NEUTRON SOURCES

N. A. Vlasov

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
2. General properties of neutron sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
3. Reactions \((\alpha, n)\) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4. Reactions \((d, n)\) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5. Fission of high-energy deuterons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6. Reactions \((p, n)\) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7. Reactions \((\gamma, n)\) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8. The pile as a neutron source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

1. INTRODUCTION

The use of neutrons has now entered the practice of a large number of laboratories and makes it possible to solve a whole series of important problems in the physics of the atomic nucleus and in other branches of science. In nuclear physics, neutrons are an indispensable means for investigating the laws of interaction of nuclear particles and the system of nuclear levels. Artificial radioactivity produced with the aid of neutrons is being applied ever more widely in studies in physics, chemistry, biology, and medicine. Alongside X-rays, and sometimes with even greater success, neutrons are used for the quantitative and structural analysis of various materials.

The diversity of requirements for neutrons calls for diversity in the methods of producing them. The technique of obtaining neutrons has undergone great development, especially in recent years, in connection with the improvement of old and the invention of new accelerators of charged particles and with the realization of the chain fission reaction. The intensive process of nuclear transformations in uranium piles is of great importance in this respect not only as a powerful neutron source in itself, but also as a method for obtaining isotopes that do not exist in nature (for example, \(\mathrm{H}^3\)) and that find application as targets when bombarded by charged particles for the production of neutrons. In connection with this

systematic study of the most important neutron sources is becoming necessary for a large circle of workers not only in science, but also in technology. In the present article the main features are set forth of the nuclear transformations used for obtaining neutrons, and the characteristics of the most important neutron sources.

It is known that neutrons, together with protons, are constituents of atomic nuclei and, consequently, make up a good half of all terrestrial matter. But in the free state neutrons are practically not encountered in nature, if one does not count the very small quantity of them produced by cosmic rays, and also emitted by heavy nuclei in the process of spontaneous fission. These natural sources of neutrons are so weak and diffuse that they have no practical significance.

The absence of free neutrons is explained above all by their tendency to enter into combination with atomic nuclei. Under terrestrial conditions the lifetime of a free neutron is measured in microseconds and is limited precisely by acts of capture of neutrons by nuclei. In rarefied cosmic space, where collisions with nuclei are very rare, a free neutron, apparently, likewise cannot exist for long, but for another reason—namely, because it must undergo radioactive decay and be transformed into a proton and an electron.

Theoretically one can predict the period of radioactive decay of the neutron[^1]. It should be not less than 10 minutes and not more than a year.

Obviously, under terrestrial conditions the neutron must, in the overwhelming majority of cases, be captured by nuclei and only very rarely decay. Therefore experimental proof of neutron decay requires very delicate and painstaking investigations. Such investigations have been carried out over several years[^5], and their latest results[^83],[^84] apparently may be regarded as evidence for the decay of the free neutron with a period approximately equal to 20 minutes. If this is so, then there can be no question of a stock of free neutrons not only on the Earth, but also anywhere in cosmic space. Only as constituents of atomic nuclei can neutrons exist, and do exist, for an indefinitely long time. Therefore free neutrons can be obtained only as a result of nuclear transformations.

Any atomic nucleus, with the exception of the nucleus of ordinary hydrogen (the proton), contains neutrons in its composition and can emit them if a sufficiently strong action is applied for this purpose. Neutrons are rather firmly bound in nuclei, and to tear even one of them out of a nucleus it is necessary to expend energy of the order of several MeV. Table I gives the values of the neutron binding energies in some nuclei, calculated on the basis of the mass table given in the appendix to Gudden’s book[^2].

Table I

Neutron binding energy in some nuclei (in MeV)

Nucleus	Energy	Nucleus	Energy	Nucleus	Energy
H³	2.18	F¹⁹	10.17	A³⁶	14.82
H³	6.15	Ne²⁰	16.73	A³⁸	11.76
He⁴	20.51	Ne²¹	7.50	A⁴⁰	8.22
Li⁶	5.17	Ne²²	9.42	K³⁹	12.67
Li⁷	7.15	Na²³	11.78	K⁴⁰	7.10
Be⁹	1.63	Mg²⁴	8.31	Ca⁴³	7.19
Be¹⁰	6.69	Mg²⁵	8.31	Ti⁴⁷	9.61
B¹⁰	8.33	Mg²⁶	23.58	Ti⁴⁸	9.80
B¹¹	11.42	Al²⁷	11.10	Ti⁴⁹	6.92
C¹²	18.67	Si²⁸	16.04	Ti⁵⁰	10.63
C¹³	4.88	Si²⁹	8.31	Cr⁵²	10.17
C¹⁴	8.16	Si³⁰	11.47	Cr⁵³	8.31
N¹⁴	10.52	P³¹	11.11	Fr⁵⁷	8.12
N¹⁵	10.75	S³²	16.70
O¹⁶	15.57	S³³	9.14
O¹⁷	4.12	S³⁴	11.01
O¹⁸	7.94	Cl³⁵	9.64
		Cl³⁷	9.52

Numerous experiments show that a neutron can be emitted by any nucleus if energy exceeding the neutron binding energy $\varepsilon_n$ is imparted to that nucleus. In this case the method of excitation of the nucleus, i.e., the method of transferring energy to it, proves to be immaterial. Consequently, any method of obtaining sufficiently strongly excited nuclei is suitable for obtaining free neutrons. In practice, such methods are either bombardment of nuclei by protons, deuterons, $\alpha$-particles and other charged and even uncharged particles, as well as gamma rays, or the realization of the process of nuclear fission, as a result of which strongly excited fragments are emitted, or, finally, the production in one way or another of radioactive decays as a result of which nuclei with excitation energy exceeding the neutron binding energy may be formed (delayed neutrons in fission or “neutron decay” of the nucleus N¹⁷). Thus, the variety of processes leading to the production of free neutrons is fairly great.

If, in addition, one takes into account that bombardment by only one kind of particle—for example $\alpha$-particles—of only one definite nucleus can yield neutrons by means of various transformations, for example $(\alpha,n)$, $(\alpha,2n)$, $(\alpha,pn)$, etc., then it is evident that the variety of processes becomes practically boundless. We shall confine ourselves to considering the simplest nuclear transformations $(\alpha,n)$, $(d,n)$, $(p,n)$, $(\gamma,n)$ and fission, which are of the greatest importance as neutron sources and have been studied to a sufficient degree.

2. GENERAL PROPERTIES OF NEUTRON SOURCES

a) Yield and Cross Section

The most important characteristic of a neutron source is its intensity, quantitatively determined by the number of neutrons emitted per unit time (per second). If in each nuclear transformation in the source \(\nu\) neutrons are emitted, then the intensity of the source is equal to the product of \(\nu\) and the number of transformations per second. For example, in a uranium pile \(\nu\) is approximately equal to 2.5 for each act of fission\(^{3}\), and the intensity of the pile as a neutron source is numerically 2.5 times greater than the number of fission acts occurring per second. This number, in turn, is proportional to the energy power released in the pile. Consequently, the neutron intensity of the pile is also proportional to its power.

In the simplest nuclear reactions, one neutron is emitted for each nuclear transformation; consequently \(\nu = 1\), and the intensity of the source is equal to the number of corresponding transformations per second. It is obvious that the number of transformations is proportional to the number of bombarding particles and, consequently, the intensity of the source is proportional to the intensity of the flux of bombarding particles.

For example, in charged-particle accelerators the neutron intensity of the source is proportional to the current through the target produced by the accelerated particles. This dependence is trivial and requires no discussion.

The number of nuclear transformations per bombarding particle (called the yield of the reaction) depends on the character of the transformation and on the energy of the bombarding particle. If the energy for all bombarding particles has one definite value \(E\), then the yield \(B\) is determined by the expression

\[ B = \sigma n x, \]

where \(\sigma\) is the effective cross section of the reaction, \(n\) is the number of nuclei in \(1\ \mathrm{cm}^{3}\) of the target, and \(x\) is the thickness of the target in \(\mathrm{cm}\).

This expression is valid only for a thin target, i.e., for a target of such thickness in which the loss of energy of the bombarding particles is small. Otherwise the condition of equality of the energies of the bombarding particles is not satisfied, and the expression for the yield becomes more complicated, since the reaction cross section \(\sigma\) depends on the energy \(E\).

Thus, the yield of neutrons (or of the reaction) for a thin target is proportional to the reaction cross section and to the thickness of the target. From the point of view of yield it is advantageous to use a thick target, i.e., one in which the bombarding particles lose energy ...

to complete stopping and, consequently, the thickness of which exceeds the range of the bombarding particles. However, in a thick target the bombarding particles can enter into reaction with any energy from the initial energy down to zero. Meanwhile, the energy of the neutron emitted in the reaction also depends on the energy of the bombarding particle. Therefore the use of a thick target is possible only in those cases where the energy of the neutrons obtained is immaterial. To obtain neutrons of a definite energy, i.e. monochromatic neutrons, it is necessary to use a thin target; moreover, it is obvious that the thinner the target, the more monochromatic the neutrons obtained.

The yield from a thick target can be calculated if the dependence of the reaction cross section on the particle energy and the law of energy loss within the thickness of the target are known. In a thin layer \(dx\), situated at depth \(x\), the yield is

\[ dB=\sigma(x)n\,dx. \]

The total effective depth of the target is equal to the range of the bombarding particles \(R\), and, consequently, the total yield is equal to

\[ B=n\int_{0}^{R}\sigma(x)\,dx. \]

Usually the reaction cross section \(\sigma\) is known (if it is known at all) as a function of the energy \(E\) of the bombarding particle; therefore it is more convenient to transform the integral to the variable \(E\).

If we introduce under the integral \(\sigma(E)\) instead of \(\sigma(x)\), then in place of \(dx\) one must substitute

\[ \frac{dx}{dE}\,dE=\frac{1}{\dfrac{dE}{dx}}\,dE \]

and correspondingly change the limits of integration, taking into account that to depth \(0\) there corresponds the energy \(E_0\), and to depth \(R\) the energy \(E=0\). Then

\[ B=n\int_{E_0}^{0}\frac{\sigma(E)}{\dfrac{dE}{dx}}\,dE = n\int_{0}^{E_0}\frac{\sigma(E)}{-\dfrac{dE}{dx}}\,dE. \]

The stopping law, expressed by the quantity \(-\dfrac{dE}{dx}\), is known for most particles used for bombardment over a wide interval of energies, and if \(\sigma(E)\) is known, then the yield from a thick target can be calculated for any value of the initial energy \(E_0\).

From the expression obtained it follows that the yield \(B(E_0)\), being a function of the initial energy of the bombarding particles, is such that

the greater this energy \(E_0\), the larger the cross section \(\sigma\) is, and the smaller the energy losses \(-\dfrac{dE}{dx}\) (ionization losses) are.

In many cases \(\sigma(E)\) is not known at all, or is known only for individual values of the energy. Then the yield from a thick target cannot be calculated, and it is determined directly from experiment.

On the other hand, knowledge of the yield \(B(E_0)\) as a function of the energy of the bombarding particles \(E_0\) makes it possible to determine \(\sigma(E_0)\). Indeed, differentiating with respect to the upper limit \(E_0\), we find:

\[ \sigma(E)=\frac{1}{n}\frac{dE}{dx}\frac{dB}{dE}. \]

Consequently, measurement of the yield from a thick target for different energies of the bombarding particles makes it possible to determine the dependence of the reaction cross section on energy. However, the derivative of an experimental curve is always known less accurately than the curve itself; therefore this method of determining the cross section proves to be rather crude.

The effective cross sections of various nuclear reactions vary within very wide limits and, moreover, for each reaction exhibit a rather complicated dependence on the energy of the bombarding particles. The general character of the dependence of \(\sigma\) on \(E\) is essentially determined by the energy of the reaction. The reaction energy \(Q\) is considered positive for exothermic reactions, i.e., reactions proceeding with the release of energy (the sum of the kinetic energies of the reaction products is greater than the sum of the kinetic energies of the reacting particles), and negative for endothermic reactions.

Exothermic nuclear reactions producing neutrons are possible when nuclei are bombarded by charged particles. Reactions \((\gamma,n)\) are always endothermic. Exothermic nuclear reactions are possible at any energy of the bombarding particles. But a charged particle with low kinetic energy encounters the potential barrier of the nucleus, and the reaction cross section under these conditions is determined by the penetrability of the barrier. The penetrability of the barrier is different from zero at any, even very small, energy \(E\); therefore the cross section of an exothermic reaction is also different from zero already at very small values of \(E\). Since the penetrability of the barrier increases exponentially with the particle energy, the cross section also increases exponentially. When the particle energy reaches values close to the barrier energy \(E_b\), the increase of the cross section ceases and it reaches a certain value, almost unchanged with further increase of energy. The absolute value of the cross section in this region is close to the geometrical cross section of the nucleus \(\pi R^2=\pi r_0^2 A^{2/3}\) \((R=r_0 A^{1/3}\) is the radius of the nucleus, \(A\) is the mass number), if the reaction with emission of a neutron

is the only possible one, i.e., if there are no competing processes of nuclear decay. In the presence of competing processes, the geometrical cross section \(\pi R^2\) is divided among them, and the cross section of the reaction with neutron emission may be significantly less than \(\pi R^2\).

A characteristic example\(^3\) of the dependence of \(\sigma\) on the energy \(E\) for an exothermic reaction is presented in Fig. 1 (the reaction \(\mathrm{D}(dn)\mathrm{He}^3\)). In the region of small energies the graph is not sufficiently clear, but it is known that the reaction begins already at deuteron energies below \(50\ \mathrm{keV}\), and as the energy increases the yield rapidly grows.

At \(E_d \simeq 1.5\ \mathrm{MeV}\) the cross section reaches “saturation” and proves to be equal to \(0.1\) barn, and then remains of this magnitude up to

Fig. 1. Cross section of the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\).

\(E_d = 3.6\ \mathrm{MeV}\). The competing process in this case is proton emission in the reaction \(\mathrm{D}(dp)\mathrm{H}^3\). The total cross section is divided approximately in half between these two processes.

The dependence of \(\sigma\) on \(E\) is analogous also for other exothermic reactions, with only the difference that in the case of heavier nuclei the barrier is higher, and the region of saturation of the cross section is shifted to the right, toward larger values of \(E\).

In some cases the indicated smooth dependence is disturbed by resonance effects. If the excitation energy \(E^*\) of the compound nucleus formed from the bombarded nucleus and the bombarding particle is equal to the energy of one of its quantum quasistationary states, then the reaction cross section increases resonantly at the corresponding value of \(E\). Since the excitation energy of the compound nucleus is equal to the sum \(\varepsilon + E\), where \(\varepsilon\) is the binding energy

of the bombarding particle in the compound nucleus, and \(E\) is its initial kinetic energy, then the resonance condition \(\varepsilon+E=E^*\) is satisfied only for one value of \(E\) for each level. An example of the resonance dependence of the cross section on \(E\) is the reaction \(\mathrm{Be}^9(\alpha n)\mathrm{C}^{12}\) (Fig. 2).

Endothermic reactions are possible only when the energy of the bombarding particle exceeds a certain limit, called the reaction threshold \(\Pi\). Consequently, for \(E<\Pi\) the reaction cross section is zero. The threshold value \(\Pi\) is not equal to the absolute value of the reaction energy \(Q\), but always exceeds it, since part of the kinetic energy of the bombarding particle is transferred to the compound nucleus, which receives its momentum. The relation between \(\Pi\) and \(Q\) is expressed by the simple formula following from the laws of conservation of energy and momentum,

\[ \Pi=-\frac{M+m}{M}\,Q = -\left(1+\frac{m}{M}\right)Q, \]

where \(M\) and \(m\) are the masses of the bombarded nucleus and of the bombarding particle, respectively. The smaller the ratio \(\frac{m}{M}\), the closer the threshold value is to the value of the reaction energy.

Fig. 2. Cross section of the reaction \(\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}\).

For values \(E>\Pi\) the reaction cross section increases with energy, generally speaking, considerably more rapidly than for exothermic reactions. The character of the increase depends on the relation between the threshold and the barrier \(E_b\). If \(\Pi<E_b\), then the cross section rapidly increases to values determined by the penetrability of the barrier, and then changes in the same way as the cross section of an exothermic reaction, i.e. it reaches saturation or even a maximum at \(E\simeq E_b\), the value at saturation being either equal to the geometrical cross section of the nucleus or smaller than it if competing decay processes exist. If \(\Pi>E_b\), then the cross section very rapidly reaches saturation beyond the threshold. In any case, in endothermic reactions the cross section and yield increase very sharply beyond the reaction threshold; therefore the threshold is determined very distinctly and with good accuracy. This is widely used for calibrating high-voltage accelerators—

...accelerators not equipped with sufficiently reliable apparatus for measuring the absolute value of the accelerating voltage.

The indicated general character of the dependence of the cross section on the energy for endothermic reactions, as also in the case of exothermic reactions, can be substantially distorted by resonance phenomena.

As an example one may point to the dependence of the cross section on the proton energy in the reaction \(\mathrm{Li}^{7}(pn)\mathrm{Be}^{7}\) (Fig. 22, p. 230). Immediately beyond the threshold \((\Pi = 1.882\ \text{MeV})\) the cross section rises very steeply, reaching 0.24 barn and then remaining constant. But at a proton energy close to \(2.2\ \text{MeV}\) a resonance maximum of the cross section is observed, due to a quasi-stationary state of the compound nucleus \(\mathrm{Be}^{8}\) with excitation energy \(E^* = 19.1\ \text{MeV}\).

A second example is the cross section of the reaction \(\mathrm{T}^{3}(pn)\mathrm{He}^{3}\) (Fig. 30, p. 237). (Here and below the symbol \(\mathrm{T}^{3}\) denotes the superheavy isotope of hydrogen \(\mathrm{H}^{3}\)—tritium.) The threshold of this reaction is \(\Pi = 1.019\ \text{MeV}\). Beyond the threshold the cross section increases sharply, but, despite the fact that the energy of the bombarding proton exceeds the energy of the potential barrier, the growth of the cross section continues up to a proton energy equal to \(2.4\ \text{MeV}\). The absence of saturation of the cross section in this case indicates the existence of a resonance peak of the cross section at a proton energy exceeding \(2.4\ \text{MeV}\), which corresponds to an excited state of the compound nucleus \(\mathrm{He}^{4}\) with an energy of about \(20\ \text{MeV}\).

Taking into account the general regularities in the dependence of reaction cross sections on the energy of the bombarding particles, one may draw the following conclusions concerning the neutron yield. The yield from a thin target, proportional to the reaction cross section at the corresponding energy, undergoes irregular fluctuations from one reaction to another, and also changes for a given reaction as a function of the energy of the bombarding particles. At a given, not very large, particle energy, the yield is the greater the smaller the nuclear potential barrier is for the bombarding particle. Therefore, light substances with a small nuclear charge are more advantageous and are usually used as targets. True, for light substances the geometrical cross section is smaller than for heavy ones, but this does not lead to a decrease in the yield, since the geometrical cross section is proportional to the number of particles \(A\) to the power \(2/3\)

\[ (\pi R^2 = \pi r_0^2 A^{2/3}). \]

Meanwhile the stopping power of the target for the bombarding particles, on the basis of which the thickness is chosen, is approximately proportional to the number of electrons in the target, i.e. proportional to the nuclear charge \(Z\), and consequently also to the number of particles in the nucleus \(A\). Owing to this, for the same stopping power, i.e. for the same value of the mean energy losses, a target made of a lighter element contains a larger number of nuclei and the total...

the geometric cross section of all nuclei of the target is proportional to \(A^{-1/3}\), i.e. it is greater for light substances than for heavy ones.

The yield from a thick target, determined by the integral dependence given above on \(\sigma\) and \(\dfrac{dE}{dx}\), obviously always increases monotonically with the energy of the bombarding particles. In those cases where the cross section may be regarded as independent of energy, for example when the energy of the bombarding particles is much greater than the barrier energy and the cross section remains constant over a large energy interval, the yield may be regarded as proportional to the range of the bombarding particles. In fact, taking \(\sigma\) out from under the integral, we obtain:

\[ B = n \int_{0}^{E_0} \frac{\sigma}{-\dfrac{dE}{dx}}\, dE = n\sigma R. \]

For the most significant energy interval, of the order of several MeV, the range of bombarding charged particles may be considered proportional to \(E^{3/2}\); consequently, the yield from a thick target under bombardment by particles with energy greatly exceeding the barrier energy is approximately proportional to \(E^{3/2}\).

b) Neutron energy

The second essential characteristic of a source is the energy spectrum of the neutrons emitted by it. The crudest spectral characteristics of neutrons are expressed by the established concepts of “fast neutrons” and “slow neutrons.” There is no strict boundary between these concepts: it may vary depending on the point of view. One may regard as slow those neutrons whose wavelength

\[ \lambda = \frac{\lambda}{2\pi} \]

is much greater than the dimensions of the nucleus, i.e. define slow neutrons from the condition

\[ \lambda \gg R, \]

where \(R\) is the radius of the nucleus. The essence of such a definition amounts to the fact that neutrons are considered slow if their capture cross sections, which in the case of resonance reach values \(\sigma_{\text{res}}=\pi\lambda^2\), may be much greater than the geometric cross section of the nucleus.

Proceeding from this definition, neutrons with energies less than 50–20 keV, including, of course, thermal neutrons, may be considered slow. Thus, for example, Bethe understands by slow neutrons “neutrons with energy not exceeding several thousand electron-volts.” Neutrons with higher energies are correspondingly regarded as fast.

The overwhelming majority of sources directly emit fast neutrons. To obtain slow neutrons, the source is placed inside some moderator, in which the neutrons lose energy as a result of multiple collisions—

NEUTRON SOURCES

with nuclei. In this case, obviously, the spectrum of fast neutrons directly emitted by the source is significant only insofar as the optimal moderation conditions depend somewhat on it. For example, the greater the energy of the primary neutrons, the greater the thickness of the moderator must be. In general, however, the moderation process may be said to impose no requirements on the primary neutron spectrum; therefore, for obtaining slow neutrons, only the yield and intensity of the source are important, and any primary spectrum is admissible.

