On the Absorption and Diffusion of Slow Neutrons*

E. Amaldi and E. Fermi

Abstract. The study of the absorption of slow neutrons has made it possible to establish the existence of absorption bands corresponding to fairly narrow energy intervals, which makes it possible to isolate relatively homogeneous groups of neutrons. In this work experiments will be described and some considerations presented concerning the absorption and diffusion of various groups of neutrons.

Contents: 1. Introduction. 2. Measurements and their treatment. 3. Selective absorption. 4. Groups of slow neutrons. 5. Systematic measurements of absorption. 6. Albedo. 7. Scattering of neutrons of one group. 8. Mean free path of thermal neutrons. 9. Groups as a function of distance from the source; energy relations. 10. Transition of neutrons from one group to another. 11. Behavior of groups near the surface of paraffin. 12. Width of absorption bands corresponding to groups. Total number of neutrons. 13. Summary and discussion.

1. Introduction

The purpose of this work is to give a systematic account of the studies we have carried out on the absorption and diffusion of slow neutrons[^1].

It is known that the simplest theory[^2] of the capture of a neutron by a nucleus leads to the result that the effective capture cross section for small neutron velocities is inversely proportional to the velocity, while the coefficient of proportionality may vary within wide limits from one element to another.

If this were true, then thicknesses of two different elements, inversely proportional to their coefficients of proportionality, would be exactly equivalent as absorbers of slow neutrons, independently of the neutron velocity and of the substance used as detector. That matters are not so simple had already been observed earlier by various authors[^3], who showed that, as a rule, the absorption by an element proved to be greater in the case when the slow neutrons were detected by means of activity induced in that same element.

Similar conclusions are also reached from the experiments of Moon and Tillman and others[^4], made at different temperatures. From these experiments

* Phys. Rev. 50, 899, 1936. Translated by A. Arsen’eva-Heil.

it follows that different detectors show different sensitivity when the temperature of the paraffin is changed. Finally, later, Rasetti, Pegram, Segrè, Fink, and Dunning[^5] measured, by means of a mechanical arrangement, the dependence of the absorption coefficient of thermal neutrons on their velocity and found that, at least in some cases—for example for Cd—the law \(\frac{1}{v}\) is not valid. In a systematic study of these phenomena we found that the selective absorption of slow neutrons appears more sharply if they are first filtered through a layer of Cd of sufficient thickness[^6]; this fact, independently observed by Szilard[^7], indicates the possible existence of relatively narrow absorption bands characteristic of various elements.

A systematic description of these phenomena can be obtained by resolving slow neutrons into groups sufficiently homogeneous with respect to their absorbing power[^8]. Such a separation may be carried out by resolving the absorption curves into exponential curves. We also studied various groups of neutrons by their scattering in substances containing hydrogen.

In Section 2 we describe the methods used in carrying out the experiments and in processing the results. In Sections 3, 4, and 5 some systematic data are collected on neutron absorption and on the intensity of the activities produced by various groups in different detectors. In Section 6 the measurement of the reflection coefficient (albedo) for neutrons of various groups is described; in Sections 7 and 8 we study the mean free path and the mean number of collisions for a neutron of a given group in paraffin. In Section 9 we show how the intensity of the activity due to different groups changes with change of the distance from the source in a large mass of water; from this relation we can derive the sequence of the groups with respect to energy and estimate the ratio of the energies of the different groups. In Sections 10 and 11 some experiments concerning diffusion are described. In Section 12 a method is described for determining the width of the energy bands corresponding to the various groups, and finally, in Section 13, all the results are collected and discussed.

In this paper we shall often use theoretical conclusions made by one of us[^9], and in what follows we shall denote this theoretical work by \(F\).

2. Measurements and Their Treatment

As a neutron source we always used small glass ampoules about \(4\ \mathrm{mm}\) in diameter and \(15\ \mathrm{mm}\) long, containing emanation (up to \(800\ \mathrm{mC}\)) and beryllium powder. Since our investigations continued over several months, it was obvious that it was necessary to establish a certain stan-

standard method for comparing results obtained with different sources and at different times.

We shall describe here the criteria used for this purpose. The activity measurements were made with an ionization chamber, shown schematically in Fig. 1, where the scale is also given. The chamber was filled with carbon dioxide at a pressure of 3 atm and was closed at the top by an aluminum window 0.1 mm thick and 7 cm in diameter. Since the aluminum window bulged strongly under the pressure, it was covered with a cellophane film, which had two purposes: to protect the chamber from possible contamination and to provide a flat surface on which the detectors were placed.

The ionization was measured by means of an Edelmann electrometer, the filament of which was projected onto a scale; it was possible to vary the sensitivity and the magnification of the filament image, so that we obtained sensitivities from 5 to 250 scale divisions per 1 V. In order to check the readings, we used a uranium oxide preparation enclosed in an aluminum packet \(5 \times 5\ \mathrm{cm}^2\). We shall denote this preparation and its activity by \(U\).

Ionization produced in our chamber by the preparation \(U\) was equivalent to the ionization obtained from a solution of \(0.96\ \mathrm{g}\) of uranyl nitrate \(\mathrm{UO}_2(\mathrm{NO}_3)_2 + 6\mathrm{H}_2\mathrm{O}\) in \(25\ \mathrm{cm}^3\) of water, placed in an aluminum tray 0.1 mm thick with a base of \(5 \times 5\ \mathrm{cm}^2\). Introducing a correction to the activity of this solution for absorption in the solution itself and in the aluminum, we find that it is equal to \(0.066\ \mathrm{g}\) of uranium element, i.e. 840 disintegrations per second.

Fig. 1.

In order to take readings more rapidly and to be able to use different electrometer sensitivities and different intervals of the scale, we constructed a nomogram. It allowed us, directly from the given electrometer sensitivity and the number of divisions obtained in the reading, to find the time during which \(U\) would have produced the same ionization. In this way, the correction for the nonlinearity of the instrument and for the dependence of the capacitance on the sensitivity was obtained automatically. Of course, the readings were corrected for the zero effect, which amounted to approximately 15% of \(U\).

The activity of a given detector, placed in a definite position relative to the source and surrounding objects, is obviously proportional to the number of neutrons emitted by the source. In order to have numbers that are mutually comparable, it is necessary always to divide the activity by the intensity of the source, measured in neutrons (neutron intensity). In order to determine it, ...

To determine it, it is not enough to assume that it is proportional to the amount of emanation contained in the source, which can be measured, for example, by \(\gamma\)-activity; in reality, small differences in the size of the beryllium grains or in the preparation of the source produce significant changes in the number of neutrons emitted. It is therefore convenient to measure the neutron intensity directly by means of the radioactivity produced in the corresponding detector. Our usual procedure was as follows: a rhodium plate \(5 \times 6.6\ \mathrm{cm}^2\) and \(10.15\ \mathrm{g}\) was activated when placed at the center of the upper surface of a paraffin cylinder

\[ 25\ \mathrm{cm}\ \text{in diameter},\ 15\ \mathrm{cm}\ \text{high}; \tag{1} \]

on the rhodium was placed a second paraffin cylinder \(13\ \mathrm{cm}\) in diameter and \(10\ \mathrm{cm}\) high; the source was placed on the axis of the lower cylinder at a distance of \(3\ \mathrm{cm}\) from its upper plane.

We shall denote by \(S\) the initial activity of rhodium with a period of \(44\ \mathrm{sec.}\), obtained under these conditions after an infinitely long irradiation. We shall call the neutron intensity \(I\) the ratio of this initial activity \(S\) to \(U\).

\[ I=\frac{S}{U}. \tag{2} \]

It should be noted that this definition depends, though not very noticeably, on the ionization chamber used. The strongest source that we used had a neutron intensity of \(144.9\). A neutron intensity equal to unity corresponds to a source containing from \(5\) to \(6\ \mathrm{mC}\) of emanation.

We are now in a position to determine what we shall call the activability \(A\) of a given detector (referred, if necessary, to one of its periods), placed in a definite position with respect to the source and the surrounding objects.

Let \(a\) be the initial activity of a detector irradiated for an infinitely long time; we define the activability \(A\) of this detector in the given position by the expression

\[ A=\frac{1000a}{S}=\frac{1000a}{IU}. \tag{3} \]

In practice, of course, we calculated the activability by means of this last formula: the neutron intensity was measured once for each source with very high accuracy, and from this value the intensity for any subsequent time was calculated according to the decay period of the emanation.

The initial intensity \(a\) was measured after irradiation for a finite interval of time and was then reduced to infinite time by well-known formulas.

In the case of elements with a short period it is convenient to establish a standard for measuring the initial activity, so as to be as independent as possible, at least for relative measurements, of possible errors in the value of the period. Thus, in the case of rhodium (44 sec.) and silver (22 sec.), which we used as detectors in a large number of experiments, we irradiated for 1 min.; in the case of silver the counts were made beginning from 20 to 80 sec. after the end of irradiation, and the activity measured in this way, multiplied by the corresponding factor, gave the initial activity. In fact, such a measurement gives not only the activity with a period of 22 sec., since there is a superposition of activity with a period of 2.3 min.; this activity was not very large, and usually no correction was introduced for it (see also Section 5). In the case of rhodium we always irradiated for 1 min. and made counts approximately from 30 to 60 sec. after the end of irradiation; the initial activity was rapidly calculated with the aid of a nomogram. In this case as well we neglected the insignificant activity with a period of 4.2 min.

In the course of this work we made a large number of measurements of the absorption of slow neutrons in various substances. The measurements were carried out under the following conditions: the measurements were made outside the paraffin, because otherwise a large fraction of the neutrons passes through the absorber many times (Section 6). The source, i.e. the emanation \(+\) Be, was usually placed in a paraffin cylinder (1) 3 cm below the center of the upper plane, on which the various detectors were placed. Most of the detectors were squares with side 5 cm.

TABLE 1

Activity \(A_n\) of various detectors

Substance	Period	Weight in g	Area in cm²	\(A_n\)
MnO\(_2\)	2.5 h	30	26	30
Cu	5 min.	13	38	2
Ga	20 min.	2.6	7	1.3
As	26 h	7	20	24
NaBr	18 min.	26	25	9.2
Rh	44 sec.	10.15	28	282
Ag	22 sec.	8	27	135
In	54 min.	0.76	9	21.2
In	16 sec.	0.76	9	14.2
J	25 min.	25	25	16.9
Ir	19 h	14.5	25	78
Au	2.7 d	6.6	25	26

We shall call the activability of a detector \(A_n\) under these conditions its normal activability. This gives an indication of the efficiency of various detectors. As an example, in Table 1 we give the normal activabilities of some of the detectors used by us.

To measure absorption, absorbers in the form of as thin as possible layers were placed between the paraffin and the detector.

In order to correct the error due to the distance \(h\) of the detector above the surface of the paraffin, we measured the change in activity of a radium detector as a function of \(h\). The empirical rule found for reducing the activability to \(h=0\) consists in dividing the results of the measurements by \(1-\dfrac{h}{7}\), where \(h\) is the height in centimeters; this rule is fulfilled sufficiently well for \(h\) up to values somewhat greater than \(1\ \mathrm{cm}\).

Strictly speaking, the correction for height should depend on the absorber used and on the group of neutrons with which we are working. At present we neglect these refinements. Even if the neutrons leaving the paraffin were homogeneous, the absorption curves obtained under the conditions described above would not be exponential, since neutrons emerging at different angles \(\theta\) traverse different thicknesses of the absorber. In calculating absorption coefficients from experimental data it is necessary to take these facts into account; for a long time we calculated this correction on the assumption that the neutrons leave the paraffin according to the cosine law.

Under this assumption the absorption curve with a thin detector (\(K=\) the neutron absorption coefficient and \(\delta=\) the thickness of the absorber), instead of the exponential \(e^{-K\delta}\) (curve \(a\) in Fig. 2), will have the form

\[ b(K\delta)=\int_0^1 e^{-\frac{K\delta}{x}}\,dx . \tag{4} \]

This function is represented by curve \(b\) in Fig. 2. At present we believe that for neutrons of group \(C\) (Section 6) it should be assumed that the number of neutrons emitted per unit solid angle, instead of being proportional to \(\cos\theta\), is approximately proportional to the expression

\[ \cos\theta+\sqrt{3}\cos^2\theta . \tag{5} \]

If this angular distribution of neutrons is adopted, then the absorption curve is given by the expression

\[ c(K\delta)=\frac{2}{2+\sqrt{3}}\int_0^1 e^{-\frac{K\delta}{x}}\left(1+\sqrt{3}x\right)\,dx, \tag{6} \]

represented in Fig. 2 by curve \(c\). This curve probably represents the course of absorption of neutrons \(C\) fairly well; for neutrons of other groups it is likely that the absorption curve will lie between \(b\) and \(c\); however, all calculations were made with curve \(c\). In the case of a thick detector, in other words, when there is considerable absorption of neutrons in the thickness used, the numbers obtained from curve \(c\) were subjected to a further correction.

The differences between the absorption coefficients given in this work and those cited in the preliminary communications are due for the most part to the fact that the calculations were made with curve \(c\) instead of curve \(b\) in Fig. 2.

TABLE 2.

