EXPERIMENTS WITH MONOCHROMATIC SLOW NEUTRONS*)

N. A. Vlasov

ABSORPTION AND SCATTERING OF NEUTRONS

The most important experimental problem solved with the aid of selectors is the investigation of the dependence of the cross section for the capture and scattering of neutrons by various nuclei on the neutron energy or, in other words, the investigation of neutron absorption spectra. Since not a single element can be characterized, with respect to its interaction with neutrons, by one value of the cross section, because this cross section is different for neutrons of different energies, it seems expedient to speak of the neutron absorption spectrum of a given element, understanding by this spectrum the curve of the dependence of the capture cross section on the neutron energy.

Measurements of absorption spectra with the aid of selectors are carried out as follows. In a well-collimated neutron beam, approximately midway between the source and the detector, a sample of the substance under investigation is placed. The intensity of neutrons of different flight times is measured in the absence of the sample, \(I_0\), and in the presence of the sample, \(I\). Possible changes in the total intensity of the neutron source are taken into account by measurement with a special monitor detector placed outside the beam. The readings of the detector in the beam are referred to identical monitor readings.

The transmission of the sample

\[ T=\frac{I}{I_0}=e^{-n\sigma} \]

determines the total cross section for the interaction of neutrons with matter according to the formula given, in which \(n\) is the number of atoms per \(1\ \mathrm{cm}^2\) of the sample area. It is clear that the attenuation of the neutron beam by the sample is due both to absorption and to scattering, and measurement of the transmission does not in principle make it possible to separate the effects of absorption and scattering, but analysis of the results in many cases makes it possible to estimate the magnitude of both effects.

*) Conclusion. See UFN, Vol. XXXV, No. 3, p. 352.

The theoretical predictions on the dependence of absorption and scattering cross sections on neutron energy\(^ {24}\), which have been fairly well confirmed by experiments, reduce to the following.

The neutron-capture cross section in the resonance region follows the well-known Breit–Wigner formula. In the simplest case of a single resonance level this formula may be written in the form

\[ \sigma(E)=\sigma_0\left(\frac{E_0}{E}\right)^{1/2} \frac{\Gamma^2}{4(E-E_0)^2+\Gamma^2}. \]

Here \(\sigma(E)\) is the capture cross section of a neutron with energy \(E\), \(\sigma_0\) and \(E_0\) are the resonance values of the cross section and energy, and \(\Gamma\) is the half-width of the resonance level. According to this formula, the capture cross section increases strongly for neutrons with energy \(E\) close to \(E_0\), and reaches its maximum value \(\sigma(E_0)=\sigma_0\) at \(E=E_0\). On the other hand, in the region of low neutron energies, namely when \(E \ll E_0\), the dependence of \(\sigma\) on energy is determined not by the resonance term in the denominator, but by the factor \(E^{-1/2}\), which also determines the law \(1/v\). As we shall see below, experiment very well confirms the Breit–Wigner formula both in the resonance region and in the region of low (thermal) energies, and proves the universality of the \(1/v\) law. Only in those cases where the resonance is close to the thermal region is the \(1/v\) law not observed, but this also follows from the Breit–Wigner formula. The number of such cases is small, since the width of the thermal region does not exceed \(0.5\) eV, whereas the mean distance between resonance levels is of the order of several tens of eV, and the probability of coincidence of a resonance with the thermal region is small.

If there is no such coincidence and, consequently, the \(1/v\) law is valid over a finite energy interval, then one may experimentally estimate the value of the scattering cross section. This estimate is based on the assumption, which is theoretically sufficiently well justified, that far from resonance the scattering cross section is practically independent of energy. If this is so, then the attenuation of a neutron beam in the region of low energies is due to two factors, one of which—capture—follows the law \(1/v\), while the other—scattering—is constant. Then the total cross section measured experimentally may be represented as a function of the neutron time of flight \(t\) in the following form:

\[ \sigma_{\text{tot}}=\sigma_{\text{capt}}+\sigma_{\text{scat}}=at+b \qquad (a \text{ and } b \text{ are constants}). \]

Consequently, the experimentally measured total cross section should be represented as a function of \(t\) by a straight line cutting off on the ordinate axis \((t=0)\) a segment \(b=\sigma_{\text{scat}}\), representing the value of the scattering cross section. Thus, the scattering cross section can be determined from the results of measurements of the total cross section by constructing the diagram \(\sigma(t)\) and extrapola-

by extrapolating the straight line from the region where the \(1/v\) law is obeyed to its intersection with the ordinate axis. The difference between the total scattering cross section and the scattering cross section obtained in this way is equal to the capture cross section.

In the resonance region, however, the situation is somewhat more complicated. Alongside potential scattering, which is the only kind far from resonance and is characterized by a cross section independent of energy, resonance scattering with a very large cross section is possible. Recently such resonance scattering has been found\(^{26,27,28}\) in special experiments for manganese, cobalt, silver, indium, and other elements. The presence of resonance scattering does not allow the experimentally observed peak of the total cross section in the resonance region to be attributed wholly to capture, and a simple experiment with attenuation of the beam does not make it possible to separate the roles of capture and scattering. This should be borne in mind when considering the measurement results given below.

The transmission of a specimen

\[ T=\frac{I}{I_0}=e^{-n\sigma} \]

depends on the product of the (total) cross section by the specimen thickness, expressed as the number of atoms \(n\) per unit area. From the standpoint of measurement accuracy, the absolute magnitude of the transmission and, consequently, the specimen thickness, are not immaterial. Therefore the selection of a suitable optimal specimen thickness becomes an essential task for the experimenter. Its solution depends on the nature of the variation of \(\sigma\) with energy. In those cases where \(\sigma\) changes smoothly with energy (outside resonance) and errors associated with resolving power are not significant, the accuracy of the measurements is determined, on the one hand, by the transmission, i.e. by the thickness of the specimen, and, on the other, of course, by the total duration of the measurements and, for a given total duration of the measurements, by the ratio \(x\) of the duration of measurements with the specimen in the path of the beam to the total duration of the measurements. A calculation of the dependence of the measurement accuracy on \(T\) and on the ratio \(x\) is given in the work of Rainwater and Havens\(^{16}\). According to this calculation the optimum values are

\[ T=0.0761 \quad \text{and} \quad x=\frac{1}{1+\sqrt{T}}. \]

This means that, to obtain the greatest measurement accuracy, the specimen thickness should be such that 7–8% of the neutrons pass through it, and the duration of the measurements with the specimen in the path of the beam should be about 75% of the total duration of the measurements, i.e. should be approximately three times greater than the duration of the measurements without the specimen. But neither value (neither \(T\) nor \(x\)) is critical, since for not very large deviations of \(T\) and \(x\) from the optimum the accuracy decreases slowly; therefore one and the same specimen can be used for transmission measurements over a fairly wide interval of energies outside resonance.

In the resonance region the cross section varies sharply with energy, and for the study of different parts of the resonance curve, obviously,

it is necessary to use specimens of different thicknesses. If the transmission is optimal at the edges of the resonance curve, then in the middle it exhibits a flat minimum, from which it is impossible to determine the true transmission (since it is very small), and consequently also the true cross section in the resonance region. It is evident that, for a better measurement of the peak value of the transmission (and consequently of the cross section), thinner specimens must be used. However, owing to the sharpness of the resonance and the finite resolving power of the apparatus, the experimentally measured peak value of the cross section will only in rare cases be close to the true one. Such a case, for example, is absorption in cadmium, in which the resonance appears at a low neutron energy (0.176 eV), where the resolving power is sufficiently high for a detailed investigation of the entire resonance curve. But already at resonance energies of about 1 eV and at maximum resolving power, as is seen from Table II, the interval of recorded energies is of the order of 0.05 eV, which amounts to about half the width for most of the known resonance lines. The curves in Figs. 7 and 8 show that the experimental peak value of the transmission may differ substantially from the actual one when the resolving power is not sufficiently high.

The method of analyzing experimental data by comparing them with the Breit–Wigner formula, taking into account the thickness of the specimen and corrections for resolving power, developed by Havens and Rainwater, is described in their 1946 paper.¹⁷ We refer those interested in the details to the original paper, and here we shall merely note that this comparison makes it possible to determine fairly well the quantity \(\sigma_0 \Gamma^2\), which enters as a constant in the Breit–Wigner formula and determines the area of the resonance curve. As regards the quantities \(\sigma_0\) and \(\Gamma\) themselves, analysis can establish only a lower limit for \(\sigma_0\) and an upper limit for \(\Gamma\). The experimental results given below are, in most cases, directly measured transmission curves referred to neutron times of flight. In some cases, a curve corresponding to the Breit–Wigner formula with selected, most suitable constants \(E_0\), \(\sigma_0\), and \(\Gamma\) has been drawn through the experimental points. These constants are given in the corresponding cases. Where the results are sufficiently reliable, the transmission has been recalculated into a cross section, and curves are given for the dependence of the cross section on neutron energy.

Let us proceed to a discussion of these results.

ABSORPTION IN BORON

Soon after the discovery of the neutron, the supposition was expressed that the cross section for neutron capture by boron is inversely proportional to their velocity, i.e.

\[ \sigma_0 \sim \frac{1}{v}. \]

This supposition was confirmed by widely known indirect experimental data (see, for example,

review by Bethe[^21]), which left no room for serious doubt as to the validity of this law, but also did not provide unconditional certainty in it. Meanwhile, many experiments were based on this law, and, of course, its collapse would have meant an upheaval in a very extensive field of research with slow neutrons. Therefore a direct and fairly precise proof of the applicability of this law to boron, provided by investigations using selectors, is a very significant fact. It is also natural that every new technique of research with slow neutrons, in particular selectors, beginning with the very first mechanical ones, was applied first of all to the measurement of the boron cross section and its dependence on energy.