But for a whole series of very interesting and important problems of nuclear physics, not only knowledge of the spectrum of fast neutrons of the source is required, but also the possibility of controlling it. Thus, the study of cross sections for the interaction of neutrons with nuclei, which reveals the system of nuclear energy levels, requires the use of monochromatic neutrons of variable energy. In connection with this and similar problems, the question of the neutron spectrum acquires decisive importance, and the choice of source is determined by its spectral characteristic, usually even at the expense of yield.

The energy of the neutrons emitted by a source depends primarily on the energy of the corresponding reaction \(Q\) and on the kinetic energy of the bombarding particles \(E\). (For the time being we shall leave aside the question of the energy of neutrons in fission and discuss it below.) This dependence can be established from the laws of conservation of energy and momentum. Let \(E, p, v\), and \(m\) denote the energy, momentum, velocity, and mass of the bombarding particle; \(E_a, P_a, V_a\), and \(M_a\) the corresponding quantities for the bombarded nucleus (note that \(E_a\) and \(P_a\) may be taken to be zero); \(E_b, P_b, V_b, M_b\) those for the product nucleus; and \(E_n, p_n, v_n, m_n\) those for the neutron.

Then the law of conservation of energy may be written in the form

\[ E + Q = E_b + E_n, \]

and the law of conservation of momentum in the form

\[ P_b^2 = p^2 + p_n^2 - 2pp_n \cos \theta, \]

where \(\theta\) is the angle between the direction of emission of the neutron and the direction of the bombarding particle.

Using the relation

\[ E = \frac{P^2}{2M} \]

and eliminating \(E_b\) from both equations, it is easy to obtain the following formula:

\[ Q = \left(1 + \frac{m_n}{M_b}\right) E_n - \left(1 - \frac{m}{M_b}\right) E - \frac{2}{M_b} \sqrt{m m_n E E_n}\cdot \cos \theta, \tag{a} \]

which expresses, in implicit form, the dependence of the neutron energy on the reaction energy \(Q\), the energy of the bombarding particle \(E\), the neutron emission angle \(\theta\), and the mass ratios of the particles participating

in the reaction. The solution of this equation with respect to \(E_n\) in its general form is rather cumbersome and inconvenient for analysis. The character of the dependence of \(E_n\) on the indicated quantities is better traced in special cases.

The dependence of \(E_n\) on the angle of emission \(\theta\) can be established by reducing formula (a) to the following form:

\[ \sqrt{E_n}\left(\sqrt{E_n}-a\cos\theta\right)=A. \tag{б} \]

Here

\[ a=\frac{2\sqrt{m m_n}}{M_b+m_n}\sqrt{E} \]

and

\[ A=\frac{M_b-m}{M_b+m_n}E+\frac{M_b+m_n}{M_b}Q \]

are quantities constant for the given reaction and the specified energy \(E\) of the bombarding particles. It follows from formula (б) that \(\sqrt{E_n}\), and consequently \(E_n\), decreases with increasing angle \(\theta\), since the constants \(a\) and \(A\) are essentially positive. This, incidentally, is also evident from simple physical considerations. The dependence of \(E_n\) on the angle of emission is determined by the velocity of the compound nucleus. In the center-of-mass coordinates the velocity of the neutron does not depend on the angle of emission. The velocity of the neutron in the laboratory coordinate system is equal to the vector sum of its velocity in the center-of-mass system (i.e., of the compound nucleus) and the velocity of the center of mass in the laboratory system. The latter is always directed forward, i.e., in the direction of the velocity of the bombarding particle \((\theta=0)\). Owing to this, the velocity, and consequently the energy, in laboratory coordinates is maximal for neutrons flying forward at the angle \(\theta=0\), and decreases with increasing angle \(\theta\). The scale of this change of velocity and energy with the angle \(\theta\) is determined by the magnitude of the coefficient

\[ a=\frac{2\sqrt{m m_n}}{M_b+m_n}\sqrt{E}. \]

It is evident that the greater the mass of the bombarding particle in comparison with the mass \(M_b\) of the product nucleus (and, consequently, also of the bombarded nucleus), the wider is the range of variation of the energy with angle. If, for example, the same product nuclei arise as the result of two reactions \((\alpha,n)\) and \((p,n)\), then, for equal energies of the bombarding particles, the neutron energy will vary with angle considerably more in the first reaction than in the second, since the mass of the \(\alpha\)-particle is four times greater than the mass of the proton, and consequently the coefficient \(a\) multiplying \(\cos\theta\) is also greater.

For one and the same reaction, the dependence of the neutron energy on the angle \(\theta\) is the stronger, the greater the energy \(E\) of the bombarding particle, and hence the greater the velocity of the compound nucleus. Examples of the angular dependence of \(E_n\) may be found below in the description of the characteristics of individual sources.

The change of \(E_n\) with the angle \(\theta\) offers great practical convenience, since it permits, in one and the same reaction, at un-

changed energy of the bombarding particles, neutrons of different energies may be obtained. This circumstance was widely used in work with the source \( \mathrm{D}(\mathrm{dn})\mathrm{He}^3 \).

The dependence of the neutron energy \(E_n\) on the energy of the bombarding particles \(E\) and on the reaction energy \(Q\) is conveniently followed by putting \(\theta = 90^\circ\) (\(\cos \theta = 0\)), i.e., for neutrons emitted at a right angle to the beam of bombarding particles. Since the dependence of \(E_n\) on the angle is always smooth and monotonic, \(E_n(90^\circ)\) represents a certain mean value of the energy of the neutrons obtained in each reaction. Under the condition \(\cos \theta = 0\), equation (a) can be rewritten in the form

\[ E_n(90^\circ) = \frac{M_b - m}{M_b + m_n} \, E + \frac{M_b}{M_b + m_n} \, Q . \tag{b} \]

We see that \(E_n(90^\circ)\) always increases with the energy of the bombarding particles. The case \(M_b - m < 0\), for which this conclusion would be incorrect, is impossible in reactions with neutron emission. A monotonic increase of \(E_n\) with increasing \(E\) is also observed for all angles \(\theta < 90^\circ\). However, for \(\theta > 90^\circ\), in particular for \(\theta = 180^\circ\), the changes of \(E_n\) with increasing \(E\) are more complicated. In some cases a decrease of \(E_n(180^\circ)\) with increasing \(E\) may be observed. These cases can be clarified by analyzing equation (a) in detail, but we shall not dwell on this. We shall indicate only, as an example, the dependence of \(E_n(180^\circ)\) on \(E\) in the reactions \(\mathrm{D}(\mathrm{d}, \mathrm{n})\mathrm{He}^3\) and \(\mathrm{T}^3(\mathrm{d}, \mathrm{n})\mathrm{He}^4\) (Figs. 11, 16, pp. 205 and 215).

In any case, for neutrons in the forward hemisphere, and sometimes also for neutrons flying backward, the energy increases with increasing energy of the bombarding particles. Owing to this, in one and the same reaction and in one and the same direction relative to the beam of bombarding particles (for example, forward), neutrons of different energies can be obtained if it is possible to control the energy of the bombarding particles. This technique is now used very widely in experiments with monochromatic fast neutrons.

The dependence of \(E_n(90^\circ)\) on \(Q\) is also evident from formula (b). The energy of the neutrons is the greater, the greater the reaction energy. This rule is valid for any angles and any values of the energy \(E\).

However, for one and the same reaction the energy \(Q\) may assume several different values if the product nucleus is formed as a result of the reaction not only in the ground state, but also in excited states. In that case, for each value of the energy of the bombarding particles and of the angle of emission there correspond several different values of the neutron energy. Consequently, under otherwise identical conditions, monochromatic neutrons cannot be obtained in a reaction having several values of \(Q\). In the case ...

case the spectrum may turn out to be line-like. It should be noted that the number of possible values of \(Q\) in each reaction depends on the energy of the bombarding particles. Indeed, negative values of \(Q\) correspond to a certain reaction threshold; therefore they can appear only in the case when \(E > P\). As a result, the number of possible values of \(Q\) may increase with the energy of the bombarding particles, and it may happen that, for example, a reaction suitable for obtaining monochromatic neutrons of low energy proves unsuitable for high energies, since an admixture of a new group with lower energy will be added to the original group of neutrons.

This circumstance proves to be very important. It limits the possibility of obtaining monochromatic neutrons of high energy or, more precisely, establishes a certain upper limit for the energy of monochromatic neutrons obtained in nuclear reactions. In fact, every nucleus produced as the result of a reaction has excited levels, which will inevitably be formed as soon as the energy of the bombarding particles exceeds the corresponding reaction threshold. At the same time one should expect that the transition to an excited level will be more probable than to the ground level, since the emitted neutron has a lower energy and, consequently, as a rule, one closer to the mean energy of the particles in the excited nucleus. According to the statistical theory of the nucleus (see, for example,\(^6\)) the spectrum of particles emitted by an excited nucleus has a maximum in the region of energies close to the mean excitation energy per nuclear particle. Although the statistical theory can be applied to light nuclei only with a certain caution, its main conclusions are all the more applicable the higher the excitation energy. At the same time, numerous experiments show that, even in reactions with light nuclei, the emission of fast particles carrying away the entire excitation energy of the nucleus is always of low probability.

For this reason, an unlimited increase in the energy of neutrons obtained in ordinary nuclear reactions, by increasing the energy of the bombarding particles, proves in general to be impossible. True, at present physical laboratories already have artificially produced neutrons with energies of the order of \(100\) MeV, but their spectrum is continuous and the mechanism of formation is quite different. We shall discuss it below.

c) Angular distribution of intensity

The angular distribution is usually understood as the dependence of the radiation intensity—in the present case, neutron radiation—on the angle \(\theta\) between the direction of the bombarding particles and the direction from the source to the point of observation. It goes without saying that po-

the concept of angular distribution has meaning only when the direction of the bombarding particles is specified. If, however, the bombarding particles fall on the substance being bombarded in any direction with equal probability, as, for example, in Ra + Be sources, then the neutron flux is distributed uniformly in all directions, independently of the character of the angular dependence in the elementary acts of the reaction.

The practical significance of the angular distribution is obvious. Knowledge of it makes it possible to choose the most advantageous irradiation conditions and to estimate the intensity of a neutron beam in any direction. The angular distribution may be characterized either by relative numbers proportional to the neutron flux density, i.e., to the number of neutrons propagating in the given direction per unit solid angle, or by the differential effective cross section.

It is practically convenient to define the differential effective cross section as a quantity equal to what the total cross section would be if the intensity in all directions were the same as at the given angle. With this definition the total cross section proves to be the result of averaging the differential cross section over angles. It may happen that to some angle \(\theta=\theta_1\) there corresponds a differential cross section, for example, numerically twice as large as the total cross section. This means that the flux intensity in this direction is twice the average over all angles. In some cases the quantity called the differential cross section is one \(4\pi\) times smaller.

For the theoretical analysis of the angular distribution, for the sake of simplicity and clarity, one usually uses a coordinate system associated with the center of inertia of the reacting particles. In this coordinate system the total momentum is zero, and the velocity of this system itself relative to the laboratory (in laboratory coordinates) is determined by the relation

\[ \mathbf{V}_c=\frac{m}{M_a+m}\mathbf{v}, \]

where \(\mathbf{V}_c\) is the velocity vector of the center of inertia, \(m\) is the mass of the bombarding particle, \(M_a\) is the mass of the bombarded nucleus, and \(\mathbf{v}\) is the velocity vector of the bombarding particle in laboratory coordinates.

The velocity of the bombarded nucleus in the laboratory system is here taken to be zero. To pass from the angular distribution \(I(\theta)\) in the laboratory system, directly observed in experiment, to the angular distribution \(I_0(\theta_0)\) in the center-of-inertia system, one may use the relation

\[ I_0(\theta_0)\,d\omega_0 = I(\theta)\,d\omega, \]

where \(d\omega_0 = 2\pi\sin\theta_0\,d\theta_0\) and \(d\omega = 2\pi\sin\theta\,d\theta\) are elements of solid angle in the center-of-inertia system and in the laboratory system

with increasing energy of the bombarding particle, the angle \(\theta\) at the vertex of the cone containing all emitted neutrons increases and reaches \(90^\circ\) when the energy reaches the value

\[ E_k=-\frac{M_a+m+m_n}{M_a-m_n}Q =-\frac{M_b}{M_b-m}Q =\frac{M_a(M_a+m-m_n)}{(M_a-m_n)(M_a+m)}\Pi =\frac{M_b(M_b-m+m_n)}{(M_b-m)(M_b+m_n)}\Pi . \]

At energies \(E>E_k\), neutrons can already be emitted in all directions, including at angles \(\theta>90^\circ\).

Thus, for \(E_k>E>\Pi\), the neutron emission angle does not exceed a certain limiting value \(\theta_m\), which at \(E=\Pi\) is equal to zero, and with increasing \(E\) grows and reaches \(90^\circ\) at \(E=E_k\). We note that in this energy interval \(E\), for each direction of emission there correspond two values of the neutron energy. The larger of them corresponds to neutron emission forward in the center-of-inertia system, and the smaller—to emission backward. For \(E>E_k\), the dependence between the neutron energy and the emission angle \(\theta\) is single-valued.

It is therefore obvious that, in an endothermic reaction, monochromatic neutrons of arbitrarily small energy cannot be obtained at the angle \(\theta=0\). Already at a bombarding-particle energy \(E\) equal to the threshold \(\Pi\), the neutron energy \(E_n\) is equal to

\[ E_n=\frac{m m_n}{(M_a+m)^2}\Pi , \]

and with increasing \(E\) two groups of neutrons arise. The energy of one of them increases with \(E\), while that of the other decreases, reaching zero at \(E=E_k\). However, at angles \(\theta>90^\circ\) only one energy group of neutrons is always emitted, and its energy is equal to zero at \(E=E_k\), and then increases monotonically with further increase of \(E\).

3. REACTIONS \((\alpha,n)\)

Reactions of the type \((\alpha,n)\) led to the discovery of the neutron and, consequently, historically proved to be the first neutron sources. For a fairly long time they were the only sources. Only with the development of the technology of artificial acceleration of charged particles, especially by means of cyclotrons, did other possibilities for obtaining neutrons appear, in particular by bombarding various nuclei with deuterons. But even after this, \((\alpha,n)\) reactions did not lose their significance. Neutron sources in which beryllium is bombarded by \(\alpha\)-particles from naturally radioactive elements \((\mathrm{Ra}+\mathrm{Be},\ \mathrm{Rn}+\mathrm{Be})\) are the most widespread even at the present time, owing to the stability, compactness, and ease of transport of the sources.

In most cases \((\alpha,n)\) reactions turn out to be endothermic, i.e., they proceed with absorption of energy. This is also to be expected

expected if one takes into account that the bombarding \(\alpha\)-particle is a very strongly bound system with a large binding energy, while the bombarded nucleus is usually stable. Meanwhile the final nucleus differs from the bombarded one by two protons and only one neutron, i.e., it has an excess of protons and therefore proves to be weakly bound and usually radioactive. It is evident that the transformation of strongly bound systems into less strongly bound ones requires an expenditure of energy. However, among light nuclei there are some for which bombardment by \(\alpha\)-particles leads to exothermic reactions. One of the most important examples of such reactions is the reaction \(\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}\). Here the final nucleus \(\mathrm{C}^{12}\) proves to be not only stable, but also one of the most tightly bound; therefore the reaction proceeds with the release of energy.

The yield of the \((\alpha,n)\) reaction is substantially determined by the penetrability of the barrier, which for \(\alpha\)-particles is large even in light nuclei. For example, already for the \(\mathrm{B}^9\) nucleus the barrier height for an \(\alpha\)-particle is about \(4\ \text{MeV}\). In heavy nuclei, at \(\alpha\)-particle energies of the order of \(5\)—\(10\ \text{MeV}\), the reactions are practically impossible, since the barrier penetrability is very small (see \(^{4}\)). Therefore, to obtain neutrons in \((\alpha,n)\) reactions, only light nuclei are used as targets.

The most widespread and important neutron source of this type is the reaction \(\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}\). It is carried out by bombarding metallic beryllium with \(\alpha\)-particles from artificially radioactive elements.

Depending on which of the active elements is used, the source receives one of the following designations: Ra + Be, Rn + Be, Po + Be, and so on. For example, the designation Ra + Be corresponds to a source in which beryllium is bombarded by \(\alpha\)-particles of radium and of its decay products.

a) The Ra + Be source

At present, the following procedure for preparing Ra + Be preparations is generally accepted \(^{8}\). A fine powder of metallic beryllium, in an amount of \(3\)—\(5\ \text{g}\) per \(1\ \text{g}\) of radium (pure), is moistened with a solution of a radium salt (usually \(\mathrm{RaBr}_2\)). The solvent is then evaporated, and the beryllium powder with the precipitate of radium salt, distributed over the surface of the grains, is triturated and poured into a glass or metal ampoule and sealed. In such a mixture, \(\alpha\)-particles of Ra and of its decay products, penetrating the powder, bombard Be nuclei and in some cases form neutrons, which are emitted from the ampoule in all directions.

Glass ampoules are simpler to manufacture, but are not strong and require careful handling. Metal ampoules (for example, copper) are more convenient. For most experiments the best is a spherical metal ampoule. \(1\ \text{g}\) Ra

with the appropriate amount of Be powder can be packed into a spherical ampoule 2.5–3 cm in diameter. In those cases when one seeks to obtain a preparation of very small dimensions, before packing into the ampoule the powder is pressed under high pressure[^8]. Owing to this, the density of the powder increases, and the same amount of mixture can be enclosed in a smaller volume. In addition, pressing the powder proves desirable for certain purposes also because it transforms the mixture into a solid body that preserves both its shape and the relative arrangement of the components, thereby ensuring a more reliable stability of the source.

The optimum amount of beryllium is chosen on the basis of the following considerations. The neutron yield will obviously be the greater, the greater the relative concentration of beryllium in the source. Collisions with Ra or Br atoms lead only to the slowing down of the α-particles and reduce the probability of the reaction. Since the stopping power of a substance is approximately proportional to its mass, the neutron yield from a Ra + Be source may be considered proportional to the fraction

\[ \frac{M_{\mathrm{Be}}}{M_{\mathrm{Be}}+M_{\mathrm{RaBr_2}}}, \]

where \(M\) denotes the masses of the corresponding components.

Thus, the yield per gram of Ra is the greater, the greater the amount of beryllium. However, the dimensions of the source are then also greater. Since compactness of the source is very desirable for many purposes, one usually confines oneself to the ratio \(M_{\mathrm{Be}} : M_{\mathrm{Ra}} = 5\), i.e., 5 g of Be per 1 g of pure Ra.

Bretscher, Cook, Martin, and Wilkinson[^27] proposed an interesting method of preparing from radium and beryllium a neutron source distinguished by a comparatively small yield, but convenient as a standard. They introduced radium chemically into the composition of crystalline fluorite \( \mathrm{RaBeF_4} \). In such a crystal the distribution of the components is geometrically ideal; therefore the neutron yield is, of course, very well reproducible and proportional to the amount of radium, if the overall size of the crystal is considerably larger than the range of the α-particles, so that the fraction of particles leaving the crystal by passing out through its surface is small. However, the relative concentration of beryllium in such a source is very small (about 3% by weight), and therefore the yield is small. According to the authors’ measurements, 1 g of \( \mathrm{RaBeF_4} \) gives on average \(1.84 \cdot 10^6\) neutrons per second (with an accuracy of about 10%), whereas an ordinary source gives 4–6 times more. Let us note that in the crystalline source \( \mathrm{RaBeF_4} \), neutron formation proceeds not only in the reaction \( \mathrm{Be^9(\alpha,n)C^{12}} \), but also in the reaction \( \mathrm{F^{19}(\alpha,n)Na^{22}} \).

A \( \mathrm{Ra}+\mathrm{Be} \) source gives neutrons as a result of the reaction \(\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}\), induced by \(\alpha\)-particles both from Ra itself and from the products of its decay*). In the radioactive uranium family, beginning with Ra, there are the following five \(\alpha\)-emitters:

1. Ra \(E_\alpha=4.791,\quad T=1590\) years, \(A_n=5.2\%\).
2. Rn \(E_\alpha=5.486,\quad T=3.825\) days, \(A_n=11.1\%\).
3. RaA \(E_\alpha=5.998,\quad T=3.05\) min., \(A_n=18.1\%\)

(further, the \(\beta\)-emitters RaB—27 min. and RaC—23 min.).

1. RaC′ \(E_\alpha=7.680,\quad T=1.5\cdot10^{-4}\) sec., \(A_n=56.5\%\)

(further, the \(\beta\)-emitter RaD—25 years).

1. Po \(E_\alpha=5.298,\quad T=140\) days \(A_n=9.1\%\).

If all the decay products are in equilibrium, then each of them undergoes the same number of decays per second as does radium, and, consequently, the total number of \(\alpha\)-particles is five times greater than that given by radium itself. Thus, for each gram of Ra the source emits not \(3.7\cdot10^{10}\), but \(1.85\cdot10^{11}\) \(\alpha\)-particles per second.

The neutron yield in the reaction \(\mathrm{Be}^9(\alpha,n)\mathrm{C}^{12}\), and in particular from a \(\mathrm{Ra}+\mathrm{Be}\) source, has been investigated in a large number of experiments. In Fig. 2 is shown the curve of the dependence of the reaction cross section on the residual range, or on the energy, of \(\alpha\)-particles\(^9\), obtained by bombarding a thin \((0.22\ \mathrm{mg}/\mathrm{cm}^2)\) beryllium target with polonium \(\alpha\)-particles. Similar curves had also been obtained earlier\(^{10-13}\). The cross section has an appreciable value already at \(\alpha\)-particle energies below 1 MeV and increases with increasing energy, as should be expected for an exothermic nuclear reaction. At an \(\alpha\)-particle energy of about 1.5 MeV a resonance maximum is observed, corresponding to an excited state of the compound nucleus \(\mathrm{C}^{13}\) with an energy of 13.8 MeV.