Detector	Absorber	Absorber
Detector	Rh	Ag
Rh	54	67
Ag	68	45

Fig. 2. Absorption curves constructed according to different laws. Curves: upper — \(a\),

\[ (K\delta)=e^{-K\delta}; \]

lower — \(b\),

\[ (K\delta)=\int_{0}^{1} e^{-\frac{K\delta}{x}}; \]

middle — \(c\),

\[ (K\delta)=\frac{2}{2+\sqrt{3}}\int_{0}^{1} e^{-\frac{K\delta}{x}}\left(1+\sqrt{3}\,x\right)\,dx . \]

3. Selective absorption

We have already mentioned that, as was established by various authors, the absorption of an element generally becomes greater when the same element is used as the detector. Thus, for example, Table 2 gives (in percent) the residual activity produced by neutrons that have passed through \(0.39\ \text{g}/\text{cm}^2\) of rhodium and \(0.96\ \text{g}/\text{cm}^2\) of silver, if a rhodium plate of \(0.36\ \text{g}/\text{cm}^2\) or a silver plate of \(0.80\ \text{g}/\text{cm}^2\) was used as the detector. Data of this kind, relating to a large number of elements, have also been published by other authors\(^{10}\). These facts suggest the existence of selective absorption for slow neutrons. The best method for studying this selective absorption consists in measuring and analyzing absorption curves. Fig. 3 gives the absorption curves of cadmium, taken—one with a rhodium detector (\(0.36\ \text{g}/\text{cm}^2\)), the other with a silver detector (\(0.057\ \text{g}/\text{cm}^2\)). From these curves it is evident that the radiations activating both of these detectors are inhomogeneous, but consist of two parts, one of which is absorbed very strongly by cadmium, while the other only very weakly.

The fact should be borne in mind that, in our geometrical arrangement, the neutrons crossed the absorber at various

angles, as indicated in Section 2, one can calculate the absorption coefficient of the strongly absorbed part from each of these curves.

Thus we obtain, from the curve with the silver detector, \(K = 16\ \mathrm{cm^2/g}\), and from the curve with the rhodium detector, \(K = 13.5\ \mathrm{cm^2/g}\). The absorption coefficient of the hard part is decidedly less than 0.01 of these values; thus, for example, in the case of the rhodium detector we find \(K = 0.05\ \mathrm{cm^2/g}\). From consideration of the curves in Fig. 3

Fig. 3. Absorption curves for Cd, taken with rhodium and silver detectors

one can see that in our silver detector one half of the activity is due to the component that is strongly absorbed by cadmium, and one half to the hard component. In our rhodium detector, 82% of the entire activity is due to the soft component and the remainder to the hard component.

Thus absorption in cadmium makes it possible to divide the slow neutrons emerging from the paraffin block, in which the neutron source is located, into two parts; that one of them which is strongly absorbed by cadmium we shall in what follows call group C. In § 5 we shall examine in detail the absorption of neutrons of this group.

We shall now study the properties of radiations that pass through a thickness of cadmium sufficient to absorb group C completely. For this purpose we repeated some experiments on absorption of radiations that had passed through \(0.27\ \mathrm{g/cm^2}\) Cd, using rhodium and silver as absorbers and detectors[^11]. Table 3, analogous to Table 2, is given as an example of the results of such experiments; the results collected in Tables 2 and 3 were obtained with the same absorbers and detectors.

Comparison of Tables 2 and 3 shows that the selectivity appears more sharply for radiation that has passed through cadmium; this indicates the inhomogeneity of the component weakly absorbed by cadmium. To study further the properties of the radiation that had passed through cadmium, we measured the absorption curve of silver with a silver detector \((2.2\ \mathrm{g}/28\ \mathrm{cm^2})\) for unfiltered radiation and for radiation filtered by \(0.27\ \mathrm{g/cm^2}\) of cadmium. These curves are given in Fig. 4, where on the abscissa axis is plotted the thickness of the silver absorber, and on the ordinate axis—the ac-

tivity of the detector. The upper curve refers to unfiltered radiation, the lower one to radiation that has passed through \(0.27\ \mathrm{g/cm^2}\) of cadmium. On both curves it is seen that the radiation activating silver contains a component that is strongly absorbed in the element itself; this component is present with the same intensity both in the filtered and in the unfiltered radiations. This means that this component is very little absorbed by cadmium. We shall call it group \(A\). The coefficient

Absorption of radiations that have passed through \(0.27\ \mathrm{g/cm^2}\) Cd

TABLE 3

Detector	Absorber	Absorber
Detector	Rh	Ag
Rh	33	84
Ag	88	29

of absorption of neutrons of group \(A\) in silver

\[ K = 20\ \mathrm{cm^2/g}. \]

Curves similar to those in Fig. 4 were also obtained with a thicker detector; however, in this case the percentage of activity due to radiation strongly absorbed in silver is, of course, smaller, since saturation occurs in a detector thickness of less than \(0.1\ \mathrm{mm}\).

Fig. 4. The upper curve is the absorption curve for Ag, for unfiltered radiation. The lower curve is the absorption curve for Ag for radiation that has passed through \(0.27\ \mathrm{g/cm^2}\) of cadmium. On the ordinate axis is plotted the activity of the silver detector. The dotted curve represents the difference between the upper and lower curves.

The dotted curve in Fig. 4 is the difference of the two other curves; thus it represents the behavior of that part of the silver activity which is absorbed by \(0.27\ \mathrm{g/cm^2}\) of cadmium, i.e. this is the absorption curve in silver with a silver detector, only for neutrons of group \(C\). From it we find the value of the absorption coefficient of neutrons of group \(C\) in silver \(K = 0.3\ \mathrm{cm^2/g}\).

4. Groups of Slow Neutrons

From the examples of the preceding section we have seen that analysis of absorption curves allows us to divide slow neutrons into groups which are absorbed differently in different elements. Although

It is well known that the decomposition of absorption curves into exponential curves can be regarded as valid only in the case when the accuracy of the individual measurements is very high and when the geometrical arrangement of the absorber and detector with respect to the source satisfies conditions that can be exactly taken into account. In our case neither of these conditions was exactly fulfilled, and thus we had no possibility of guaranteeing the complete homogeneity of each component of the radiation. Therefore in what follows we have confined ourselves to describing the properties of approximately homogeneous groups of neutrons.

The differences in behavior in going from one group to another are so noticeable that each of them may be regarded as individual, despite a small internal inhomogeneity. The decomposition into groups was possible when the following circumstances were taken into consideration:

a) for some absorbers there are very large differences in the absorption coefficients in passing from group to group;

b) it is possible to use different absorbers and detectors, which behave quite differently;

c) it is possible to filter out slow neutrons by means of absorbers in order to reduce the inhomogeneity.

The most obvious supposition concerning the physical nature of the difference between the groups is that it is due to a difference in velocity. In a paraffin block, fast neutrons emitted by the source undergo gradual slowing down as a result of successive collisions with hydrogen atoms, until they reach an energy corresponding to thermal motion; thus at every moment there are neutrons in the paraffin with all possible velocities—from the energy of thermal motion to the energy with which they left the source. Accordingly, neutrons with all these velocities diffuse out of the paraffin block. Hence we may think that different groups correspond to neutrons of different energy intervals. Selective absorption must be interpreted as an irregular variation of the absorption coefficient as a function of neutron velocity.

The possibility of an irregular dependence of the absorption coefficient on neutron velocity was recently shown by the arguments of Bohr, Breit, and Wigner[^12]. We shall return to the discussion of the physical nature of the neutron groups after describing the experiments on absorption and scattering of the various groups.

Already in the preceding section we called the radiation strongly absorbed by cadmium \((K = 13.5—16\ \text{cm}^2/\text{g})\) group \(C\); and the radiation strongly absorbed by silver \((K = 20\ \text{cm}^2/\text{g})\), group \(A\).

We also saw that only 50% of the activity of our silver detector \((0.057\ \text{g}/\text{cm}^2)\) is due to group \(C\); of the remaining 50%, about half is due to group \(A\), while the remainder is due to radiation that is only weakly absorbed by cadmium and se-

edge-on (Fig. 4). We have called this radiation group \(B\); it is probably complex. Until now we have not been able to decompose it even into approximately homogeneous components, since we have not found an element with a high absorption coefficient for group \(B\).

In the preceding paragraph we showed that \(72\%\) of the activity of our rhodium detector \((0.36\ \mathrm{g}/\mathrm{cm}^2)\) is due to group \(C\); the remaining \(28\%\) constitute a new group, which we called group \(D\), and which is very strongly absorbed by cadmium. Studying a detector made of \(\mathrm{PbJ}_2\), \((19\ \mathrm{g}/\mathrm{cm}^2,\) normal activability \(11.4)\), we found that only about \(25\%\) of its activity is due to group \(C\), and less than \(10\%\) to group \(A\); the remainder is due to a component which is weakly absorbed by all the elements so far studied by us, with the exception of iodine; this radiation, to which about \(70\%\) of the activity of our detector is due, was called group \(J\).

Frisch, Hevesy, and McKay\(^{13}\) studied absorption in gold with the aid of a gold detector, using neutrons which had passed through cadmium, and obtained results similar to those which we obtained for silver (Fig. 4). We tried to resolve the activity of a gold detector \((1.75\ \mathrm{g}/25\ \mathrm{mm}^2)\) into components corresponding to the groups already mentioned; we found that the activity of gold is partly due to groups \(C\), \(A\), and \(B\); but the high absorption coefficient of gold with a gold detector is probably due to radiation different from all the preceding groups.

5. Systematic measurements of absorption

Group \(C\). Of all the groups, group \(C\) is the best known, because in almost all detectors a large percentage of the activity is due to it; there are many grounds for supposing (Section 13) that this group consists to a considerable extent of neutrons having the energy of thermal motion; on this basis we shall often refer to it as the thermal group. The study of group \(C\) is especially easy, because a cadmium thickness of \(0.3\text{--}0.4\ \mathrm{g}/\mathrm{cm}^2\) absorbs it completely, while not appreciably absorbing any of the other groups. In order to obtain the part of the activity due to group \(C\), it is sufficient to take the difference between the activity obtained without a cadmium filter and with a cadmium filter of the above thickness. In Table 4 we give the absorption coefficients in \(\mathrm{cm}^2/\mathrm{g}\) for neutrons of group \(C\), obtained with various detectors (always taking the difference between the activity obtained without a filter and the activity obtained with a cadmium filter and different absorbers). It should be noted that the absorption coefficients in some elements, measured with different detectors, remain approximately the same. The differences are undoubtedly in part due to errors of measurement; however, we

TABLE 4

Absorption coefficients in cm²/g for neutrons of group C
(those which are strongly absorbed by cadmium)

Detector	Rh	Ag	Cd	In	Ir	Au	Hg
Mn 2.5 hr.	0.9	0.3	—	—	—	—	—
Rh 44 sec.	0.6	0.2	13.5	0.6	1.0	0.2	0.7
Ag 22 sec.	0.7	0.3	16	—	0.9	0.25	0.9
In 54 min.	0.6	0.2	14	0.9	1.0	0.2	0.6
Ir 19 hr.	1.0	0.3	—	—	—	—	0.6

we think that they cannot be entirely attributed to them.

Nevertheless, if we recall the large differences in the absorption coefficients of one and the same element for neutrons of different groups, we see that group C does indeed possess remarkable internal homogeneity.

Group D. Group D is usually observed when rhodium or indium is used as the detector; both these elements are convenient for carrying out precise measurements.

From the fact that group D can be determined with either one or the other of these two detectors, it follows that the corresponding absorption bands almost coincide; of course, there is as yet no reason to suppose that the coincidence must be complete. In § 9 we shall see that, among all the groups we have studied, group D has the lowest energy, with the exception of group C.

We have already said that 28% of the activity of our rhodium detectors is due to group D. In an indium detector of 0.76 g/cm², 42% of the activity of the 54-minute period is due to group D, the remaining 58% being due to group C. Thus, for studying group D it is sufficient to use one of these detectors, filtering the radiation with 0.3–0.4 g/cm² of cadmium in order to eliminate group C. Table 5 gives some absorption coefficients in cm²/g for D-neutrons.

Group A. This group is characterized by a large value of the absorption coefficient in silver \((K = 20\ \text{cm}^2/\text{g})\); it is usually studied with the aid of a thin layer of silver as detector.

As was mentioned above, 25% of the normal activation of a silver detector of thickness 0.057 g/cm² is due to A-neutrons. To isolate this group it is sufficient to form the difference

TABLE 5

Absorption coefficients in cm²/2 for neutrons of group D (those which are strongly absorbed by Rh)

Detector	Absorber	Absorber	Absorber	Absorber	Absorber	Absorber	Absorber	Absorber
Detector	B	Rh	Ag	Cd	In	Ir	Au	Hg
Rh 44 sec. . . . .	4.7	2.0	0.06	0.05	3.0	1.0	0.03	0.07
In 54 min. . . . .	—	1.6	0.09	—	3.8	1.0	0.04	0.04

of the activities of a silver detector without a silver filter and with one sufficiently thick to absorb group A completely, but not so thick as to absorb the other groups appreciably. In practice it is convenient to eliminate group C by means of a cadmium filter.

Group A can also be detected with the aid of gold. A gold detector 1.535 g/25 cm² thick has a normal activity of 21, of which 13% is due to group A; the absorption coefficient in gold for this group is equal to 4 cm²/2. The absorption coefficient for group A in boron is equal to 3 cm²/2.

With respect to the other groups mentioned in the preceding section, not very much is known. We made a small number of measurements of absorption coefficients, which we give here:

Group B: boron \(K = 2.3\ \text{cm}^2/2\); gold \(K = 1.7\ \text{cm}^2/2\).

Group J: boron \(K = 1\ \text{cm}^2/2\); iodine \(K = 0.7\ \text{cm}^2/2\).

We have seen that, whereas group C almost always accounts for more than 50% of the activity of various detectors placed outside the paraffin, the other groups are found in a smaller number of detectors and smaller percentages of the activity are due to them.