We have already said above that the first mechanical selectors of Dunning et al., as well as Alvarez’s first selector-modulator, made it possible only to establish qualitatively that the boron cross section increases as the neutron velocity decreases. Better results were unattainable at that time because of the low resolving power of the instruments.

The work carried out somewhat later by Ferteil et al. for the same reason even led the authors to erroneous conclusions about the non-observance of the law \(1/v\). The first quantitatively reliable results of investigations by means of a selector were published simultaneously by two groups of authors in the same May issue of Physical Review for 1946. One of these papers belongs to Manley, Haworth, and Luebke[^14], the other to Bacher, Becker, and McDaniel[^11]. Both works were performed at the beginning of the war and published considerably later. The second of them contains more precise results. We shall begin our review with it.

Bacher, Becker, and McDaniel investigated absorption in boron for neutrons with energies from \(0.028\ \mathrm{eV}\) to \(50\ \mathrm{eV}\). The absorber samples were prepared from a fine powder of boron carbide \(B_4C\). To obtain thin uniform layers, the boron carbide powder was sprayed from a height of \(1.5\ \mathrm{m}\) onto flat-bottomed aluminum dishes, into which a weak solution of cellophane in ether had first been poured. The amount of cellophane used as a cementing material did not exceed \(2\%\) by weight. The thickness of the absorber was determined by weighing before and after spraying. The five absorbers used had thicknesses \(0.273,\ 0.1197,\ 0.0585,\ 0.0487\) and \(0.0318\ \mathrm{g/cm^2}\) of boron. In measurements with the first two, thickest absorbers, the distance from source to detector used was \(3\ \mathrm{m}\), the width of the opening pulses, the same for source and detector, was \(20\ \mu\mathrm{sec}\), and the carrier frequency was \(2500\) cycles. The neutron beam was then filtered through a cadmium layer \(0.9\ \mathrm{g/cm^2}\). For the remaining absorbers the carrier frequency was \(500\) cycles, the pulse width \(50\text{--}80\ \mu\mathrm{sec}\), the source—detector distance \(2\ \mathrm{m}\), and cadmium was absent. A significant distortion of the results was found as a consequence of “contamination” of the beam by neutrons from preceding cycles—slower and longer-lived in para-

fine. To eliminate this effect, the “threshold” described above, consisting of a Cd layer and thin paraffin, was used; it shortens the neutron lifetime, and a thin \(B_4C\) filter \((0.0205\ \mathrm{g/cm^2}\ B)\) was also used, which removed the slowest neutrons. The results of the measurements are presented in Fig. 12 in the form of a diagram showing the cross section as a function of the flight time. The experimental points, within the limits of error, lie on a single straight line, which satisfies the \(1/v\) law. The straight line drawn by the method of least squares has a slope of \((1.55 \pm 0.035)\times 10^{-24}\ \mathrm{cm^2/\mu sec/m}\). Hence the boron cross section as a function of energy in eV may be expressed by the formula:

\[ \sigma_B=\left[(112\pm3)E^{-1/2}+(4.4\pm1.4)\right]\times 10^{-24}\ \mathrm{cm^2}. \]

Fig. 12. Effective boron cross section according to the measurements of Bacher, Becker, and McDaniel.

The number 4.4 in square brackets corresponds to the energy-independent scattering cross section obtained by extrapolating the straight line to \(t=0\).

Rainwater and Havens\(^{16}\) extended the measurements to an even larger neutron-energy interval—from 0.01 to 250 eV. Here four different absorber thicknesses were used. To prepare the specimens, \(B_2O_3\) was dried for 48 hours at a temperature of \(140^\circ\mathrm{C}\), then packed tightly into aluminum boxes. One of the specimens contained \(1.092\ \mathrm{g/cm^2}\ B_2O_3\) and \(0.418\ \mathrm{g/cm^2}\ Al\), another—\(2.464\ \mathrm{g/cm^2}\ B_2O_3\) and \(0.418\ \mathrm{g/cm^2}\ Al\), the third—\(0.310\ \mathrm{g/cm^2}\ B_2O_3\) and \(0.418\ \mathrm{g/cm^2}\ Al\). As the fourth specimen, several layers of Pyrex of total thickness \(0.955\ \mathrm{g/cm^2}\) were used, which, according to analysis, contained \(12.9\%\ B_2O_3\) (the remainder being mainly \(SiO_2\)).

The measurements were made at distances between source and detector of 5.2 and 5.4 m. The leading frequency for slow neutrons was 200 cycles (period \(5000\ \mu\mathrm{sec}\)); for fast neutrons, 1000 cycles \((1000\ \mu\mathrm{sec})\). The width of the opening pulses for different sections is illustrated by the triangles in Figs. 13 and 14, which show the measurement results in the same form as Fig. 12. Fig. 14 shows, on an enlarged scale, the part of the diagram of Fig. 13 close to the origin of coordinates, i.e. corresponding to the maximum measured neutron energies. To record this section, a ...

sample of thickness \(2.464\ \mathrm{g/cm^2}\) of \(B_2O_3\), and the maximum resolving power. Although, as is evident from the figure, it is still not sufficiently large, this should not strongly affect the results, as follows from the considerations of Becker and Bacher set forth in the chapter on resolving power. It is nevertheless strange that the straight line goes to the origin of the coordinates and not higher. This is probably the result of an insufficiently accurate allowance for scattering in oxygen and aluminum, which are contained in the sample. The corrections for scattering in Al, O, and \(SiO_2\) (in the case of Pyrex) were adopted from the calculation that the cross sections are constant and equal to:

\[ \begin{aligned} \sigma_O &= 4.1 \cdot 10^{-24}\ \mathrm{cm^2};\\ \sigma_{Al} &= 1.5 \cdot 10^{-24}\ \mathrm{cm^2};\\ \sigma_{Si} &= 2.5 \cdot 10^{-24}\ \mathrm{cm^2}. \end{aligned} \]

In the greater part of the results, of course, these corrections are only weakly reflected, since the boron capture cross section is large, but at very small boron cross sections in the region of high neutron velocities it is precisely they that limit the accuracy. We again see excellent agreement of the experimental data with the law \(\frac{1}{v}\).

Fig. 13. Effective cross section of boron according to the measurements of Rainwater and Havens.

The authors give the following expression for the boron cross section:

\[ \sigma_B = (118 \pm 4)\,E^{-1/2}\cdot 10^{-24}\ \mathrm{cm^2} \qquad (\text{energy in eV}). \]

Within the indicated errors it agrees with the results of the preceding authors.

Sutton, McDaniel, Anderson, and Lavatelli\(^{21}\) carried out measurements analogous to the preceding ones in an even larger energy interval from \(0.01\) to \(1000\ \mathrm{eV}\), with a distance between source and detector of \(7.6\ \mathrm{m}\). Four samples of different thickness were used. Of these, three consisted of \(B_4C\), and the fourth, the thinnest, of gaseous \(BF_3\) in a cuvette. The results of this work are presented in Fig. 15 in the form of a diagram showing the product \(\sigma_B v\) as a function of neutron energy. If the law \(\frac{1}{v}\) is valid, then \(\sigma_B v\) should remain

constant. The figure shows that the experimental points do indeed group rather well near a horizontal straight line, although the energy changes by a factor of \(10^5\). Analysis of the results leads to the following conclusions. If the total cross section of boron is represented in the form \(\sigma_{\text{full}}=\alpha t+\beta\), assuming the capture cross section \(\sigma_{\text{capt}}=\alpha t\) to be proportional to the flight time, and the scattering cross section \(\beta\) to be constant, then after introducing the correction for scattering in fluorine (taken as \(4.1\cdot10^{-24}\ \mathrm{cm}^2\)), carbon (\(4.85\times10^{-24}\ \mathrm{cm}^2\)), oxygen (\(4.2\cdot10^{-24}\ \mathrm{cm}^2\)) and lead (\(9.6\cdot10^{-24}\ \mathrm{cm}^2\)) (one of the filters contained lead borate), which for the \(B_4C\) filter amounts to \(2.2\times10^{-24}\ \mathrm{cm}^2\) per boron atom, it is found that the scattering cross section in boron is \(\beta=4.2\cdot10^{-24}\ \mathrm{cm}^2\), and \(\alpha=1.58\cdot10^{-24}\ \mathrm{cm}^2\mathrm{m}/\mu\mathrm{sec}\) (if \(t\) is in \(\mu\mathrm{sec}/\mathrm{m}\)). Expressing this through the neutron energy, we obtain:

Fig. 14. Effective cross section of boron in the region of high energies.

\[ \sigma_B=(114\,E^{-1/2}+4.2)\cdot10^{-24}\ \mathrm{cm}^2. \]

Comparing with the preceding results, we see that this latter is intermediate and, within the limits of experimental accuracy, agrees with both. The last authors did not indicate the errors. Relatively

Fig. 15. Product of the boron cross section by neutron velocity \(\sigma\cdot v\) according to the measurements of Sutton et al.

EXPERIMENTS WITH MONOCHROMATIC NEUTRONS

relative to the law \(1/v\), they note that the product \(\sigma v\) remains constant within \(\pm 10\%\) up to \(1000\) eV, if the assumption \(\beta=\mathrm{const}\) is correct.

Fermi, Marshall, and Marshall\(^5\), with the aid of a semi-mechanical selector, also investigated the cross section for the interaction of boron with neutrons having velocities from \(1700\) to \(5000\ \mathrm{m/sec}\) \((0.015—0.13\ \mathrm{eV})\). As the sample they used well-purified gaseous \(\mathrm{BF}_3\) at different pressures. Without describing the experiments in detail, the authors state that, within the accuracy of the method, the boron cross section varies as \(1/v\), and for neutrons with velocity \(2200\ \mathrm{m/sec}\) (energy \(kT\) at \(T=293^\circ\mathrm{K}\)) is equal to \(699\cdot 10^{-24}\ \mathrm{cm}^2\). To check this value they carried out an experiment with an absorber consisting of a solution of \(\mathrm{Na}_2\mathrm{B}_4\mathrm{O}_7\) in heavy water, enclosed in a thin aluminum vessel. Another identical vessel with heavy water, but without \(\mathrm{Na}_2\mathrm{B}_4\mathrm{O}_7\), was used for the control measurement. This experiment gives \(\sigma_{\mathrm{B}}\) for neutrons of the same velocity equal to \(700\cdot 10^{-24}\ \mathrm{cm}^2\).