For \(E_\alpha>3\) MeV the cross section again increases. A break in the curve in the region \(E_\alpha=4\) MeV indicates the possibility of a second resonance level. At the full energy of the Po \(\alpha\)-particles, equal to 5.3 MeV, the cross section reaches a value of 0.4 barn. At higher energies the cross section continues to increase.

In Fig. 3, curve I shows\(^{13}\) the neutron yield from a thick beryllium target bombarded by ThC′ \(\alpha\)-particles. Curve II represents the beginning of curve I on an enlarged scale. Curve III shows the ratio of the yield of slow neutrons to the yield of fast ones. The presence of an intense group of slow neutrons has been noted by many authors. Their origin is attributed to the reaction \(\mathrm{Be}^9(\alpha,n)\,3\mathrm{He}^4\), which begins at an energy of the bombarding particle of about 4.8 MeV.

*) In addition to \(\alpha\)-particles, neutrons are also formed by \(\gamma\)-rays through the reaction \(\mathrm{Be}^9(\gamma,n)\mathrm{Be}^8\). But the number of these photoneutrons is so small that they may be neglected.

Amaldi and Fermi \(^{14}\) found that the yield from a thick target is at least three times greater for \(\alpha\)-particles from RaC′ than for \(\alpha\)-particles from Rn or RaA.

The relative yield of neutrons from various \(\alpha\)-emitters—decay products of radium—according to Birch \(^{12}\), is given above in the list of \(\alpha\)-emitters in the form of the quantity \(A_n\), expressed as a percentage of the total yield.

Fig. 3. Neutron yield in the reaction \(\mathrm{Be}^9(\alpha,n)\). \(I\)—fast neutrons, \(II\)—the same on an enlarged scale, \(III\)—ratio of the yield of slow neutrons to the yield of fast ones.

The absolute neutron yield of a Ra + Be source, according to various measurements \(^{15—18}\), ranges from 6700 to 25,000 neutrons per sec. per 1 mg Ra. Such a large scatter in the results of different measurements is explained not only, and apparently not so much, by differences in the sources used themselves as by experimental errors.

The method of these measurements in most cases amounted to determining the radioactivity of manganese dissolved in a large tank of water and absorbing practically all the neutrons emitted by a source placed at the center of the tank. When equilibrium is established, the activity of the manganese, i.e., the number of decays occurring per second, is equal to the number of neutron-capture events by manganese nuclei, and consequently to the number of neutrons emitted by the source. The technique of these measurements is described in the papers cited. Later measurements, carried out

with greater reliability, give the following results. Anderson and Feld\(^8\) give a formula for determining the yield of pressed \( \mathrm{RaBr_2}+\mathrm{Be} \) sources, according to which the yield is

\[ 1.7\cdot 10^7\, \frac{M_{\mathrm{Be}}}{M_{\mathrm{Be}}+M_{\mathrm{RaBr_2}}} \]

neutrons per second per \(1\,g\) of Ra.

For a ratio of \(5\,g\) Be per \(1\,g\) Ra \((1.66\,g\,\mathrm{RaBr_2})\), the yield of a pressed source prepared according to the procedure described in detail in \(^8\) turns out to be \(1.28\cdot 10^7\) neutrons per second per \(1\,g\) Ra.

For a source prepared by the usual method\(^ {20}\) from \(504\,mg\) Ra (in the form of \(\mathrm{RaBr_2}\)) and \(3\,g\) Be, on the basis of measurements reducing to the determination of the amount of helium formed in the reaction \(\mathrm{B}^{10}(n,\alpha)\mathrm{Li}^7\) when all the source neutrons are captured by boron, a yield value of \((5.5\pm0.4)\cdot 10^6\) neutrons per second was obtained\(^ {20}\), which is equivalent to \((1.09\pm0.08)\cdot 10^7\) neutrons per second per \(1\,g\) Ra. Even in these measurements the accuracy does not exceed \(8\%\), since the measurements are rather difficult. On the basis of the latest data, without exceeding the limits of their accuracy, it may be assumed that for ordinary \(\mathrm{Ra}+\mathrm{Be}\) sources the yield is \(10^7\) neutrons per second per \(1\,g\) Ra, or \(10^4\) per \(1\,mg\), i.e. one neutron per 18,500 \(\alpha\)-particles, or approximately \(5\cdot 10^{-5}\) neutrons per one \(\alpha\)-particle.

If a \(\mathrm{Ra}+\mathrm{Be}\) source is made from radium freed from decay products, as is usually the case, then its neutron activity proves to be a rather complicated function of time, quite unlike the exponential with a period of 1580 years that characterizes the decay of radium itself. On the basis of Bjerge’s data\(^ {12}\) on the relative neutron yield from individual groups of \(\alpha\)-particles, one can establish approximately the following character of the change of neutron activity with time.

As long as the source contains pure radium without decay products, which have not yet had time to accumulate, its activity will amount to \(5.2\%\) of the full equilibrium activity. The \(\alpha\)-emitters following radium—Rn, RaA, and RaC′—will accumulate with the emanation period of 3.8 days, since the decay periods of all the products from RaA to RaC′, inclusive, are small. Within 25 days their activity reaches \(99\%\) of the equilibrium value, increasing according to the law \(1-e^{-\lambda t}\), where \(\lambda\) is the decay constant of the emanation. In this case the neutron activity of the source increases from \(5.2\) to \(90.9\%\) of the equilibrium value. The remaining \(9.1\%\) of the equilibrium neutron activity, accounted for by the \(\alpha\)-particles of Po, will increase with a period of 25 years, which belongs to the predecessor of polonium—RaD. Consequently, a \(\mathrm{Ra}+\mathrm{Be}\) source made from radium purified from decay products proves to be far from stable. True, a month after preparation the increase in activity becomes rather slow, but nevertheless amounts to a value of the order of

0.2% for each year close to the time of manufacture; in precise measurements with a source this should be taken into account.

The neutron spectrum of the Ra + Be source proves to be continuous and rather complex. Formula (a), which determines the neutron energy, for the present case (assuming \(m_n=1\), \(m=4\), \(M_b=M_c^{12}=12\)) may be written in the form

\[ Q=\frac{13}{12}E_n-\frac{8}{12}E_\alpha-\frac{1}{3}\sqrt{E_nE_\alpha}\cdot \cos \theta . \tag{a_1} \]

First of all, let us note that the reaction energy \(Q\) has several values corresponding to different excitation states of the final nucleus \(\mathrm{C}^{12}\). The scheme of excited states of the \(\mathrm{C}^{12}\) nucleus remains insufficiently well studied. According to the review of Lauritsen and Fowler \(^{21}\), the energies of the excited states are 3 MeV, 4.3 MeV, 7.1 MeV, 9.5 MeV, etc. The transition of \(\mathrm{C}^{13}\) to the ground state corresponds to the value of the reaction energy \(Q_0=5.75\) MeV; and to the listed excitation states, respectively, \(Q_1=2.75\) MeV, \(Q_2=1.45\) MeV, \(Q_3=-1.35\) MeV, \(Q_4=-3.75\) MeV. If a thin layer of beryllium were bombarded by a monochromatic beam of \(\alpha\)-particles, then in any direction relative to this beam several groups of neutrons would be observed with energies corresponding to the given values of \(Q\) according to formula \((a_1)\). In other words, the neutron spectrum in this case would be linear.

However, in actual Ra + Be sources there are, first of all, five groups of \(\alpha\)-particles with different energies. In addition, inside the beryllium powder all \(\alpha\)-particles are slowed down until their energy is completely lost and, consequently, enter the reaction with any energy from the maximum down to zero. Finally, the directions of the \(\alpha\)-particles are arbitrary, and consequently the angles of neutron emission are arbitrary.

All these factors lead to the smearing of the spectral lines into broad bands, which undoubtedly overlap, and as a result the neutron spectrum of Ra + Be proves to be certainly continuous and rather complex. In fact, the dependence of the neutron energy on the emission angle alone is already sufficient to turn a line spectrum into a continuous one. Table II gives the values of the energies of neutrons emitted at angles \(\theta=0\) (forward) and \(\theta=180^\circ\) (backward) when bombarding a thin layer of be—

Table II

Energy of Po + Be neutrons at angles
\(\theta=0\) and \(\theta=180^\circ\)

\(Q\) in MeV	\(E_n(0^\circ)\)	\(E_n(180^\circ)\)	Relative intensity
5.8	11.8	6.8	\(10^{-3}\)
2.8	7.8	4.4	10
1.4	6.5	3.4	3.5
\(-0.6\)	4.1	1.8	1

radium by $\alpha$-particles; polonium for different values of the reaction energy.

As is evident from the table, the energy range in each group is rather large, so that the groups overlap appreciably.

In the fourth column are given the relative intensities of the various groups according to Bernardini’s determination^22. A direct investigation of the neutron spectrum was carried out by Dunning^23, who studied the distribution, according to range, of recoil protons knocked out by neutrons from a thin layer of paraffin. It is true that Dunning used an $\mathrm{Rn}+\mathrm{Be}$ source, and not $\mathrm{Ra}+\mathrm{Be}$, but with respect to the neutron spectrum these two sources do not differ substantially. The results of the measurements are shown in Fig. 4. The number of recoil protons decreases as their range increases. In places there are not very sharp breaks in the curve, indicating the presence of separate energy groups of neutrons. Since the protons knocked out in the forward direction were studied, their energy is equal to the energy of the corresponding neutrons, and from the positions of the breaks, using the energy–range curves, one can find the energies of the corresponding neutron groups. However, the appearance of these groups should rather be regarded with surprise than be assigned serious significance. Dunning’s curve should be considered only as a rather rough characteristic of the $\mathrm{Ra}+\mathrm{Be}$ neutron spectrum. It indicates, in particular, the presence of a large number of low-energy neutrons, apparently arising in the reaction $\mathrm{Be}^9(\alpha,n)3\mathrm{He}^4$.

Fig. 4. Absorption curve of recoil protons from neutrons of $\mathrm{Rn}+\mathrm{Be}$.

A paper by Teucher^24 has recently been published, investigating the $\mathrm{Ra}+\mathrm{Be}$ neutron spectrum by means of recoil protons produced in a thick-layer photographic emulsion. In these measurements recoil protons directed forward within an angle of $10^\circ$, with energy not less than $0.5$ MeV, were recorded. The energy distribution of the protons is shown in Fig. 5 by black dots. From the proton spectrum one can pass to the neutron spectrum by using the well-known dependence of the neutron scattering cross section on a proton on energy. The neutron spectrum obtained on the basis of the corresponding calculation is shown in the same figure by circles. The hatched rectangles characterize the statistical errors of the measurements. The accuracy of the results is not great, but they confirm

an obvious conclusion that the neutron spectrum is continuous, extending to energy values above 11 MeV, with the greatest intensity falling in the energy interval from 3 to 7 MeV.

Recently a whole series of works has been published\(^{76—81}\) devoted to the study of the spectrum of neutrons and gamma rays in the reaction Be\(^9(\alpha,n)\)C\(^{12}\). In most of them an excited level of the C\(^{12}\) nucleus with an energy

Fig. 5. Neutron spectrum of Ra + Be, measured from the ranges of recoil protons in a photoemulsion. •—protons, ○—neutrons.

about 4.5 MeV is convincingly and distinctly found, to which there corresponds a reaction energy \(Q = 1.3\) MeV, but with respect to other excited levels the results do not agree, and their existence still remains doubtful. On the basis of these data, one may regard as unquestionable the presence in the neutron spectrum of Be\(^9(\alpha,n)\)C\(^{12}\) of two groups corresponding to two values of the reaction energy: 5.8 and 1.3 MeV. The results of earlier measurements of the neutron spectrum\(^{23,24}\), apparently, can quite well be reconciled with the assumption of the absence of other groups.

b) Rn + Be source

This source has been widely used and is still used because of the simplicity of its preparation when an emanation apparatus is available. It can be made even more compact than a (Ra + Be) source. Its disadvantage is that it is considerably less stable than a (Ra + Be) source and decays rapidly. The source is usually prepared in a glass

ampoule. The ampoule is filled with beryllium powder, soldered to an emanation apparatus, evacuated, and then filled with emanation, which is distributed in the pores between the beryllium grains. After filling with emanation the ampoule is soldered off, and the source is ready for use. The dimensions of the ampoule may be very small. For example, a source containing 500 mC of emanation can be made in the form of an ampoule 3–5 cm long and 3–6 mm in diameter.

The neutron yield is approximately the same as from a \((\mathrm{Ra}+\mathrm{Be})\) source, since the missing groups of \(\alpha\)-particles of Ra and Po are the least effective, while the other three groups are identical. On the other hand, the relative concentration of beryllium in this source is always greater. The neutron spectrum practically coincides with the neutron spectrum from a \((\mathrm{Ra}+\mathrm{Be})\) source.

The dependence of the neutron activity on time, however, is quite different. It is clear that the emanation is introduced into the ampoule in pure form, without subsequent decay products. In the first minutes after preparation the neutron activity of \(\mathrm{Rn}+\mathrm{Be}\) is due to only one group of \(\alpha\)-particles of the emanation itself. Subsequently RaA with a period of 3 minutes and RaC′ with a period of about 1 hour accumulate (RaB—27 min., RaC—23 min.). During the first hours the neutron activity increases, reaching a maximum, and then decreases with the half-life period of the emanation (3.8 days).

c) Source \(\mathrm{Po}+\mathrm{Be}\)

This source\(^{77,78}\) is distinguished by a relatively small neutron yield, and therefore is used less often than the preceding ones. But in comparison with them it is more convenient in those cases where monochromatic \(\alpha\)-particles must be used, since polonium emits only one group of \(\alpha\)-particles with an energy of 5.298 MeV. In addition, the decay of polonium is accompanied by very weak \(\gamma\)-radiation (of the order of \(10^{-5}\) quanta per decay), whereas Ra and Rn are strong sources of \(\gamma\)-rays. In those cases where it is desirable to have a neutron source without strong accompanying gamma radiation, the \(\mathrm{Po}+\mathrm{Be}\) source proves more convenient than the preceding ones. To prepare the source, polonium is deposited as a thin layer on a metallic surface, and then is brought either into direct contact with metallic beryllium, or placed at some distance from a beryllium target, depending on the purpose of the source. The activity of the source decreases with time in the same way as the \(\alpha\)-activity of polonium, i.e., with a period of 140 days.

Besides the reaction \(\mathrm{Be}^{9}(\alpha,n)\) considered, any other \((\alpha,n)\) reaction may also serve as a neutron source, i.e., a reaction with any other nucleus as the target. It is known, however, that

the neutron yield with other targets is considerably smaller. The greatest yield after beryllium is obtained when boron is bombarded with $\alpha$-particles in the reaction $B^{11}(\alpha,n)N^{14}$. The dependence of the cross section of this reaction on the energy of the $\alpha$-particles${}^{25}$ is presented in Fig. 6. The yield from a thick target is approximately 10 times smaller than in the reaction $Be^9(\alpha,n)C^{12}$.

Fig. 6. Cross section of the reaction $B^{10,11}(\alpha,n)N^{13,14}$.

Bombardment of other elements by $\alpha$-particles gives a yield tens of times smaller; therefore, for obtaining neutrons it is practically not used.

4. REACTIONS $(d,n)$

With the development of the technique of artificial acceleration of charged particles, the reaction $(d,n)$ has become extremely widespread as a source of neutrons. Among other reactions it is distinguished by a large neutron yield. In the overwhelming majority of cases, deuterons are used to obtain neutrons on cyclotrons and other accelerating devices.

The reaction $(d,n)$ amounts to the transfer of the proton from the bombarding deuteron to the bombarded nucleus, as a result of which a neutron is released. The reaction energy is equal to the difference between the binding energy of the proton in the final nucleus and in the deuteron. Since the binding energy of the proton (and neutron) in the deuteron is very small ($2.18$ MeV), and in most other nuclei considerably greater, the difference, and consequently the reaction energy $Q$, is usually positive. Therefore reactions $(d,n)$ are, as a rule, exothermic and can be observed at any, even very small, deuteron energy $E_d$. The yield at small $E_d$ is determined by the penetrability of the potential barrier for the deuteron. Since the charge of the deuteron is equal to unity and

is half the charge of the \(\alpha\)-particle, the potential barrier is also approximately half as low, and its penetrability is considerably greater. Therefore, generally speaking, the yield is considerably greater when nuclei are bombarded with deuterons than with \(\alpha\)-particles of the same energy. For all nuclei with charge \(Z<12\), the height of the potential barrier does not exceed \(3\ \text{MeV}\). Consequently, for deuterons accelerated even in a small cyclotron, the potential barrier is not at all an obstacle hindering nuclear interaction with light nuclei.

Besides the \((d,n)\) reaction, a source of free neutrons in the bombardment of nuclei by deuterons may be the process of simple splitting of the deuteron in the Coulomb field of the nucleus into a proton and a neutron, i.e. the \((d,pn)\) process, in which the bombarded nucleus remains unchanged.

Landau and Lifshitz\(^{36}\) carried out a theoretical calculation of the effective cross section of the \((d,pn)\) process and showed that in some cases it is even several times larger than the cross section of the \((d,n)\) and \((d,p)\) processes. It is possible that the neutrons obtained as a result of bombarding nuclei with deuterons are due to both processes, i.e. both \((d,n)\) and \((d,pn)\).

However, in practically important cases the \((d,pn)\) process has not been observed, and its role is apparently small, although it has not been studied definitively.

To obtain deuterium, electrolytic decomposition of heavy water in a special apparatus is usually used. Gaseous deuterium is passed through a narrow opening into an ion source, in which ionization is produced in one way or another. The deuterium ions formed in the ion source are drawn by an electric field into the accelerating device.

In this case, however, not only free deuterons are obtained, but also molecular ions \(D_2^+\), the number of which may even exceed the number of deuterons. If the ions are accelerated in a high-voltage tube and, consequently, all pass through one and the same accelerating potential, then the energy of the deuterons at the exit from the tube will depend on whether they have passed through the accelerating field singly or as a pair. Indeed, the charge of an individual deuteron and of a molecular pair is the same; therefore one and the same energy is imparted sometimes to one deuteron, sometimes to a pair of deuterons. Owing to this, a mixed atomic-molecular beam of deuterons accelerated in the tube will be nonuniform in energy. If this phenomenon is undesirable, for example in those cases when it is necessary to obtain monochromatic neutrons, then it is necessary to analyze the beam with the aid of, say, a magnetic field. In cyclotrons and other resonance accelerators such analysis takes place automatically, since the resonance conditions for the same charge are satisfied for only one value of the mass.

a) The Reaction \(D(d,n)\mathrm{He}^3\)

Bombardment of deuterium by deuterons has been very widely used, and is still used at present, for obtaining neutrons. The main positive features of the reaction \(D(d,n)\mathrm{He}^3\) are:

1) a fairly large neutron yield already at very low energies of the bombarding deuterons, for example at \(E_d = 50\) kev,

2) monochromaticity of the beam of the neutrons obtained in any direction with a thin target, and in some cases also with a thick target,

3) a strong dependence of the neutron energy on the angle of emission, making it possible to obtain monochromatic neutrons over a wide energy interval even with an unchanged deuteron energy.

With respect to neutron yield at small \(E_d\), the reaction \(D(d,n)\mathrm{He}^3\) until very recently was the most advantageous. None of the known other reactions gave so large a yield at deuteron energies \(E_d < 700\) kev. Owing to this, on accelerators giving particles with energies of several hundred kev (for example, 200–300 kev), neutrons in quantities sufficient for many experiments were obtained only in this reaction. For almost an entire decade the reaction \(D(d,n)\mathrm{He}^3\) was practically the only source of monochromatic neutrons of variable energy.

At the present time another reaction has been found\(^3\)—\(T^3(d,n)\mathrm{He}^4\), giving an even larger neutron yield at low deuteron energies. In those cases where there is a sufficient quantity of tritium for preparing a target, and the spectrum of the neutrons obtained is not important, the reaction \(D(d,n)\mathrm{He}^3\) loses its exclusive advantages, yielding first place to the reaction \(T^3(d,n)\mathrm{He}^4\). However, the isotope \(T^3\) is radioactive and is obtained artificially in small quantities; it therefore proves to be rather expensive and by no means generally available. In addition, in the reaction \(T^3(d,n)\mathrm{He}^4\) very fast neutrons with energy above 13 Mev are obtained. Therefore, for obtaining monochromatic neutrons with energy from 2 to 13 Mev, the reaction \(D(d,n)\mathrm{He}^3\) still remains the most advantageous.

A deuterium target is usually prepared in the form of a layer of heavy ice frozen onto a metallic backing, cooled during bombardment by liquid air. More advantageous, of course, is a gas target containing pure deuterium. With the same stopping power, the yield from a gas target should be approximately 5 times greater than from a target made of heavy ice, since the relative concentration of deuterium in the gas is 5 times greater than in \(D_2O\). But the use of a gas target requires the introduction of foil or

NEUTRON SOURCES

the film separating the target from the vacuum chamber of the accelerator. In passing through this foil, the deuterons are, naturally, slowed down. The loss of energy in the foil leads to a decrease in yield. Since the foil cannot be made very thin, at low initial deuteron energy the slowing in the foil reduces the yield so much that it deprives the gas target of its advantages. Therefore, when deuterons are accelerated to energies of 200–500 keV, a target of heavy ice is more often used than a gas target. Other compounds with heavy hydrogen are used as targets only very rarely. In some experiments, for example, “heavy paraffin” was used. In it the relative concentration of deuterium is somewhat higher than in heavy water ($\mathrm{CD}_2$ instead of $\mathrm{D}_2\mathrm{O}$), but it is more difficult to prepare than heavy ice. For greater stability, $\mathrm{D}_2\mathrm{O}(\mathrm{P}_2\mathrm{O}_5)$ is sometimes used, but with such a target the yield is considerably lower.