This is not because, in different elements, the absorption coefficient for neutrons of group C is greater than for neutrons of the other groups; we have already encountered numerous examples contradicting this. Most likely this is due to the fact that the number of C-neutrons which emerge from the paraffin is considerably greater than the number of neutrons of the other groups.

Thus a very important number for characterizing a group is its intensity. We shall define this quantity for practical purposes as follows: the intensity \(N_0\) of any group is the number of neutrons emitted in 1 sec through 1 cm² per unit neutron intensity of the source, taken as the average over 25 cm² at the center of the upper surface of the paraffin (\(I\)) containing the source (neutron intensity—unity), at a distance of 3 cm below the center of this surface.

The principle on which the measurement of the group population is based is the following: we place at the center of the paraffin block a thin detector with surface \(s\); the number of neutrons which strike this detector is equal to

\[ N_0/s, \]

If we denote by \(K\) the absorption coefficient of the detector, assuming that its thickness \(\delta\) is very small, then it can be seen that the number of neutrons absorbed by the detector will be equal to

\[ \sqrt{3}\,N_0/sK\delta, \]

where the factor \(\sqrt{3}\) is due to the fact that the neutrons strike at different angles, which is taken into account according to equation (5).

If we denote by \(\eta\) the efficiency of the ionization chamber, i.e. the ionization current from one disintegration per second from the surface, then the normal activity \([see equation (3)]\) of our detector will be

\[ A = 1000\sqrt{3}\,sN_0K\delta\,\frac{K_1}{K}\,\frac{\eta}{840\,\eta_U}, \tag{7} \]

where \(\eta_U\) is the efficiency of the chamber for the \(\beta\)-rays emitted by our uranium preparation, which, as has already been said, corresponds to 840 disintegrations per second.

The factor \(\frac{K_1}{K}\) is introduced in order to take into account the circumstance that, if the detector absorbs neutrons of a given group (with absorption coefficient \(K\)), then it is possible that some of these neutrons produce either nonradioactive isotopes or radioactive ones with a period different from that under study; we have denoted by \(K_1\) the absorption coefficient due only to that process which produces the activity under study; obviously, the coefficient \(K\) determines the absorption of neutrons in the detector, while \(K_1\) determines the activity of the detector.

From (7) we find the population of the group under study

\[ N_0 = 0.485\,\frac{\eta_U}{\eta}\,\frac{K}{K_1}\,\frac{A}{sK\delta}. \tag{8} \]

In this expression for \(N_0\), the quantities \(A\), \(s\), \(K\), and \(\delta\) are directly measurable; in most cases it is possible to take \(K_1\) equal to \(K\); in other cases, where the element has more than one period, the ratio of their activities, corrected for absorption of \(\beta\)-rays and for the different efficiencies \(\eta\), will make it possible to estimate \(\frac{K}{K_1}\).

A considerable error in the measurement of \(N_0\) is due to insufficient knowledge of the ratio of efficiencies for \(\beta\)-rays of different hardness. From the relatively small variations which were found in the computed populations for different detectors under the pred-

From the constancy of the efficiency \(\eta\), one may conclude that this quantity does not vary within wide limits. In the following calculations we always set \(\eta = \eta_U\).

In Table 6 we give the population values for various groups, calculated using different detectors; for the absorption coefficients we used in the calculations the mean of the numbers given in the preceding tables. The values given in Table 6 were calculated by means of a formula similar to (8), in which, however, the absorption of neutrons and electrons in the detector was taken into account. We also introduced a small correction to account for the reflection of electrons from the upper layers of the detector.

TABLE 6

Population \((N_0)\) for various groups of neutrons

Detectors	C	D	A	J
Rh	43	9	—	—
Ag	51	—	0.6	—
In	31	6	—	—
J	—	—	—	5
Au	33	—	0.5	—

The data for indium were calculated taking into consideration two periods—16 sec. and 54 min.—and neglecting the weak period of 4 hours. For rhodium and for silver we performed the calculation taking into account not only the periods of 44 and 22 sec., but also the periods of 4.2 and 2.3 min.

If we try to calculate the population of group \(C\) from the activity of iridium (19 hours), we find a value of about 10; this value seems to us too small for it to be attributed to measurement errors. We therefore tried to find another activity strong enough to justify the preceding small value of the population. In fact, this suspicion was quite well founded, since we found a second activity of iridium with a period of 68 days, whose activability, although not yet measured, is of the same order of magnitude as the activability of the 19-hour period. The \(\beta\)-particles of this new activity have a very small penetrating power.

It is very interesting to determine, for elements having two or more periods due to slow neutrons, whether activities of different periods are produced by the same groups or by different ones.

For this purpose we studied the following detectors: rhodium (\(0.36\ \mathrm{g/cm^2}\), 44 sec., 4.2 min.), silver (\(0.057\ \mathrm{g/cm^2}\), 22 sec., 2.3 min.), indium

\((0.065 \ \text{g}/\text{cm}^2;\ 16 \text{ sec.},\ 54 \text{ min.})\), sodium bromide \((1 \ \text{g}/\text{cm}^2;\ 18 \text{ min.},\ 4.22 \text{ sec.})\), and compared, for each of them, the activabilities of the two periods obtained with the corresponding filters and without such filters.

In the case of rhodium, the ratios of the activabilities of both periods, at 44 sec. and at 4.2 min., without a filter and with a cadmium filter sufficiently thick to absorb group \(C\), remain the same (about 10)\(^{14}\). In the case of silver, however, we found a considerable difference in the behavior of the two periods; for group \(C\) the activability of the long period amounts to 24% of the total activity; for group \(B\), 19%; finally, for group \(A\) we found 3%, which within the limits of experimental error may be regarded as equal to zero; therefore we may say that group \(A\) characterizes the 22-sec. silver period.

In the case of indium, the percentages of activation of the two periods due to groups \(C\) and \(D\) are approximately equal. Likewise, for bromine the cadmium filter does not noticeably change the ratio of the activabilities of the two periods.

We also replaced the paraffin cylinder with which the measurements were ordinarily made by an identical cylinder filled with water and, using detectors of rhodium, silver, and lead iodide and suitable filters, determined the percentages of activation due to the various groups. In this way we were able to show that the ratios of the abundances of the various groups are the same with water and with paraffin. The absolute values also do not change appreciably. It should be noted, however, that the ratios of the intensities of the various groups change at different distances from the source; this will be discussed in more detail in Section 9.

Fig. 5. Increase of activity as a function of time

Finally, we tried to establish whether the ratio of the intensities of the various groups changes when different sources of fast neutrons are used\(^{15}\). For this purpose we determined the activability of rhodium and silver detectors with cadmium and se-

silver filters and without them, using a source that had just been prepared, in which radium C was not yet in equilibrium, and followed the increase of activability during the formation of radium C. We found no radiation, for the silver and rhodium group, during the formation of radium C, although the activability increased from 40 (the effect is due to the particles of RaEm + RaA) to 100 (the effect is due to the particles of RaEm + RaA + RaC). In Fig. 5 we give the curve of the increase in activity as a function of time. It should be noted that the $\alpha$-particles of radium C are highly effective in producing neutrons from beryllium; their effectiveness is three times greater than the effectiveness of the $\alpha$-particles of the emanation and RaA.

6. Albedo

Measurements of the normal activability, as well as the absorption measurements of which we have spoken up to now, were made outside the paraffin block. In this section we shall study the properties of slow neutrons inside paraffin.

The detector (rhodium $10\ \text{g}/28\ \text{cm}^2$ or silver $2.2\ \text{g}/38.5\ \text{cm}^2$) was placed at the center of the upper surface of the normal paraffin block 1, and a second, identical paraffin cylinder was placed on it. The detector could be placed between the corresponding filters.

In order quickly to denote the relative position of the source, paraffin, detectors, and absorbers, it is convenient to adopt the following convention: we shall denote the source by the letter $S$, paraffin by the letter $P$, the detector by the letter $R$, and by Cd and Ag the cadmium ($0.27\ \text{g}/\text{cm}^2$) and silver ($0.057\ \text{g}/\text{cm}^2$) absorbers. Thus, $SPCdRCdP$ denotes an experiment in which above the source ($S$) there are $3\ \text{cm}$ of paraffin ($P$), then a layer of cadmium (Cd), the detector ($R$), a second layer of cadmium (Cd), and, finally, the second paraffin cylinder ($P$).

TABLE 7

Activabilities for various arrangements $S$—source, $P$—paraffin, Cd—cadmium, Ag—silver, $R$—detector

Arrangement	Silver	Rhodium
$SPR$	66.1	282
$SPCdR$	32.2	80
$SPCdAgR$	17.5	78
$SPRP$	417	1025
$SPCdRCdP$	66.5	140
$SPCdRP$	92.2	244
$SPRCdP$	113.1	371
$SPCdAgRAgCdP$	34.5	144
$SPCdAgRCdP$	45.6	—
$SPCdRAgCdP$	54.8	—

In Table 7 we give the activabilities of the two detectors—the silver and the rhodium—in various experiments. If we compare the experiments $SPR$ and $SPRP$, we notice a strong increase in activity due to the placement of the second paraffin block. This large increase occurs because neutrons, both fast and slow, which in the absence of the second block would have gone away, can now be reflected back from this second block.

By placing the detector between the corresponding absorbers, it is possible to decompose the radiation activating it into groups. Thus, for example, a comparison of the experiments \(SPRP\) and \(SPCdRCdP\) shows that in the case of a silver detector the fraction

\[ \frac{417-66.5}{417}=0.84 \]

of the activity is due to group \(C\), and in the case of a rhodium detector the fraction

\[ \frac{1025-140}{1025}=0.86 \]

is due to this same group. It should be noted that the ratio of group \(C\) to the other groups is much greater inside the paraffin than outside it; we shall soon see the reason for such behavior.

To check the correctness of the decomposition into groups of the detector activity also inside the paraffin, we doubled the thickness of the cadmium absorber in the experiment \(SPCdRCdP\) and found only a very small further decrease in activity. In the case of a silver detector, adding silver absorbers to the cadmium ones causes the activity to decrease by half (experiments \(SPCdRCdP\) and \(SPCdAgRAgCdP\)). This fact shows that also inside the paraffin one can speak of group \(A\), strongly absorbed by silver. In agreement with what was obtained outside the paraffin, adding thin silver absorbers to the cadmium ones does not produce a further decrease in the activity of rhodium.

We shall now study the behavior of group \(C\). Let us consider the results of the measurements \(SPRP\), \(SPCdRP\), \(SPRCdP\), and \(SPCdRCdP\); this last measurement indicates what part of the activity is caused not by group \(C\), so that, if we subtract this value from the results of the other three measurements, we obtain the part of the activity due only to group \(C\).

The difference between the measurements \(SPCdRP\) and \(SPRCdP\) can be easily explained, taking into account the circumstance that the density of slow neutrons inside the paraffin decreases with increasing distance from the source in such a way that the layer of cadmium which shields the detector on the side of the source stops more neutrons than the same layer on the other side. The mean value obtained from these two experiments may be regarded as the value which would be obtained in the case of a uniform distribution of the slow-neutron density inside the paraffin, if the detector were covered on one side with cadmium.

Thus we find for the silver detector the activability due only to group \(C\),

\[ A_c=417-66.5=350.5. \]

The activability due only to group \(C\), if the silver is covered with a layer of cadmium only on one side, is

\[ B_c=\frac{99.2+113.1}{2}-66.5=36.1. \]

The corresponding values for rhodium are \(A_c=885,\ B_c=167.5\). The ratio \(\frac{A_c}{B_c}\) is 9.7 for silver and 5.3 for rhodium.

A simple argument allows us to understand the significance of this ratio.

We shall call \(\beta\) the probability that a neutron which falls on a plane surface bounding a very large block of paraffin will leave it after first undergoing several deflections in the paraffin.

We shall call this quantity \(\beta\), which corresponds to the coefficient of diffuse reflection of slow neutrons from the surface of paraffin, the albedo.

In reality the albedo depends on the angle of incidence of the slow neutrons and, obviously, is the smaller, the smaller the angle of incidence (see F, section 5).

In our elementary argument we shall neglect this dependence.

We shall call \(\zeta\) the probability that a neutron, crossing the detector, will be absorbed by it. For \(\zeta\) also we shall neglect the dependence on the angle of incidence.

The method for measuring the albedo is based on comparing the activities \(A_c\) and \(B_c\) defined above.

Let \(N\) be the number of \(C\)-neutrons which fall on a detector of group \(C\), when on one side it is covered by a layer of material completely absorbing this group. The number of neutrons absorbed by the detector is \(N\zeta\), and its activity \(B_c\) will be proportional to \(N\zeta\).

Let us now remove the absorber and calculate the increase in the activity of the detector.

First, we must take into account the circumstance that neutrons can strike the detector from both sides; this gives a factor 2.

But a considerably more important increase is due to the fact that, when a layer which absorbs all neutrons is placed directly behind the detector, a neutron of group \(C\) striking the detector crosses the latter only once; if the absorber is absent, the neutron may pass through the detector many times.

