na}^{24}\)).
The theoretical value of the ratio \(\frac{a}{b}\), according to the data of Guth and Mullin, is 1.65, which is in agreement with experiment.
If the dependence of the interaction energy on the direction of the neutron spin is neglected, then a sharp discrepancy with the experimental data appears. In this case, for the ratio \(\frac{a}{b}\) one obtains:
\[ \frac{a}{b}=\frac{2}{3}\left(1+\frac{2\sigma_{P\to S}}{\sigma_{P\to D}}\right). \]
At a \(\gamma\)-quantum energy of \(2.76\,\text{MeV}\), from the last formula we have \(\frac{a}{b}=0.72\) instead of the experimental value 1.5.
In agreement with the experimental data of earlier years^3,14, and also with theoretical predictions^4,5,6,7,8, in the case when the \(\gamma\)-quantum energy is close to the disintegration threshold, the angular distribution of the emitted neutrons is spherically symmetric.
ELECTRON DISINTEGRATION OF BERYLLIUM
Experimentally, the splitting of the \(\mathrm{Be}^{9}_{4}\) nucleus by fast electrons was investigated by Collins, Waldman, and Polye^18, and later by Collins, Waldman, and Guth^19.
The reaction of electron disintegration proceeds according to the scheme
\[ \mathrm{Be}^{9}_{4}+e_{v}=\mathrm{Be}^{8}_{4}+n+e_{v'}, \]
where \(e_v\) and \(e_{v'}\) are the incident and scattered electrons with velocities \(v\) and \(v'\), respectively. As in the case of photodisintegration, splitting takes place beginning with an electron energy of \(1.63\,\text{MeV}\), which corresponds to the binding energy of the neutron in the \(\mathrm{Be}^{9}_{4}\) nucleus.
The theory of the splitting of the beryllium nucleus by electrons was given by Mamasakhlisov^20 in 1943.*)
*) Cardirole^21, in his work of 1947, considered it necessary to derive anew the formula for the effective cross section of the electron disintegration of beryllium, completely disregarding the conclusions available in Mamasakhlisov’s work of 1943. Cardirole’s article does mention Mamasakhlisov’s work, but this is done in such a way that the reader will not discover that the derivation of the formulas for electron disintegration in fact belongs not to Cardirole, but to Mamasakhlisov.
The transition of the nuclear system \(\mathrm{Be}_4^8+n\) from one state to another is due, first, to the Coulomb interaction and, second, to the effect of absorption by the nucleus of the \(\gamma\)-quantum emitted by the decelerated electron.
Considering the interaction that is connected with the presence in the system of an electric dipole moment, for the differential effective cross section of the process one may write:
\[ d\sigma=-\frac{2\pi}{\hbar}\frac{E_0}{c^2p_0}|H_{01}|^2\rho'\,dE'\,d\Omega', \]
where \(p_0\) and \(E_0\) are the momentum and energy of the incident electron, \(H_{01}\) is the matrix element corresponding to the transition of the nuclear system \(\mathrm{Be}_4^8+n\) from one state (bound neutron) to another (free neutron), with the direction of scattering of the neutron lying within the solid angle \(d\Omega'\), and the neutron energy lying in the interval from \(E'\) to \(E'+dE'\); \(\rho'\) is the number of final states of the neutron, equal to
\[ \rho'=\frac{p'E'}{8\pi^3\hbar^3c^3}, \]
where \(p'\) is the momentum of the scattered neutron.
Using the Møller potentials \(^{22}\), as Bethe and Peierls \(^{1}\) did in the theory of the deuteron, one can calculate the effective cross section for the electron disintegration of beryllium, regarding it as the system \(\mathrm{Be}_4^8+n\).