The reaction $\mathrm{D}(d,n)\mathrm{He}^3$ has been investigated rather well. Like most $(d,n)$ reactions, it is exothermic, the reaction energy being $Q = 3.28$ MeV. For clarity the reaction scheme is usually written as

\[ \mathrm{D}+\mathrm{D}\to \mathrm{He}^3+n+3.28\ \text{MeV}, \]

introducing the value of the reaction energy $Q$ as a term on the right-hand side. Negative values of $Q$ will correspond to endothermic reactions.

Along with the reaction $\mathrm{D}(d,n)\mathrm{He}^3$, when deuterons are bombarded by deuterons a second reaction, $\mathrm{D}(d,p)\mathrm{H}^3$, occurs, with energy $Q = 4.0$ MeV. The cross sections of both reactions are approximately the same over a large interval of deuteron energies $E_d$; therefore, sometimes the intensity of one of them is judged by observing the other.

The $\mathrm{D}(d,n)$ reaction gives an appreciable yield of neutrons already at a deuteron energy equal to 50 keV. According to Zinn and Seely$^{28}$, a mixed atomic–molecular beam of deuterons with energy 60 keV, bombarding a heavy-ice target, gives a number of neutrons equivalent to 125 millicuries of $\mathrm{Rn}+\mathrm{Be}$ for each milliampere. When the deuteron energy is increased by 20 keV, the yield doubles. Amaldi, Hafstad, and Tuve$^{29}$ investigated the yield as a function of $E_d$ in the interval from 0.300 to 1.0 MeV when various targets, including heavy ice, were bombarded with deuterons. Their results are presented in Fig. 7. It should be noted that in this figure the value of the yield, expressed in the absolute number of neutrons, is apparently overstated by approximately a factor of two, since the neutron yield from the $\mathrm{Rn}+\mathrm{Be}$ source adopted as the standard was taken, according to the data of Amaldi and Fermi, to be equal to 25,000 neutrons per second per millicurie, which is approximately twice the currently generally accepted value. On the basis of the data of Amaldi, Hafstad, and Tuve one may conclude that the reaction $\mathrm{D}(d,n)\mathrm{He}^3$ with a $\mathrm{D}_2\mathrm{O}$ target is

Fig. 7. Neutron yield in the reaction \((d, n)\) with various targets.

Horizontal axis: deuteron energy, keV.

Left vertical axis: number of neutrons per second per milliampere.

Right vertical axis: \((p_{11}+\mathrm{Be})\)-equivalent \((\mu\mathrm{cu})\).

Curve labels: \(D+\mathrm{Li}\), \(D+\mathrm{Be}\), \(D+D_2O\), \(D+D_2O(P_2O_5)\), \(D+C\).

best with respect to neutron yield at deuteron energies below 700 keV.

Figure 8 shows the curve of neutron yield when a thick heavy-water target is bombarded by deuterons with energies from 100 to 300 keV[^13].

The results of experiments with thin targets make it possible to determine the effective cross section of the reaction.

The most detailed study of the reaction for deuteron energies from 0.5 to 3.7 MeV was carried out by Gunter and Richardson[^30]. The curve they obtained for the dependence of the cross section on \(E_d\), on the basis of their own data compared with others, is presented in Fig. 1.

In the energy region below 0.5 MeV, the various data are not in very good agreement with one another; but there is no doubt that the cross section increases monotonically with energy, reaching a maximum value of 0.1 barn at \(E_d \simeq 1.5\) MeV, and then decreases very slowly with increasing deuteron energy. At \(E_d = 10\) MeV, according to measurements by Erickson, Fowler, and Stovall[^31], the reaction cross section is equal to 0.07 barn.

Fig. 8. Neutron yield in the reaction \(D(d,n)\) from a thick \(D_2O\) target.

The angular distribution of neutrons changes rather complicatedly with the energy of the bombarding deuterons. Already at energies of the order of 200 keV it differs noticeably from a spherically symmetric distribution in center-of-inertia coordinates. The differential cross section in these coordinates is, naturally, always symmetric with respect to the angle \(90^\circ\), i.e., the same for the angles \(90^\circ+\varphi\) and \(90^\circ-\varphi\), since the bombarding and bombarded deuterons move toward one another with equal velocities and are indistinguishable. At any energy the differential cross section is larger at angles \(0\) and \(180^\circ\) than at an angle of \(90^\circ\). The ratio of the corresponding values of the cross section increases with deuteron energy.

The results of detailed studies of the angular distribution, carried out by Gunter and Richardson[^30], are presented in Fig. 9. Along the abscissa is plotted the angle \(\theta\), in center-of-inertia coordinates; along the ordinate, the differential cross section of the reaction. The different curves refer to different deuteron energies. With increasing energy, the deviation from a spherically symmetric distribution becomes stronger. At \(E_d > 1.5\) MeV the monotonic increase of the cross section from the angle \(90^\circ\) toward \(0^\circ\) and \(180^\circ\) is broken; minima appear at angles approximately \(60^\circ\) and \(120^\circ\).

Fig. 9. Angular distribution of neutrons in the reaction D(d,n)He³ at various deuteron energies (in center-of-inertia coordinates).

Such a complex dependence of the angular distribution on the energy of the deuterons is explained by the fact that the reaction proceeds as a result of collisions of deuterons with different values of the angular momentum, and the relative role of the different values of the angular momentum changes with the deuteron energy. The differential cross section can be represented in the form of an expansion in a series of spherical functions \(P_l(\cos \theta_0)\):

\[ \sigma(\theta_0)=A_0P_0 + A_2P_2 + A_4P_4 + A_6P_6+\ldots \]

The expansion coefficients \(A_l\) determine the probability of collisions with different values of the angular momentum \(l\). It turns out that, in the investigated energy interval, all four coefficients \(A_l\) must be considered different from zero, and only in this case can the series expansion be brought into satisfactory agreement with the experimental data. In other words, even collisions with so large a value of the angular momentum as \(l=6\) have an appreciable probability. The values of \(A_l\) corresponding to different energies \(E_d\) are given in the work of Hunter and Richards\({}^{30}\).

Usually the angular distribution of neutrons is represented in the form of the more convenient expansion in a series in cosines

\[ \sigma(\theta_0)=K(1+A\cos^2\theta_0+B\cos^4\theta_0+C\cos^6\theta_0+\ldots). \]

In this series the coefficients \(K, A, B, C\) are also functions of the energy; their form is given graphically in \({}^{30}\). The coefficient \(A\) changes sign at \(E_d \simeq 1.4\ \text{MeV}\); the remaining coefficients are positive and increase monotonically with energy.

The angular distribution in laboratory coordinates, which is of greater interest for practical purposes, is shown in Fig. 10. The neutron flux density, proportional to the differential effective cross section shown in the figure, is always greater at the angle \(\theta=0\), i.e., in the direction of the deuteron beam.

At \(E_d=3.69\ \text{MeV}\) the forward flux density is approximately 8 times greater than at an angle of \(90^\circ\); at lower deuteron energies the ratio is smaller.

The energy of the neutrons produced in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) can be calculated from formula (a), which, upon substitution of the corresponding mass values, reduces to the form

\[ Q=\frac{4}{3}E_n-\frac{1}{3}E_d-\frac{2\sqrt{2}}{3}\sqrt{E_dE_n}\cdot\cos\theta. \]

The reaction energy \(Q\) has a single value, \(3.28\ \text{MeV}\) (according to some data \(3.30\ \text{MeV}\)), for deuteron energies up to \(10\ \text{MeV}\). Owing to this, monochromatic neutrons can be obtained in this reaction if a thin target is used.

To calculate the neutron energy for different values of \(E_d\) and \(\theta\), one may solve the last equation as a quadratic in \(\sqrt{E_n}\).

The solution has the form

\[ \sqrt{E_n}=0.3535\cos\theta\cdot\sqrt{E_d}\pm \]

\[ \pm\sqrt{(0.125\cos^2\theta+0.250)E_d+2.475}. \]

The results of calculations of \(E_n\) by this formula are presented graphically in Fig. 11, where the angle of neutron emission \(\theta\) in degrees is plotted along the abscissa, and the neutron energy \(E_n\) along the ordinate.

Fig. 10. Angular distribution of neutrons in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) in laboratory coordinates.

Different curves correspond to different values of the energy of the bombarding deuterons, from \(0.2\) to \(4\ \mathrm{MeV}\).

As can be seen from the graph, in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) one can obtain neutrons with energies from \(1.65\ \mathrm{MeV}\) and higher. The maximum neutron energy, corresponding to the angle \(\theta=0\), for a given deuteron energy is approximately equal to \(E_d+Q\). This equality is fulfilled the more accurately, the larger \(E_d\) is. The energy of neutrons emitted backward, i.e., at an angle \(\theta\) close to \(180^\circ\), varies weakly and nonmonotonically with the deuteron energy. When \(E_d\) changes from \(0\) to \(5\ \mathrm{MeV}\) (approximately), \(E_n(180^\circ)\) slowly decreases from \(2.20\ \mathrm{MeV}\) to \(1.65\ \mathrm{MeV}\), and then begins to increase just as slowly. Obviously, at large angles, for which the depen-

dependence of \(E_n\) on \(E_d\) is weak even with a thick target; the neutrons will be monochromatic. As is seen from the course of the curves, to each value of \(E_d\) there corresponds a certain value of the angle \(\theta\), at which \(E_n\) is almost independent of \(E_d\). When \(E_d\) changes from 0 to 5 MeV, this value of \(\theta\) increases from 90 to 180°. Very monochromatic

Fig. 11. Neutron energy in the reaction \(D(d,n)\mathrm{He}^3\) as a function of the emission angle for different values of the deuteron energy.

neutrons with an energy of about 2.5 MeV can be obtained even with a thick target at an angle close to 90°, if \(E_d\) is small, for example not more than 200 keV. For neutrons emitted at an angle \(\theta = 90^\circ\), the energy is determined by a very simple expression, which follows from the general one if one puts \(\cos\theta = 0\):

\[ E_n = \frac{3}{4}Q + \frac{1}{4}E_d . \]

In particular, if \(E_d\) changes, owing to slowing down in the thick target, from 200 keV to 0, then \(E_n\) changes from 2.47 MeV to 2.52 MeV, i.e. by 50 keV, or by 2%.

The range of variation of the neutron energy with the angle \(\theta\) is the wider, the larger \(E_d\) is. For example, at \(E_d = 4\) MeV, \(E_n\) varies from 1.65 to 7.30 MeV. True, in this case the intensity of the neutron beam also varies strongly with angle (Fig. 10), and, generally speaking, in order to obtain neutrons of different energies it is more advantageous to vary the deuteron energy, always observing the neutrons going forward.

But if the possibilities for varying the deuteron energy are limited, then even at fixed energy, observation at different angles supplies neutrons with an energy varying over a very wide interval.

The reaction \(\mathrm{D}(\mathrm{d},\mathrm{n})\mathrm{He}^3\) is especially important as a source of monochromatic neutrons with energies from 2 to 10 MeV. There are practically no other sources for this energy interval. It should be noted, however, that at high deuteron energies, along with the monochromatic neutrons of the reaction \(\mathrm{D}(\mathrm{d},\mathrm{n})\mathrm{He}^3\), inevitable admixtures appear of neutrons of other energies, arising from collisions of deuterons with various elements of the installation—diaphragms, films, residual gas. The need to exclude the effect of these admixtures requires special precautions and complicates the corresponding experiments.

b) The reaction \(\mathrm{Be}^9(\mathrm{d},\mathrm{n})\mathrm{B}^{10}\)

Bombardment of beryllium by deuterons is the usual method of obtaining neutrons by means of cyclotrons[^83]. From the time the first cyclotrons appeared and up to the appearance of uranium boilers, i.e. for approximately a decade, the reaction \(\mathrm{Be}^9(\mathrm{d},\mathrm{n})\mathrm{B}^{10}\) was the most intense and very widespread source of neutrons. But even up to the present time, despite the development of chain-reaction technology, which supplies very intense fluxes and beams of neutrons, the cyclotron as a neutron source has not lost its significance. For example, for the very important field of research—the study of the interaction of monochromatic slow neutrons with matter by means of selectors with a “flashing beam”[^73]—the cyclotron is the most convenient and practically the only source of neutrons. And in this and in other applications, to obtain neutrons on a cyclotron the reaction \(\mathrm{Be}^9(\mathrm{d},\mathrm{n})\mathrm{B}^{10}\) is usually used.

The beryllium target of a cyclotron is usually made of pure metallic beryllium in the form of a plate or a layer of powder pressed onto a backing of another metal, for example copper. The target is soldered or fastened to a copper backing, cooled during bombardment by running water. Cooling is necessary because the deuteron beam delivers to the target a rather large energy, which is transformed almost exclusively into heat. A cyclotron of medium size can give

a beam of deuterons with an energy of 8 MeV and an intensity inside the chamber (between the dees) of the order of 100 μA. The power of such a beam is evidently equal to

\[ 8\cdot 10^6\cdot 100\cdot 10^{-6}=800\ \text{W}, \]

i.e., greater than the power of an ordinary electric hot plate. Almost all this power is released as heat in the surface layer of the target, up to 0.5 mm thick and with an area of the order of 1 cm², if the target is internal, and of several cm² if the target is external (behind the deflecting plate). Without cooling, the target would very quickly melt or even evaporate. Even with water cooling, a low-melting target proves unsuitable for a cyclotron. Beryllium is a rather refractory and weakly sputtering metal, and therefore a beryllium target withstands bombardment very well. From Fig. 7 it is clear that bombardment of lithium with deuterons gives a greater neutron yield than bombardment of beryllium. However, metallic lithium is very low-melting (\(t_{\text{melt.}}=179^\circ\text{C}\)) and volatile, and also chemically very active; therefore its use as a cyclotron target is practically excluded. If stable lithium compounds are used, the yield decreases. Therefore beryllium, used in pure metallic form, also proves to be the most advantageous target material with respect to yield.

The neutron yield in the reaction \(\mathrm{Be}^9(d,n)\mathrm{B}^{10}\) at deuteron energies up to 1 MeV, according to the measurements of Amaldi, Hafstad, and Tuve \(^{29}\), is presented in Fig. 7.

The yield increases rather rapidly, though smoothly, with the deuteron energy \(E_d\), and at \(E_d=1\) MeV reaches approximately 7 g (Ra + Be)-equivalent per microampere of deuteron current.

As regards the yield at higher deuteron energies, there are data obtained on various cyclotrons and electrostatic generators and summarized by Livingston \(^{32}\). These data are not exact and are not fully compatible with one another, but they are of interest for lack of more systematic ones. Below we give an extract from Livingston’s table, in which neutron yields are given in gram-equivalents of Ra + Be per microampere of deuteron current from a thick beryllium target.

The same data are represented graphically in Fig. 12.

Kornog and Libby \(^{33}\) indicate that the 60-inch cyclotron at Berkeley, at a deuteron energy equal to 16 MeV, gives 1 neutron for every 200 deuterons from a thick beryllium target.

The discrepancy between the figures corresponding to cyclotrons and electrostatic generators at identical deuteron energies is probably due, at least in part, to

Table III

Neutron yield from a thick beryllium target at various installations per 1 μA of deuteron current

Installation	Cyclotrons: California University	Cyclotrons: Harvard University	Cyclotrons: Rochester University	Cyclotrons: Cornell University	Electrostatic generators: Cornell Institute	Electrostatic generators: Cornell Institute
Energy in MeV	16	8–12	3–7	1–2	3–5	1–2
Beam intensity in μA	200	20–100	4–50	25	15–50	10
Yield in gram-equivalents Ra + Be per 1 μA	6000	3000	200	40	100	7

because on the generators an unanalyzed mixed atomic-molecular beam was used; therefore the energy of the deuterons is in reality lower than indicated.

All the data were obtained in measurements of the intensity of the neutron beam directed forward, along the direction of the deuteron beam.

Fig. 12. Neutron yield from a thick beryllium target of a cyclotron according to Livingston.

Livingston notes that the increase of the yield with deuteron energy proceeds considerably faster than the depth of penetration of deuterons into the target, which is proportional to \(E^{3/2}\). Thus, from 3.5 to 16 MeV the actual yield increases six times faster than \(E^{3/2}\). This discrepancy may be explained by two causes. First, the intensity of the neutron beam directed forward increases with increasing deuteron energy faster than the in-

integral; secondly, some of the neutrons may be formed as a result of the simple breakup of the deuteron in the field of the nucleus, i.e., the process \((d,pn)\), whose probability increases with increasing energy.

The angular distribution of the neutron-flux intensity, according to observations on the California cyclotron, does not differ very strongly from a spherically symmetric one. Thus, at a deuteron energy of \(16\) MeV, the neutron flux in the forward direction is only \(1.7\) times more intense than in the backward direction. However, later observations by Falk, Kreitz, and Zaitz \(^{34}\) indicate a rather strong concentration of the neutron beam within a small interval of small emission angles. The curve they obtained for the angular distribution of neutrons produced by bombardment of a thick target of \(\mathrm{LiBO}_2\) with \(15\) MeV deuterons is shown in Fig. 13.

Fig. 13. Angular distribution of neutrons upon bombardment of \(\mathrm{LiBO}_2\) with deuterons of energy \(15\) MeV.

For a thick beryllium target, as the authors indicate, the distribution is approximately the same and, in general, depends only weakly on the material of the target. As can be seen from the figure, the angular distribution is characterized by a sharp forward directionality, since at the angle \(\theta = 0\) the curve has a rather sharp maximum. For a beryllium target, the angular width of the beam between the directions corresponding to intensities half as large as the maximum (at \(\theta = 0\)) is only \(26^\circ\). Already at the angle \(\theta = 40^\circ\), as can be seen from Fig. 13, the intensity falls to approximately \(10\%\) of the maximum. It is evident that at lower deuteron energies one should expect smaller concentrations of the neutron flux in the region of small angles; therefore the integral intensity, of course, increases with energy much more slowly than the intensity at small angles.

The energy of neutrons produced in the reaction \(\mathrm{Be}^9(d,n)\mathrm{B}^{10}\) can be calculated on the basis of the formula

\[ Q = 1.1 E_n - 0.8 E_d - 0.28 \sqrt{E_d E_n}\cdot \cos \theta . \]

The reaction energy \(Q\) has several values. Studies of the neutron spectrum for a thin target and low deuteron energy

(up to 1 MeV)\(^{35,36,37}\) found four groups of neutrons corresponding to four values of \(Q\). Later experiments\(^{38,39}\) with deuterons up to 1.6 MeV reveal a fifth group and give the following five values of the reaction energy \(Q\): 4.33; 3.70; 2.19; 0.73 and \(-0.74\) MeV. The first value of \(Q\) corresponds to the ground state of the \(B^{10}\) nucleus; the others to excited states, with the following excitation energies: 0.69; 2.20; 3.66 and 5.13 MeV. The ratio of the intensities of the different neutron groups changes rather strongly with the change in the deuteron energy; therefore, on the basis of these data it is impossible to predict the spectrum of neutrons produced at deuteron energies of several MeV.

As the deuteron energy increases, one should expect the appearance of new neutron groups corresponding to negative values of \(Q\), i.e., to higher excitation levels of the final nucleus \(B^{10}\).

Table IV

Neutron energies (in MeV) produced in the reaction \(Be^9(d,n)B^{10}\) at \(E_d = 8\) MeV for various values of the reaction energy \(Q\) and emission angles \(\theta\)

\(Q\) in MeV	\(E_n\) (0°)	\(E_n\) (90°)	\(E_n\) (180°)
4.4	12.4	9.8	7.8
3.7	11.7	9.2	7.5
2.2	10.2	7.8	6.0
0.7	8.5	6.5	5.0

In any case there is no doubt that, at any deuteron energy, the neutron spectrum in the \(Be^9(d,n)B^{10}\) reaction with a thin target must be line-like and consist of four or more groups, while with a thick target it must be continuous and rather complex.

A general idea of the neutron energy and its dependence on the angle of emission for four values of \(Q\) is given by Table IV, which lists the energies of neutrons emitted at angles \(\theta = 0, 90\), and \(180^\circ\) for a deuteron energy equal to 8 MeV, calculated from the formula given above.

Considering Table IV, it is easy to understand that in the case of a thick target the separate neutron groups, being greatly broadened, overlap one another and form a continuous spectrum with a limiting energy approximately equal to \(Q + E_d\).

c) Reactions \(Li^7(d,n)2He^4\) and \(Li^7(d,n)Be^8\).

Bombardment of lithium with deuterons, as is seen from Fig. 7, gives a very good yield of neutrons and is not very widely used only because of the difficulties of preparing the target. The reaction producing neutrons in this case proceeds by two different paths, shown in the heading of the paragraph. In the first case the compound nucleus \(Be^9\) immediately splits into three particles: a neutron and two

α-particle; in the second case—into two: a neutron and \(\mathrm{Be}^8\). The reaction energy for both variants has rather large values—15.05 Mev and 14.91 Mev, respectively; therefore the neutrons are produced with very high energy. It was precisely for obtaining neutrons with energies of the order of 15–20 Mev that the bombardment of lithium by deuterons was usually employed.

The spectrum of neutrons formed in the reaction \(\mathrm{Li}+d\) depends on the path by which the reaction proceeds. In the first case, i.e. in fission immediately into three particles, the neutron spectrum must be continuous. Indeed, the laws of conservation of energy and momentum can be satisfied by assigning to one of the three particles any energy from zero up to some maximum. Consequently, the formation is possible of both a neutron and an α-particle with any energy, and as a result of many different cases both kinds of particles are formed with a continuous distribution in energy.

In the second case the splitting of the compound nucleus occurs into two particles: a neutron and a \(\mathrm{Be}^8\) nucleus. It is true that the \(\mathrm{Be}^8\) nucleus is unstable and rather rapidly decays in turn into two α-particles. But if this decay occurs after the splitting of \(\mathrm{Be}^9\), then the neutron spectrum cannot be continuous at a definite deuteron energy. As in all the preceding cases, here the neutron energy is uniquely determined by the deuteron energy, the angle of emission, and the reaction energy \(Q\).

If \(Q\) has several values, the neutron spectrum must be a line spectrum; if, however, there is only one value of \(Q\), then the neutron energy also has one value for each deuteron energy and definite angle of emission.