Indeed, the first time the neutron has probability \(\zeta\) of being absorbed and probability \((1-\zeta)\) of passing through the detector; the probability that the neutron will return to the detector will thus be \(\beta(1-\zeta)\), and the probability that it will be absorbed on the second passage is \(\beta(1-\zeta)\zeta\), etc. Hence we find that the probability of absorption will be

\[ \zeta+\zeta\beta(1-\zeta)+\zeta\beta^2(1-\zeta)^2+\cdots = \frac{\zeta}{1-\beta(1-\zeta)} = 1-\frac{\beta}{\zeta+\beta}. \]

The number of absorbed neutrons will be

\[ \frac{2N\zeta}{1-\beta+\beta\zeta}. \]

And hence the ratio \(\dfrac{A_c}{B_c}\) will be

\[ \frac{A_c}{B_c}=\frac{2}{1-\zeta+\beta\zeta}. \tag{9} \]

If \(\zeta\) is very small, i.e., in the case of a very thin detector, this expression reduces to

\[ \frac{A_c}{B_c}=\frac{2}{1-\beta}. \tag{10} \]

Thus measurement of the ratio \(\dfrac{A_c}{B_c}\) allows us to determine the albedo \(\beta\).

In the case of our silver detector \(\zeta\) is very small, but not so small that it can be neglected. We can estimate it from the thickness of the detector and the absorption coefficient, taking into account, of course, the oblique incidence of the neutrons. In this way we find \(\zeta=0.03\), and hence \(\beta=0.82\).

The rhodium detector is considerably thicker; for it \(\zeta\) is about \(0.35\); it is clear that the greatest error in this large value would give a large error in the value of \(\beta\). Therefore we cannot use measurements with rhodium to determine \(\beta\), but we can calculate for this detector the ratio \(\dfrac{A_c}{B_c}\), using the albedo value found in the measurements with silver. We have \(\dfrac{A_c}{B_c}=4.3\), in sufficient agreement with the experimental value \(5.3\).

We have established that the concept of albedo, in the form in which we introduced it, is not fully definite, since part of the neutrons reflected from the surface of the paraffin depends on the angular distribution of the incident neutrons. In order to make the concept of albedo more precise, we can determine \(\beta\) with the aid of equation (10), which is valid in the case of an infinitely thin detector (experimental albedo; see \(F\), section 7).

The experimental albedo depends on the number \(N\) of free paths which a \(C\)-neutron can make on average before it is absorbed by protons in the paraffin.

It is possible to show (\(F\), formula 63) that the relation connecting these two quantities has the following form:

\[ \beta=1-\frac{2}{\sqrt{N}}. \tag{11} \]

This equation is valid under the assumption that the energy of thermal motion is very small (it may be neglected) in comparison with the quantum \(h\nu\) of the elastic binding of the hydrogen atom in paraffin; otherwise a small correction must be introduced (section 13).

From the value \(\beta=0.82\) we find from equation (11) \(N=124\).

We carried out several experiments, similar to those just described, in order to determine the albedo for neutrons of group \(A\), using a silver detector of \(0.010\ \mathrm{g/cm^2}\), so that \(\zeta\) would not be too large for group \(A\).

The behavior of group \(A\) in this respect is entirely different from that of group \(C\), since we found that the albedo of group \(A\) is practically equal to zero. This fact should not be interpreted to mean that neutrons of group \(A\) are not reflected at all by paraffin; rather, it shows that when \(A\)-neutrons are reflected, they undergo such a change in velocity that they cease to belong to group \(A\).

Because of the low albedo of group \(A\), the activity produced by \(A\)-neutrons, when the second block of paraffin is placed on the detector, increases only by about a factor of 2. This increase is not due to albedo, but to the fact that neutrons can enter the detector from both sides.

Groups \(B\), \(D\), and \(J\) also have a very small albedo. It follows from this that for all groups, with the exception of group \(C\), the change in velocity caused by a collision is usually sufficient to remove the neutron from the energy band corresponding to the given group (Section 12).

The high value of the albedo of group \(C\), as compared with the albedo of the other groups, accounts for the fact that inside the paraffin the part of the activity due to group \(C\) is always much greater than outside the paraffin.

In the same way one may also explain Tilman’s experiments\(^{16}\) concerning the different behavior of various detectors placed on a block of paraffin, depending on the thickness of the second paraffin layer placed on top. A strong relative increase in activity should be expected for detectors very sensitive to group \(C\); of course, in interpreting these experiments one must also take into account the \(\zeta\) of the detector according to relation (9).

7. Scattering of Neutrons of Individual Groups

From the results of the preceding section we concluded that \(C\)-neutrons can make a large number of free paths. Therefore their motion in paraffin is analogous to diffusion, provided only that one takes into account that the neutrons disappear as a result of their capture by protons. Let \(N\) be the mean number of free paths for a neutron of group \(C\), and let \(\lambda\) be the mean length of its free path; obviously, the mean distance over which this neutron diffuses will be of the order of \(\lambda\sqrt{N}\); thus diffusion experiments enable us to measure this quantity\(^{17}\). We shall call the expression

\[ l = \lambda \left(\frac{N}{3}\right)^{\frac{1}{2}} . \tag{12} \]

the “diffusion length.”

It can be obtained directly from the measurements which we shall now describe.

Let us consider a C-neutron inside a paraffin block bounded by a plane surface, and let \(x\) be the distance of the neutron from this surface. On the basis of diffusion theory (\(F\), Section 3) it can be shown that the probability \(p(x)\) that the neutron will emerge from the paraffin before it is absorbed is equal to

\[ p(x)=e^{-\frac{x}{(D\tau)^{1/2}}}, \tag{13} \]

where \(D\) is the diffusion coefficient for thermal neutrons in paraffin, and \(\tau\) is their mean lifetime for the capture process. In the case of thermal neutrons the kinetic energy is small in comparison with the quanta of the elastic vibrations of hydrogen in paraffin. We shall therefore make the assumption that the hydrogen atoms may be regarded as immobile centers of isotropic scattering, and that the mean free path \(\lambda\) does not depend on the velocity (see \(F\), Sections 10, 11). In reality these assumptions are not entirely correct, as will be discussed in detail in Section 13.

In the case of isotropic diffusion the diffusion coefficient \(D\) is given by the expression

\[ D=\frac{1}{3}\lambda v, \tag{14} \]

where \(v\) is the mean velocity of the neutron; moreover we have

\[ N=\frac{v}{\lambda}; \tag{15} \]

hence

\[ D\tau=\frac{1}{3}\lambda^{2}N. \tag{16} \]

Finally we find (see \(F\), formula 34)

\[ p(x)=e^{-\frac{x}{\left(\lambda^{2}\frac{N}{3}\right)^{1/2}}} = e^{-\frac{x}{l}}. \tag{17} \]

By determining the probability \(p(x)\), we shall be able to measure \(l\), and hence also \(\lambda^{2}N\).

The determination of \(p(x)\) is based on the following principle: let the detector of group C be sufficiently large to cover the whole plane surface of the paraffin block from which the neu-

rons \(C\). We shall assume that this detector absorbs all emerging thermal neutrons in a thickness small in comparison with the absorption of \(\beta\)-rays. The activity of this detector is proportional to the number of neutrons emerging from the paraffin, and does not depend on their angular distribution. If we now destroy \(Q\) thermal neutrons at a depth \(x\) below the surface of the paraffin, the activity of the detector will decrease by an amount proportional to \(Qp(x)\); to destroy \(Q\)-neutrons we may place an absorber of thermal neutrons at depth \(x\), and then we shall be able to obtain the quantity \(Q\) from the activity produced in the absorber.

The experiment was carried out in such a way that the detector was placed on a paraffin cylinder with a diameter of \(24\) cm and a height of \(15\) cm. The source was located on the axis of the cylinder at a distance of \(3.5\) cm below the upper plane. The upper part of the cylinder was cut into layers so that the absorber could be placed at different depths. As the detector we used a rhodium plate (\(10\) g/\(28\) cm\(^2\)); since this plate covered only a small part of the upper paraffin plate, we made measurements by placing the detector successively in nine different positions, and added the results.

As the absorber of thermal neutrons we used a plate of a Cd—Sn alloy containing \(0.019\) g/cm\(^2\) of cadmium. Since the small thickness of cadmium practically absorbs only neutrons of group \(C\), the decrease in the detector activity is due only to the absorption of these neutrons, which can be easily verified. Owing to the fact that absorption of neutrons in cadmium does not activate this element, we could not directly measure the number \(Q\) of absorbed neutrons; therefore we measured the activity caused by neutrons \(C\) in a rhodium plate having, for group \(C\), the same absorption as our cadmium absorber. We measured the activity of both sides of this rhodium plate and added the results in order to take into account that neutrons enter from both sides, as explained in Section 11.

The absorber was placed at distances \(x = 0, 1, 2\), and \(3\) cm. Table 8 gives the results of the measurements. In the first column are given

TABLE 8

Data for determining the probability \(p(x)\)

\(x\) in cm	Decrease in activability	\(Q\)	Ratio	Corrected ratio
0	89	358	0.249	0.207
1	139	1135	0.123	0.123
2	128	1580	0.081	0.081
3	98	1950	0.050	0.050

depths at which the absorber was placed; in the second—the decrease in the activation of the detector due to the presence of the absorber; the third column gives the sum of the activations of the rhodium plate, equivalent to the absorber, measured on both sides only for group \(C\); the fourth gives the ratio of the second column to the third. In the fifth column the values of these ratios are corrected taking into account the circumstance that our detector does not satisfy the condition of complete absorption of \(C\)-neutrons in a thickness small compared with the absorption of electrons (absorption coefficient for electrons \(=7.3\ \mathrm{cm}^2/\mathrm{g}\) in rhodium; absorption coefficient for neutrons \(0.7\ \mathrm{cm}^2/\mathrm{g}\); thickness \(0.36\ \mathrm{g}/\mathrm{cm}^2\)). It follows from this that our detector is more sensitive to neutrons emerging at a large angle. The absorber, when it is in the position \(x=0\), absorbs more intensely those neutrons which emerge with a large inclination; therefore the decrease in the activity of the detector in this case is relatively greater than in the case when the absorber is placed deep inside the paraffin, since in this latter case, owing to collisions, there is no coherence between the direction of the emerging neutrons and the direction which they had when crossing the absorber. In order to take this factor into account, we must reduce the value of the ratio for \(x=0\), so that it may be compared with the other ratios.

A quantitative calculation shows that for this correction the first value must be multiplied by \(0.832\).

The values of the corrected ratios, given in the last column, are proportional to \(p(x)\); it may be seen that they depend, to a good approximation, exponentially on \(x\).

The “diffusion length” is equal to the reciprocal of the coefficient of this exponential function; thus we find

\[ l = 2.1\ \mathrm{cm}. \]

From this we obtain, according to (12),

\[ \lambda^2 N = 13\ \mathrm{cm}^2. \]

We carried out the same experiments for groups \(D\) and \(A\). In the case of group \(D\) the source was at a distance of \(3.4\ \mathrm{cm}\) below the center of the upper plane of the paraffin cylinder, similar to that used in the preceding experiments. In the case of group \(A\) the distance was \(2.4\ \mathrm{cm}\).

As detectors we used, for groups \(D\) and \(A\) respectively, indium \((0.065\ \mathrm{g}/\mathrm{cm}^2)\) and silver \((0.057\ \mathrm{g}/\mathrm{cm}^2)\), both covered with cadmium filters about \(0.5\ \mathrm{g}/\mathrm{cm}^2\) thick. As absorbers we used, for group \(D\), rhodium \((0.36\ \mathrm{g}/\mathrm{cm}^2)\), and for group \(A\)—silver \((0.057\ \mathrm{g}/\mathrm{cm}^2)\).

In Table 9 the results of these measurements are collected; in the first column we give the depth \(x\) at which the absorber was placed; columns two and five give respectively, for groups \(D\) and \(A\), the decrea-

TABLE 9

Data for \(D\)- and \(A\)-neutrons. \(x\)—the depth at which the absorber is located

\(x\)	Group \(D\): decrease in activity	Group \(D\): activ. absorber	Group \(D\): activ. lower side	Group \(A\): decrease in activity	Group \(A\): activ. absorber	Group \(A\): activ. lower side
0	12.0	102	68	15.3	27.3	17.4
0.16	9.6	105	71	11.6	31.2	(18)
0.34	6.4	124	75	8.3	34.6	(19)
0.87	2.7	173	93	3.8	51.5	(22)
1.87	0.5	244	101	0.5	61.5	(23.1)

activity of the detectors; the third column gives the sum of the activities measured at different depths on both sides of the rhodium absorber, covered with cadmium in order to measure the activity due only to group \(D\); the sixth column contains the corresponding data for the silver absorber of group \(A\). The fourth and seventh columns give the activities of the absorber, measured on the lower side and due only to neutrons of group \(D\) or group \(A\), which enter the absorber from the lower side (for the method of carrying out this measurement see section 11). The numbers in parentheses in the last column were interpolated, using certain results similar to those mentioned in section 11.

From the low value of the albedo for groups \(D\) and \(A\) we already concluded in the preceding paragraph that the neutrons of these groups generally make only one free path. Therefore we cannot apply diffusion theory to these groups. Nevertheless, it is still useful to treat the measurement data for groups \(D\) and \(A\) by the same method that we used for group \(C\). For this purpose we examine the ratios of the second column to the third and the ratios of the fifth to the sixth as functions of \(x\). It is then found that in both cases the ratios decrease exponentially to a good approximation, and in both cases the reciprocal of the coefficient of the exponential function is equal to 0.42. We note the analogous behavior of these two groups, in contrast to the behavior of group \(C\).

However, it is more correct to treat the data of Table 9 under the assumption that neutrons of groups \(D\) and \(A\) can make only one free path. Assuming this, we can determine the mean free path of the neutrons of these two groups. If this assumption is incorrect, the values that we obtain will represent only an upper limit for the mean free path.