The formula for the effective cross section, obtained by V. Mamasakhlisov \(^{20}\), has the form*)
\[ \sigma=\frac{4}{3}\left(\frac{e^3}{\hbar c}\right)^2 \left(\frac{Z\mu}{M}\right)^2 \frac{1}{a^2} \int_0^{T_0/\varepsilon_0} \frac{(\gamma-1)^{1/2}}{\gamma^4} F^2(\gamma)\{A\lg B-C\}\,d\gamma, \tag{8} \]
where
\[ A=\frac{(T_0+1)^3(T_0+1-\varepsilon_0\gamma)^2}{T_0^2+2T_0}, \qquad C=\frac{3}{2}\, \frac{\{(T_0-\varepsilon_0\gamma)^3+2(T_0-\varepsilon_0\gamma)\}^{1/2}} {(T_0^2+2T_0)^{1/2}}, \]
\[ B=\frac{ T_0(T_0+1-\varepsilon_0\gamma)+T_0-2\gamma+ (T_0^2+2T_0)^{1/2}\{(T_0-\varepsilon_0\gamma)^3+2(T_0-\varepsilon_0\gamma)\}^{1/2} }{\varepsilon_0\gamma}, \]
\(T_0\) is the kinetic energy of the incident electron in units \(m_ec^2\), \(F(\gamma)\) is determined from formulas (4) and (5), and \(\gamma\) denotes the ratio of the kinetic energy of the electron to the binding energy of the neutron, i.e.
\[ \gamma=\frac{T_0}{\varepsilon_0}. \]
*) After correction of an algebraic error that had crept into the formula of Bethe and Peierls and was repeated by Mamasakhlisov in work \(^{20}\).
In the case when \(\gamma \simeq 1\), with sufficient accuracy one may write:
\[ F(\gamma)=2b\left(1+\frac{1+\alpha r_0}{2}\gamma\right), \]
where
\[ b=-2\left(\frac{\beta}{\alpha}\right)^{3/2} a\cdot \sin \beta r_0 . \]
Expanding the polyintegral function in the expression for the total cross section in a series in powers of \(\gamma-1=x\ll 1\) and restricting ourselves to the first term of the expansion, we obtain the final expression for the total cross section in the form\({}^{20}\)
\[ \sigma=\frac{32}{27}\left(\frac{e^2}{\hbar c}\right)^2 \left(\frac{Z\mu}{M}\right)^3 \frac{b^3}{a^2}\, f(0)\left(\frac{T_0-\varepsilon_0}{\varepsilon_0}\right)^{3/2}, \tag{9} \]
where
\[ f(0)=\left(1+\frac{1+\alpha r_0}{2}\right)^2(A\lg B-C)\big|_{\gamma=1}. \]
Substituting \(r_0=5\cdot 10^{-13}\,\mathrm{cm}\) and \(T_0=3.37\) (which corresponds to an energy of the incident electrons of \(1.72\,\mathrm{MeV}\)), we obtain \(\sigma\simeq 0.21\cdot 10^{-31}\,\mathrm{cm}^2\). This value is rather close to the experimental value found by Collins, Waldman, and Guth\({}^{19}\), and also by Wiedenbeck\({}^{23}\). According to the latter, at an energy of the incident electrons of \(1.75\,\mathrm{MeV}\), \(\sigma\simeq 0.71\cdot 10^{-31}\,\mathrm{cm}^2\).
As shown by Guth and Mullin\({}^{11}\), the discrepancy with experiment is removed if, in deriving the effective cross section of electron disintegration, one takes into account the dependence of the depth of the interaction well on the value of the total angular momentum of the nuclear system.
In the works of Mamasakhlisov\({}^{9}\), Guth and Mullin\({}^{11}\), and Cardirola\({}^{21}\), the photomagnetic decay of the beryllium nucleus is also considered. Taking into account the interaction of the dipole magnetic moment of the nuclear system \(\mathrm{Be}^{8}_{4}+n\) with the radiation, the effective cross section of the photomagnetic decay of the beryllium nucleus is found. It is shown, moreover, that for the values of the \(\gamma\)-quantum energy used in the experiments the magnetic effective cross section is considerably smaller than the effective cross section of the photoelectric disintegration of beryllium.
It is of interest to consider the process inverse to photodisintegration, namely, the capture of neutrons by \(\mathrm{Be}^{8}_{4}\) nuclei with emission of \(\gamma\)-quanta. Since, as indicated above, the lifetime of the \(\mathrm{Be}^{8}_{4}\) nucleus with respect to decay into two \(\alpha\)-particles is considerably greater than the time of interaction of this nucleus with a neutron\({}^{24}\), it is quite possible that the \(\mathrm{Be}^{8}_{4}\) nucleus, before decaying into two \(\alpha\)-particles, manages to capture a neutron, turning into a stable-
the compound nucleus \(\mathrm{Be}_{4}^{8}\). It is easy to calculate the probability of this process if one uses the known thermodynamic relation between the effective cross sections of capture and disintegration.