The third experiment for measuring the same cross section is based on the measurement of the boron cross section for the resonance neutrons of indium. As the boron absorber, gaseous \(\mathrm{BF}_3\) is used in a steel cylindrical vessel \(30\ \mathrm{cm}\) long under a pressure of \(3—4\) atmospheres (from \(44\) to \(68\ \mathrm{lb}/\mathrm{dm}^2\)). To separate out the effect of the resonance neutrons, the indium indicator was surrounded, first, by cadmium, and second (in measuring the background), by indium itself. Taking, from the data of Havens and Rainwater (see below), the resonance energy of indium as \(1.44\ \mathrm{eV}\), and introducing corrections for scattering by boron (it was assumed that \(\sigma=2\cdot 10^{-24}\ \mathrm{cm}^2\)) and fluorine \((3.7\cdot 10^{-24}\ \mathrm{cm}^2)\), the authors obtain from this experiment

\[ \sigma_{\mathrm{B}}=710\cdot 10^{-24}\ \mathrm{cm}^2. \]

As the average of the three experiments they take

\[ \sigma_{\mathrm{B}}=703\cdot 10^{-24}\ \mathrm{cm}^2 \]

for \(v=2200\ \mathrm{m/sec}\).

Recalculating to neutron energy, we obtain

\[ \sigma_{\mathrm{B}}=\left(112E^{-\frac12}+2\right)\cdot 10^{-24}\ \mathrm{cm}^2. \]

The number \(2\) in the parentheses corresponds to the value adopted by Fermi et al. (probably on the basis of their own experiment, although nothing is said about this) for the scattering cross section of boron. This number does not agree with the preceding data \((4.2\cdot 10^{-24}\ \mathrm{cm}^{2\,19})\), and, since Fermi et al. do not discuss this question, there are no data for judging the correctness of one or the other value.

In conclusion, for convenience of comparison, we shall list the data cited in Table IV.

All the values of the cross section given refer to the natural mixture of boron isotopes, in which, according to Seaborg’s tables, there is contained \(\mathrm{B}^{10}\)—\(18.4\%\), \(\mathrm{B}^{11}\)—\(81.6\%\). The absorption of neutrons by boron proceeds through the process \(\mathrm{B}^{10}(n,\alpha)\mathrm{Li}^7\), whereas \(\mathrm{B}^{11}\) practically does not participate in the absorption. Recently, enrichment of boron

N. A. VLASOV

Table IV

Boron cross sections according to various authors

Authors	Cross section in \(10^{-24}\ \mathrm{cm}^2\)
Bacher, Becker, MacDaniel	\((112 \pm 3)E^{-\frac{1}{2}} + (4.4 \pm 1.4)\)
Rainwater, Havens	\((118 \pm 4)E^{-\frac{1}{2}}\)
Sutton, MacDaniel, Anderson, Lavatelli	\(114E^{-\frac{1}{2}} + 4.2\)
Fermi, Marshall, Marshall	\(112E^{-\frac{1}{2}} + 2\)
Mean (without taking account of weight)	\(114E^{-\frac{1}{2}} + 3.5\)

by the isotope \(B^{10}\), and enriched boron is used for filling neutron-registering instruments. The average cross section per atom of enriched boron will obviously be greater by as many times as the content of \(B^{10}\) has been increased. For pure \(B^{10}\) the capture cross section should be 5.43 times larger than that given in Table IV, i.e.

\[ \sigma_{B^{10}} = 620E^{-\frac{1}{2}}. \]

As to the scattering cross section, this, of course, cannot be said, since it may be of the same order both for \(B^{10}\) and for \(B^{11}\).

Thus, all the experiments listed confirm the \(1/v\) law for the absorption of neutrons by boron in the energy interval from \(0.01\) to 1000 eV. The absolute value of the cross section may be considered definite—

Fig. 16. Permeability of a sample of \(4.85\ \mathrm{g/cm^2}\) LiF and the cross section of lithium. The straight line is according to the equation
\[ \sigma = (11.5E^{1/2} + 1.7)\cdot 10^{-24}\ \mathrm{cm}^2. \]

nym with an accuracy of up to 2%. Using these now sufficiently reliable data, one can apply the boron-absorption method for measuring neutron resonance energies in a considerably simplified form. The above-mentioned experiment by Fermi et al. with indium serves as an illustration of this.

ABSORPTION IN LITHIUM

Everything that has been said with regard to boron applies, to a known extent, also to lithium, which, along with boron, was considered to absorb neutrons according to the law \(1/v\) and often, especially in early works, was used for neutron detection, owing to the fact that absorption proceeds through the process \(\mathrm{Li}^6(n,\alpha)\mathrm{H}^3\), which gives an \(\alpha\)-particle and an \(\mathrm{H}^3\) nucleus with a total energy of about 4.6 MeV.

The lithium cross section was investigated by Havens and Rainwater \({}^{17}\) in the energy interval from 0.02 to 250 eV. The absorber samples were made of LiF, dried and enclosed in aluminum boxes. The thicknesses of the samples were: \(4.85\ \mathrm{g/cm^2}\) LiF and \(0.418\ \mathrm{g/cm^2}\) Al for one, and \(0.862\ \mathrm{g/cm^2}\) LiF and \(0.418\ \mathrm{g/cm^2}\) Al for the other.

In the figure: “Effect Al and F”; “Constant fraction of the beam”; “Transmission”; “Li”; “Cross section \(10^{-24}\ \mathrm{cm^2}\)”; “Energy (eV)”; “Time of flight (\(\mu\)sec/m)”.

Fig. 17. Transmission of a sample of \(0.862\ \mathrm{g/cm^2}\) LiF and the lithium cross section. The straight line is the same as in Fig. 16.

The measurement results presented in Figs. 16 and 17 confirm the \(1/v\) law for the capture cross section and can be expressed in the form

\[ \sigma_{\mathrm{Li}}=\left[(11.5\pm0.2)E^{-1/2}+(1.7\pm0.2)\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

Since in the natural mixture the isotope \(\mathrm{Li}^6\) is present in an amount of only 7.5%, the capture cross section for the \(\mathrm{Li}^6\) nucleus will be equal to

\[ \sigma_{\mathrm{Li^6}}=153E^{-1/2}\cdot 10^{-24}\ \mathrm{cm^2}. \]

N. A. VLASOV

ABSORPTION IN CADMIUM

Cadmium, along with boron, is widely known as a strong absorber of neutrons. But, unlike boron, it has a sharp absorption boundary; namely, it absorbs thermal neutrons very strongly and much more weakly the faster ones, with energies of the order of 1 eV and higher. In this connection cadmium is used as an absorber of thermal neutrons, and the very concept of “thermal neutrons” is inseparably linked with cadmium. It is precisely the neutrons absorbed by cadmium that are considered thermal. Hence it is clear that knowledge of the dependence of the cadmium cross section on neutron energy is of exceptional interest.

The capture of a neutron by cadmium leads to the formation of a stable isotope of cadmium and is accompanied only by γ-radiation. This was established already in the first experiments with neutrons. Recently it has been shown\(^{35}\) that the neutron-absorbing isotope is Cd\(^{113}\), contained in natural cadmium in an amount of 12.3%. As a result of neutron capture the stable isotope Cd\(^{114}\) is formed.

Radiative capture of neutrons, i.e., capture accompanied by the emission of γ-rays, must have a resonant character if the compound nucleus excited as a result of capture possesses a discrete spectrum of states. From this point of view the properties of cadmium could be explained by the fact that it exhibits resonant capture of neutrons with energy close to thermal. Assuming the applicability of the Breit–Wigner formula to the cross section for neutron capture by cadmium, it was already possible, by the boron absorption method, to determine the position and width of the resonance line. G. Hoffman and Livingston\(^{29}\), proceeding in precisely this way, found that the resonance energy of neutrons for cadmium is 0.18 eV, and the width of the resonance line is \(\Gamma = 0.15\) eV. These values, as we shall see, are very close to those obtained later by direct and precise experiments with selectors.

But the assumptions underlying experiments of the Hoffman and Livingston type were not unconditionally reliable, since they had no experimental confirmation. Only with the aid of selectors was it possible directly and accurately to measure the cadmium cross section as a function of neutron energy and to prove the validity of these assumptions and the applicability of the Breit–Wigner formula. And it was precisely cadmium that proved to be the most suitable element for this purpose, since in the region of small neutron energies, where the resolving power of selectors is sufficiently good, the resonance curve can be studied in detail.

The first measurements were carried out by Ferti et al.\(^{19}\), but owing to the low resolving power of their instrument the results turned out to be rather crude and do not merit discussion.

Considerably more accurate results were obtained by Baker and Bacher\(^{10}\). They discovered an entirely distinct peak in the capture cross section at neutron energies of about 0.14 eV, a sharp decrease

in the region of large energies and a weaker decrease in the region of small energies down to a minimum \((\sigma_{\min}=2500\cdot10^{-24}\ \mathrm{cm}^2)\) at an energy of about \(0.03\ \mathrm{eV}\), and then an increase of the cross section for still slower neutrons. We shall not discuss these measurements in detail, since more accurate results are now available, but we must note that it was they which first reliably demonstrated resonance

Fig. 18. Total cross section of cadmium, \(o\)—for a specimen of \(43.3\ \mathrm{mg/cm^2}\), \(\times\)—for a specimen of \(219\ \mathrm{mg/cm^2}\). The solid curves are according to the Breit–Wigner formula for \(E_0=0.180\ \mathrm{eV}\), \(\sigma_0=7800\cdot10^{-24}\ \mathrm{cm}^2\), \(\Gamma=0.114\ \mathrm{eV}\) (upper curve) and \(\Gamma=0.108\ \mathrm{eV}\) (lower).

absorption in cadmium and established agreement between the Breit–Wigner theory and experiment.