Let us compare the activity of the detector of group \(D\) for the case when an absorber is located at depth \(x\) and when it is absent. Both activities are sums of two parts: (a) the activity due to neutrons of group \(D\) that have not crossed the plane at depth \(x\), and (b) the activity due to \(D\)-neutrons that have crossed this plane.

From our assumption that neutrons of the \(D\)-group can make only one free path, it follows that all neutrons responsible for part (b) of the activity have crossed the plane \(x\) from below. Part (a) has not changed in the presence of the absorber in the plane \(x\), whereas part (b) has decreased in the presence of the absorber. The total decrease of the activity of the counter is equal to the decrease of part (b) alone. For simplicity, in our discussion we shall call the neutrons of group \(D\) that arrive at the plane \(x\) from below neutrons of class \(b\).

The decrease of the detector activity, given in the second column of Table 9, can therefore be calculated by taking into account only neutrons of class \(b\). From our assumption it follows that the layer of paraffin of thickness \(x\), which neutrons of class \(b\) must traverse in order to reach the detector, should be regarded as an absorbing (but not as a scattering) layer having absorption coefficient \(\frac{1}{\lambda_D}\).

If there is no absorber, the activity of the detector due only to neutrons of class \(b\) will be proportional to

\[ N_b c\left(\frac{x}{\lambda_D}\right), \]

where \(N_b\) is the number of neutrons of class \(b\), and \(c\) is the function defined by equation (6) (the absorption curve with allowance for the oblique incidence of neutrons).

We have implicitly made the assumption that the angular distribution of neutrons of class \(b\) does not depend on the depth \(x\); in reality this assumption is not quite correct, but we think that it will not introduce any appreciable error.

When the absorber is located in the plane \(x\), the activity of the detector due only to neutrons of class \(b\) is proportional to

\[ N_b c\left(\frac{x}{\lambda_D}+K_D\delta\right), \]

where \(\delta\) is the thickness of the absorber and \(K_D\) is its absorption coefficient for \(D\)-neutrons.

Thus the difference of the two activities is proportional to

\[ N_b\left[c\left(\frac{x}{\lambda_D}\right)-c\left(\frac{x}{\lambda_D}+K_D\delta\right)\right]. \]

This expression corresponds to the data of the second column of Table 9,

In the fourth column is given the activability of the absorber, due only to neutrons of class \(b\); it is, obviously, proportional to \(N_b\) (Section 11).

Therefore the ratios of the corresponding data in the second and fourth columns will be proportional to

\[ c\left(\frac{x}{\lambda_D}\right) - c\left(\frac{x}{\lambda_D} + K_D\delta\right). \tag{18} \]

For our rhodium absorber \(K_D = 1.8\ \text{cm}^2/\text{g}\); \(\delta = 0.36\ \text{g}/\text{cm}^2\), so that \(K_D\delta = 0.65\). Similar considerations are also valid for group \(A\) \((K_A = 20\ \text{cm}^2/\text{g}\) and \(\delta = 0.057\ \text{g}/\text{cm}^2)\).

In Fig. 6 the curves \(^{18}\) for groups \(A\) and \(D\) are plotted under the assumption that their values at \(x = 0\) are equal to 100; the points represent the experimental values of the ratios of the numbers in the second and fourth

Fig. 6. Ratio of activities (equation 18). The points represent experimental values

columns of Table 9. The abscissae are calculated under the assumption that \(\lambda_D = \lambda_A = 1.1\ \text{cm}\). Thus it appears that the neutrons of these two groups have the same mean free path, which is slightly greater than \(1\ \text{cm}\). We have already noted that this method is based on the assumption that neutrons of groups \(D\) and \(A\) make only one free path. Although this assumption is quite realistic for group \(A\), it is more doubtful for group \(D\) (§ 12). However, the fact that we have found approximately the same value for \(\lambda_D\) and \(\lambda_A\) may serve as an argument in favor of the validity of this assumption.

8. Mean Free Path of Thermal Neutrons

In the preceding paragraph we found the value of the mean free path for neutrons of groups \(D\) and \(A\). The value found depends on the assumption that neutrons of these groups make only one free path (Section 12).

In order to verify this assumption, it would have been possible to measure directly the mean free path of these groups. Up to now we have not been able to carry out such a measurement because of the low intensity. However, we were able to measure directly the mean free path \(\lambda\) of thermal neutrons in paraffin.

This result, together with the measured value of \(\lambda^2 N\) from the preceding paragraph, makes it possible to calculate the mean number of free paths of a thermal neutron. We must recall that the albedo measurements also make it possible to calculate the value of \(N\) (Section 6).

The measurement of the mean free path \(\lambda\) can be carried out by placing, between the source and the detector, different thicknesses of paraffin, whose dimensions must be such as just to screen the detector from the source, and if the source and the thermal-neutron detector are situated at a distance, as large as possible in comparison with their dimensions. Under ideal geometrical conditions, in which the solid angles under which the detector is seen from the source and the source from the detector were very small, the activity would be proportional to \(e^{-x/\lambda}\), where \(x\) is the thickness of the paraffin scatterer.

In order to obtain an intensity not too small, we had to work under conditions very different from the ideal ones, and therefore it was necessary to correct the results in order to take into account the imperfection of the geometry.

The apparatus which we used was as follows.

Source. A paraffin cylinder 12 cm in diameter and 13 cm high contained the RaEm + Be source at a distance of 2 cm below the center of the upper base. This cylinder was completely covered with a cadmium layer of \(0.5\ \mathrm{g/cm^2}\); in the center of the upper base an opening \(5 \times 5.5\ \mathrm{cm^2}\) in size was made in the cadmium, which could be opened and closed. If we measure the activity of a slow-neutron detector placed outside the paraffin cylinder, once with the opening open and once with it closed, and take the difference of the two activities so measured, we obtain the part of the activity due only to neutrons of group \(C\) emerging through the opening. This difference is thus equivalent to the activity that would be due to a source consisting only of \(C\)-neutrons, having the same position and dimensions as the opening. In this sense, in what follows we shall speak of a source of \(C\)-neutrons, or simply of source \(C\).

Detector. In order to have sufficiently good geometrical conditions and sufficient intensity, we constructed several small cylindrical ionization chambers\(^{18}\) with an internal diameter of 3.5 cm and 10 cm long, filled with oxygen at a pressure of 75 atm. The element which was to be irradiated (Rh \(0.125\ \mathrm{g/cm^2}\)) was placed inside the chamber in the form of a cylindrical layer 5 cm high and 3.5 cm in diameter and itself served as the electrode at high potential. The other electrode,

connected to the electrometer, was a metallic rod placed on the axis of the chamber. The walls of the chamber were made of steel 4 mm thick and practically did not absorb slow neutrons, which, passing through the walls of the chamber, could reach the detector.

This arrangement had the advantage, in comparison with the methods we had used until then, that, owing to the high pressure, almost all the energy of the β-rays was expended inside the chamber; at the same time the electrode had a large useful surface and the geometrical conditions were very good. In fact, with this arrangement it is possible to obtain almost the same sensitivity of observation as with counters, and, in addition, it has the advantage of greater stability, characteristic of ionization chambers.

The chamber, connected to the electrometer, was irradiated for 2 min. 15 sec.; after the source had been removed, we began the count, i.e. measured the number of scale divisions traversed in 2 min.

We made two series of measurements: in the first series the distance from the source \(C\) to the axis of the ionization chamber was 20 cm; in the second—10 cm; in both series the paraffin scatterers were placed at half the distance between the source \(C\) and the detector. To avoid accidental scattering, the ionization chamber and the entire path between the source \(C\) and the detector were shielded by a layer of cadmium of thickness \(0.5\ \mathrm{g/cm^2}\).

Table 10 gives, in arbitrary units, the results of the two series of measurements (each value is the mean of three counts); the error is approximately one unit.

TABLE 10

Measurements of activity for different thicknesses \((x)\) of paraffin with the detector at a distance of 10 and 20 cm from the source

\(x\)	20 cm	20 cm	10 cm	10 cm
	without Cd	with Cd	without Cd	with Cd
0	45.1	23.6	175	92.5
0.047	40.5	22.1	162	89.5
0.099	36.8	21.7	158	90
0.203	29.9	19.3	140.5	87
0.38	24.8	17.8	127	86
0.68	21.2	16.7	116.5	87

In the first column are given the thicknesses of the paraffin scatterers (density 0.9) in centimeters. In the second and third columns are given the activities at a distance of 20 cm, respectively: with open and

closed cadmium window of the source. The activity due only to the \(C\)-neutrons emerging from the window, as we have already noted above, is the difference between the values in these two columns. The fourth and fifth columns are analogous to the second and third, but refer to the case when the distance between the source and the detector is \(10\) cm.

To obtain from these measurements the value of the mean free path of thermal neutrons, it is necessary to take into account corrections for the imperfection of the geometry, which are especially large in the measurements at a distance of \(10\) cm. The calculation of corrections for thick scatterers is rather doubtful, whereas for thin ones a sufficiently accurate estimate of the corrections can be obtained.

For a distance of \(20\) cm and a thin scatterer the correction was calculated as follows: the useful transverse cross section of the detector was \(17.5\ \text{cm}^2\), and the area of the scatterer \(22.3\ \text{cm}^2\); the neutrons reaching the detector consisted partly of those which had undergone no collision in the scatterer, and partly of those which had undergone one collision in it (multiple scattering may be neglected if the thickness \(x\) of the scatterer is small).

The first number of neutrons is, obviously, proportional to the area \((17.5\ \text{cm}^2)\) of the detector and to the probability \(e^{-x/\lambda}\) that the neutron has not undergone a collision in the scatterer. For small \(x\) the number of these neutrons is proportional to

\[ 17.5\left(1-\frac{x}{\lambda}\right). \]

We neglect the small effect arising from oblique paths of neutrons in the scatterer. On the other hand, the number of neutrons which strike the scatterer is proportional to four times its area, because its distance from the source is equal to half the distance of the detector from the source, i.e., this number is proportional to \(4\cdot 22.3 = 89.2\). A fraction \(\frac{x}{\lambda}\) of these neutrons undergoes a collision in the scatterer; assuming that the \(C\)-neutrons are scattered isotropically by the hydrogen atoms in paraffin (Section 13 and \(F\), Sections 10 and 11), we find that the fraction of scattered neutrons which strike the detector is equal to the ratio of the detector area to the surface of a sphere with radius equal to the distance between the scatterer and the detector \((10\ \text{cm})\).

The total number of scattered neutrons which strike the detector will therefore be proportional to

\[ 89.2\left(\frac{x}{\lambda}\right)\cdot\left(\frac{17.5}{4\pi\cdot 10^2}\right). \]

Adding this number to the number of neutrons which strike the detector without undergoing collisions in the scatterer, we obtain the total

the number of neutrons incident on the detector as a function of \(x\). This number is proportional to

\[ 17.5\left(1-0.929\,\frac{x}{\lambda}+\cdots\right). \]

Thus, taking the activity for \(x=0\) as unity, we find that the activity of the detector as a function of the thickness of the scatterer \(x\) (for a thin scatterer) is given by the expression

\[ 1-0.929\,\frac{x}{\lambda}+\cdots . \]

From this relation we obtain that the tangent to the diffusion curve at the point \(x=0\) intersects the abscissa axis at the point

\[ x_1=\frac{\lambda}{0.929}\ \mathrm{cm}. \]

From the graph of the experimental data we found the tangent for \(x=0\); it intersects the abscissa axis at the point \(x_1=0.29\ \mathrm{cm}\). Thus we find

\[ \lambda=0.29\cdot 0.929=0.27\ \mathrm{cm}. \]

We also made sure that the entire course of the experimental diffusion curve could be brought into agreement with the calculated curve also for thick scatterers (the calculation, the details of which we do not give here, was carried out by means of the methods explained in \(F\), section 5).

The calculation of corrections for experiments with a distance of \(10\ \mathrm{cm}\) is somewhat more complicated, because in this case one cannot neglect the circumstance that the neutrons are incident obliquely.

The calculation was made by means of a numerical procedure, and from it we found the expression for the initial form of the diffusion curve

\[ 1-0.84\,\frac{x}{\lambda}+\cdots . \]

From the experimental curve, in the same way as above, we found \(x=0.84\ \mathrm{cm}\), so that from this measurement we have

\[ \lambda=0.84\cdot 0.4=0.34\ \mathrm{cm}. \]

From the totality of these two measurements we may obtain, as the most probable value of the mean free path of thermal neutrons (section 13),

\[ \lambda=0.3\ \mathrm{cm}. \]

9. Neutron groups for different distances from the source; energy ratios

In a previous paper \(^{19}\) we measured the activity of a rhodium detector as a function of the distance from the source in a tank of water. The numbers given there correspond to a mixture of groups \(C\) and \(D\), both of which produce activity in rhodium.

It is obviously of interest to study the behavior of each separate group as a function of the distance from the source. The measurements were made in a cylindrical vessel with water (95 cm deep and 90 cm in diameter); the following detectors were used: for groups \(C\) and \(D\), an Rh plate (\(0.36\ \mathrm{g/cm^2}\)) with an area of \(5 \cdot 5\ \mathrm{cm^2}\). This detector, shielded by \(0.5\ \mathrm{g/cm^2}\) Cd, is sensitive only to \(D\)-neutrons; the difference between the activities measured without Cd and with Cd gives the activity due only to group \(C\). A silver detector \(0.057\ \mathrm{g/cm^2}\) thick, with an area of \(5 \cdot 5\ \mathrm{cm^2}\), shielded by Cd, was used as a detector for group \(A + B\), which were not studied separately. Finally, for group \(J\) we used a PbJ detector, \(0.068\ \mathrm{g/cm^2}\), shielded by Cd, with an area of \(5 \cdot 5\ \mathrm{cm^2}\). The activities of the detectors were measured on both sides for various distances from the source; the sum of the two activities is proportional to the number of neutrons of the given group entering the detector (Section 11).