Experiments have shown that in reality both reaction paths occur. The continuous spectrum of α-particles formed in the reaction \(\mathrm{Li}^7(d,n)2\mathrm{He}^4\) was observed in 1933 by Oliphant, Kinsey, and Rutherford\(^{40}\). The neutron spectrum has been studied by a number of investigators\(^{41,42,74,75}\), and a continuous neutron spectrum was found, extending from very small energies up to 14 Mev, together with a homogeneous group of neutrons with an energy of about 14 Mev, corresponding to the second variant of the reaction. In Fig. 14 is shown the spectrum of forward-directed neutrons obtained\(^{42}\) in the bombardment of a thin lithium target by deuterons with an energy of 0.7 Mev. With increasing deuteron energy, the energy of the homogeneous group of neutrons increases, but its relative intensity decreases. Neutrons with an energy of 20 Mev can be obtained at a deuteron energy of about 6 Mev. The formula for calculating the neutron energy in the present case can be represented in the form

\[ Q=\frac{9}{8}E_n-\frac{3}{4}E_d-\frac{1}{4}\sqrt{2E_dE_n}\cos\theta . \]

d) The Reaction \(T^3(d,n)He^4\)

In 1948–1949 a whole series of papers\(^{3,44,45,46}\) was published devoted to nuclear reactions with the superheavy isotope of hydrogen—tritium. Tritium is radioactive and therefore is not found in the composition of natural hydrogen. It decays with a period of about 12 years, emitting electrons with a maximum energy of about 19 kev. At the present time tritium can be obtained in various nuclear reactions\(^{43}\), for example in the reaction \(Li^6(n,\alpha)T^3\), in quite appreciable quantities.

Among the nuclear reactions between tritium and other particles, the reaction \(T^3(d,n)He^4\) proved especially promising. It can

Fig. 14. Neutron spectrum in the \(Li^7(d,n)\) reaction at angle \(\theta=0\) for \(E_d=0.7\) Mev.

be carried out, and in fact has been carried out, both by bombarding deuterium with tritium ions and, conversely, by bombarding tritium with deuterons. In the first case, to obtain ions, gaseous tritium, like other gases, was introduced into an ion source, from which the ions entered the accelerating apparatus.

As a target in bombardment by deuterons, tritium is used either in the form of a gas enclosed in a special chamber separated by a thin foil from the vacuum space of the accelerating apparatus, or in the form of a layer adsorbed by a metallic plate. Metals such as tantalum or zirconium\(^{3}\) absorb hydrogen, and consequently tritium as well, very well and retain it under bombardment. A gas target is, of course, more advantageous with respect to the neutron yield, since in it the tritium concentration is higher; but at low energy of the bombarding deuterons their slowing in the foil may lead to a decrease in the yield and thereby deprive the gas target of its advantages.

The technique of using a tritium gas target has certain special features connected with the necessity of economical use of small quantities of the costly gas. One of the installations adapted for this purpose is described in \(^{44}\). A solid target with adsorbed tritium is simpler and more convenient to handle. Concerning the neutron yield from thick targets, the following data are given \(^{3}\): a gas target at a deuteron energy of \(600\ \text{keV}\) gives \(5\cdot 10^{8}\) neutrons per microcoulomb of deuterons; a zirconium target at an energy of \(200\ \text{keV}\) gives \(10^{8}\) neutrons per microcoulomb. The use of a target of superheavy ice \(\mathrm{T_2O}\) proves practically inconvenient, since the target is rapidly contaminated by ordinary water, always present in the vapors in the various elements of the apparatus.

The reaction \(\mathrm{T^3}(d,n)\mathrm{He^4}\) is strongly exothermic, since as a result of it, at the expense of the two weakly bound nuclei \(\mathrm{T^3}\) and \(\mathrm{D^2}\), the very strongly bound nucleus \(\mathrm{He^4}\) is obtained. The reaction energy is \(Q=17.6\ \text{MeV}\). Owing to this, the reaction is in principle possible at an arbitrarily small deuteron (or tritium ion) energy. Indeed, experiments have shown that an appreciable neutron yield is obtained already at \(E_d\) of the order of several \(\text{keV}\). In the first experiments \(^{45}\), tritium ions were accelerated and a strong resonance at low energies was discovered. Subsequently \(^{46}\), measurements were carried out with deuterons as the bombarding particles and over a wider energy interval. The method of measuring the reaction cross section proves in this case to be rather simple and reliable, owing to the fact that, along with the neutron, a fairly fast \(\alpha\)-particle is formed in the reaction (with an energy of the order of \(3\)–\(4\ \text{MeV}\)). It is obvious that the total number of \(\alpha\)-particles flying out of a thin target is equal to the total number of neutrons. Meanwhile, the absolute number of \(\alpha\)-particles can be counted by means of a proportional counter or an ionization chamber with a high accuracy, which is very difficult to attain by registering directly the flux of the neutrons themselves. The results of measurements of the reaction cross section are systematized in the review of Hanson, Taschek, and Williams \(^{3}\). The dependence of the cross section on the neutron energy \(E_d\) is best expressed by the following formula:

\[ \sigma=\frac{58}{E_d}\cdot \frac{e^{-\frac{1.72}{\sqrt{E_d}}}} {1+\frac{(E_d-0.096)^2}{0.174^2}} . \]

(\(E_d\) is in \(\text{MeV}\), \(\sigma\) is in barns.)

The formula is the product of two factors, of which one

\[ \left(E_d^{-1/2} e^{-1.72\cdot E_d^{-1/2}}\right) \]

determines the penetrability of the potential barrier, while the second determines the resonance dependence of the cross section on

energy, which follows from the well-known Breit–Wigner dispersion formula. The constants in this formula were chosen from the condition of best agreement with the experimental data. The constant \(0.096\ \mathrm{MeV}=96\ \mathrm{keV}\) corresponds to the resonance value of the deuteron energy, and \(0.174\ \mathrm{MeV}\) to the width of the resonance.

In Fig. 15 the dependence of \(\sigma\) on \(E_d\) is presented graphically. The points correspond to the experimental data, and the solid curve to the formula given.

As is seen from the formula and Fig. 15, the resonance is observed at a very small deuteron energy, while the cross section reaches a rather large value at the maximum—more than 4 barns. At the same deuteron energy the cross section of the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) is at least 100 times smaller, and for \((d,n)\) reactions with other elements still more so. Hence it is evident that the reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) is a very powerful source of neutrons, and a large yield can already be obtained at deuteron energies of the order of \(100\ \mathrm{keV}\); such deuterons can be produced with the aid of a standard x-ray apparatus. Apparently, such a neutron source has a great future. There is no doubt that it will become very widespread as soon as the active, but fairly long-lived, isotope of hydrogen—tritium—becomes available to a large number of laboratories.

Fig. 15. Cross section of the reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\).

The energy of the neutrons produced in the reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) can be calculated on the basis of the formula

\[ Q=17.6\ \mathrm{MeV}=\frac{5}{4}E_n-\frac{1}{2}E_d-\frac{1}{2}\sqrt{2E_nE_d}\cdot\cos\theta . \]

Since the reaction energy \(Q\) has the value \(17.6\ \mathrm{MeV}\), the neutron energy is rather large. Obviously, for \(E_d\) close to zero,

\[ E_n=\frac{4}{5}Q=14\ \mathrm{MeV}, \]

and with increasing \(E_d\) the neutron energy also increases, distribu-

spreading in the forward hemisphere, especially at small angles \(\theta\).

Figure 16 shows the dependence of the neutron energy on the angle \(\theta_n\) for different values of \(E_d\). As can be seen from the figure, with deuteron energies of about \(3.5\ \mathrm{MeV}\) in the reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) one can obtain neutrons with energies from 12 to \(20\ \mathrm{MeV}\). Since the final nucleus \(\mathrm{He}^4\) has no excited levels with energy below

Fig. 16. Neutron energy in the reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\).

\(20\ \mathrm{MeV}\), the reaction energy \(Q\) has the single value \(17.6\ \mathrm{MeV}\), and the neutron energy depends only on the angle \(\theta\). In the case of a thin target, a monochromatic group of neutrons with high energy propagates in each direction. However, at a small deuteron energy, even a thick target does not lead to strong violations of neutron monochromaticity, since their energy is large.

The reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) is a source of the fastest monochromatic neutrons. The possibility of obtaining monochromatic neutrons with energies above \(20\ \mathrm{MeV}\) in this reaction depends on the presence or absence of excited states in the final nucleus \(\mathrm{He}^4\). In the cited works there are

indication^3 of the presence of an excited state with an energy above 29 MeV. If such a level actually exists and the transition of the final nucleus He\(^4\) in the reaction T\(^3(d,n)\)He\(^4\) to this level is possible, then at deuteron energies above 3–4 MeV, along with the main group of neutrons, a considerably slower one should appear. Nevertheless, the reaction does not lose its significance as a source of very fast monochromatic neutrons,

Fig. 17. Dependence of the neutron emission angle \(\theta_n\) on the emission angle of the \(\alpha\)-particle \(\theta_\alpha\) in the reaction T\(^3(d,n)\)He\(^4\).

since a possible second group, owing to the very large difference in energy, can be separated by filters, and in some experiments may not be taken into account at all.

The angular distribution of neutrons can be established on the basis of the results of measurements^43,3 of the angular distribution of \(\alpha\)-particles. It is obvious that, for each value of the deuteron energy \(E_d\), there exists a one-to-one relation between the neutron emission angle \(\theta_n\) and the emission angle of the \(\alpha\)-particle \(\theta_\alpha\). In Fig. 17 the group of descending curves represents this relation. The different curves correspond to different values of the deuteron energy, indicated in the figure.

(in Mev). Since the reaction energy is large, both kinds of particles can be emitted at any angles from 0 to 180°. The smaller the angle of emission of the α-particle, the larger the angle of emission of the neutron, and conversely.

In the same figure the rising curves show the ratio \(k_{n\alpha}\)—the neutron flux density at the angle \(\theta_n\) to the α-particle flux density at the angle \(\theta_\alpha\) corresponding to \(\theta_n\) in the sense of the relation established between these angles by the preceding curves of Fig. 17.

In other words,

\[ k_{n\alpha}=\frac{\sigma_n(\theta_n)}{\sigma_\alpha(\theta_\alpha)} \]

is the ratio of the differential cross section of the reaction with formation of a neutron emitted at the angle \(\theta_n\) to the differential cross section of the reaction with formation of an α-particle emitted at the corresponding angle \(\theta_\alpha\). With the aid of the curves of Fig. 17 one can find the angular distribution of neutrons on the basis of the measured angular distribution of α-particles, which is shown in Fig. 18.

Fig. 18. Angular distribution of α-particles in the reaction \(T^3(d,n)He^4\).

The upper half of Fig. 18 shows the angular distribution of α-particles in the laboratory coordinate system. The angles \(\theta_n\) and \(\theta_\alpha\) in Fig. 17 correspond to the same coordinate system. Although α-particles are emitted with greater probability at small angles than at large ones, and large \(\theta_n\) correspond to small \(\theta_\alpha\), the angular distribution of neutrons also turns out to be condensed more in the region of small angles, since the ratio \(k_{n\alpha}\) in Fig. 17 increases with increasing \(\theta_\alpha\). To construct the angular distribution of neutrons using the upper half of Fig. 18, one must, for each value of the angle \(\theta_\alpha\) shown on the abscissa, find the corresponding value of \(\theta_n\) from Fig. 17, and multiply the ordinate of the curve of Fig. 18 by the corresponding value of the ratio \(k_{n\alpha}\). The number obtained in this way will be equal to the differential cross section for formation of a neutron emitted at the angle \(\theta_n\).

The second (lower) half of Fig. 18 depicts the angular distribution of α-particles in the center-of-mass system. At low deuteron energy the angular distribution is close to spherically symmetric. As \(E_d\) increases, the distribution becomes more complicated, and its deviations from spherical symmetry do not increase monotonically with energy. Thus, for example, at \(E_d = 1\) MeV the deviations are stronger than at \(E_d = 1.5\) MeV, and at \(E_d = 2\) MeV they are stronger than at \(E_d = 2.5\) MeV.

d) Other \((d, n)\) reactions

In addition to the reactions \((d, n)\) listed above, many others are possible and known, with various elements serving as targets. However, with respect to neutron yield, not one of these reactions can stand comparison with those listed—the most important neutron sources. Consequently, at present the other \((d, n)\) reactions can be of interest only as sources of monochromatic neutrons with energies lying in an interval that is in practice inaccessible to the principal sources. As is seen from the consideration of the preceding sources, monochromatic neutrons with continuously variable energy in the interval from 2 to 6 MeV can be obtained in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\), if deuterons with energies up to 3–4 MeV, accelerated by electrostatic generators, are used. The use of cyclotrons for this purpose proves practically inconvenient, since the beam of the deuterons themselves in a cyclotron is insufficiently monochromatic; and for varying the deuteron energy in a cyclotron there is practically only one method—moving the target along the radius inside the chamber between the dees. The operating conditions with an internal target as a neutron source in most cases prove unsatisfactory, because the possibility of bringing the measuring apparatus close to the target is limited, and the presence of massive structural elements of the cyclotron (the walls and lids of the chamber, the magnet poles, etc.) in the immediate vicinity of the target leads to unavoidable scattering of neutrons—elastic and inelastic—as a result of which the neutron spectrum becomes more complicated.

Monochromatic neutrons with energies up to 2 MeV can be obtained in the \((p,n)\) reactions described below. Thus, the neutron energy interval from very low values up to 5–6 MeV proves quite accessible to modern sources of monochromatic neutrons. The reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) gives neutrons with energies above 12 MeV. Thus the energy interval from 5–6 to 12 MeV remains inaccessible. To fill this interval it is desirable to find reactions with a \(Q\)-value having the single value of the order of 5 MeV. One such reaction is \(\mathrm{N}^{14}(d,n)\mathrm{O}^{15}\),

for which the energy is \(Q = 5.1\) MeV. No other values have been found at deuteron energies up to \(1\) MeV \({}^{47,?}\).

The neutron yield in this reaction is 2–3 times smaller than in the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) at \(E_d = 1\) MeV, and the angular distribution is more uniform. Thus, in this reaction one can obtain monochromatic neutrons with energies of \(5\) MeV and higher. However, the possibility of the appearance of slower groups at deuteron energies above \(1\) MeV is not excluded, and there are even indications \({}^{48}\) of the presence of a value \(Q = 1.1\) MeV; therefore the question of the possible interval of accessible energies requires additional special investigations.

Hanson, Taschek, and Williams \({}^{3}\) also point to the reaction \(\mathrm{C}^{12}(d,n)\mathrm{N}^{13}\), with energy \(Q = -0.26\) MeV, as a convenient source of monochromatic neutrons with energies up to \(2\) MeV. In comparison with \((p,n)\) reactions, it is more advantageous in that it has a lower threshold and a higher cross section.

5. BREAKUP OF DEUTERONS OF HIGH ENERGY

To obtain very fast neutrons it is quite natural to strive to increase the energy of the bombarding particles, in particular deuterons. However, nuclear reactions, even with very fast bombarding particles, apparently cannot yield very fast neutrons, since even at very strong excitations the nucleus emits, with greatest probability, particles with energies close to the share of the excitation energy falling on average to one particle of the nucleus; and when the number of particles in the nucleus is large this share proves to be small. In other words, the neutron emitted as a result of a nuclear reaction with a very fast bombarding particle will, with greatest probability, carry away only an insignificant fraction of the energy of the bombarding particle. From this point of view, the problem of obtaining very fast neutrons appears practically solvable only by using very light nuclei, for example deuterons, as targets.

However, experiments with very fast deuterons with energies up to \(200\) MeV, obtained on the 184-inch phasotron \({}^{49,50}\), showed that bombardment of any target leads to the formation of fast neutrons. It was found that the fast neutrons propagate forward in a rather narrow cone around the direction of the beam of bombarding deuterons, and the opening of this cone, as well as the neutron yield, depends only weakly on the target material. This phenomenon is difficult to explain from the standpoint of the usual ideas about nuclear reactions as processes passing through the obligatory stage of formation of a compound nucleus. Indeed, if bombardment led to the capture of the deuteron by a not very light nucleus with mass number, for example, not less than \(100\), then the compound nucleus would have

would be excitation energy, distributed approximately uniformly among all the particles, and could emit neutrons with small energy and practically in any directions.

Apparently, the mechanism of formation of fast neutrons as a result of bombardment of nuclei by fast deuterons is quite different. The works of Serber51, 52 and Dankov53 are devoted to discussion of this mechanism. According to the ideas developed in Serber’s works, the formation of fast neutrons upon collision of fast deuterons with a nucleus is possible either as a result of the direct knocking-out by the deuteron of one of the neutrons of the nucleus, or as a result of such a splitting of the deuteron itself upon collision with the nucleus, when the neutron is freed from the proton but is not captured by the nucleus.

The first case—the knocking-out of a neutron from the nucleus—is possible when the neutron is located at the edge of the nucleus and, having received an impact from the deuteron, flies out practically without interacting with the remaining particles. However, the average energy transferred by a fast deuteron to a neutron, as Serber points out, cannot be very large and is approximately equal to 25 MeV; therefore very fast neutrons cannot appear in this way even when deuterons collide with neutrons located at the edge of the nucleus. If, however, the deuteron collides with a neutron that subsequently must still pass through the nucleus, then in collisions with other particles of the nucleus the neutron will lose even this relatively small energy, and thus the result of such a collision is reduced to the formation of a strongly excited nucleus, which must then decay in the usual manner, emitting neutrons or other particles of small energy.

As a consequence of this, apparently, the main mechanism for the formation of fast neutrons in the collision of fast deuterons with nuclei is the breakup of the deuteron itself. Since the binding of the particles in the deuteron is very weak, the rupture of this bond can occur even at a small kinetic energy of the deuteron and is all the more probable when the deuteron energy is about 200 MeV. The cause of the breakup may be either the electrostatic or the nuclear interaction of the deuteron with the nucleus. In other words, breakup may occur both in the Coulomb field of the nucleus and in the field of nuclear forces.

The cross section for deuteron breakup in the Coulomb field of the nucleus was calculated by Dankov53, and in the field of nuclear forces by Serber51. A comparison of the results obtained shows that breakup in the Coulomb field proves to be considerably less probable than in the nuclear field, even in bombardment of the heaviest nuclei. For example, for a uranium nucleus the breakup cross section in the Coulomb field is approximately 25% of the breakup cross section in the nuclear field. As the charge of the nucleus decreases, the probability of breakup in the Coulomb field decreases much more rapidly than in the nuclear field.

Experimental data are in complete agreement with these theoretical conclusions. The energy spectrum and angular distribution of fast neutrons agree well with the results of Serber’s calculations, but do not agree with the results of Dankov’s calculations, which take into account the Coulomb interaction.

Of all possible cases of the interaction of a deuteron with a nucleus by means of nuclear forces, Serber takes into account only those cases in which the deuteron strikes the edge of the nucleus. The deuteron may be represented as a system similar to a dumbbell, consisting of two particles—a proton and a neutron—located at a distance \(r\), which, generally speaking, changes with time, but during the collision of a very fast deuteron with the nucleus may be regarded as constant. A collision in which only one of the particles enters the region of the effective cross section of the nucleus, while the other passes outside this region, can evidently lead to the breakup of the deuteron with the liberation of one of the particles (either the proton or the neutron). The particle that has collided with the nucleus will either be captured by it or strongly scattered. In this case the second particle, bound to the first rather weakly (binding energy \(2.18\) MeV), will pass by the nucleus, experiencing a relatively small acceleration at the moment of breakup, since its kinetic energy is much greater than \(2.18\) MeV.

If the nucleus is represented as an opaque sphere of radius \(R\), then the effective cross section \(\sigma_{\mathrm{пр}}\) of such collisions will be equal to the area of a circular layer \(2\pi Rl\), where \(l\) is the value, averaged over all possible orientations, of the projection of the mean radius of the deuteron \(\bar r\) onto a plane perpendicular to the initial trajectory of the deuteron. It is easy to calculate that \(l = \frac{1}{2}\bar r\), and consequently, \(\sigma_{\mathrm{пр}} = \pi R\bar r\). If the Coulomb interaction of the deuteron with the nucleus is not taken into account, then one may expect that the number of cases of neutron liberation as a result of breakup will be equal to the number of cases of proton liberation; therefore the cross section for the formation of a fast neutron \(\sigma_n\) will be half as large as \(\sigma_{\mathrm{пр}}\), i.e.

\[ \sigma_n = \frac{1}{2}\pi R\bar r, \]

The Coulomb repulsion acting on the proton should, generally speaking, lead to an increase in the probability of finding the proton at the moment of collision farther from the nucleus and, consequently, to a relative decrease in the probability of neutron liberation as a result of breakup. However, the energy of the Coulomb interaction is small in comparison with the kinetic energy of the particles if the deuteron energy is about \(200\) MeV; therefore the probability of formation of a free neutron is approximately equal to the probability of formation of a free proton.

Taking \(\bar r = 2.1 \cdot 10^{-3}\,\text{cm}\) and \(R = 1.5 \cdot A^{\frac{1}{3}} \cdot 10^{13}\,\text{cm}\), Serber obtains the following expression for the effective cross section for the formation of a fast neutron as a result of the breakup of a deuteron at the edge of a nucleus with number of particles \(A\):

\[ \sigma_{\mathrm n}=\frac{1}{2}\,\pi R\bar r =5\cdot A^{\frac{1}{3}}\cdot 10^{-26}\,\text{cm}^2. \]

Thus, the cross section depends rather weakly on the mass of the target nucleus. Thus, for a Be nucleus it is equal to \(0.1\) barn, and for a U nucleus, \(0.3\) barn. This is in complete agreement with the experimentally established weak dependence of the yield of fast neutrons on the target material.

Knowing the value of the cross section \(\sigma_{\mathrm n}\), it is easy to calculate the yield of fast neutrons from any target. Thus, for example, the yield from a beryllium target \(1.25\,\text{cm}\) thick, in which deuterons with initial energy \(190\,\text{MeV}\) lose \(20\,\text{MeV}\) to ionization, amounts to \(2\%\), i.e. one neutron for every 50 deuterons.