More accurate results were subsequently obtained by Rainwater and Havens[^15], and then by the same authors jointly with Wu and Dunning[^16]. We shall discuss the latter work as the most accurate, and from the former we shall give only the curve (Fig. 18), which represents the cross section as a function of time of flight over a wide interval and gives a general idea of the absorption spectrum of cadmium. The results of the latest measurements are shown in Figs. 19, 20, and 21 in the form of curves corresponding to definite portions of the absorption spectrum and being more accurate. These curves represent the cross section of cadmium as a function of the neutron energy in the interval from \(0.008\) to \(2.1\ \mathrm{eV}\). For higher neutron energies the absorption has also been investigated, and the results are presented in Fig. 22 in the form of a diagram showing the transmissivity as a function of time of flight.

Cadmium, used in the measurements, according to spectrographic analysis (arc spectrum) contained less than 0.05% Pb and Te, less than 0.005% Al, Ag, and Cu, and traces of Ca, Cr, Fe, and Sn. The remaining elements could be present in amounts less than 0.05% if they are not detected in the arc spectrum, and less than \(10^{-5}\%\) if they are detected. To make the absorbers, cadmium was rolled with rollers. Four samples were used—of thickness \(438\ \mathrm{mg/cm^2}\), \(219\ \mathrm{mg/cm^2}\), \(4.47\ \mathrm{g/cm^2}\), and \(11.03\ \mathrm{g/cm^2}\). The resolving power is characterized by the intervals between the experimental points—the distance between them does not exceed the recorded energy interval. The statistical accuracy is of the order of 1–3%. The very arrangement of the points indicates the high accuracy of the measurements.

Fig. 19. Total cross section of cadmium in the region of low neutron energies. The solid curve is according to the Breit–Wigner formula for \(E_0 = 0.176\ \mathrm{eV}\), \(\Gamma = 0.115\ \mathrm{eV}\), \(\sigma_0 = 7200 \cdot 10^{-24}\ \mathrm{cm^2}\).

Fig. 20. Total cross section of cadmium in the resonance region (continuation of Fig. 19).

The smooth curve represents the Breit–Wigner formula with optimally chosen constants. Its actual agreement with the experimental points over the whole investigated interval, where the energy changes by a factor of 100 and the cross section by a factor of 1000, is convincing proof of the validity of the theory of resonance capture of neutrons proposed in its simplest form by Breit and Wigner and developed later in a whole series of works (see, for example, \(^{30}\)). On this basis, the analysis of experimental data by means of the Breit–Wigner formula proves to be fully justified, and the constants of this formula acquire a physical meaning grounded in experiment.

In the case of cadmium, the quantities that best agree with the experimental data are:

\[ \begin{aligned} \text{resonance energy}\quad &E_0 = 0.176 \pm 0.002\ \text{eV},\\ \text{line width}\quad &\Gamma = 0.115 \pm 0.002\ \text{eV},\\ \text{resonance cross section}\quad &\sigma_0 = (7200 \pm 200)\cdot 10^{-24}\ \text{cm}^2,\\ \text{scattering cross section}\quad &\sigma_{\text{sc}} = (5.30 \pm 0.7)\cdot 10^{-24}\ \text{cm}^2. \end{aligned} \]

The last quantity was obtained from measurements at large neutron energies (see Fig. 22) and was adopted as a constant correction over the entire energy interval. We have already noted that in such cases this is not lawful, since the scattering cross section in resonance may have large values. However, in the case of cadmium there are grounds to suppose that the scattering cross section in resonance is considerably smaller than the absorption cross section, although the authors do not refer to this.

According to Bethe1, knowing the peak cross section \(\sigma_0\) and the resonance energy \(E_0\), one can estimate the relative role of absorption and scattering in the following way. The level width \(\Gamma\) obtained in the experiment, inversely proportional to the lifetime \(\tau\) of the compound nucleus in the corresponding excited state,

\[ \left(\Gamma = \frac{h}{\tau}\right), \]

is the total width corresponding to all processes of transition of the nucleus from this state to another. To each transition process there may be assigned its own partial width

\[ \Gamma_i = \frac{h}{\tau_i}. \]

Fig. 21. Total cross section of cadmium at energies greater than the resonance energy (continuation of Figs. 19 and 20).

In the present case (as in many other cases with the capture of a slow neutron), two transition processes compete: emission of \(\gamma\)-rays and emission of a neutron, i.e. resonance elastic scattering (inelastic nuclear scattering at small neutron energies is impossible). Accordingly, the total width will be equal to \(\Gamma = \Gamma_r + \Gamma_n\), where \(\Gamma_r\) is the radiation width and \(\Gamma_n\) is the neutron width. The physical meaning of the partial (including neutron) width consists in the fact that it deter—

... determines the lifetime of the nucleus with respect to the corresponding transition process and, consequently, is proportional to the effective cross section of the given process.

The quantities \(\sigma_0\), \(E_0\), and the ratio \(\dfrac{\Gamma_n}{\Gamma}\) are connected by Bethe’s formula

\[ \sigma_0 = 1.30 \cdot 10^{-18} \left(1 \pm \frac{1}{2i+1}\right) \frac{1}{E_0}\frac{\Gamma_n}{\Gamma}, \]

where \(i\) is the spin of the capturing nucleus, in the present case the nucleus \(\mathrm{Cd}^{113}\), and the sign in the parentheses depends on whether the neutron spin is added to it or subtracted from it in the formation of the compound nucleus*). If one assumes \(i=\frac{1}{2}\), then, depending on the sign in the parentheses, \(\dfrac{\Gamma_n}{\Gamma}\) is equal either to \(0.53\cdot 10^{-2}\) or to \(1.6\cdot 10^{-2}\). With other assumptions about the spin of \(\mathrm{Cd}^{113}\), as is easy to see, intermediate ratios are obtained.

Fig. 22. Permeability of a cadmium sample of \(11.03\ \mathrm{g/cm^2}\) and the total cross section for neutrons of high energies.

Meanwhile, this ratio (with an accuracy up to the interference of resonance scattering with potential scattering, which apparently is not very important in the present case) is proportional to the ratio of the resonance cross sections of scattering and capture. Consequently, \(\sigma_{\text{scat}}/\sigma_{\text{capt}}\) in resonance is of the order of \(1/100\).

Recently an experimental confirmation of this conclusion has been obtained.

Biman \(^{31}\) finds that resonance scattering in Cd has a cross section

\[ (40 \pm 15)\cdot 10^{-24}\ \mathrm{cm^2}. \]

The sensitivity of comparing the experimental results for Cd with the Breit–Wigner formula is such that, for a specified \(\sigma_0=7200\), a change of \(E_0\) or \(\Gamma\) by \(\pm 0.001\) gives noticeably worse agreement. If \(\sigma_0\) is changed by \(\pm 200\), and \(\Gamma\) is adjusted, then the agreement does not extend to the region of small energies (Fig. 19). From these considerations the indicated errors were determined.

Thus, experiment establishes that cadmium resonantly absorbs neutrons with energy \(E_0=0.176\ \mathrm{eV}\), having for this energy a cross section

*) The formula is valid for \(\Gamma_n \ll \Gamma_\gamma\); consequently, for \(\Gamma \simeq \Gamma_\gamma+\Gamma_n=\Gamma\), it is valid also in the present case.

\(7200 \cdot 10^{-24}\ \text{cm}^2\). Toward higher energies the cross section drops sharply (see Fig. 21). Toward lower energies the cross section at first decreases, reaching a minimum \(\sigma_{\min}=2200 \cdot 10^{-24}\ \text{cm}^2\) at an energy of \(0.03\text{--}0.04\ \text{eV}\) (the mean thermal energy at normal temperature), and then increases in accordance with the factor \(E^{-1/2}\sim 1/v\) in the Breit--Wigner formula. Despite this minimum in the region of thermal neutrons, the cadmium cross section remains sufficiently large and, of course, does not undermine cadmium’s reputation as a good absorber of thermal neutrons. But the absorption boundary is determined by the thickness of the absorber, since even at an energy of \(0.5\ \text{eV}\) the cross section exceeds \(100 \cdot 10^{-24}\ \text{cm}^2\), while \(10 \cdot 10^{-24}\ \text{cm}^2\) is reached only at \(1.5\ \text{eV}\).

All cross sections are given as averages per atom of normal cadmium with its natural isotopic content. Since the absorbing isotope \(\mathrm{Cd}^{113}\) is present in an amount of \(12.3\%\), for it the actual resonance cross section is
\[ \sigma_0(\mathrm{Cd}^{113})=58500 \cdot 10^{-24}\ \text{cm}^2 . \]
We used this value in calculating the ratio \(\frac{I_n}{I}\).

RESONANCE ABSORPTION IN OTHER ELEMENTS

1. Indium

Indium is known as a good selective absorber of so-called \(D\)-neutrons and has been widely used as a neutron indicator for thermal (\(C\)) and \(D\)-neutrons, since neutron capture leads to the formation of the radioactive isotope \(\mathrm{In}^{116}\) with half-lives of 13 sec and 54 min. (two isomeric states).

The study of the absorption spectrum by means of selectors was first carried out by Becker and Bothe\(^{10}\), then by Becker, Bothe, and McDaniel\(^{11}\), Havens and Rainwater\(^{17}\), McDaniel\(^{12}\), and finally by Havens, Wu, Rainwater, and Meaker\(^{19}\).