In order to make as small as possible the errors associated with the finite dimensions of the vessel, the measurements were carried out in such an arrangement that the midpoint between the source and the detector was at the center of the cylinder of water.

The dimensions of our detectors, and also of the ampoule containing RaEm + Be, were not so small that the distance between the centers of the source and detector could be regarded as equal to the distance between these objects, especially for small distances. We therefore used mean distances. The experimental results for groups \(C\), \(D\), \(A + B\), and \(J\) are given in Tables 11, 12, 13, and 14. The numbers represent the sum of the intensities measured on both sides of the detector. The data for groups \(C\), \(D\), and \(A + B\) were obtained as the mean values from 3 series of measurements for each side. For group \(J\) we made only one series of measurements for each side, since the period of iodine is sufficiently long to obtain sufficiently accurate results.

TABLE 11

Group \(C\) (difference in the activity of the Rh detector without Cd and with Cd)

\(r\) (cm)	2.5	3.6	5.4	7.3	10.2	15.1	20.1	25	30	35	40
Activity	2121	1898	1441	1006	523	167	58	23	9	3.3	1.7

TABLE 12

Group \(D\) (Rh detector, shielded with cadmium)

\(r\) (cm) . . .	2.5	3.6	5.4	7.3	10.2	15.1	20.1
Activity	306	252	166	96	40	10.6	3.8

TABLE 13

Groups \(A+B\) (Ag detector, shielded with cadmium)

\(r\) (cm) . . .	2.5	3.6	5.4	7.3	10.2	15.1
Activity	144	119	73	39.6	15.7	4.3

TABLE 14

Group \(J\) (\(\mathrm{PbJ}_2\) detector, shielded with cadmium)

\(r\) (cm) . . .	2.62	3.36	5.1	7.0	9.79
Activity	24.9	19.7	12.0	6.28	2.46

In Fig. 7 the activities, multiplied by \(r^2\) (\(r\) is equal to the distance from the source), are plotted as a function of \(r\). The data are taken from Tables 11, 12, 13, 14; the units of the ordinates were chosen so that each curve enclosed the same area. Although the differences between

Fig. 7. Ordinates—activation \(\times r\). Abscissae—\(r\). The units of the ordinates are chosen so that the curves enclose the same area

the curves \(D\), \(A+B\), and \(J\) are small, the accuracy of the measurements is high enough to make it possible to notice their difference from one another. In studying these curves one can see that the activity due to various

groups, decreases with \(r\) according to different laws; the activity decreases more rapidly for the groups in the following order: \(C, D, A+B\), and \(J\). This fact enables us to arrange the groups in order of increasing energy. Let us note that, on the average, neutrons which have undergone a larger number of collisions have lost more energy; on the other hand, neutrons which have suffered a larger number of impacts, on the average, diffuse to greater distances from the source. Thus it follows that the groups for which the intensity decreases more rapidly with distance \(r\) have the greater energy. The order of increasing energy for the groups we have investigated is therefore as follows:

\[ C,\ D,\ A+B,\ J. \]

These considerations may be made quantitative in the following way: a convenient parameter for characterizing the rapidity of decrease of the various groups as a function of distance is the mean value of the square of the distance from the source for neutrons of the different groups, \((r^2)_{\mathrm{av}}\).

\((r^2)_{\mathrm{av}}\) was calculated for each group by means of the equation

\[ \frac{\displaystyle \int_{0}^{\infty} f(r)\,r^2\,dr} {\displaystyle \int_{0}^{\infty} f(r)\,dr}, \]

where \(f(r)\) is the curve shown in Fig. 7 and referring to the group under consideration. The values of \((r^2)_{\mathrm{av}}\) are given in Table 15.

TABLE 15

Mean values of \(r^2\)

Group	\((r^2)_{\mathrm{av}}\) (in \(\mathrm{cm}^2\))
\(C\)	326.7
\(D\)	276.6
\(A+B\)	270.6
\(J\)	262.2

Although the activity curves as a function of distance were measured very accurately, we could not determine the intensity of the groups \(D, A+B\), and \(J\) out to large distances. In calculating \((r^2)_{\mathrm{av}}\) we therefore extrapolated the intensities of these

groups to large distances. These extrapolations are relatively good, since for large distances the law of decrease becomes quite similar for all groups, and one may represent the satisfactorily final part of the curves in Fig. 7 by an exponential curve with coefficient \(0.106\ \mathrm{cm}^{-1}\). Nevertheless, a considerable error may be caused by this extrapolation.

It can be shown (\(F\), section 2) that for groups with energy greater than the quantum \(h\nu\) of elastic vibrations of hydrogen in paraffin (in practice, for groups with energy greater than 1 V), the difference \((r'^2)_{\mathrm{cp}}-(r''^2)_{\mathrm{cp}}\) of the mean squares of the distances of two groups is related to the ratio \(\dfrac{W''}{W'}\) of the corresponding energies by the equation

\[ \lg \frac{W''}{W'}=\frac{(r'^2)_{\mathrm{cp}}-(r''^2)_{\mathrm{cp}}}{6l_g^2}, \tag{19} \]

where \(\lambda_g\) is the mean free path of the neutrons of both groups in paraffin. It is assumed that \(\lambda_g\) is the same for both groups (section 7).

The preceding equation can easily be understood qualitatively down to the numerical factor in the denominator; the left-hand side represents the average number of collisions necessary to reduce the energy from \(W''\) to \(W'\). On the other hand, the mean of the square of the displacement of a neutron during a free path is \(2\lambda_g^2\) (the factor 2 is due to the fact that we take the mean of the square, and not the square of the mean). If the orientations of successive free paths were incoherent,* \((r^2)_{\mathrm{cp}}\) would increase by \(2\lambda_g^2\) for each collision, and then \(\lg \dfrac{W''}{W'}\) collisions would produce an increase of \(2\lambda_g^2 \lg \dfrac{W''}{W'}\); this assumption would then give a formula similar to (19), with a factor 2 instead of 6. The factor 6 is obtained by taking into account the coherence of successive free paths, and also the fluctuations in the number of collisions necessary to reduce the energy from \(W''\) to \(W'\). We can use (19) to determine the ratio of the energies of any two groups \(n\) (excluding group \(C\), for which this formula is not valid). This method, however, is very inaccurate, since small errors in \(\lambda_g\) and in \((r^2)_{\mathrm{cp}}\) cause considerable changes in the ratio \(\dfrac{W''}{W'}\).

Since we measured \((r^2)_{\mathrm{cp}}\) in water, and not in paraffin, we must substitute for \(\lambda_g\) the length of the mean free path in water; we can obtain it from its value in paraffin, \(1.1\ \mathrm{cm}\) for nonthermal neutrons (section 7), assuming that the length of the mean free path is inversely proportional to the concentration of hydrogen; thus we find \(\lambda = 1.27\ \mathrm{cm}\).

* This word is used here to denote the degree of coincidence of the directions of motion of the primary and scattered neutrons.

Taking this value, we find from (19) that the difference \((r^2)_{\mathrm{cp}}\) corresponding to the energy ratio \(e\) is equal to \(6\lambda_g^2 = 9.7\ \mathrm{cm}^2\).

From Table 15 we obtain the following energy ratios for the groups \(J\), \(A + D\), and \(D\):

\[ W_J : W_{A+D} : W_D = 4.4 : 1.85 : 1. \]

As we have already said, the error in these ratios may be very large.

It may be better to obtain from (19) the values of the mean free path in water, assuming that the energies of the groups are inversely proportional to the square of the absorption coefficient in boron20 (see also Section 13).

Taking the following absorption coefficients in boron: \(K_J = 1\); \(K_B = 2.3\); \(K_A = 3\); \(K_D = 4.7\); \(K_C = 38\ \mathrm{cm}^2/g\), we obtain the following ratios between the energies:

\[ W_J : W_B : W_A : W_D : W_C = 1440 : 270 : 160 : 65 : 1 \]

and hence, assuming21 \(W_C = kT = 0.025V\), we shall have:

\[ W_J = 36;\quad W_B = 7;\quad W_A = 4;\quad W_D = 1.6V. \]

From these values, by means of equation (19), we find the mean free path in water \(\lambda_g = 0.87\ \mathrm{cm}\) instead of \(1.27\ \mathrm{cm}\). The difference between the two values lies within the limits of experimental errors.

We further note that \((r^2)_{\mathrm{cp}}\) for group \(C\), given in Table 15, differs considerably from the values of \((r^2)_{\mathrm{cp}}\) for the other groups. For example:

\[ (r_C^2)_{\mathrm{cp}} - (r_D^2)_{\mathrm{cp}} = 50\ \mathrm{cm}^2. \]

This difference is partly due to the large number of free paths of neutrons when their energy has already reached thermal energy; the magnitude due to this diffusion process (Section 7) will be

\[ 2\lambda^2 N = 26\ \mathrm{cm}^2. \]

The remaining \(24\ \mathrm{cm}^2\) correspond to the process of slowing down neutrons from the energy corresponding to group \(D\) to the energy of thermal excitation. This process cannot be easily calculated, since it is complicated by the effect due to the chemical binding of hydrogen (\(F\), Section 19).

10. Transition of neutrons from one group to another

In the preceding section we arranged the groups in order of decreasing energy. It is obvious that a neutron belonging to a group with high energy, after several collisions, will have

On the Absorption and Diffusion of Slow Neutrons

lower energy and can, in this way, pass into the group with low energy. We must therefore expect that neutrons of all groups, after a certain number of collisions, become thermal neutrons.

We therefore tried to observe the transformation of \(D\)-neutrons into \(C\)-neutrons[^22].

The principle underlying these experiments can be easily understood if we suppose that we have a detector \(R\), activated only by \(C\)-neutrons, and an absorber \(A\), absorbing only \(D\)-neutrons. We place the detector \(R\) at the center of the upper base of a paraffin cylinder containing the source \(S\); if now an absorber is placed between the paraffin and the detector, no decrease in the activity of the detector will occur, since the \(C\)-neutrons emerging from the paraffin are not absorbed by the absorber \(A\). If, however, we place the absorber \(A\) inside the paraffin, 1 or 2 cm below the upper base, it may happen that it will absorb some of the \(D\)-neutrons which might have been transformed into \(C\)-neutrons as a result of collisions in the paraffin before reaching the detector.

If \(D\)-neutrons are indeed transformed into \(C\)-neutrons, we must expect that the absorber \(A\) will produce no effect if it is placed between the paraffin and the detector, but that it will reduce the activity if it is placed inside the paraffin, 1 or 2 cm below the upper base of the cylinder.

In reality the experiment cannot be performed under such ideal conditions, because we do not have a detector only for group \(C\) and an absorber only for group \(D\). We therefore proceeded as follows. We used as detector a rhodium plate \(0.36 \text{ g/cm}^2\) thick; about 70% of its activity is due to group \(C\), and the remainder to group \(D\). To measure the activity produced only by group \(C\), all measurements were made with a cadmium absorber \(0.54 \text{ g/cm}^2\) thick, which was placed directly under the rhodium plate, and without it. The difference between the activities without cadmium and with cadmium gives the activity produced only by group \(C\).

As an absorber for group \(D\) we used an indium plate \(1.83 \text{ g}/28.5 \text{ cm}^2\); the absorption coefficient of indium for group \(D\) is \(3.4 \text{ cm}^2/\text{g}\), and for group \(C\), \(0.7 \text{ cm}^2/\text{g}\). It follows from this that the absorption of \(C\)-neutrons in our indium plate is far from negligible, especially if we take into account the large number and albedo of this group. To introduce a correction for this, we prepared a plate of an Sn—Cd alloy, which was equivalent to the indium plate with respect to the absorption of group \(C\) and practically did not absorb \(D\)-neutrons. Taking the difference in the activities produced by group \(C\) in the detector, once with the indium plate and once with the Sn—Cd plate, we obtain the effect produced by the absorption of group \(D\) in indium.

Table 16 gives the activities obtained in the rhodium plate

E. AMALDI AND E. FERMI

TABLE 16

Activities induced in an Rh plate in various arrangements. \(S\)—source, \(P\)—paraffin (cm of thickness), detector

Arrangement	Activity
\(SP\ (3,5)\ R\)	\(280,4 \pm 0,7\)
\(SP\ (3,5)\ \mathrm{In}R\)	\(230,2 \pm 0,7\)
\(SP\ (3,5)\ \mathrm{InCd}R\)	\(45,6 \pm 0,4\)
\(SP\ (3,5)\ \mathrm{Cd}R\)	\(68,7 \pm 0,5\)
\(SP\ (3,5)\ \mathrm{cd}R\)	\(254,8 \pm 0,7\)
\(SP\ (2,5)\ \mathrm{cd}P(1)\ R\)	\(253,0 \pm 0,7\)
\(SP\ (2,5)\ \mathrm{In}P(1)\ R\)	\(241,5 \pm 0,7\)
\(SP\ (2,5)\ \mathrm{cd}P(1)\ \mathrm{Cd}R\)	\(71,1 \pm 0,5\)
\(SP\ (2,5)\ \mathrm{In}P(1)\ \mathrm{Cd}R\)	\(64,9 \pm 0,5\)
\(SP\ (1,5)\ \mathrm{cd}P(2)\ R\)	\(254,0 \pm 0,7\)
\(SP\ (1,5)\ \mathrm{In}P(2)\ R\)	\(248,9 \pm 0,7\)
\(SP\ (1,5)\ \mathrm{cd}P(2)\ \mathrm{Cd}R\)	\(69,2 \pm 0,5\)
\(SP\ (1,5)\ \mathrm{In}P(2)\ \mathrm{Cd}R\)	\(68,1 \pm 0,5\)

in different experiments; the different arrangements are denoted in the same way as in Section 6; the numbers in parentheses after the designation \(P\) for paraffin give, in centimeters, the thickness of the paraffin layers; since we used two different cadmium layers—one of thickness \(0,54\ \mathrm{g/cm^2}\) for complete absorption of group \(C\), and another, \(0,0036\ \mathrm{g/cm^2}\), equivalent to an indium layer with respect to the absorption of neutrons of group \(C\), we denoted these two absorbers respectively by Cd and cd. The numbers given in the table are the averages of ten measurements.