The effective cross section for the capture of a neutron by the nucleus \(\mathrm{Be}_{4}^{8}\) with emission of a \(\gamma\)-quantum whose energy is close to the binding energy of the unpaired neutron in the nucleus \(\mathrm{Be}_{4}^{8}\), according to Mamasakhlisov\({}^{9}\), has the form
\[ \sigma_c=\frac{16\pi}{9}\frac{e^2}{\hbar c} \left(\frac{Z\mu}{M}\right)^3 \frac{b^2M_0\varepsilon_0}{a^2\mu M_1c^2} \left(\frac{\varepsilon_0}{E}\right)^{1/2} \left(1+\frac{1+\alpha r_0}{2}\right)^2, \tag{10} \]
where \(E\) is the kinetic energy of the incident neutron in the center-of-inertia system, and \(M_0\) is the neutron mass.
For slow neutrons with energies from 1 to 100 ev the effective cross section proves to be of the order \(10^{-27}\)—\(10^{-28}\ \mathrm{cm}^2\). As the formula given above shows, the dependence of the effective cross section on the energy of the incident neutron in the case of photoelectric capture of a neutron by the nucleus \(\mathrm{Be}_{4}^{8}\) turns out to be the same as in the photomagnetic capture of a neutron by a proton (according to Fermi).
The effective cross section for the capture of a neutron by the nucleus \(\mathrm{Be}_{4}^{8}\) for slow neutrons takes values that lie within the range of possible observation. Unfortunately, there are no experimental data in this direction.
In summary, it should be noted that, as the data presented above show, the representation of the excess neutron in the beryllium nucleus as being in the field of the nuclear core leads to a satisfactory description of all experimental facts concerning the photodisintegration and electronic disintegration of this nucleus.
CITED LITERATURE
- H. Bethe and R. Peierls, Proc. Roy. Soc. A 148, 116 (1935).
- Z. Szillard and Chalmers, Nature 134, 494 (1934).
- L. I. Rusinov and A. N. Sagaidak, ZhETF 6, 866 (1936).
- V. J. Mamasachlisof, Phys. Zeits. Sow. Un. B 10, 214 (1936).
- V. I. Mamasakhlisov, Communications of the Academy of Sciences of the Georgian SSR, 1, 509 (1940).
- V. I. Mamasakhlisov, Communications of the Academy of Sciences of the Georgian SSR, 3, 515 (1949).
- V. I. Mamasakhlisov, Proceedings of the Institute of Physics and Geophysics of the Academy of Sciences of the Georgian SSR 10, 1 (1947).
- V. I. Mamasakhlisov, ibid., 9, 1 (1946).
- V. I. Mamasakhlisov, ibid., 9a, 1, 17 (1946).
- V. I. Mamasakhlisov, ibid., 10, 23 (1947).
- E. Guth and C. Mullin, Phys. Rev. 76, 234 (1949).
- Chadwick and Goldhaber, Nature 135, 65 (1935).
- J. Tandberg, The Absorption of Hard \(\gamma\)-rays. Uppsala, 1937.
- Korsunskii, Nikolaevskaia and Bak, ZhETF 9, 517 (1939).
- B. Russel, D. Sachs, A. Wattenberg and R. Fields, Phys. Rev. 73, 545 (1948).
- A. Borsellino, Nuovo Cimento 5, No. 4 (1948).
- B. Hamermesh, M. Hamermesh, A. Wattenberg, Phys. Rev. 76, 611 (1949).
- Collins, Waldmen and Polye, Phys. Rev. 55, 412 (1939).
- Collins, Waldmen and Guth, Phys. Rev. 56, 876 (1939).
- V. J. Mamasachlisof, Journ. of Physic. 7, 239 (1943).
- P. Cardiriola, Journ. de Phys. et red. 8, 155 (1947).
- C. Moller, Zeits. f. phys. 70, 786 (1931).
- M. Wudenbeck, Phys. Rev. 69, 235 (1945).
- G. C. Boldwin, Phys. Rev. 76, 182 (1949).
- T. Goloborodko, ZhETF 10, 835 (1940); Dokl. Akad. Nauk 31, 857 (1941).