The last two, most accurate, works are in mutual agreement and prove the presence of a strong resonance max-

Fig. 23. Transmission of \(0.193\ \text{g}/\text{cm}^2\) of indium. Three resonance minima at energies \(8.6\), \(3.8\), and \(1.44\ \text{eV}\).

of the maximum of the cross section at an energy of 1.44 eV and of weaker maxima at energies of 3.8 and 8.6 eV. In Figs. 23 and 24 the results of measurements3 are given in the form of permeability curves for samples of various thicknesses. At the right a scale of cross sections is given in units of \(10^{-24}\ \mathrm{cm}^2\).

Fig. 24. Permeability of \(0.03865\ \mathrm{g/cm^2}\) indium. Krivaya—according to the Breit–Wigner formula for \(E_0 = 1.44\ \mathrm{eV}\), \(\sigma_0\Gamma^2 = 210\) and \(\Gamma = 0.09\ \mathrm{eV}\).

The authors summarize the results by the following numerical data*:

1. First resonance
   \[ E_0 = 1.44 \pm 0.02,\qquad \sigma_0\Gamma^2 \sim 210 \]
   (if \(\Gamma \sim 0.09\), then \(\sigma_0 \sim 26000\)).

2. Second resonance
   \[ E_0 = 3.8 \pm 0.2,\qquad \sigma_0\Gamma^2 \sim 120 \]
   (if \(\Gamma \sim 0.1\), then \(\sigma_0 \sim 1200\)).

3. Third resonance
   \[ E_0 = 8.6 \pm 0.4,\qquad \sigma_0\Gamma^2 \sim 300 \]
   (if \(\Gamma \sim 0.1\), then \(\sigma_0 \sim 3000\)).

The cross section for thermal neutrons, according to MacDaniel,[^12] varies according to the law \(1/v\) and for an energy of \(0.025\ \mathrm{eV}\) is equal to \(191 \cdot 10^{-4}\ \mathrm{cm}^2\).

Since indium has only two stable isotopes (\(\mathrm{In}^{113}\) and \(\mathrm{In}^{115}\)), at least two of the levels found belong to one of them, and consequently the distance between two neighboring levels in this case is less than 8 eV.

* In what follows all energies and widths are given in eV, and cross sections in \(10^{-24}\ \mathrm{cm}^2\).

2. Silver

Silver is known as a selective absorber of \(A\)-neutrons and was used as an indicator of neutrons possessing strongly excited activities with periods of 22 sec and 2.3 min. The absorption spectrum was investigated by Becker, Bacher, and McDaniel\({}^{11}\), Havens and Rainwater\({}^{17}\), and Rainwater, Havens, Wu, and Dunning\({}^{18}\).

Fig. 25, 26. Transmission of \(7.84\ \text{g}/\text{cm}^2\) and \(0.276\ \text{g}/\text{cm}^2\) of silver.

Figures 25 and 26 give the transmission curves from the latter work. A strong resonance maximum of the cross section (minimum of transmission) appears at an energy of 5.1 eV, and two weaker ones at energies of 16 and 45 eV. Since the resolving power in the region of these energies is small, all three experimental peak values of the cross section are lower than the true ones, and the cross-section scale is valid only in the region of smooth changes of transmission.

Numerical results:

1. First resonance
   \[ E_0 = 5.1 \pm 0.2, \qquad \sigma_0\Gamma^2 = 300 \pm 50. \]

2. Second resonance
   \[ E_0 = 16 \pm 1, \qquad \sigma_0\Gamma^2 = 24. \]

3. Third resonance
   \[ E_0 = 45 \pm 4, \qquad \sigma_0\Gamma^2 = 700. \]

In the region of thermal energies the total cross section varies according to the law \(1/v\), and it can be represented in the form

\[ \sigma_{\text{therm}} = \left[(9.05 \pm 0.30)E^{-1/2} + (6.6 \pm 0.5)\right]\cdot 10^{-24}\ \text{cm}^2 . \]

Silver, like indium, has two stable isotopes, \(\mathrm{Ag}^{107}\) and \(\mathrm{Ag}^{109}\), and consequently at least two of the levels found here belong to one of them.

Fig. 27. Permeability of gold, \(0.210\ \text{g}/\text{cm}^2\). Resonance at \(4.8\ \mathrm{eV}\).

3. Gold

Gold is known as a strong absorber of \(A\)-neutrons and has been used as a neutron indicator with a period of 2.7 days. It was investigated by Havens and Rainwater\(^{17}\), Havens, Rainwater, Wu, and Muehlhause\(^{19}\) in the resonance region, and by Brill and Lichtenberger\(^{6}\) and McDaniel\(^{21}\) in the thermal region.

The results are presented in Figs. 27 and 28. A strong single resonance level appears at the energy \(4.8\ \mathrm{eV}\). The experimental peak value of the cross section is lower than the true one. If one assumes \(\Gamma \sim 0.1\), then \(\sigma_0 \sim 60\,000\). In the thermal region the cross section varies according to ...

Fig. 28. Total cross section of gold in the region of low energies.

according to the law \(1/v\) and is equal to \(100 \cdot 10^{-24}\ \text{cm}^2\) for neutrons with velocity \(2200\ \text{m/sec}\). Mack, Daniel, and others give the following formula for the thermal region:

\[ \sigma_{\text{therm}}=\{(14.9 \pm 0.3)E^{-1/2}+(9.0 \pm 0.2)\}\cdot 10^{-24}\ \text{cm}^2. \]

4. Manganese

From experiments by the boron-absorption method it was known that manganese absorbs, in addition to thermal neutrons, a resonance group of neutrons with an energy of several tens of eV, for example, according to the measurements of Goldsmith and Rasetti—60 eV. Manganese was used as a neutron absorber and indicator with a period of 2.59 hours. In experiments for determining the absolute intensity of neutron sources it is convenient as an absorber and indicator, since in the form of a solution it can be introduced into a large volume of water.

Measurements with selectors revealed a resonance at a considerably higher energy than indirect experiments had indicated, and thereby showed the inaccuracy of the boron-absorption method for large resonance energies, as should have been expected. Manganese was investigated by Rainwater, Havens, Wu, and Dunning2. The results of the investigation are presented in Figs. 29 and 30.

Fig. 29. Transmission of \(16.24\ \text{g/cm}^2\) of manganese. In the thermal region the law is \(1/v\).

\[ \sigma = 2.24\,E^{-1/2}+2.26. \]

A strong resonance was found at energy \(E_0 = 300 \pm 40\ \text{eV}\). Owing to the low resolving power, the experimental peak cross section is, of course, considerably lower than the true one. For neutron energies below 30 eV the cross section may be represented in the form

\[ \sigma=[(2.24 \pm 0.05)E^{-1/2}+(2.2 \pm 0.4)]\cdot 10^{-24}\ \text{cm}^2. \]

The constant term 2.2 corresponds to the scattering cross section far from resonance. In the resonance region, as we have already noted above, the cross section

scattering, according to special experiments by Langsdorf et al.\(^{26}\), turns out to be very large and exceeds the resonance-capture cross section. The curve in Fig. 30 shows the transmission for a neutron beam, and therefore corresponds to both effects: absorption and scattering.

Fig. 30. Transmission of \(16.24\ \text{g/cm}^2\) of manganese in the region of high energies. Resonance at an energy of about \(300\ \text{eV}\).

5. Antimony

Antimony was investigated by Havens and Rainwater\(^{17}\) and by Rainwater, Havens, Wu, and Dunning\(^{18}\). Three resonance levels were found (two stable isotopes) at energies \(5.8 \pm 0.15\ \text{eV}\), \(15\ \text{eV}\), and \(21\ \text{eV}\). The results of the measurements are presented in Fig. 31 in the form of a transmission curve. In the thermal region the cross section may be represented in the form

\[ \sigma_{\text{therm}}=\left[(0.64\pm0.04)E^{-1/2}+(4.2\pm0.5)\right]\cdot10^{-24}\ \text{cm}^2. \]

Consequently, the scattering cross section far from resonance is

\[ \sigma_{\text{scat}}=(4.2\pm0.5)\cdot10^{-24}\ \text{cm}^2. \]

6. Iodine

Iodine is known as an absorber and indicator of \(J\)-neutrons with a period of 25 min. Measurements by the method of absorption in boron indicated the presence of a very broad resonance level \((\Gamma \sim 15\ \mathrm{eV})\) at an energy of the order of 100 eV (for example, according to Goldsmith and Rasetti—140 eV).

Fig. 31. Permeability of \(6.84\ \mathrm{g/cm^2}\) (upper curve) and \(22.4\ \mathrm{g/cm^2}\) (lower curve) of antimony.

Investigations with the aid of selectors were carried out by Bø, Rainwater and Havens\(^{20}\), and by Johns\(^{22}\). The latter work is more detailed and is devoted especially to iodine. The results of the measurements, presented in Figs. 32 and 33, we take from the first work, where they are given more clearly. The first resonance level appears at an energy of \(20.6 \pm 0.4\ \mathrm{eV}\). According to Johns, \(E_0 = 20.3\), \(\sigma_0 > 40 \cdot 10^{-24}\ \mathrm{cm^2}\), \(\Gamma < 0.8\ \mathrm{eV}\).

At an energy of about 37 eV a broad absorption band has been found, extending from 30 to 45 eV (according to Johns). Rainwater and Havens suppose here two unresolved levels at energies \(32 \pm 2\ \mathrm{eV}\) and \(42 \pm 2\ \mathrm{eV}\). There are not very reliable indications of levels at energies of about 80 eV, 200 eV, and 600 eV.