The question of the energy of the breakup neutrons is resolved in the following way. If the neutron were very weakly bound in the deuteron, then at the moment of breakup its kinetic energy would not change and would remain equal to that fraction of the deuteron energy which belongs to the neutron, i.e. approximately (to within the difference in the neutron and proton masses) half the deuteron energy; in this case monochromatic neutrons with energy

\[ E_{\mathrm n}=\frac{1}{2}E_{\mathrm d}. \]

would be obtained. However, the binding of the deuteron cannot be neglected. The influence of the binding on the spectrum of breakup neutrons can be taken into account by considering the motion of the neutron relative to the center of mass of the deuteron. If at the moment of breakup the momentum of the neutron in the coordinates of the deuteron center of mass was \(p_{\mathrm n}\), then the momentum in laboratory coordinates after breakup will be equal to the sum of \(p_{\mathrm n}\) and the momentum of the center of mass \(p_0\). Owing to the motion of the neutron inside the deuteron, its energy after breakup may turn out to be either greater or less than \(\frac{1}{2}E_{\mathrm d}\), depending on whether the direction of \(p_{\mathrm n}\) coincides with the direction of \(p_0\) or, conversely, their directions are opposite. Since the directions of \(p_{\mathrm n}\) are arbitrary and their absolute value has various values, the spectrum of breakup neutrons must be continuous. In other words, in addition to neutrons with the average energy \(\frac{1}{2}E_{\mathrm d}\), both faster and slower ones may also be formed, and the relative spread in energies should be the greater, the greater the ratio of the deuteron binding energy \(\varepsilon_{\mathrm d}\) to its kinetic energy \(E_{\mathrm d}\). Using the wave function of the deuteron to find \(p_{\mathrm n}\), Serber comes to the conclusion that the neutron spectrum must be represented by a bell-shaped curve, hav—

maximum at \(E_n = \dfrac{1}{2} E_d\) and a half-width \(\Delta E_n = 1.533 \sqrt{E_d \varepsilon_d}\).

For \(E_d = 190\) MeV the half-width is \(\Delta E = 31\) MeV.

The shape of the spectrum is represented by the following formula, which gives the effective cross section \(\sigma\) as a function of the neutron energy:

\[ d\sigma = \frac{1}{4}\pi R \bar r\, \frac{E_d \varepsilon_d} {\left[\left(E_n-\frac{1}{2}E_d\right)^2+E_d\varepsilon_d\right]^{3/2}} . \]

The dependence of the cross section on the neutron energy has the character of a resonance, since the formula given differs from the resonance formula only by the power \(3/2\) instead of 1 in the denominator. To obtain the total effective cross section \(\sigma\), the formula must be integrated over energy from 0 to \(E_d\).

This conclusion agrees quite well with the experimental data. It is true that direct measurement of the neutron spectrum appears rather difficult, but the neutron spectrum can be judged from the spectrum of the protons that are also produced in the process of deuteron breakup. To within the comparatively weak influence of the Coulomb field, the spectra of neutrons and protons from breakup should be identical.

The spectrum of protons produced by bombardment of a copper target 19 mm thick with deuterons of energy 190 MeV was studied\({}^{50}\) in the following way. Detectors of fast protons were placed at various distances from the target inside the phasotron chamber. The protons emerging from the target are, naturally, bent by the magnetic field of the phasotron and describe approximately circular trajectories, the radius of curvature depending on the proton energy. Owing to this, the farther the detector stood from the target, the higher the energy of the protons that could reach it. The proton energy was determined from the known distance and the value of the magnetic-field strength. The detectors were graphite plates, in which the reaction \(C^{12}(p,pn)C^{11}\) produces positron-active carbon \(C^{11}\) with a half-life of 20.5 minutes. The cross section of the reaction \(C^{12}(p,pn)C^{11}\) remains practically constant for protons with energy above 50 MeV,\({}^{54}\) and therefore the activities of all the detectors served as the same measure of the intensity of the proton flux. In order that the detectors should not interfere with the main beam of accelerated deuterons and should not be irradiated by it, they were placed below the dee; consequently the protons entering them in fact traveled not along circumferences but along helical lines.

The results of the measurements are presented in Fig. 19, where along the abscissa axis is plotted the radius of curvature of the proton trajectories (lower scale) or their energy (upper scale), and along the ordinate axis—the relative intensity. The points represent the experimental

results, and the curve is the result of Serber’s calculations. As can be seen from Fig. 19, the proton spectrum is indeed continuous and, in shape, rather close to that calculated by Serber. The maximum of the spectrum corresponds to an energy value close to \(\frac{1}{2}E_d\). The mean value of \(E_d\) is below \(190\ \mathrm{MeV}\) owing to energy losses in the thickness of the target. The half-width of the spectrum is of the order of \(40\ \mathrm{MeV}\).

It follows from this that Serber’s theoretical calculations are quite suitable for determining the spectrum of breakup neutrons, and therefore they are in fact used.

Fig. 19. Proton spectrum in the breakup of deuterons with an energy of \(190\ \mathrm{MeV}\). The curve was calculated theoretically (according to Serber).

The angular distribution of fast neutrons was studied\(^ {49}\) directly by means of carbon detectors activated by fast neutrons as a result of the reaction \( \mathrm{C}^{13}(n,2n)\mathrm{C}^{11}\). The detectors were placed in various positions outside the phasotron chamber in such a way that the angle between the directions from the target to them and the direction of the deuteron beam varied over wide limits. The activity of each detector was taken to be proportional to the intensity of the neutron flux in the corresponding direction.

Figure 20 presents the results of measurements of the angular distribution for beryllium and uranium targets. The points correspond to the experimental data, and the curves to the results of Serber’s calculations.

As can be seen from the curves, the fast neutrons are emitted forward in a rather narrow cone. The intensity maximum is naturally observed in the direction of the deuteron beam. The intensity falls to one half of the maximum at an angle of only about \(5^\circ\) for the beryllium target and about \(6.5^\circ\) for the uranium

targets and to 0.1 of the maximum at angles less than \(15^\circ\) for both targets.

Calculations of the angular distribution, carried out by Serber, are analogous to the calculations of the spectrum. The angle of emission of a neutron is determined by the magnitude of the transverse component of the neutron momentum relative to the deuteron center of mass, which can also be found from the deuteron wave function. To obtain

Fig. 20. Angular distribution of neutrons in the breakup of deuterons with an energy of 190 MeV. The curves are theoretical (according to Serber).

the width of the angular distribution, Serber gives the following expression:

\[ \Delta \theta = 1.601\theta_0 = 1.601 \sqrt{\frac{\varepsilon_d}{E_d} \left(1-\frac{E_d}{8Mc^2}\right)}, \]

where \(M\) is the mass of the proton; the remaining notation is as before. The form of the angular distribution is given by the expression

\[ d\sigma = \frac{\overline{R} r}{\pi(1+\zeta^2)^{3/2}} \left\{ 1-\frac{1}{2\zeta^2} \left[ \frac{(1+\zeta^2)}{\operatorname{tg}\zeta} -\zeta \right] \right\} d\Omega, \]

where \(\zeta=\dfrac{\theta}{\theta_0}\), \(\theta\) is the angle of neutron emission, \(\theta_0\) is the angle determined by the preceding expression for \(\Delta\theta\), and \(\Omega\) is the solid angle.

The good agreement of Serber’s computational results with experiment makes it possible to use his formulas also for finding the angular distribution of neutrons. It should be noted, however, that in Serber’s formulas somewhat different factors appear depending on whether the nucleus is assumed to be opaque to deuterons or partially transparent. The accuracy of the experimental results given above is insufficient to give preference to one of these two theoretical assumptions. The available data on the scattering of fast neutrons indicate that, for particles with energies of the order of 100 MeV and higher, the nucleus should be regarded as partially transparent. For a completely transparent nucleus, Serber gives the following formulas:

Neutron spectrum:

\[ d\sigma = \frac{\sqrt{E_d \varepsilon_d}} {\pi \left[\left(E_n-\frac{1}{2}E_d\right)^2 - E_d \varepsilon_d\right]} \, dE_n . \]

Half-width of the spectrum:

\[ \Delta E_n = 2\sqrt{E_d \varepsilon_d}. \]

Angular distribution:

\[ d\sigma = \frac{R\bar r} {\pi(1+\zeta^2)^{1/2}} \, d\Omega . \]

Half-width of the angular distribution:

\[ \Delta \theta = 1.533\,\theta_0 . \]

The spectral distribution proves to be broader for a transparent nucleus, while the angular distribution, on the contrary, is broader for an opaque one. Moreover, the difference in the angular distribution is small, as is already evident from comparing the coefficients in the expression for \(\Delta \theta\) (1.601 for an opaque nucleus, 1.533 for a transparent one).

Thus, the experimental results obtained in bombarding various targets with deuterons of energy 190 MeV, and the theoretical formulas of Serber that agree with them, establish that bombardment of nuclei by very fast deuterons gives an intense beam of fast neutrons directed forward in a narrow cone whose angular width is the smaller the greater the deuteron energy. The neutron energy is, on average, equal to half the deuteron energy, but fluctuates about this value within rather wide limits; consequently, the neutron spectrum is continuous.

Besides stripping neutrons, bombardment of nuclei by fast deuterons must undoubtedly give slower neu-

neutrons emitted by highly excited nuclei. As a result of each collision with a deuteron, including a collision leading to the breakup of the deuteron, the nucleus should be strongly excited and emit neutrons or other particles. The angular distribution of such neutrons should differ by a significantly smaller forward directionality, and their total number may even exceed the number of breakup neutrons.

Complex disintegrations of nuclei with the emission of one or several charged particles under bombardment by fast deuterons have been observed. In these observations neutrons were not registered, but this does not indicate their absence. On the contrary, there is no reason to doubt that they are also emitted by nuclei.

In the mentioned measurements of the angular distribution of breakup neutrons, neutrons emitted by excited nuclei could have remained unnoticed, since the reaction \(C^{12}(n, 2n)C^{11}\) in the detector has a high threshold (above \(20\) MeV) and is excited only by very fast breakup neutrons.

6. REACTIONS \((p, n)\)

The reaction \((p, n)\) amounts to the replacement, in the bombarded nucleus, of one neutron by a proton. The final nucleus is an isobar of the initial, bombarded one. Stable isobars, as is known, are very rare; therefore one of the nuclei—the bombarded or the final one—must be radioactive.

If the bombarded nucleus is stable, then the final nucleus proves to be radioactive. As a rule, the radioactivity of the final nucleus is positron activity (or \(K\)-capture); therefore, as a result of the decay of the final nucleus, it again spontaneously transforms into the initial bombarded nucleus. Since spontaneous radioactive decay cannot be an endothermic process, the reverse process [i.e. the reaction \((p, n)\)] must be endothermic. Thus, if the bombarded nucleus is stable, the reaction \((p, n)\) must be endothermic and have a threshold.

Like any endothermic reaction, the reaction \((p, n)\), with respect to neutron yield, is less favorable than exothermic reactions. Nevertheless, the use of the reaction \((p, n)\) as a neutron source is practiced fairly often, especially in recent times.

The reason for the broad use of the reaction \((p, n)\) as a neutron source is the possibility of obtaining monochromatic neutrons whose energy can conveniently be varied over wide limits. In particular, in this reaction it is possible to obtain monochromatic neutrons with very small energies, of the order of several keV.

As a source of monochromatic neutrons the reaction \((p, n)\) is used most often on accelerators of the generator type.

Van de Graaff generator. The technique of monochromatizing a proton beam, and of controlling, stabilizing, and precisely measuring the accelerating voltage, has recently been very well developed in a number of laboratories. The use of very thin targets in combination with this technique makes it possible to obtain very well monochromatized neutron beams of precisely known energy. Moreover, it is possible to vary smoothly the energy of the protons and, consequently, of the neutrons. All this makes the Van de Graaff generator an almost ideal source of monochromatic neutrons, permitting detailed investigations of the dependence of the cross sections for the interaction of neutrons with nuclei on energy. Examples of investigations of this type are the measurements of the total cross section of aluminum for neutrons with energies from 10 to 1000 kev, in which more than ten resonance maxima were found⁵⁵, and measurements of the scattering cross section of sulfur for neutrons with energies from 16 to 250 kev, which for the first time convincingly demonstrated the interference of potential and resonance scattering, leading to a characteristic dependence of the cross section on energy⁵⁶.

The most widely used reaction of this type is the reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \).

a) The reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \)

The bombardment of lithium by protons, as is known, gave the historically first nuclear reaction induced by artificially accelerated particles. This was the reaction \( \mathrm{Li}^7(p,\alpha)\mathrm{He}^4 \), which has a positive energy \(Q\) of the order of 17 Mev, begins already at very small proton energies of the order of 10 kev, and is accompanied by gamma rays with an energy of about 17 Mev.

The reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \), like all \((p,n)\) reactions, is endothermic, with a threshold \( \Pi = 1.882 \) Mev and energy \( Q = -1.647 \) Mev. The energy of the neutrons produced in the reaction is, in a rough approximation, equal to \( E_p - \Pi \).

To obtain monochromatic neutrons, which is the principal task of the \( \mathrm{Li}^7(p,n) \) source, in addition to monochromatizing the proton beam—which is achieved to a high degree by applying magnetic analysis at the exit of the accelerating tube—it is necessary to use a very thin lithium target. In this case it is more convenient to express the target thickness not in grams per \(1\ \mathrm{cm}^2\), as usual, but directly by the number determining the average energy loss of a proton in the target thickness. This number characterizes the degree of monochromaticity of the protons entering the reaction and the related degree of monochromaticity of the neutrons. In some experiments the target thickness reached up to 1 kev, which corresponds to approximately 10 micrograms of lithium per \(1\ \mathrm{cm}^2\).

The target is usually prepared by evaporating metallic lithium in vacuum onto a backing of a heavy metal, for example—

tantalum$^{3}$. The thickness of the backing is in principle immaterial, since in it protons with energies of several MeV do not cause any nuclear transformations because of the great height of the potential barrier of heavy nuclei. The choice of the backing thickness is dictated by the conditions of weighing, which is necessary for determining the thickness of the lithium layer, and by the conditions of cooling the target. In addition, it is desirable to design the target so that in its immediate vicinity there are no massive parts capable of strongly scattering neutrons and thereby disturbing not only their angular distribution, but also their monochromaticity.

Fig. 21. Neutron-yield curves in the reaction Li$^{7}$ (p, n) Be$^{7}$ at the angle $\theta = 0$, used to determine the target thickness.

Determining the target thickness by weighing in vacuum gives satisfactory results only for not very small thicknesses. The difficulty of determining small thicknesses is connected above all with the inevitable contamination of the target. There exists, however, a direct and quite convenient method of measuring the target thickness in kilovolts. It is based on the fact that when $E_p$, close to the reaction threshold, is used, the neutrons propagate only forward; therefore the intensity of the neutron beam at the angle $\theta = 0$ increases very sharply at the threshold and reaches a rather distinct maximum at a proton energy exceeding the threshold by an amount numerically equal to the target thickness in energy measure. In Fig. 21 curves are given illustrating this method of measuring the thickness. The thickness is determined as the difference between the proton energy corresponding to the maximum of the curve and the energy corresponding to its

beginning (threshold). Three curves refer to two different target thicknesses—9 keV (curve 1) and 2 keV (curve 2—for a freshly prepared target, curve 3—for the same target after 4 weeks of use). A comparison of curves 2 and 3 illustrates the accumulation of contamination with time.

It should be noted that the maximum of the forward neutron yield shown in Fig. 21 is not at all connected with any resonance dependence of the reaction cross section on energy, but arises

Fig. 22. Full cross section of the reaction Li\(^7\)(p, n)Be\(^7\).

only as a consequence of the features of the angular distribution of neutrons in endothermic reactions near threshold, discussed above.

Since metallic lithium is chemically very active, the preparation of the target and its installation in the working position must be carried out in vacuum or in an atmosphere of a noble gas.

A fairly large number of published works have been devoted to the study of the reaction Li\(^7\)(p, n)Be\(^7\). The principal results obtained approximately by the middle of 1949 are briefly set forth in the review by Hanson, Taschek, and Williams,\(^3\) from which we borrow most of the following information.

The full reaction cross section was measured\(^57\) by comparing the neutron flux from the target with the flux produced by a Ra + Be source calibrated to an accuracy of 5%.

For comparison, the target and the Ra source were placed alternately in a large tank with a solution of MnSO$_4$ in water. The solution was irradiated with neutrons each time for several hours, then thoroughly stirred, after which a standard sample was taken from it, whose activity under specified conditions was measured with a beta counter.

By this method the total neutron flux is recorded, and the result does not depend on the spectrum if the volume of the tank with the solution is sufficiently large. But measurements with a tank prove to be

Fig. 23. Neutron yield in the Li$^7$(p, n)Be$^7$ reaction at angle $\theta = 0$ from a target of thickness 40 keV. The dashed curve is the neutron energy.

rather lengthy and inconvenient, since the activity of the solution decays with a period of 2.5 hours (Mn$^{56}$), and to begin a new measurement before the activity from the preceding one has decayed is inconvenient—this leads to a decrease in the accuracy of the measurements. Therefore, in fact, comparison by manganese activity was carried out only for two values of the proton energy, while the positions of the remaining points of the curve were determined from the gamma activity of Be$^7$ accumulated in the target as a result of each irradiation.

The results of measurements of the total cross section of the Li$^7$(p, n)Be$^7$ reaction are shown in Fig. 22. The cross section rises sharply beyond the threshold, reaches a value of 0.24 barn at an energy $E_p = 1.92$ MeV, then remains constant up to $E_p = 2.05$ MeV, and, finally, forms

Fig. 24. Yield and energy of neutrons in the reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \) at angles \(\theta = 120^\circ\) and \(\theta = 135^\circ\).

Fig. 25. Angular distribution of neutrons in the reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \) in laboratory coordinates.

a broad resonance line with a maximum in the region of 2.23 MeV, where it reaches 0.5 barn.

The differential cross section, proportional to the intensity of the neutron flux at some emission angle \(\theta\), varies quite differently from the total one, especially at proton energies close to the threshold, again owing to the peculiarities of the angular distribution in endothermic reactions. Figure 23 shows the curve of neutron yield at the angle \(\theta = 0\) (forward), obtained with a target 40 keV thick (at \(E_p = 1.9\) MeV). The first maximum, as was already noted, is not connected with the resonance; the second corresponds to the resonance and here is expressed much more sharply than on the curve of the total cross section. The yield at large angles (120 and 135°) is shown by the two upper curves in Fig. 24. In contrast to the preceding curve, both of these curves do not have the first maximum, as was to be expected, and the resonance maximum proves to be considerably more smeared out.

The angular distribution of neutrons for various \(E_p\) is shown in Fig. 25. As is seen from the curves, it is rather complex and varies nonmonotonically in the resonance region. Taschek and Hemmendinger, analyzing the angular distribution, come to the conclusion that, in center-of-mass coordinates, it may be represented in the form of an expansion

\[ \sigma(\theta_0, E_p) = \]

\[ = A_0 P_0 + A_1 P_1 + A_2 P_2 \ldots, \]

where \(P_l\) is the Legendre polynomial of order \(l\) in \(\cos \theta_0\), \(\theta_0\) is the neutron emission angle in center-of-mass coordinates, and \(A_l\) is an expansion coefficient depending on the proton energy. Graphs of the coefficients \(A_0, A_1, A_2\) are presented in Fig. 26. The coefficient \(A_1\) for the first-order polynomial, as is seen from the figure, changes sign in passing through the resonance; the others always remain positive. The analysis applies to the proton-energy interval from the threshold to 2.55 MeV.

Fig. 26. Coefficients in the expansion of the angular distribution of neutrons in the reaction \(\mathrm{Li}^7(p,n)\) in a series of spherical functions as functions of the proton energy.

Fig. 27. Nomogram for determining the neutron energy in the reaction \(\mathrm{Li}^{7}(p,n)\mathrm{Be}^{7}\).

The energy of the neutrons produced in the reaction \( \mathrm{Li}^7(p,n)\mathrm{Be}^7 \) can be calculated on the basis of the formula

\[ Q=-1.647\ \text{MeV}=\frac{8}{7}E_n-\frac{6}{7}E_p-\frac{2}{7}\sqrt{E_nE_p}\cos\theta . \]

In particular, at the angle \(\theta=90^\circ\),

\[ E_n=\frac{3}{4}E_p-1.882\ \text{MeV}. \]

Fig. 27 shows a nomogram that makes it possible to find the neutron energy for any values of \(E_p\) (from threshold to \(4\ \text{MeV}\)) and \(\theta\) (from \(0\) to \(180^\circ\)). Let us consider the method of using the diagram with one example. Suppose \(E_p=2500\ \text{keV}\) and it is necessary to find \(E_n\) for the angle \(\theta=30^\circ\). We find the value \(2500\ \text{keV}\) on the \(E_p\) scale and follow the corresponding dashed circle to the point of its intersection with the solid ray \(30^\circ\). From the point of intersection we drop an arc parallel to the solid circumferences to the intersection with the \(E_n\) axis (the lower horizontal axis). The point of intersection gives the value of \(E_n\) on the scale on the \(E_n\) axis, in the present case \(750\ \text{keV}\). Conversely, one can find \(E_p\) from given \(E_n\) and \(\theta\), using the dashed rays.

Examining the diagram, it is not difficult to establish that for \(E_p<1.92\ \text{MeV}\) neutrons cannot propagate into the backward hemisphere, while each value of the angle \(\theta\) in the forward hemisphere corresponds to two values of \(E_n\), as was to be expected on the basis of the considerations set forth in Section 2,c. The energy of neutrons emitted at the angle \(\theta=0\), corresponding to the threshold value \(E_p=1.882\ \text{MeV}\), is \(29\ \text{keV}\). As \(E_p\) increases from \(1.882\) to \(1.92\ \text{MeV}\), the energy of one group of neutrons at \(\theta=0\) increases from \(29\) to \(80\ \text{keV}\), while that of the other decreases from \(29\ \text{keV}\) to zero.