The first five measurements were made to check the identity of the thin cadmium layer (Cd) and indium In with respect to absorption of \(C\)-neutrons. Their result showed that the absorption in the indium plate is probably slightly greater than the absorption in the cadmium layer; the difference \(280,4 - 68,7 - (230,2 - 45,6) = 27,1 \pm 1,2\) represents the activity of the rhodium detector due to that part of group \(C\) which is absorbed by the indium plate; the difference \(280,4 - 245,8 = 25,6 \pm 1\) is the analogous activity for the thin cadmium layer.

The next four measurements were made with an absorber (Cd or In) at a distance of \(1\ \mathrm{cm}\) inside the paraffin. The difference \(253,0 - 71,1 = 181,9 \pm 0,9\) represents the activity of the detector due to group \(C\), when the cadmium absorber (Cd) is located at a distance of \(1\ \mathrm{cm}\) inside the paraffin. The difference \(241,5 - 64,9 = 176,6 \pm 0,9\) gives the analogous activity with an indium absorber. The difference between these two activities, equal to \(5,3 \pm 1,2\), is considerably greater than what we could have expected from the small difference between the two absorbers if they are placed close against the detector.

The four measurements following these are analogous to the preceding ones, with the only difference that the absorber was placed at a distance of \(2\ \mathrm{cm}\) below the surface of the paraffin. In this series of measurements as well there is a difference analogous to that observed in the preceding series; this difference is now \(4,0 \pm 1,2\).

Considering these numbers, one may conclude that the observed effect gives an indication of the conversion of neutrons of group \(D\) into neu-

neutrons of group \(C\), and that it is 2.3 times greater than the mean square error.

Although we have sufficient confidence in the reality of the effect in this sense, we cannot draw a final conclusion from this experiment. An estimate of the effect that was to be expected in this experiment shows that it is indeed of the same order of magnitude as the observed effect; the estimate was made as follows.

We measured the activities induced in an indium plate in the experiments described above, placed inside the paraffin \(2\ \mathrm{cm}\) below the upper base, separately for groups \(C\) and \(D\), and found, respectively, 257 and 47.5. These numbers are proportional to the numbers of neutrons of these two groups absorbed in our indium layer.

Let \(p_1\) be the probability that a neutron \(C\), which is in the paraffin at the site of the indium absorber, will leave through the upper plane and be absorbed by the rhodium detector; similarly, let \(p_2\) be the probability that a neutron \(D\), which is at the site of the indium absorber, will leave through the upper plane already transformed into a neutron \(C\) and be absorbed by the rhodium.

The decreases in the rhodium activities caused by group \(C\) and due respectively to the absorption of \(D\)- and \(C\)-neutrons in the indium layer will be in the ratio \(\dfrac{47.5p_2}{257p_1}\). Assuming that \(p_1\) and \(p_2\) are equal, or at least of the same order of magnitude, we find that these decreases are in the ratio \(\dfrac{47.5}{257}\), and since the decrease due to the absorption of \(C\)-neutrons is 26.9 (Table 16), the effect which we should expect in the preceding experiment is equal to 5, i.e. of the same order of magnitude as the effect actually observed.

Similar experiments were made by Preiswerk and Galban²³, who found a similar ratio between groups \(J\) and \(A\).

11. Behavior of the Groups Near the Surface of Paraffin

In Section 9 we studied the activity produced by the various groups inside so large a tank of water that we could neglect effects due to the finiteness of its dimensions. We shall now study the behavior of the activity produced by the different groups near the surface of paraffin.

First of all we shall consider the behavior of group \(C\). From the theory of diffusion of thermal neutrons it follows that, irrespective of the position of the source, the density of thermal neutrons near the surface bounding the paraffin decreases in the direction toward the boundary of the paraffin (\(F\), Section 6, formula 57); it can moreover be shown that if we denote by \(x\) the depth below the surface of the paraffin,

if so, then the density \(n(x)\) for small \(x\) is approximately proportional to

\[ x+\frac{\lambda}{\sqrt{3}}; \]

therefore, if we extrapolate the curve \(n(x)\) in the direction toward the boundary of the paraffin \((x>0)\), then \(n(x)\) will be equal to zero at a distance from the surface of the paraffin equal to

\[ x_0=-\frac{\lambda}{\sqrt{3}}. \]

This is precisely the distribution of the density of thermal neutrons near the surface which determines the law of the angular distribution (5) of group \(C\) neutrons emerging from the paraffin. If the density were constant near the surface of the paraffin, the cosine law would hold.

To measure \(n(x)\), it is necessary to use a detector of group \(C\) so thin as not to disturb the neutron distribution. Its activity in this case will be proportional to the density \(n(x)\) of thermal neutrons.

We used two detectors, which were obtained by electrolytic deposition of rhodium on nickel plates (the latter served only as nonactivated substrates) with an area of \(29\ \text{cm}^2\). The weight of the rhodium deposited in these two detectors was respectively \(0.087\) and \(0.169\ \text{g}\), taking into account the absorption coefficient of rhodium for group \(C\) \((0.7\ \text{cm}^2/\text{g})\). The \(K\delta\) values for these two detectors were respectively equal to \(0.002\) and \(0.004\); these values correspond to average probabilities \(\zeta\) of absorption of a thermal neutron passing through the detector of about \(0.004\) and, respectively, \(0.008\) (§ 6). These probabilities are very small, and therefore the detectors used may be regarded as very thin.

The measurements were made with a paraffin cylinder \(24\ \text{cm}\) in diameter and \(10\ \text{cm}\) high, containing the source at a distance of \(3.2\ \text{cm}\) below the upper base. The layer of paraffin \(3.2\ \text{cm}\) thick between the source and the upper base was cut into thin plates so that the detector could be placed at different depths.

To find the activity due only to group \(C\), we always took the difference between the activities of the detector obtained without screens and with two cadmium screens \(0.44\ \text{g}/\text{cm}^2\) thick, between which the detector could be placed.

In Fig. 8 the activity of our two detectors, due only to group \(C\), is plotted as a function of the depth \(x\). We also give parts of the curves extrapolated in the direction of negative \(x\). Their intersection with the abscissa axis lies near the point

\[ x_0=-0.18\ \text{cm}, \]

From this value we find

\[ \lambda = 0.31\ \mathrm{cm} \]

in good agreement with the values obtained from direct measurements (§ 8).

We have already noted that this experiment has a very simple explanation if the detector is very thin. A thick detector, on the contrary, strongly disturbs the density of thermal neutrons, so that the curve of the dependence of the activity on \(x\) differs greatly from the preceding one.

Thus, using a rhodium detector of thickness \(0.36\ \mathrm{g/cm^2}\) and adding, in order to obtain data comparable with those from the preceding experiment, the activities induced by group \(C\), we found that the activity curve, extrapolated in the direction of negative \(x\), intersects the \(x\)-axis at the point \(x_0=-0.4\ \mathrm{cm}\), i.e., at a distance more than twice as large as in the case of the thin detector.

Fig. 8. Activity of group \(C\), plotted as a function of the depth \(x\) in paraffin

In this experiment we added the activities measured on both sides of the detector, since this sum is proportional to the total number of thermal neutrons entering the detector. Let \(N_1\) and \(N_2\) be the numbers of neutrons entering the detector from sides 1 and 2, respectively; the activities measured on sides 1 and 2 are given by the expressions:

\[ A_1=N_1a+N_2b, \]

\[ A_2=N_1b+N_2a, \tag{20} \]

where \(a\) and \(b\) are constants characterizing the detector and depending on its absorption coefficients for neutrons and electrons and on its thickness. The ratio \(\frac{a}{b}\) can be obtained by measuring the ratio \(\frac{A_1}{A_2}\) when the detector is placed outside the paraffin, so that \(N_2=0\).

For a rhodium detector of thickness \(0.35\ \mathrm{g/cm^2}\), \(\frac{a}{b}=\frac{4}{3}\) for group \(C\) and \(\frac{2}{1}\) for group \(D\). This difference depends on the large value of the absorption coefficient of rhodium for neutrons of group \(D\),

For a thin detector one would have \(a=b\). Summing both equations (20), we find

\[ A_1 + A_2 = (a+b)(N_1+N_2), \]

and therefore the sum of the activities measured on both sides of the detector is always proportional to the total number \((N_1+N_2)\) of neutrons crossing the detector. From (20) we find

\[ N_1=\frac{aA_1-bA_2}{a^2-b^2},\quad N_2=\frac{aA_2-bA_1}{a^2-b^2}, \tag{21} \]

i.e., by measuring \(A_1\) and \(A_2\) and the ratio \(\frac{a}{b}\), we can find values proportional to \(N_1\) and \(N_2\) (this method was used in Section 7 for measuring \(N_1\) and \(N_2\) separately). Table 17 gives

TABLE 17

Numbers of neutrons entering
the detector from side (1) and (2);
\(x\)—paraffin thickness

\(x\)	Group \(C\)	Group \(C\)	Group \(D\)	Group \(D\)
\(x\)	\(N_1\)	\(N_2\)	\(N_1\)	\(N_2\)
0	1	0	1	0
0.16	1.10	0.28	1.07	0.07
0.34	1.15	0.63	1.17	0.14
0.53	1.28	0.91	1.25	0.28
0.87	1.60	1.19	1.31	0.45
1.87	2.19	2.00	1.35	1.00

the values \(N_1\) and \(N_2\), measured with our rhodium detector of thickness \(0.36\ \mathrm{g/cm^2}\), separately for groups \(C\) and \(D\). In order to compare these two series of numbers, we took as unity for both groups the value of \(N_1\) at \(x=0\). The source in these experiments was at a distance \(x=2.4\ \mathrm{cm}\). The data for group \(C\) do not admit of a simple interpretation, since the detector cannot be regarded as very thin.

For group \(D\), however, the detector, although thick, does not disturb the distribution of \(D\)-neutrons, since these neutrons make on average only one free path (Sections 6, 7, 12), and therefore each of them strikes the detector only once.

Extrapolating the curve \(N_1+N_2\) for group \(D\) toward negative values of \(x\), we find its intersection with the \(x\)-axis at the point \(x_0=-0.9\).

The fact that this intersection lies considerably farther from the surface of the paraffin than for group \(C\) (\(x_0=0.18\ \mathrm{cm}\) with a thin

detector), is due, on the one hand, to the greater length of the mean free path and, on the other hand, to the coherence in the orientation of successive free paths, which is very considerable for neutrons with energy greater than the quantum of the elastic bond of hydrogen in paraffin.

Comparing the second and fourth columns of Table 17, we note that the course of the curve \(N_1\) near the surface for our detector does not differ very much for groups \(C\) and \(D\), whereas at great depths group \(C\) increases considerably more rapidly than group \(D\). This can be understood by taking into account the increase in the albedo of the paraffin layer placed on the detector as a function of the thickness of the layer.

12. Width of the energy bands corresponding to the groups. Total number of neutrons

In this paragraph we shall consider a method for determining the width of the energy bands1 that correspond to the different groups, or, more precisely, the ratio

\[ \frac{W_{\max}}{W_{\min}} \]

of the maximum energy to the minimum energy that bound the band. In the calculation it is assumed that the bands have sharp boundaries. It would also be possible to make similar calculations assuming that the band has a resonance form; however, this is hardly worth doing in view of the low accuracy of the experimental data.

In the last part of this paragraph we shall give an estimate of the total number of neutrons emitted by the \(\mathrm{RaEm} + \mathrm{Be}\) source.

The method for determining the width of the band corresponding to a group with energy greater than \(1\text{ V}\) is based on comparison of the following two quantities: the activation \(A_g\) of the detector, due to neutrons of the group \(g\) under consideration, and the activation \(B_C\) of the same detector, due to thermal neutrons. This latter activity is obtained by shielding the detector on one side with a layer of cadmium sufficiently thick to absorb completely the thermal neutrons striking it; \(B_C\) can, of course, be obtained as the difference of the activations of our detector with a cadmium layer on only one side or on both sides.

If the distribution of fast neutrons inside the large paraffin block were uniform, then it is clear that \(A_g\) and \(B_C\) would have values independent of the position and orientation of the detector. However, the source of fast neutrons is small, and therefore we must compare the mean values of \(A_g\) and \(B_C\), taken over all positions and orientations inside the paraffin; i.e., we must compare

\[ \int \widetilde{A}_g\,d\tau;\qquad \int \widetilde{B}_C\,d\tau, \]

where \(\widetilde{A}_g\) and \(\widetilde{B}_c\) are the mean values of the detector activations measured in two opposite orientations; these mean values are practically identical with the means taken over all directions. By using the indicated integrals we eliminate all the complications arising from the nonuniform distribution of neutrons.