In the thermal region the cross section may be represented in the form

\[ \sigma_{\mathrm{therm}} = \left[(1.12 \pm 0.05)E^{-1/2} + (3.8 \pm 0.2)\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

According to Johns,

\[ \sigma_{\mathrm{therm}} = \left[1.06E^{-1/2} + 3.6\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

Consequently, the scattering cross section far from resonance is

\[ \sigma_{\text{scat}}=(3.7\pm0.2)\cdot 10^{-24}\ \text{cm}^2 . \]

But at energies below 0.04 eV the total cross section changes twice in jumps—

Fig. 32. Transmission of \(18.86\ \text{g}/\text{cm}^2\) of iodine. From 0.04 to 3 eV \(\sigma = 1.12E^{-1/2}+3.8\).
Diffraction jumps at wavelengths 1.46 and 2.37 Å.

Fig. 33. Transmission of \(18.86\ \text{g}/\text{cm}^2\) of iodine. Near 37 eV there are probably two levels at energies 32 and 42 eV.

jumps, as is seen from Fig. 32. These jumps of the cross section are connected with the diffraction of neutron waves in crystalline iodine. They are observed

at wavelengths for which Bragg reflection becomes impossible \((\lambda > 2d \sin \theta)\). These neutron wavelengths, at which jumps in the cross section are observed, differ greatly among various authors. Rainwater and Havens give \(1.46\ \text{Å}\) and \(2.37\ \text{Å}\), and Jones \(3.9\ \text{Å}\) and \(4.8\ \text{Å}\), although in both cases pure crystalline iodine was used. In any event, the very fact of a stepwise change of the cross section is shown quite clearly, especially by Jones. It should be noted that iodine has only one stable isotope; consequently, all the levels discovered correspond to excited states of \(J^{128}\).

7. Mercury

The transmissibility of mercury, investigated by Havens and Rainwater\(^{17}\), reveals an interesting feature: at energies of \(10\ \mathrm{eV}\) and below it decreases with energy faster than would be expected if the cross section varied according to the law \(\frac{1}{v}\). Such a behavior is possible only in the case when the resonance level corresponds to a very small positive or negative value of the resonance energy of the neutron. A negative kinetic energy of the neutron, of course, has no physical meaning, but the corresponding resonance level is quite possible. Indeed, if the resonance energy (kinetic) of the neutron is \(E_n\), then the corresponding excitation energy of the compound nucleus is

\[ E^* = \varepsilon_n + E_n, \]

where \(\varepsilon_n\) is the binding energy of the neutron in the compound nucleus.

Fig. 34. Transmissibility of \(3.25\ \mathrm{g/cm^2}\) of mercury. The solid curve is according to the Breit–Wigner formula at \(E=-2.0\ \mathrm{eV}\); the dashed curve corresponds to \(\sigma \sim \frac{1}{v}\).

Visible labels in the figure: “Transmissibility”; “Scattering constant”; “Hg”; “Energy (eV)”; “Time of flight (\(\mu\) sec/m)”; \(0.01, 0.04, 0.1, 0.4, 1.0\); \(0, 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440\); \(0, 100, 200, 300, 400, 500, 600\); \(10, 5, 1, 0.5, 0.25, 0.10, 0.05, 0.025\).

But nuclear levels for which \(E^* < \varepsilon_n\) are also quite possible. In such a case, the quantity \(E_n\) determined from the equality will be negative.

sign. If \(E_n\) is negative but small in absolute value, i.e., if the neutron binding energy is only slightly greater than the excitation energy of the corresponding nuclear level, then in the region of small neutron energies there will fall the tail of the resonance curve, in which the cross section increases with decreasing velocity faster than \(1/v\). From this point of view Havens and Rainwater analyzed the experimental transmission curves for mercury, comparing them with the Breit–Wigner formula. The best agreement of the experimental points with the Breit–Wigner curve is obtained if one assumes \(E_0=-2\ \mathrm{eV}\) and a certain constant cross-section term equal to \(15\cdot 10^{-24}\ \mathrm{cm}^2\).

Fig. 35. Transmission of \(12.9\ \mathrm{g/cm}^2\) of mercury. The dashed curves are for \(E_0=-2.25\ \mathrm{eV}\) and \(E_0=-1.75\ \mathrm{eV}\), the solid one for \(E_0=-2.0\ \mathrm{eV}\). The straight line: \(\sigma=64E^{-1/2}+15\). The constant term was selected by trial and error.

The results of the measurements and comparisons are presented in Figs. 34, 35, 36. Each figure corresponds to a definite absorber thickness; moreover, the greater the absorber thickness, the more accurately the region of high energies was investigated. In Figs. 35 and 36 the extreme dashed curves correspond to \(E_0=-2.25\ \mathrm{eV}\) and \(-1.75\ \mathrm{eV}\), while the middle curve, which agrees best, corresponds to \(E_0=-2.0\ \mathrm{eV}\). The adopted constant cross-section value \(15\cdot 10^{-24}\ \mathrm{cm}^2\), shown in the figure by the horizontal straight line, has no physical meaning and was chosen as giving the best agreement of the curves with the experimental points. Changing it in either direction gives a systematic divergence of the results from the Breit–Wigner curve. At energies of about \(30\ \mathrm{eV}\) the authors assume the presence of several unresolved resonance levels. If their influence in the low-energy region is taken into account, one can obtain a negative resonance energy somewhat smaller in absolute value

chine, but in any case not less than 1.25 eV. Assuming, however, that the influence of positive levels on the course of the curve at energies below

Fig. 36. Transmission of \(34.4\ \mathrm{g/cm^2}\) of mercury. The notation is the same as in Fig. 35. The disagreement of the data at \(E > 10\ \mathrm{eV}\) indicates the presence of positive levels.

10 eV is insignificant, the authors give the following final results:

\[ -\,E_0 = 2.0 \pm 0.2,\qquad \frac{\sigma_0\Gamma^2}{4E_0^{3/2}} = 64 \pm 3,\qquad \sigma_{\mathrm{const}} = 15 \pm 1. \]

The presence of more than one level near 25 eV is possible.

8. Iridium

Iridium was investigated by Rainwater, Havens, Wu, and Dunning2. The results are presented in Figs. 37 and 38. The authors summarize them by the following numerical data:

1. \[ \sigma_{\mathrm{therm}}=\left[(64\pm2)E^{-1/2}+(14\pm6)\right]\cdot10^{-24}\ \mathrm{cm^2}. \]

2. First resonance

\[ E_0=0.64\pm0.015,\qquad \sigma_0\Gamma^2=(\text{from }5\text{ to }20), \]

\[ \sigma_0 \gtrsim 4500\ \text{(probably)},\qquad \Gamma \leq 0.07\ \text{(probably)}. \]

1. Second resonance

\[ E_0=1.27\pm0.04,\qquad \sigma_0\Gamma^2=(\text{from }5\text{ to }20), \]

\[ \sigma_0 \gtrsim 4000\ \text{and}\ \Gamma \leq 0.07\ \text{(probably)}. \]

Fig. 37. Permeability of \(3.08\ \mathrm{g/cm^2}\) of iridium. In the thermal region
\[ \sigma=(64.5E^{-1/2}+14)\cdot 10^{-24}\ \mathrm{cm^2}. \]

Fig. 38. Permeability of \(3.08\ \mathrm{g/cm^2}\) (left) and \(0.0424\ \mathrm{g/cm^2}\) of iridium.

1. For \(2.2\ \mathrm{eV} \leq E \leq 3.5\ \mathrm{eV}\), \(\sigma = 25 \pm 4\).

2. Third resonance

\[ E_0 = 5.2 \pm 0.2,\qquad \sigma_0 \Gamma^2 = 55 \text{ (roughly).} \]

1. Fourth resonance

\[ E_0 = 8.7 \pm 0.3,\qquad \sigma_0 \Gamma^2 = 50 \text{ (roughly).} \]

1. Near \(E = 14\ \mathrm{eV}\), \(\sigma = 18 \pm 3\).

2. Fifth resonance \(E_0 = 25 \pm 5\).

3. The course of the curve for \(E > 25\ \mathrm{eV}\) indicates the possible presence of several levels here.

These at least five levels belong to two isotopes, since iridium has two stable isotopes, \(\mathrm{Ir}^{191}\) and \(\mathrm{Ir}^{193}\).

9. Tantalum

Tantalum was investigated by Havens, Wu, Rainwater, and Macklin \({}^{19}\). The results are presented in Figs. 39, 40. The authors summarize them with the following numerical data:

1. Below \(1\ \mathrm{eV}\)

\[ \sigma = \bigl[(3.0 \pm 0.1)E^{-1/2} + (7.2 \pm 0.4)\bigr]\cdot 10^{-24}\ \mathrm{cm}^2. \]

1. The first and principal resonance appears at

\[ E_0 = 4.1 \pm 0.1,\qquad \sigma_0 \Gamma^2 \sim 44 \]

(if the region \(1/v\) is assigned only to one level, \(4.1\ \mathrm{eV}\), then \(\sigma_0\Gamma^2 = 96\)).

1. Second strong resonance

\[ E_0 = 10.0 \pm 0.3,\qquad \sigma_0 \Gamma^2 \sim 25. \]

1. Third resonance

\[ E_0 = 13 \pm 0.5,\qquad \sigma_0 \Gamma^2 \sim 3. \]

1. Fourth resonance

\[ E_0 = 22 \pm 2,\qquad \sigma_0 \Gamma^2 \sim 18. \]

1. Fifth resonance

\[ E_0 = 37 \pm 3,\qquad \sigma_0 \Gamma^2 \sim 400. \]

1. There are indications of the presence of more than one level near \(100\ \mathrm{eV}\), near \(300\ \mathrm{eV}\), and at still higher energies.

Fig. 39. Transmission of \(22.4\ \mathrm{g/cm^2}\) of tantalum. In the thermal region \(\sigma=(3.0E^{-1/2}+7.2)\cdot 10^{-24}\ \mathrm{cm}^2\).

It is noteworthy that tantalum has only one stable isotope, Ta\(^{181}\); consequently, all these levels refer to Ta\(^{182}\). The authors

Fig. 40. Transmission of 9.98 g/cm\(^2\) (upper curve) and 22.44 g/cm\(^2\) (lower curve) of tantalum.

consider the influence of impurities in the tantalum sample on these results to be incredible.