Fig. 28. Energy of neutrons in the reaction \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\) as a function of the excess proton energy above threshold \(\Delta E_p=E_p-I\) for various emission angles.

Monochromatic neutrons with energy \(E_n<80\ \text{keV}\) can be obtained only at angles \(\theta>90^\circ\). In this case the energy

of protons must be greater than \(1.92\) MeV, since at smaller values of \(E_p\) there are no neutrons in the backward hemisphere.

The values of \(E_n\) for \(\theta=0\) and various \(E_p\) are shown by the dotted curve in Fig. 23, and for \(\theta=120^\circ\) and \(135^\circ\)—by the two lower curves in Fig. 24. In Fig. 28 the values of \(E_n\) are given for \(\Delta E_p=E_p-P\) (the excess of \(E_p\) above the threshold) at various values of \(\theta\).

In the review\(^3\) it is stated that attempts to establish the presence of other groups of neutrons corresponding to a reaction energy \(Q\) different from \(-1.664\) MeV led to a negative result. Subsequent works\({}^{58,59}\), however, reliably established the presence of a second group of neutrons, corresponding to the transition of the final nucleus \(\mathrm{Be}^7\) into an excited state with energy \(435\) keV at \(E_p=3.36\) and \(3.91\) MeV. Fig. 29 presents the spectrum of recoil protons produced in a photographic emulsion by neutrons from the source \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\). Along with the principal group of neutrons, a second one is undoubtedly observed as well, although weaker in intensity. Its relative intensity at \(E_p=3.91\) MeV is \(9\%\) for \(\theta=0^\circ\) and \(16\%\) for \(\theta=60^\circ\); at \(E_p=3.31\) MeV and \(\theta=0\)—\(17\%\).

Fig. 29. Neutron spectrum in the reaction \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\).

This circumstance is very vexing, since it deprives a very widespread and widely used source \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\) of its chief advantage—the monochromaticity of the neutron beam. This may not have affected the results of earlier work, since the intensity of the second group is small; but in further work its presence must be taken into account.

It is curious to note that the excited level of \(\mathrm{Be}^7\) has almost the same energy as the well-known level of the “mirror” nucleus \(\mathrm{Li}^7\) (478 keV).

b) Reaction \(\mathrm{T}^3(p,n)\mathrm{He}^3\)

Owing to the possibility of obtaining tritium in quantities sufficient for preparing targets, reactions with it may be regarded as quite feasible in practice, in particular for obtaining neutrons. Along with the strongly exothermic reaction \(\mathrm{T}^3(d,n)\mathrm{He}^4\) considered above, as a source

of neutrons is the endothermic reaction \(T^3(p,n)He^3\), whose scheme may be written in the form

\[ T^3 + H^1 \to He^3 + n - 0.764\ \text{MeV}. \]

The reaction energy is \(Q=-0.764\ \text{MeV}\), the threshold \(P=1.019\ \text{MeV}\). The reaction \(T^3(p,n)\) is of value as a source of monochromatic neutrons and is more convenient than the reaction \(Li^7(p,n)\), since it, first, has a considerably lower threshold and, consequently, can be carried out with protons of lower energy; second, it has a somewhat larger cross section and, consequently, makes it possible to obtain a greater yield of neutrons; and, finally, third, it gives monochromatic neutrons over a considerably wider energy interval.

Fig. 30. Total cross section of the reaction \(T^3(p,n)He^3\).

The technique for preparing the target, described above in the description of the reaction \(T^3(d,n)\), is also suitable in the present case. However, the effect of the target thickness on the relative monochromaticity of the neutron beam is, evidently, stronger here; therefore the use of a thick target with a layer of tritium adsorbed on a metal backing is apparently undesirable. A gas target is undoubtedly more convenient, since its thickness (in grams per \(1\ \text{cm}^2\) of target) can easily be regulated and measured very well. The stopping properties of the foil separating the gas target from the vacuum in the accelerating apparatus are not very significant, since the protons must have an energy above \(1\ \text{MeV}\). It is important to know only the absolute value of the mean loss of proton energy in this foil and to take into account the dependence of these losses on energy, since with increasing proton energy the losses will decrease. However, if the foil is not very thick, the change in the losses may prove insignificant.

The total effective cross section of the reaction\(^{60, 3}\), as a function of proton energy (from the threshold to \(2.4\ \text{MeV}\)), is shown in Fig. 30.

Near the threshold the cross section increases very sharply, as in the case of the reaction \(\mathrm{Li}^7(p,n)\), and then, in the proton-energy interval from \(1.2\) to \(2.4\) MeV, it increases almost linearly from \(0.17\) to \(0.56\) barn. The continued increase of the cross section up to an energy of \(2.4\) MeV is somewhat surprising, since the barrier height in the present case is not more than \(1\) MeV. Apparently, the increase of the cross section is due to a resonance at a proton energy near \(2.4\) MeV (possibly somewhat higher), which corresponds to the formation of an excited \(\mathrm{He}^4\) nucleus with an energy of order \(20\) MeV. In addition to this fact, the existence of an excited level of \(\mathrm{He}^4\) is indicated by observations\({}^{61}\) of fairly intense gamma radiation with an energy of about \(20\) MeV, appearing as a result of radiative capture of protons by tritium. The character of the increase of the cross section also indicates a large width of the level. A comparison of Fig. 30 and Fig. 22 shows that at proton energies up to \(2.4\) MeV, and apparently also higher, the cross section of the reaction \(\mathrm{T}^3(p,n)\) is always larger than that of the reaction \(\mathrm{Li}^7(p,n)\); consequently, the neutron yield is also higher for targets of equal stopping-power thickness, since the number of tritium nuclei at the same thickness is twice as large as the number of lithium nuclei.

Fig. 31. Neutron yield in the reaction \(\mathrm{T}^3(p,n)\mathrm{He}^3\) at the angle \(\theta = 0\) from a thin target.

The neutron yield at the angle \(\theta = 0^\circ\) as a function of the proton energy is shown in Fig. 31. Near the threshold, as in the case of the reaction \(\mathrm{Li}^7(p,n)\), a sharp maximum of the yield is formed, associated with the features of the angular distribution, but not with a resonance.

The energy of the neutrons formed in the reaction \(\mathrm{T}^3(p,n)\mathrm{He}^3\) can be calculated from the formula

\[ Q=-0.764\ \text{MeV}=\frac{4}{3}E_n-\frac{2}{3}E_p-\frac{2}{3}\sqrt{E_pE_n}\cos\theta . \]

In particular, at the angle \(\theta = 90^\circ\)

\[ E_n(90^\circ)=\frac{1}{2}E_p-0.573\ \text{MeV}. \]

The dependence of \(E_n\) on the emission angle is stronger here than in the reaction \(\mathrm{Li}^7(p,n)\), since the bombarded nucleus is considerably lighter and the velocity of the center of inertia in laboratory coordinates is greater. Figure 32 shows a nomogram for determining \(E_n\) from given

Fig. 32. Nomogram for determining the neutron energy in the reaction \(T^3(p,n)He^3\).

values of \(E_p\) and \(\theta\), analogous to the nomogram given above for the reaction \(\mathrm{Li}^7(p,n)\) (see Fig. 27).

Investigations of the reaction \(\mathrm{D}(d,n)\mathrm{He}^3\) have shown that the final nucleus \(\mathrm{He}^3\) is not formed in excited states at deuteron energies up to \(10\) MeV. It follows from this that excited states of \(\mathrm{He}^3\) do not occur in the excitation-energy interval from \(0\) to \(11\) MeV.

On this basis one may expect that the reaction \(\mathrm{T}^3(p,n)\mathrm{He}^3\), which also has the nucleus \(\mathrm{He}^3\) as the product nucleus, will have a single energy value \(Q=-0.764\) MeV for proton energies up to \(12\) MeV. Consequently, in the reaction \(\mathrm{T}^3(p,n)\), generally speaking, monochromatic neutrons with energies from \(0\) to \(10\) MeV, and possibly even higher, may be obtained.

The minimum energy of neutrons emitted forward, corresponding to the threshold value of the proton energy, is \(60\) keV. Neutrons of lower energy can be obtained only at large angles, where, however, the yield is considerably smaller.

Fig. 33. Angular distribution of neutrons in the reaction \(\mathrm{T}^3(p,n)\mathrm{He}^3\) in center-of-mass coordinates.

The angular distribution of neutrons in this reaction is distinguished by a strong concentration around the forward direction, i.e., at angles \(\theta\) close to zero. The dependence of the differential cross section on the angle \(\theta_0\) in the center-of-mass system is shown in Fig. 33.

The individual curves correspond to different proton-energy values indicated (in MeV) in the figure. The dependence of the angular distribution on the proton energy, as is seen from the figure, proves to be rather complicated.

c) Other \((p,n)\) reactions

A drawback of the \(\mathrm{Li}^7(p,n)\) and \(\mathrm{T}^3(p,n)\) reactions as sources of monochromatic neutrons, besides the relatively high threshold, is the difficulty of obtaining neutrons of low energy (of the order of several keV). In the direction of the proton beam in the reac-

in the reaction \(\mathrm{Li}^7(p,n)\) one can obtain neutrons with energies not lower than 29 kev, and in the reaction \(\mathrm{T}^3(p,n)\)—not lower than 60 kev. At angles \(\theta > 90^\circ\), generally speaking, arbitrarily slow neutrons can be obtained, but their intensity is very low. Scattering of the considerably more intense flux of neutrons directed forward, even in small details of the apparatus, may lead to substantial disturbances of the purity of the spectrum of neutrons observed at large angles. This is what creates difficulties.

In this connection, reactions \((p,n)\) with heavier nuclei are more convenient for obtaining neutrons of low energy, of the order of 10 kev and below. The minimum energy of neutrons emitted forward is, in this case, lower, and the angular distribution is more uniform. Owing to this, relatively slow neutrons can already be obtained by observation at an angle \(\theta = 0^\circ\), and observation at large angles is not complicated by scattering of neutrons flying forward to such an extent as in reactions with light elements.

As examples of such reactions, Hanson, Taschek, and Williams\(^3\) consider the reactions \(\mathrm{V}^{51}(p,n)\), \(\mathrm{Cr}^{51}\) \((Q = -1.50\ \mathrm{Mev})\), and \(\mathrm{Sc}^{45}(p,n)\mathrm{Ti}^{45}\) \((Q = -2.8\ \mathrm{Mev})\). The latter reaction gives a considerably higher yield near threshold, but the threshold itself proves to be rather large. In the review by Hanson et al.\(^3\), curves are given for the neutron yield from thick targets for these reactions at various proton energies. The yields are in general lower than for reactions with light elements; therefore, for obtaining fast neutrons the reactions are of no interest.

If one keeps in mind the possibility of using, as targets, long-lived radioactive isotopes that are at present produced artificially in sufficient quantities, then attention should be paid to the reactions \(\mathrm{Be}^{10}(p,n)\mathrm{B}^{10}\) and \(\mathrm{C}^{14}(p,n)\mathrm{N}^{14}\). The energies of these reactions are easy to calculate, knowing the limits of the \(\beta\)-spectra emitted by these isotopes (0.56 Mev and 0.15 Mev, respectively), and the mass difference of the neutron and proton (1.25 Mev). Both reactions prove to be endothermic, but the energies \(Q\) have small absolute values—\(-0.18\) Mev and \(-0.6\) Mev, respectively. The thresholds of both reactions are low (0.20 and 0.64 Mev); consequently, they can be carried out with accelerators having a relatively low accelerating voltage. With regard to the neutron spectrum, some assumptions can be made on the basis of known data on the level schemes of the final nuclei\(^ {62}\). The nucleus \(\mathrm{B}^{10}\) has several rather low-lying levels; the lowest of them is 0.411 Mev. Consequently, in the reaction \(\mathrm{Be}^{10}(p,n)\mathrm{B}^{10}\), only neutrons with energy not exceeding 0.5 Mev will be monochromatic. With further increase of the proton energy, apparently, slower groups of neutrons will appear, corresponding to transitions of \(\mathrm{B}^{10}\) into excited states.

For the nucleus \( \mathrm{N}^{14} \), levels with energies below 4 MeV are unknown. If they in fact do not exist, then the reaction \( \mathrm{C}^{14}(\mathrm{n},\mathrm{p})\mathrm{N}^{14} \) will prove suitable for obtaining monochromatic neutrons with energies from 3 keV to 4–5 MeV, and possibly even higher.

7. REACTIONS: \((\gamma,\mathrm{n})\)

The splitting of nuclei by gamma rays (photodisintegration) with the emission of a neutron (the photoneutron effect) was used rather widely for obtaining monochromatic neutrons, especially in the 1930s, when other sources of monochromatic neutrons were unknown or could not be realized because suitable accelerators were lacking. For example, a whole series of important works devoted to the study of the interaction of fast neutrons with matter was carried out with the aid of photoneutrons by A. I. Leipunskii and his collaborators (see, for example, \(^{61}\)).

At the present time, owing to the development of techniques for obtaining monochromatic neutrons with controllable energy in the reactions \((\mathrm{p},\mathrm{n})\) and \((\mathrm{d},\mathrm{n})\), photoneutron sources have lost their importance, unless one counts their possible applications for creating standards or reference samples. In addition, the reactions \((\gamma,\mathrm{n})\) attract the attention of modern investigators in themselves, because of the possibility of clarifying, with their aid, the mechanism of the interaction of gamma rays with nuclei (experiments on the angular distribution of neutrons) and determining neutron binding energies.

The effective cross sections for the interaction of gamma rays with nuclei are very small—considerably smaller than with electrons, since nuclei consist of heavy particles. The largest known value of the photodisintegration cross section is about \(1.5\cdot 10^{-27}\ \mathrm{cm}^2\), i.e., of the order of 0.001 barn. The cross sections for the interaction of gamma rays with electrons are of the order of a barn; therefore, when a beam of gamma rays enters matter, it is attenuated mainly through interaction with electrons, and it produces splitting of nuclei only very rarely. Thus, photoneutron sources turn out to be rather weak in intensity, and using even such a powerful source of gamma rays as a betatron to obtain neutrons proves inexpedient. Reactions with charged particles usually give a substantially higher neutron yield.

Photodisintegration reactions \((\gamma,\mathrm{n})\), naturally, are always endothermic, since their essence consists in tearing out of the nucleus a neutron bound in it. The reaction energy is equal to the neutron binding energy in the nucleus (with accuracy up to the sign).

The value of the reaction threshold is very close to the value of the reaction energy. The relation between them can be established on the basis of the laws of conservation of energy and momentum. If the energy of the quantum is \(h\nu\)

is equal to the threshold value: \((h\nu)_n=\Pi\), then one may write:

\[ \Pi=-Q+E_c, \]

\[ P=\frac{\Pi}{c}, \]

where \(E_c\) is the kinetic energy of the nucleus split off by the quantum, but moving in such a way that the relative velocity of the neutron and the residual nucleus is zero; \(P\) is the momentum of the same nucleus, equal to the momentum of the quantum.

Eliminating \(E_c\) and \(P\), we obtain:

\[ -Q=\Pi\left(1-\frac{\Pi}{2Mc^2}\right), \]

where \(M\) is the mass of the nucleus.

The second term in the parentheses is a small correction, and, using the fact that \(Q\) and \(\Pi\) are close in absolute value, one may replace \(\Pi\) in it by \(Q\). Then we obtain:

\[ \Pi=-Q\left(1+\frac{Q}{2Mc^2}\right). \]

The quantity \(Q\) is of the order of several MeV, while \(2Mc^2\) is of the order of several billion eV (several BeV); therefore it is obvious that the difference between \(\Pi\) and \(Q\) in absolute value is of the order of tenths of a percent, i.e. of the order of a kilovolt. In practice, almost always, the photodisintegration threshold may be taken as equal to the binding energy of the particle in the nucleus. Values of the binding energy of neutrons and other particles in various nuclei are given in Table I.

To knock a charged particle out of a nucleus it is not sufficient that the quantum energy be equal to the reaction threshold, since in that case the particle has zero velocity and cannot overcome the potential barrier. Owing to the influence of the barrier, the probability of knocking out a charged particle by a quantum is, generally speaking, less than the probability of knocking out a neutron for equal excesses of the quantum energy above the threshold.

Among stable nuclei, the lowest neutron binding energies are found in \(\mathrm{Be}^9\) (\(1.63\) MeV) and \(\mathrm{D}^2\) (\(2.18\) MeV), which are usually used as targets for obtaining photoneutrons. However, the concept of a “target” in the present case has a restricted meaning, since the layers of material bombarded by gamma rays may and should be rather thick. In fact, gamma rays pass through fairly large thicknesses of matter without noticeable attenuation. For example, to attenuate a beam of gamma rays with energy \(3\) MeV by a factor of \(e=2.7\), a layer of metallic beryllium about \(15\) cm thick is required. Obviously, the photoneutron yield is the greater, the thicker the layer of bombarded material, and for better utilization of gamma rays it is necessary

use layers of beryllium or heavy water several centimeters thick.

Usually, gamma rays from radioactive preparations are used to obtain photoneutrons. The preparation is placed at the center of the volume occupied by the beryllium or heavy water. The formation of photoneutrons occurs throughout the entire volume. The power of a given volume element as a neutron source depends on the intensity of the gamma-ray source \(I\), the cross section of the photoneutron effect \(\sigma\), the distance from the source \(r\), and the gamma-ray absorption coefficient \(\mu\). In the case of a point source of monochromatic gamma rays enclosed in a solid homogeneous block of material, the number of neutrons formed per second in a unit volume at a distance \(r\) is equal to

\[ b = I \frac{n\sigma}{4\pi r^2} e^{-\mu r}. \]

Here \(I\) is the number of quanta emitted by the source per second in all directions, and \(n\) is the number of nuclei of the substance undergoing disintegration per unit volume.

Fig. 34. Cross section of the reaction Be\(^9(\gamma, n)\).
The curve is theoretical.

The total number of neutrons formed in an infinite volume is equal to

\[ B = \int_{0}^{\infty} 4\pi r^2 b\,dr = I \frac{n\sigma}{\mu}, \]

i.e., the maximum photoneutron yield is determined by the ratio of the nuclear gamma-ray absorption coefficient \(\sigma\) to the total coefficient \(\mu\).

In a finite block of material the yield is, obviously, smaller. It can be determined for any particular case by taking the integral with a finite upper limit, which is determined by the size and shape of the block. Thus, for a spherical block of radius \(R\) we obtain:

\[ B = I \frac{n\sigma}{\mu}\left(1 - e^{-\mu R}\right). \]

The effective photodisintegration cross sections of deuterium and beryllium have recently been investigated in a large number of works. Summary data on the total cross sections are given, for example, in works devoted to theoretical calculations of the cross section \(^{64,65}\).

The cross section of deuterium for \(h\nu = 2.62\) MeV is \(14.8 \cdot 10^{-28}\ \text{cm}^2\), for \(h\nu = 2.76\) MeV—\(15.5 \cdot 10^{-28}\ \text{cm}^2\) (the mean of three values), and for \(h\nu = 6.2\) MeV \(11.6 \cdot 10^{-28}\ \text{cm}^2\).

The dependence of the beryllium cross section on \(h\nu\) is shown in Fig. 34. The points represent various experimental results; the solid curve is theoretical.

The energy of photoneutrons can in many cases be calculated with sufficient accuracy by the simple formula

\[ E_{\mathrm n}=\frac{M}{M+1}(h\nu-Q), \]

where \(M\) is the mass of the final nucleus (the neutron mass is taken as unity).

The dependence of \(E_{\mathrm n}\) on the emission angle \(\theta\) for photoneutrons is rather weak, since the momentum of the quantum transferred to the nucleus is small, and consequently the velocity of the center of inertia is small. The exact relation between \(h\nu\) and \(E_{\mathrm n}\) can be obtained from the energy and momentum equations

\[ h\nu=E_{\mathrm n}+E_r+Q, \]

\[ P_r^2=\left(\frac{h\nu}{c}\right)^2+p_{\mathrm n}^{\,2}-2p_{\mathrm n}\frac{h\nu}{c}\cos\theta . \]

Here \(E_{\mathrm n}\) and \(p_{\mathrm n}\) are the energy and momentum of the neutron, \(E_r\) and \(P_r\) are the energy and momentum of the final nucleus; the remaining notation is as before. Eliminating from these equations \(E_r=p_r^2/2M\), we obtain:

\[ E_{\mathrm n}=\frac{M}{M+1}\left[\left(1-\frac{h\nu}{2Mc^2}\right)-Q\right] +\frac{h\nu}{Mc}\sqrt{2E_{\mathrm n}}\cos\theta . \]

The third term in brackets determines the dependence of the neutron energy \(E_{\mathrm n}\) on the emission angle. Since it is in general small, one may substitute into it the approximate value

\[ E_{\mathrm n}=\frac{M}{M+1}(h\nu-Q). \]

Then the equation takes the form

\[ E_{\mathrm n}=\frac{M}{M+1}\left[\left(1-\frac{h\nu}{2Mc^2}\right)-Q\right] +\frac{h\nu}{M+1}\sqrt{\frac{2M(h\nu-Q)}{(M+1)^2}}\cos\theta . \]

Let us estimate the coefficient at \(\cos\theta\), which determines the magnitude of the correction \(\Delta E_{\mathrm n}\) to the mean neutron energy as a function of angle, for the reactions \(\mathrm D^2(\gamma,n)\) and \(\mathrm{Be}^9(\gamma,n)\), taking the quantum energy \(h\nu=2.6\) MeV.

In the first case \(M=1\), \(Q=2.2\) MeV and \(\Delta E_{\mathrm n}=25\) keV. Since the mean neutron energy is \(0.2\) MeV, the relative change of energy as a function of emission angle lies within \(\pm 12.5\%\). The neutron energy in the direction of the quantum is \(225\) keV, and in the opposite direction is \(175\) keV.

In the second case (for Be) \(M=8\), \(Q=-1.6\) MeV and \(\Delta E_{\mathrm n}=20\) keV.

The mean value \(E_{\mathrm n}=0.9\) MeV; the relative change of the energy with angle is \(\pm 2.2\%\).

Thus, the photoneutron beam proves to be insufficiently monochromatic even when monochromatic gamma rays are used, if the direction of the \(\gamma\)-rays is not selected.

The monochromaticity of photoneutrons, in addition to the angular dependence, is disturbed by yet another unavoidable circumstance. Since

to obtain an appreciable yield of photoneutrons one has to use thick layers of beryllium or heavy water; and since both of these substances are good moderators, any real source will emit, along with the primary neutrons arising directly in the reaction, also neutrons that have been slowed down. It is obvious that the greater the thickness of the layer of substance, the more slowed-down neutrons there will be. In this connection, to obtain monochromatic photoneutrons it is necessary to use thicknesses small in comparison with the mean free path of the neutrons.

In most work with photoneutrons[^63] ThC″ and RaC were used as gamma-ray sources. In the gamma-ray spectrum of ThC″ there is a single line with energy \((2.62\,\text{MeV})\), exceeding the photodisintegration threshold of Be and D. Therefore the sources \(\mathrm{ThC''}(\gamma)+\mathrm{Be}\) and \(\mathrm{ThC''}(\gamma)+\mathrm{D}\), for a small layer thickness, give approximately monochromatic neutrons.

In the spectrum of RaC′ the hardest line has energy \(2.193\,\text{MeV}\), the next one has energy \(1.76\,\text{MeV}\). Both energy values are above the photodisintegration threshold of Be, but the second is very close to it, and therefore the neutrons are produced practically only by the harder line. For photodisintegration of the deuteron the energy of this line is sufficient, but the cross section is very small; therefore neutrons \(\mathrm{RaC}(\gamma)+\mathrm{D}\) are detected with difficulty.

In 1947 Wattenberg[^66] used, for obtaining photoneutrons, a whole series of artificially radioactive elements obtained by irradiation in a pile and emitting hard gamma rays.

All the sources mentioned give a small set of photoneutron energy values. It is impossible to obtain monochromatic neutrons with smoothly varying energy so long as there are no methods for monochromatizing gamma rays of variable energy. The bremsstrahlung gamma radiation of the betatron has a continuous spectrum and can give photoneutrons with a continuous distribution in energy. But this is not of interest, since neutrons of a continuous spectrum can be obtained with a large yield in reactions with charged particles.

8. THE PILE AS A SOURCE OF NEUTRONS

It is known that the chain reaction in a uranium pile takes place because the capture of a slow neutron by a uranium nucleus causes fission of this nucleus, accompanied by the emission of secondary neutrons in an amount of about 2.5 on average per fission event[^67]. These secondary neutrons have energies of the order of one MeV. The results of measurements of the spectrum of fission neutrons, obtained by Zinn and Szilard[^68], are presented in Fig. 35. The largest number of secondary neutrons have energies lying in the interval from 0.5 to 3.5 MeV. Other measurements[^67] establish that \(1/8\) of the neutrons have energies above \(1.4\,\text{MeV}\), \(1/16\)—above ...

2.4 MeV, but there is a small number of neutrons with energies up to 11 MeV. There are indications that the spectrum of fission neutrons is similar to the spectrum of Ra + Be neutrons. Although all these indications are imprecise, they undoubtedly establish that the secondary fission neutrons are fast, with energies of the order of 1 MeV and with a continuous distribution in energy.

On colliding with the moderator nuclei, neutrons lose energy and in most cases reach thermal velocities before they are captured again. Consequently, inside the boiler, at distances from the uranium blocks comparable with the mean free path of fast neutrons in the moderator, there are both fast and slow neutrons, i.e., neutrons of all velocities, beginning with the smallest thermal velocities and ending with the maximum corresponding to an energy of the order of 11 MeV. The velocity distribution in the thermal region agrees fairly well with the Maxwell distribution

\[ N(v)\,dv = Kv^{2} e^{-\frac{v^{2}}{v_{0}^{2}}}\,dv, \]

where \(N(v)\) is the number of neutrons with velocities in the interval from \(v\) to \(v + dv\), \(K\) is a constant, and

\[ v_{0}=\sqrt{\frac{2kT}{m}} \]

is the root-mean-square velocity of thermal motion. In the region of large (nonthermal) velocities, \(N(v)\) is approximately inversely proportional to the fourth power of the velocity, i.e., to the square of the neutron energy,

\[ N(v)\,dv=\frac{K_{1}}{v^{3}}\,dv=\frac{K_{2}}{E}\,dv . \]

Fig. 35. Spectrum of fission neutrons according to Zinn and Szilard. Curve \(I\) — recoil He nuclei, curve \(II\) — recoil protons.

Such a character of the neutron spectrum in the nonthermal region can be readily understood if one takes into account that the number of neutrons with a given velocity is proportional to the lifetime, the magnitude of which is determined by the nature of the slowing-down process. The time between two collisions of a neutron with moderator nuclei is inversely proportional to the velocity. But, in addition, the absolute value of the loss of velocity is the greater, the greater the velocity itself; consequently, a neutron would on average pass through one and the same interval of velocities in the slowing-down process the faster, the greater its velocity, if the time between collisions were the same.

The two parts of the spectrum pass smoothly into one another in the intermediate region. The relation between the total numbers of neutrons in the thermal and epithermal regions, quantitatively expressed by the ratio of the constants \(K\) and \(K_1\) (or \(K_2\)), is determined by the mean lifetime of thermal neutrons. The greater it is, the more thermal neutrons there are and relatively the fewer epithermal ones.

Figure 36 shows the spectrum of neutrons emitted by the central region of the Argonne heavy-water pile used as moderator, measured\(^{69}\) with a crystal spectrometer. The points represent the results of measurements; the solid curve is the Maxwellian distribution at a temperature of \(400^\circ\) K (the actual temperature inside the pile is apparently below \(400^\circ\) K). The dashed curve corresponds to the law \(1/E\), with a suitable scale. The experimental points fit the theoretically predicted curves fairly well, although the effective temperature of the neutron spectrum in the thermal region proves to be somewhat higher than the actual one. Most of the neutrons in the pile, as is seen from the figure, have thermal velocities. It should be noted that the number of recoils plotted on the ordinate axis is proportional to the density of neutrons in the pile, and not to their flux, since the intensity was measured with a detector whose sensitivity is inversely proportional to the velocity. For many problems it is not the neutron densities \(N(v)\) that are essential, but the fluxes through a given surface, i.e. the quantities \(vN\).

Fig. 36. Spectrum of neutrons emerging from the depth of a uranium–deuterium pile, measured with a crystal spectrometer. The solid curve is the Maxwell distribution at \(T = 400^\circ\) K; the dashed curve is \(1/E\).

In the volume of the moderator—in points distant from the uranium blocks by distances large in comparison with the mean free path of fast neutrons, for example in thick graphite walls—

...serving as neutron reflectors, or in the so-called thermal column, which is a large block of graphite built into the wall of the Argonne pile—there can no longer be fast neutrons; only long-lived thermal neutrons diffuse there. At such points the neutron spectrum agrees very well with the Maxwell distribution\(^{70}\), and the effective temperature is very close to the actual temperature of the moderator, since the lifetime of neutrons in the moderator is very long and the thermal spectrum is not substantially distorted by the improbable process of capture.

The distribution of neutrons in space inside a pile, in the case of localized uranium blocks\(^{71}\), is a rather complicated function of the coordinates. It is obvious that inside and in the immediate vicinity of the uranium blocks the density of thermal neutrons is considerably less than in the surrounding layers of moderator, since uranium strongly absorbs neutrons and the moderator absorbs them only weakly. For a homogeneous pile, i.e., one in which the uranium and the moderator form a continuous homogeneous mixture (for example, a solution), the distribution of thermal neutrons in space can be found by solving the diffusion equations\(^{2}\).

The resulting solutions may be regarded as applicable to inhomogeneous piles representing a regular spatial lattice with a large number of cells, if by the density found one understands the density averaged over the volume of one cell. For example, for a cubic pile with side \(a\), having a structure similar to a cubic space lattice, according to Fermi\(^{71}\) the neutron density \(n\), as a function of the coordinates \(x, y, z\), can be represented in the form

\[ n(x,y,z)=n_0\sin\frac{\pi x}{a}\sin\frac{\pi y}{a}\sin\frac{\pi z}{a}, \]

where \(n_0\) is the density at the center of the pile. The origin of coordinates is taken to be the vertex of one of the corners of the cube.

The mean density for such a distribution is equal to

\[ \bar n=\frac{8}{\pi^3}\,n_0=0.26\,n_0, \]

i.e., approximately four times smaller than the maximum density \(n_0\) at the center. Along each of the axes of the cubic pile passing through the centers of opposite faces, the density varies as

\[ \sin\frac{\pi x}{a}. \]

The absolute value of the density depends on the power released in the pile. The relation between the neutron density and the power of the pile can be established in the following way\(^{71,2}\).

Let each act of fission be accompanied by the release of energy \(200\ \mathrm{MeV}\). Then, to produce a power of \(1\ \mathrm{W}\), as is easily...

to count them, \(3\cdot 10^{10}\) fission events per second are required. The total number of neutron-capture events is approximately twice as large as the number of fission events, since neutrons are captured not only by the fissioning isotope, but also by the other constituents of the pile. It may therefore be considered that, in a pile operating at a power of \(1\) watt, \(6\cdot 10^{10}\) neutron-capture events occur per second.

On the other hand, the number of captures in \(1\ \mathrm{cm}^{3}\) per second is equal to

\[ \frac{nv}{\lambda}, \]

if \(n\) is the neutron density, \(v\) the velocity, and \(\lambda\) the mean path length of a neutron before capture, or the mean free path with respect to capture, defined for a homogeneous substance by the equality

\[ \lambda=\frac{1}{N\sigma}, \]

where \(N\) is the number of atoms in \(1\ \mathrm{cm}^{3}\) and \(\sigma\) is the mean capture cross section per atom.

In the entire pile of volume \(V\), each second

\[ \frac{\overline{nv}}{\lambda}V \]

neutrons are captured (\(\overline n\) is the mean density).

If the power of the pile is \(W\) watts, then

\[ \frac{\overline{nv}}{\lambda}V=6\cdot 10^{10}W, \]

and the mean neutron flux in the pile is

\[ \overline{nv}=\frac{\lambda}{V}\,W\,6\cdot 10^{10} \]

neutrons per second per \(1\ \mathrm{cm}^{2}\), i.e. the mean flux is proportional to the power of the pile, to the mean free path with respect to capture (and consequently to the mean lifetime of a neutron \(\tau=\frac{\lambda}{v}\)), and inversely proportional to the volume of the pile.

The quantity \(\lambda\) is inversely proportional to the mean neutron-capture cross section per atom of the pile and, consequently, depends on the ratio of the amounts of uranium and moderator in the pile. For pure graphite, for example, \(\lambda\) is about \(25\ \mathrm{m}^{71}\). The greater the relative uranium content in the pile, the sooner a neutron will be captured, and hence the smaller \(\lambda\).

If for a certain particular pile, following Fermi\(^{71}\), one takes \(\lambda=350\ \mathrm{cm}\), then

\[ \overline{nv}=\frac{W}{V}\,2.1\cdot 10^{13} \]

neutrons per second per \(1\ \mathrm{cm}^{2}\).

The neutron flux at the center of a cubic pile is

\[ n_{0}v=3.9\,\overline{nv}=\frac{W}{a^{3}}\,8.2\cdot 10^{13}. \]

Seren, Friedlander, and Turkel\(^{72}\), who irradiated in the Argonne Laboratory piles a large number of elements and measured for them the capture cross section of thermal neutrons, indicate that

in the thermal column a neutron flux of the order of \(10^{11}\) could be obtained, while at the center of the pile it would be 470 times greater, i.e. about \(0.5\cdot 10^{14}\) neutrons per second per \(1\ \text{cm}^2\).

The fluxes calculated in this way should be taken into account in those cases when irradiation by thermal neutrons is carried out on a sample of small thickness, i.e. one through which the neutron flux is only slightly attenuated. In the case of thick targets the calculation becomes more complicated, but in any event an increase in thickness leads to a decrease in the flux.

Thus, a uranium pile proves to be a very intense source of moderated thermal neutrons. The flux of fast neutrons inside the pile (or emerging outward through an opening) has a relatively lower intensity. Nevertheless, a perhaps even more concentrated and intense flux of fast neutrons can be obtained by means of cyclotrons, for example in the reaction \(\mathrm{Be}^9(d,n)\).

All neutrons of the pile—both fast and slow—are characterized by a continuous distribution in energy, i.e. by a continuous spectrum. Monochromatic neutrons can be obtained from a pile only with the aid of special auxiliary devices such as a mechanical selector \({}^{73}\) or a crystal monochromator, which select from the continuous spectrum a group of neutrons with a definite energy. As a laboratory source, the pile is used chiefly for studying the properties of slow neutrons and for irradiating various substances with them, in which radioactive isotopes are formed as a result of neutron capture.

CITED LITERATURE

1. V. Berestetskii, I. Pomeranchuk, ZhETF 19, 756 (1949).
2. K. Guden, Scientific and Technical Foundations of Nuclear Energy, vol. 1, IL, 1948.
3. A. O. Hanson, R. F. Taschek, J. H. Williams, Rev. Mod. Phys. 21, 635 (1949).
4. G. A. Bethe, Nuclear Physics, Gostekhizdat, 1948, p. 88.
5. A. H. Snell a. Miller, Bull. Amer. Phys. Soc. 23, 21 (1948).
6. A. Akhiezer, I. Pomeranchuk, Problems of Nuclear Theory, OGIZ, 1948, p. 133.
7. E. Konopinski, E. Teller, Phys. Rev. 73, 882 (1948).
8. H. L. Anderson, B. F. Feld, Rev. Scient. Instr. 18, 186 (1947).
9. I. Halpern, Phys. Rev. 76, 248 (1949).
10. G. Bernardini, Zeits. f. Physik 85, 557 (1933).
11. J. Chadwick, Proc. Roy. Soc. 142, 1 (1933).
12. T. Bjerge, Proc. Roy. Soc. 164, 243 (1938).
13. E. Shullinger, Zeits. f. Physik 114, 185 (1939).
14. E. Amaldi, E. Fermi, Phys. Rev. 50, 899 (1936).
15. R. Jaeckel, Zeits. f. Physik 91, 493 (1934).
16. F. A. Paneth, H. Loliet, Nature 136, 950 (1935).
17. F. A. Paneth, Proc. Roy. Soc. 157, 412 (1936).
18. G. A. Fink, Phys. Rev. 50, 738 (1936).
19. E. Amaldi, E. Fermi, Ric. Scient. 7, 454 (1936).

1. F. Seidl, S. P. Harris, Rev. Scient. Instr. 18, 897 (1947).
2. W. F. Hornyak, T. Lauritsen, Rev. Mod. Phys. 20, 191 (1948).
3. G. Bernardini, D. Bocciarelli, Atti Ac. Lincei 29, 139 (1936), Ric. Scient. 8, 33 (1937).
4. J. R. Dunning, Phys. Rev. 45, 587 (1934).
5. M. Teucher, Zeits. f. Physik 126, 410 (1949).
6. R. L. Walker, Phys. Rev. 76, 244 (1949).
7. L. Landau, E. Lifshitz, JETP 18, 750 (1948).
8. E. Bretscher, G. B. Cook, G. R. Martin, D. H. Wilkinson, Proc. Roy. Soc. 196, 436 (1949).
9. Zinn, Seeley, Phys. Rev. 50, 1101 (1936).
10. Amaldi, Hafstad, Tuve, Phys. Rev. 51, 896 (1937).
11. G. T. Hunter, H. T. Richards, Phys. Rev. 76, 1445 (1949).
12. K. W. Erickson, J. L. Fowler, E. J. Stovall, Phys. Rev. 75, 894 (1949); 76, 1141 (1949).
13. M. S. Livingston, J. Appl. Phys. 12, 339 (1941).
14. Cornog, Libby, Phys. Rev. 59, 1046 (1941).
15. C. E. Falk, E. Creutz, F. Seitz, Phys. Rev. 76, 322 (1949).
16. T. W. Bonner, W. M. Brubaker, Phys. Rev. 50, 308 (1936).
17. C. F. Powell, G. E. Fertel, Nature 144, 115 (1939).
18. H. H. Staub, W. E. Stephens, Phys. Rev. 55, 131 (1939).
19. Evans, Malich, Risser, Phys. Rev. 75, 1161 (1949).
20. W. D. Whitehead, C. E. Mandeville, Phys. Rev. 77, 732 (1950).
21. Oliphant, Kinsey, Rutherford, Proc. Roy. Soc. 141, 722 (1933).
22. T. W. Bonner, W. M. Brubaker, Phys. Rev. 48, 742 (1935).
23. H. T. Richards, Phys. Rev. 71, 796 (1947).
24. B. V. Aivazov, M. B. Neiman, UFN 36, 145 (1948).
25. R. F. Taschek, G. A. Jarvis, A. Hemmendinger, G. G. Everhart, H. T. Gittings, Phys. Rev. 75, 1361 (1949).
26. E. Bretscher, A. P. French, Phys. Rev. 75, 1154 (1948).
27. R. F. Taschek, A. Hemmendinger, G. A. Jarvis, Phys. Rev. 75, 1464 (1949).
28. W. M. Gibson, D. L. Livesey, Proc. Roy. Soc. 60, 523 (1948).
29. Stephens, Dyanab, Bonner, Phys. Rev. 52, 1079 (1937).
30. A. C. Helmholz, E. M. McMillan, D. C. Sewell, Phys. Rev. 72, 1003 (1948).
31. W. W. Chupp, E. Gardner, T. B. Taylor, Phys. Rev. 73, 742 (1948).
32. R. Serber, Phys. Rev. 72, 1008 (1948).
33. R. Serber, Phys. Rev. 72, 1114 (1948).
34. S. M. Dancoff, Phys. Rev. 72, 1016 (1948).
35. W. Heckrote, P. Wolff, Phys. Rev. 73, 264 (1948).
36. Seagondollar, Barschall, Phys. Rev. 72, 439 (1947).
37. R. K. Adair, C. K. Bockelman, R. E. Peterson, Phys. Rev. 76, 308 (1949).
38. R. Taschek, A. Hemmendinger, Phys. Rev. 74, 373 (1948).
39. T. A. Hall, Phys. Rev. 77, 411 (1950).
40. V. R. Jonson, M. J. W. Laubenstein, H. T. Richards, Phys. Rev. 77, 413 (1950).
41. G. A. Jarvis, A. Hemmendinger, Argo, R. F. Taschek, Phys. Rev. 76, 168 (1949).
42. Argo, Gittings, Hemmendinger, Jarvis, Mayer, Taschek, Phys. Rev. 76, 182 (1949).
43. V. N. Kondrat’ev, UFN 38, 153 (1949).

1. A. I. Leipunsky, ZhETF 16, 33 (1946).
2. I. F. E. Hanson, L. Hulthen, Phys. Rev. 76, 1163 (1949).
3. E. Guth, C. J. Mullin, Phys. Rev. 76, 234 (1949).
4. A. Wattenberg, Phys. Rev. 71, 497 (1947).
5. K. Gudmen, Scientific and Technical Foundations of Nuclear Power Engineering, vol. 2, IL, 1950.
6. Zinn, Szillard, Phys. Rev. 56, 619 (1939).
7. W. J. Sturm, Phys. Rev. 71, 757 (1947).
8. E. Fermi, L. Marshall, J. Marshall, Phys. Rev. 72, 193 (1947).
9. E. Fermi, UFN 32, 54 (1947).
10. L. Seren, H. N. Friedlander, S. H. Turkel, Phys. Rev. 72, 888 (1947).
11. N. A. Vlasov, UFN 35, 352 (1948).
12. H. L. Green, W. M. Gibson, Proc. Phys. Soc. A. 62, 407 (1949).
13. H. Wäffler, Helv. Phys. Acta 23, 239 (1950).
14. C. E. Bradford, W. E. Bennett, Phys. Rev. 78, 302 (1950).
15. R. W. Pringle, K. I. Roulston, S. Standil, Phys. Rev. 78, 627 (1950).
16. J. Terrel, Phys. Rev. 79, 239 (1950).
17. P. R. Bell, W. H. Jordan, Phys. Rev. 79, 392 (1950).
18. W. H. Guier, I. H. Roberts, Phys. Rev. 79, 719 (1950).
19. B. G. Whitmore, W. B. Baker, Phys. Rev. 78, 799 (1950).
20. M. Livingston, M. Rose, M. Namias, The Cyclotron, Gostekhizdat, 1948.
21. A. H. Snell, F. Pleasonton, R. V. McCord, Phys. Rev. 78, 310 (1950).
22. J. M. Robson, Phys. Rev. 78, 311 (1950).
23. E. Kelly, C. Leith, E. Segre, C. Wiegand, Phys. Rev. 79, 96 (1950).
24. R. S. Christian, E. W. Hart, Phys. Rev. 77, 441 (1950).

PROOF CORRECTION NOTE

Neutrons with energies up to 350 MeV have recently been obtained[^85] by bombarding targets with protons of energy 350 MeV. The interaction between the neutron and the proton is of an exchange character[^86], owing to which a recharge of the proton is observed when it collides with the neutron of the nucleus. It turns out that the fast proton after colliding with the neutron is scattered with equal probability either as a proton, i.e., retaining its charge, or as a neutron, i.e., transferring its charge to the former neutron, which after the collision has become a proton.

The neutrons formed in this way have energies close to the initial energy of the protons. The results of not very precise investigations of the neutron spectrum[^85] show that, when a beryllium target 5.08 cm thick is bombarded with protons of energy 350 MeV, neutrons of a continuous spectrum propagate in the direction of the proton beam, extending from 350 MeV into the region of lower energies, in any case beyond 200 MeV. The maximum of the spectrum corresponds to an energy of approximately 260 MeV.