10. Tungsten

Tungsten was investigated in the same work as tantalum\(^{19}\). The results are presented in Figs. 41, 42, and 43. The authors summarize them with the following data:

1. Below 1 eV
   \[ \sigma=\left[(2.72\pm0.05)E^{-1/2}+(5.7\pm0.2)\right]\cdot 10^{-24}\ \text{cm}^2. \]

2. First strong resonance
   \[ E_0=(4.0\pm0.1),\qquad \sigma_0\Gamma^2\sim 13. \]

3. Second (weaker)
   \[ E_0=7.4\pm0.2,\qquad \sigma_0\Gamma^2\sim 5. \]

4. Third, the strongest
   \[ E_0=18.0\pm0.5,\qquad \sigma_0\Gamma^2\sim 3000. \]

This level may be complex (not single), since toward higher energies the curve falls more slowly than would be expected for a single level.

1. Fourth resonance (possibly complex)
   \[ E_0=45\pm2,\qquad \sigma_0\Gamma^2\sim 400. \]

1. The fifth minimum of the curve at

\[ E_0 = 180 \pm 20,\qquad \sigma_0 \Gamma^2 \sim 10000, \]

if the level is single.

Figure labels: Transmission; Energy (eV); Flight time (μ sec/m); Cross section \((10^{-24}\ \mathrm{cm}^2)\); W.

Fig. 41. Transmission of \(29.2\ \mathrm{g/cm^2}\) of tungsten. In the thermal region

\[ \sigma = 2.72 E^{-1/2} + 5.7. \]

Figure labels: Transmission; Energy (eV); Flight time (μ sec/m); Cross section \((10^{-24}\ \mathrm{cm}^2)\); W; \(1100\ \mathrm{eV}\), \(180\ \mathrm{eV}\), \(45\ \mathrm{eV}\), \(18.0\ \mathrm{eV}\), \(7.4\ \mathrm{eV}\).

Fig. 42. Transmission of \(5.06\ \mathrm{g/cm^2}\) of tungsten.

1. Another minimum near \(1100\ \mathrm{eV}\) corresponds, probably, to several levels.

Tungsten has five stable isotopes. Strong resonance absorption probably belongs to the single odd isotope \(W^{183}\), although the even isotopes \(W^{184}\) and \(W^{186}\) also capture neutrons and give known activities with periods of 77 days and 24.1 hours.

Fig. 43. Permeability of \(4.90\ \text{g}/\text{cm}^2\) of tungsten.

11. Platinum

Platinum was investigated in the same work\({}^{19}\). The results are presented in Figs. 44, 45, and 46. The authors’ conclusions from these results are as follows:

1. In the interval \(0.04\)—\(0.8\ \text{eV}\)

\[ \sigma=\left[(1.03\pm0.06)E^{-1/2}+(120\pm0.3)\right]\cdot10^{-24}\ \text{cm}^2. \]

1. At energies \(<0.035\ \text{eV}\), owing to diffraction of neutron waves by crystals of metallic platinum, there are jumps in permeability corresponding to wavelengths \(1.6\ \text{\AA}\) and \(2.4\ \text{\AA}\).

2. Resonance at

\[ E_0=11.5\pm0.4,\qquad \sigma_0\Gamma^2\sim55. \]

1. Resonance at

\[ E_0=18.2\pm1,\qquad \sigma_0\Gamma^2\sim30. \]

1. Broad minima of permeability at energies around \(100\ \text{eV}\) and \(1000\ \text{eV}\) indicate the presence of strong unresolved levels. The number of stable isotopes of platinum is 5.

Fig. 44. Transmission of a 25.8 g/cm² platinum plate.

Fig. 45. Transmission of a 25.8 g/cm² platinum plate. The minimum at 1.25 eV is probably due to an impurity of rhodium.

Fig. 46. Transmission of a 25.8 g/cm² platinum plate. Deviations from \(1/v\) are caused by interference of neutron waves on platinum crystals.

N. A. VLASOV

12. Zirconium

Zirconium was investigated in the same work as the three preceding elements3. The samples were made from \( \mathrm{ZrO_2} \), and the correction for oxygen was taken from the calculation that its scattering cross section is constant and equal to \(4.1 \cdot 10^{-24}\ \mathrm{cm^2}\).

Fig. 47. Transmission of \(6.45\ \mathrm{g/cm^2}\) \( \mathrm{ZrO_2} \). In the thermal region
\[ \sigma_{\mathrm{Zr}} = 0.74 E^{-1/2} + 6.8 . \]

Fig. 48. Transmission of \(6.45\ \mathrm{g/cm^2}\) \( \mathrm{ZrO_2} \). The minimum near \(7\ \mathrm{eV}\) is probably due to two unresolved levels at energies \(5.7\) and \(7.6\ \mathrm{eV}\).

The results are presented in Figs. 47 and 48. The authors’ conclusions are as follows:

1. Below \(0.6\ \mathrm{eV}\)
   \[ \sigma = \left[(0.74 \pm 0.10)E^{-1/2} + (6.8 \pm 0.3)\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

2. Resonance at
   \[ E_0 = 1.09 \pm 0.03,\qquad \sigma_0\Gamma^2 \sim 5. \]

1. The second resonance at

\[ E_0 = 2.30 \pm 0.07, \qquad \sigma_0 \Gamma^2 \sim 8. \]

1. There is a hint of a resonance at

\[ E_0 = 5.7 \pm 0.5, \qquad \sigma_0 \Gamma^2 \sim 10 \quad \text{(not reliable)}. \]

1. The third resonance

\[ E_0 = (7.6 \pm 0.4), \qquad \sigma_0 \Gamma^2 \sim 50. \]

Zirconium has five stable isotopes.

13. Osmium

Osmium was investigated by Bue, Rainwater, and Havens \(^{20}\). The samples were prepared from metallic powder compressed under high pressure. The results are presented in Figs. 49 and 50 and are summarized by the following data:

Fig. 49. Transmission of \(17.3\ \mathrm{g/cm^2}\) of osmium. Below \(0.1\ \mathrm{eV}\), diffraction effect in crystals.

1. In the thermal region

\[ \sigma = \left[(2.7 \pm 0.1) E^{-1/2} + (15 \pm 0.4)\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

1. There are jumps in the transmission due to diffraction, especially at \(\lambda = 1.75\ \text{\AA}\) and \(2.36\ \text{\AA}\).

2. Resonance at

\[ E_0 = 6.5 \pm 0.3, \qquad \sigma_0 \Gamma^2 \sim 10. \]

1. Resonance at

\[ E_0 = 8.8 \pm 0.3, \qquad \sigma_0 \Gamma^2 \sim 35. \]

1. Resonance at

\[ E_0 = 20 \pm 1,\quad \sigma_0\Gamma^2 \sim 25. \]

1. Probable resonance at

\[ E_0 = 28 \pm 1.5,\quad \sigma_0\Gamma^2 \sim 8 \]

(to within a factor of 10).

1. Resonance at

\[ E_0 = 42 \pm 2,\quad \sigma_0\Gamma^2 \sim 10 \]

(to within a factor of 5).

1. Resonance at \(E_0 = 84 \pm 6\) and others unresolved at higher energies.

Fig. 50. Transmission of \(17.3\ \mathrm{g/cm^2}\) of osmium. The curve was taken with maximum resolving power.

14. Cobalt

Cobalt was investigated in the same work\(^{20}\). It is of interest because a very strong resonance was found at an energy of 115 eV, the experimentally obtained value \(\sigma_0\Gamma^2 = 200\,000\) being considerably greater than that which should be expected from the course of the curve in the region where the law \(1/v\) is valid below 5 eV \((\sigma_0\Gamma^2 = 30\,000)\). The authors point to the possible influence on the scale of the curve in the \(1/v\) region of the resonance levels they found at energies of the order of 1000 and 10,000 eV, or to the possibility that the level at 115 eV is complex. In fact, apparently, the explanation of this discrepancy lies in the fact that experimentally

discovered by them is due not so much to absorption as to resonance scattering. The large cross section of resonance scattering in cobalt was later proved by the experiments of Harris, Langsdorf, and Seidl28. Along with manganese, cobalt provides an example of such a nuclear level for which the neutron width exceeds the radiative one. It should apparently be expected that such examples will not be isolated if resonance at high neutron energies is considered.

Fig. 51. Transmission of \(12.4\ \mathrm{g/cm^2}\) of cobalt.

The results of the investigations of cobalt are presented in Figs. 51 and 52 and are summarized by the following data:

1. Below \(5\ \mathrm{eV}\)

\[ \sigma = (6.4 \pm 0.15)\cdot 10^{-24}\ \mathrm{cm^2}. \]

1. A strong resonance minimum of transmission at

\[ E_0 = 115 \pm 5,\qquad \sigma_0 \Gamma^2 \sim 200\,000, \]

if the minimum is due to a single level.

If the \(1/v\) behavior is attributed to a single level at \(E_0 = 115\ \mathrm{eV}\), then

\[ \sigma_0 \Gamma^2 \sim 30\,000. \]

1. There are indications of strong unresolved levels in the region from \(1000\) to \(10\,000\ \mathrm{eV}\).

Fig. 52. Transmission of \(1.06\ \mathrm{g/cm^2}\) of cobalt.

15. Thallium

Thallium was investigated in the same work*20. The results of the measurements are presented in Figs. 53 and 54.

It is analogous to cobalt, since it has a strong resonance at \(E_0 = 270\ \mathrm{eV}\), for which \(\sigma_0 \Gamma^2 \sim 20\,000\), if it pertains to a single level.

Fig. 53. Permeability of \(52.03\ \mathrm{g/cm^2}\) of thallium.

Fig. 54. Permeability of \(52.09\ \mathrm{g/cm^2}\) of thallium in the region of the minimum, taken with maximum resolving power.

level. In the interval from 50 to \(0.5\ \mathrm{eV}\) the cross section may be represented in the form

\[ \sigma = \left[(0.6 \pm 0.2)\cdot E^{-1/2} + (9.7 \pm 0.2)\right]\cdot 10^{-24}\ \mathrm{cm^2}. \]

If this \(1/v\) behavior is due to one level at \(E_0 = 270\ \mathrm{eV}\), then

$g_l\Gamma^2$ must be 11,000. We suppose that this discrepancy, as in the case of cobalt, means that at $E_0 = 270\ \mathrm{eV}$ thallium has a large cross section for resonance scattering of neutrons. In addition, a second unresolved resonance is indicated at $E_0 \sim 1100\ \mathrm{eV}$. In the region below $0.5\ \mathrm{eV}$ there are strong diffraction effects, distorting the law $\frac{1}{v}$.

16. Columbium

Columbium was investigated in the same work.[^20] The results are presented in Figs. 55 and 56. The small resonance minimum of the transmission at an energy of about $4\ \mathrm{eV}$ is attributed to an impurity

Fig. 55. Transmission of $19.37\ \mathrm{g/cm^2}$ of columbium. Strong diffraction effect at low energies.

Fig. 56. Transmission of $19.37\ \mathrm{g/cm^2}$ of columbium. The weak minimum at $4.1\ \mathrm{eV}$ may be attributed to an impurity of tantalum $(0.4\%)$.

of tantalum in the composition of the sample. At lower energies there is a weakly expressed $\frac{1}{v}$ law

\[ [\sigma=(0.10 \pm 0.04)E^{-1/2} + (6.4 \pm 0.2)], \]

which, probably, is also due to an impurity. Columbium itself shows no resonance. In the region of low energies, however, very distinct diffraction effects appear. Especially sharp jumps in transmission correspond to wavelengths of $2.05$ and $2.67\ \text{\AA}$ (see Fig. 55).

17. Germanium

Germanium was studied in the same work[^20]. The results are presented in Figs. 57 and 58. A minimum of transmission was found, corresponding to a resonance at an energy of about 95 eV. If the level is single, then \(\sigma_0 \Gamma^2 \sim 800\). From the course of the curve one may suspect that the level is complex.

Fig. 57. Transmission of \(4.70\ \mathrm{g/cm^2}\) of germanium.

Fig. 58. Transmission of \(4.70\ \mathrm{g/cm^2}\) of germanium in the region of the minimum. The minimum at 95 eV is probably complex.

At energies below 40 eV the cross section is practically constant (\(\sim 8.3 \cdot 10^{-24}\ \mathrm{cm^2}\)). The behavior of \(1/v\) at low energies is very strongly distorted by diffraction effects in the crystal, which cause a nonuniform decrease of the cross section with decreasing energy.

18. Gadolinium

Gadolinium is known as the strongest absorber of thermal neutrons, having an average cross section, according to old data, of the order of \(30\,000 \cdot 10^{-24}\ \mathrm{cm^2}\). It was studied by Brill and Lichtenberger[^6] with po-

of the power of the semi-mechanical selector in the energy interval from 0.002 to 0.2 eV. The results are presented in Fig. 59. The smooth curve, monotonically increasing with decreasing energy, corresponds to the Breit–Wigner formula. The optimal constants are \(\sigma_0 = 45\,000 \cdot 10^{-24}\ \mathrm{cm}^2\), \(E_0 = 0.028\ \mathrm{eV}\), \(\Gamma = 0.118\ \mathrm{eV}\). The resonance is found at a very small energy \(E_0 \ll \Gamma\), and the curve has no maximum, since in the Breit–Wigner formula the factor \(E^{-1/2}\) is dominant, while the resonance denominator changes only weakly.

Fig. 59. Effective cross section of gadolinium for slow neutrons. The solid curve is according to Breit–Wigner for \(\sigma_0 = 45\,000 \cdot 10^{-24}\ \mathrm{cm}^2\), \(E_0 = 0.028\ \mathrm{eV}\), \(\Gamma = 0.118\ \mathrm{eV}\).

As can be seen from Fig. 59, for very slow neutrons the cross section reaches a value of \(120\,000 \cdot 10^{-24}\ \mathrm{cm}^2\). Since this cross section belongs to one of the seven stable isotopes of gadolinium, the actual cross section of the nucleus capturing the neutron is at least four times larger, since the content of the most abundant isotope \(Gd^{158}\) is 23.45%. A study of the radioactivities excited by neutrons\(^{34}\) shows that none of the isotopes giving activity as a result of neutron capture has a large cross section. Consequently, the strong capture cannot be attributed to \(Gd^{158}\) or \(Gd^{160}\), since the corresponding isotopes \(Gd^{159}\) and \(Gd^{161}\) are \(\beta\)-active. The most probable neutron absorbers are the odd isotopes: \(Gd^{155}\) (15.61%) or \(Gd^{157}\) (16.42%). If the resonance capture of slow neutrons really belongs to one of them, then the observed maximum cross section proves to be

of the order of \(10^5 \cdot 10^{-24} = 10^{-19}\ \mathrm{cm}^2\). Experiments by Dempster et al.\(^{35}\) with a mass spectrograph do indeed show that, as a result of intense neutron irradiation, the content of \( \mathrm{Gd}^{157} \) decreases most strongly (from 16.42% to 9.86%), then \( \mathrm{Gd}^{155} \) (from 15.6% to 13.6%), while the content of the even isotopes (158 and 156) increases.

19. Dysprosium

Dysprosium is known as a widely used indicator of thermal neutrons, giving a strongly excited 2.5-hour activity. It was investigated in the same work as gadolinium\(^{6}\). The results are presented in Fig. 60.

Fig. 60. Effective cross section of dysprosium for slow neutrons.

In the energy interval from 0.15 to 0.007 eV the cross section increases with decreasing energy faster than according to the law

\[ \frac{1}{v}. \]

This indicates the presence of a resonance at lower energies.

Studies with a crystalline neutron monochromator\(^{33}\) reveal in Dy two resonances in the region of positive ener-

... (1.7 eV and 5.5 eV) and indicate the presence of a resonance at negative energy, agreeing in this respect with the results of Brill and Lichtenberger.

Without going into an analysis of the results obtained with the aid of a crystal monochromator, we shall note only that many elements have also been investigated by this method, in particular the rare earths that strongly absorb neutrons—Gd, Sm, Eu, Dy. The crystal-monochromator method has approximately the same, if not still narrower, limits of applicability as the selector-modulator method; therefore the results are in many respects analogous to those discussed here, and in those cases where they refer to the same object, for example Gd, Dy, Ir, they agree with them within the limits of experimental accuracy.

CONCLUSION

Investigations of the interaction of neutrons with matter by means of selectors cover a comparatively narrow energy interval from \(10^{-3}\) to 100 eV and, with difficulty, up to 1000 eV. Monochromatic neutrons with energies from \(10^3\) to \(10^5\) eV are still practically inaccessible to the experimenter. Extending measurements by means of selector-modulators into this energy region encounters serious technical difficulties and, if it is possible at all, only on the basis of fundamental changes and improvements in the method.

But the energy interval investigated appears to be the most interesting, since it includes slow neutrons, in particular thermal neutrons, which, first, interact most strongly with matter, and second, if one may so express it, are most widely distributed in nature*) and play a decisive role in systems with a self-sustaining chain reaction of fission of heavy nuclei.

Of course, knowledge of the neutron absorption spectra of the elements making up the structure of a pile is of great importance for understanding and taking account of possible processes and for the rational design of the pile; for this purpose the selector is a very valuable instrument.

But the theoretical significance of the problems already solved and of those accessible to solution by means of selectors is the most substantial. Experiments with selectors for the first time directly and convincingly proved the resonant character of the interaction of slow neutrons with nuclei. They confirmed the correctness of the theoretical ideas about the character of this interaction and gave these ideas quantitative foundations. In particular, they proved that the Breit–Wigner resonance formula correctly describes the dependence of the se-

*) In the atmosphere, among neutrons of cosmic origin, or in artificially created systems, there are, of course, more thermal neutrons than any others, since they live considerably longer than fast ones.

ture capture from the neutron energy. The results of these experiments give, for the first time directly and with an accuracy up to the third decimal place, the measured width of a nuclear level, and show that this width in most cases is of the order of 0.1 eV for nuclear excitation energies approximately equal to the neutron binding energy (\(\sim 8\) MeV).

It is essential to note that for different nuclei the level width varies only very slightly. In all cases where it has been measured reliably (Cd, Gd, Ir, In, and also Sm, Eu, Rh from experiments with a crystal monochromator \(^{37}\)), it falls within the interval from 0.07 to 0.2 eV. It follows from this that the lifetime of the excited states of different nuclei is of the order of \(10^{-14}\) sec.

Experiments with selectors also show directly that the density of levels in the investigated interval of excitation energies is rather large for some nuclei (for example, in \(Ta^{182}\) there are five levels in an interval of 40 eV). Alongside this, other nuclei show no resonant interaction at all in the interval investigated. No other method gives such a detailed investigation of the spectrum of states of the nucleus.

Finally, with the aid of selectors, processes of interference of neutron waves caused by the molecular or crystalline structure of matter were also distinctly observed and can be studied. In this case, an average-sized cyclotron is used as the neutron source, giving, in comparison, say, with a boiler, a very moderate number of neutrons.

Recently Rabi, Rainwater, and Havens \(^{33}\) have attempted to apply a selector even to the study of such a subtle effect as the interaction of neutrons with electrons.

From what has been said it follows, first, that the importance of selectors in the field of neutron physics is very great, and second, that their possibilities are far from exhausted, especially if it proves possible to extend the accessible interval of the energies investigated.

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1. Bethe. ↩

2. visible reference marker in the source. ↩↩

3. Reference 19 as cited in the source page. ↩↩