We shall now assume that, in the paraffin around our detector, \(q\) fast neutrons are produced per second in \(1\ \mathrm{cm}^3\). These neutrons are slowed down in such a way that near our detector we shall have neutrons of all possible velocities.

It can be shown (\(F\), Section I) that, for energies greater than \(1\ \mathrm{V}\), the number of neutrons having velocities between \(v\) and \(v+dv\) will be

\[ \frac{2q\lambda(v)}{v^2}\,dv. \tag{22} \]

From this we can easily calculate the activity \(A_g\) as a function of the energies \(W_{\max}\) and \(W_{\min}\) bounding the energy band \(g\), of the mean free path, of the absorption coefficient of the detector \(K_g\) for the group \(g\), and of the surface \(s\) and thickness \(\delta\) of the detector. We find (\(F\), Section 8)

\[ A_g=\eta s q\lambda_g K_g \lg\frac{W_{\max}}{W_{\min}}\frac{1}{2} \left\{ \int_0^\delta b(K_g x)e^{-\mu x}\,dx+ \right. \]

\[ \left. +\int_0^\delta b(K_g x)e^{-\mu(\delta-x)}\,dx \right\}, \tag{23} \]

where \(\mu\) is the absorption coefficient for the \(\beta\)-rays of the detector in the detector itself; \(b(K_gx)\) is function (4), and \(\eta\) is the efficiency of the ionization chamber for the detector \(\beta\)-rays. The last factor in (23) represents the effect of absorption of neutrons and \(\beta\)-rays in the detector; it would be equal to \(\delta\) for a very thin detector.

Likewise, the activity \(B_c\), due to thermal neutrons, can be calculated as a function of the absorption coefficient of thermal neutrons in the detector \(K_c\) and of the “diffusion length”

\[ (D\tau)^{\frac{1}{2}}=\left(\frac{\lambda^2 N}{3}\right)^{\frac{1}{2}}, \]

taking into account that, in this case, the angular distribution of neutrons

striking the detector, is given by equation (5), we find

\[ B_c=\eta s q\lambda (N)^{\frac12} K_c \int_0^\delta c(K_c x)e^{-\mu x}\,dx, \tag{24} \]

where \(c(K_c x)\) is function (6). We have neglected the change in the absorption coefficient of our detector for thermal neutrons of different velocities. For a very thin detector the integral is equal to \(\delta\). We must now, as already mentioned, integrate (23) and (24), and we obtain

\[ \left. \begin{aligned} \int A_g\,d\tau &=\eta s Q K_g\lambda_g \lg \frac{W_{\max}}{W_{\min}}\cdot \frac12 \left\{ \int_0^\delta b(K_g x)e^{-\mu x}\,dx +\int_0^\delta b(K_g x)e^{-\mu(\delta-x)}\,dx \right\},\\ \int B_c\,d\tau &=\eta s Q K_c\lambda (N)^{\frac12} \int_0^\delta c(K_c x)e^{-\mu x}\,dx, \end{aligned} \right\} \tag{25} \]

where \(Q=\int q\,d\tau\) is the total number of neutrons emitted by the source in 1 sec. From (25) we find

\[ \lg \frac{W_{\max}}{W_{\min}} = \]

\[ = \frac{ K_c\lambda (N)^{\frac12}\displaystyle\int A_g\,d\tau }{ K_g\lambda_g\displaystyle\int B_c\,d\tau } \cdot \frac{ 2\displaystyle\int_0^\delta c(K_c x)e^{-\mu x}\,dx }{ \displaystyle\int_0^\delta b(K_g x)e^{-\mu x}\,dx + \displaystyle\int_0^\delta b(K_g x)e^{-\mu(\delta-x)}\,dx }. \tag{26} \]

We carried out experiments with groups \(D\), \(A\), and \(J\). For group \(D\) we used a rhodium detector \((0.36\ \mathrm{g/cm^2})\); for group \(A\), a silver detector \((0.057\ \mathrm{g/cm^2})\), and for group \(J\), a detector made of lead iodide \((0.76\ \mathrm{g/cm^2})\); since the small absorption coefficient in iodine for group \(C\) is not very well known, we used the data obtained with rhodium also in the calculation of \(B_c\) for group \(J\), assuming the same chamber efficiency for the \(\beta\)-rays of rhodium and iodine. For the calculation of the integrals

\[ \int \widetilde{A}_g\,d\tau,\quad \int \widetilde{B}_c\,d\tau \]

We determined \(\widetilde A_g\) and \(\widetilde B_c\) for one definite distance and obtained values for all distances with the aid of the curves in Fig. 7. In the following equations the numerical values are written in the same order in which the various quantities enter into formula (26):

Group \(D\)

\[ \lg \frac{W_{\max}}{W_{\min}} = \frac{0.7\sqrt{13}\cdot 4.58\cdot 10\cdot 2\cdot 0.108} {1.8\cdot 1.1\cdot 1.5\cdot 10^{-6}\cdot 0.087} +0.052 =0.60 \]

Group \(A\)

\[ \lg \frac{W_{\max}}{W_{\min}} = \frac{0.25\sqrt{13}\cdot 7.8\cdot 10^{-4}\cdot 2\cdot 0.048} {20\cdot 1.1\cdot 2.04\cdot 10^{5}\cdot 0.021} +0.019 =0.038, \]

Group \(J\)

\[ \lg \frac{W_{\max}}{W_{\min}} = \frac{0.7\sqrt{13}\cdot 4.16\cdot 10^{4}\cdot 2\cdot 0.108} {0.38\cdot 1.1\cdot 1.5\cdot 10^{6}\cdot 0.065} +0.049 =0.27, \]

From these values we can obtain the ratio \(W_{\max}/W_{\min}\) for each group; we shall call the quantity \(\lg \frac{W_{\max}}{W_{\min}}\) the logarithmic width. This quantity has a simple physical meaning: it represents the average number of collisions of a neutron during the time that it belongs to the given group. For example, group \(A\), which has a logarithmic width equal to 0.04, is such a narrow energy band that only 4% of the neutrons pass through it during the slowing-down process. In the case of group \(A\), the detector used cannot be regarded as thin with respect to neutron absorption, and if the resonance form of the band were taken into account, then a simple numerical estimate would show that the logarithmic width of group \(A\), calculated from (26), is too large (by a factor of about 2). For the other detectors this correction is considerably smaller.

The logarithmic width of group \(J\), and especially of group \(D\), is considerably greater, as also follows from the large population (§ 5) of these groups. Nevertheless, even for group \(D\) the probability that a neutron will belong to this group for more than one free path is relatively small and is equal to 0.27. These facts explain our results, namely that the albedo of all nonthermal groups is practically equal to zero.

On the other hand, one might have expected a slight difference in the behavior of groups \(D\) and \(A\) in the diffusion experiments described in Section 7. The fact that we did not succeed in finding such a difference may, however, be explained by the inaccuracy of our measurements, since the expected difference is rather small.

If we assume that the energies of the groups are those which were obtained in Section 9 from absorption in boron, we obtain for the widths of the groups \(D\), \(A\), and \(J\), respectively, \(1\), \(0.15\), and \(10\) V.

Finally, we can use the second equation (25) to calculate the total number of neutrons \(Q\) emitted by the source. In (25) \(B_c\) is the initial activity of the detector used. To obtain the activability (3) it is necessary to multiply \(B_c\), given in (25), by

\[ \frac{1000}{JU}=\frac{1000}{J\cdot 840\,\eta U}, \]

so that we find:

\[ \frac{Q}{J} = 0.840\,\frac{\eta U}{\eta}\, \frac{\displaystyle \int B_c\,d\tau} {\displaystyle sh(N)^{\frac12}K_c \int_0^\delta c(K_c x)e^{-\mu x}\,dx}. \tag{27} \]

The same quantities enter into this formula as those which we used in calculating the logarithmic width. Using, for example, the data on rhodium activation, we find

\[ \frac{Q}{J}=160\,000\left(\frac{\eta U}{\eta}\right). \tag{28} \]

Assuming \(\eta=\eta_U\), we have that our neutron unit corresponds to \(160\,000\) neutrons per 1 sec. Taking into account that 1 neutron unit corresponds approximately to \(6\) mC RaEm \(+\) Be, we finally have: \(27\,000\) neutrons per 1 sec. from 1 millicurie.

This value is much larger than the values found by other methods (25). This difference can be attributed only in part to the inequality of \(\eta\) and \(\eta_U\).

13. Summary and Discussion

From the results we have presented, it seems possible to conclude that, for elements sensitive to slow neutrons, the effective cross section for absorption is often an irregular function of the energy of the slow neutrons, possessing sharp maxima that constitute a kind of absorption band.

Analysis of the absorption curves of various elements with various detectors enables us to establish the existence of several distinct absorption bands (groups \(A\), \(B\), \(C\), \(D\), and \(J\)). There are various grounds for thinking that group \(C\) (radiation strongly absorbed by cadmium) corresponds, at least for the most part, to neutrons having thermal-motion velocities, whereas the other groups correspond to higher velocities.

The most direct proof of this assertion is provided by the experiment with a mechanical velocity selector[^26] (for observing only group \(C\)), which permits a direct measurement of the velocity of the \(C\)-neutrons; this velocity was found to be equal to the thermal velocity.

The same result can be drawn from the experiments of Preiswerk and Halban and others,[^27] who found that radiation filtered by cadmium is insensitive to changes in temperature; this shows that neutrons not belonging to group \(C\) have an energy greater than that of thermal motion.

Finally, our albedo measurements (Section 6) show that only neutrons of group \(C\) can make several mean free paths while they belong to this group. This fact is easily understood if one assumes that group \(C\) consists of neutrons in thermal equilibrium, since then successive collisions do not change their mean energy; on the contrary, a neutron belonging to an energy band much greater than \(kT\) has a high probability of leaving this band after a single collision.

In all the cases we investigated, the greater part of the activity (in most cases more than 50%) is due to thermal neutrons; however, this does not mean that the absorption coefficient for thermal neutrons is, as a rule, larger than the absorption coefficient for neutrons having energies of several volts. To a certain extent this is explained by the large number of thermal neutrons emerging from the paraffin block containing the source (Table 6).

The problem of determining the energy bands corresponding to the known non-thermal groups can be solved by the following method, first used by Frisch and Placzek, and by Wickes, and Livingston and Bethe. They assumed that the effective cross section for absorption in boron for a slow neutron is inversely proportional to the neutron velocity \(v\). Assuming this, we obtained from our measurements the following energies:

Group	\(C\)	\(D\)	\(A\)	\(B\)	\(J\)
Volts	0.037	1.6	4	7	36

However, the error in measurements of this kind may be rather large.

A check of the assumption underlying this determination could be made by comparing the absorption coefficients in lithium for the different groups with the absorption coefficient for the same groups in boron; in this case the ratios should be the same in view of the fact that the mechanism of absorption of slow neutrons in both elements is one and the same. From the measurements of Halban and Preiswerk[^28] one may conclude that no such agreement exists, although the differences obtained are possibly due to secondary causes.

An independent determination of the energy sequence of the groups, and also a determination of the energy ratio (the latter only for nonthermal groups), was given by us in Section 9 and is based on considering the change in intensity of the various groups as a function of distance from the source. The order of energies that we found agrees with that obtained from absorption in boron, and the values we found for the energy ratio, although smaller than those obtained from absorption in boron, are nevertheless not incompatible with the latter, in view of the rather large errors entering into these determinations.

Our method, although very inaccurate, has the advantage of being direct. We were also able to determine, by analogous methods, the width of the energy bands (more precisely, the relative width \(\Delta W/W\)) for the nonthermal groups; we found that the narrowest band corresponds to group \(A\) (radiation strongly absorbed by silver), for which \(\Delta W/W = 0.04\). The other groups (§ 12) are somewhat broader; nevertheless, the probability that a neutron remains in the same group after one free path is quite small.

Connected with the width of the absorption bands is the population of the various groups, which in essence corresponds to the number of neutrons emerging in 1 sec from the surface of a paraffin block containing the source (Section 5). The most numerous of all the groups is the thermal group (population 40), whereas the least numerous is group \(A\) (population 0.5), which, as we have seen, is also the narrowest. These facts are in qualitative agreement with the ideas of Bohr, Breit, and Wigner.

Some further data on the properties of these absorption bands can be obtained by comparing the behavior of two periods of one and the same element that are sensitive to slow neutrons. In this direction we were able to investigate only silver, rhodium, indium, and bromine. Only in the case of silver did we find different behavior for two periods of this element. In all other cases we found no difference within the limits of our accuracy. It should be noted that rhodium, indium, and bromine represent three cases in which the number of observed periods produced by slow neutrons is greater than the number of known isotopes.

Sections 6 to 11 are devoted to the study of the diffusion properties of slow neutrons in substances containing hydrogen. The diffusion process may be divided into two successive phases: first (the slowing-down phase) the neutron loses energy as a result of successive collisions until it reaches the energy of thermal motion; then (the diffusion phase) the energy on average no longer decreases, and the neutron is reflected until it is absorbed by protons or by other nuclei.

During the slowing-down phase the mean free path decreases very rapidly to a value of the order of \(1\ \mathrm{cm}\), then

remains approximately constant until the neutron energy becomes comparable with the quantum \(h\nu\) of the elastic vibrations of the hydrogen atom in paraffin.

We may say that all the groups studied, except the thermal one, belong to energy intervals where the mean free path in paraffin \(\lambda_g\) is about \(1\ \text{cm}\).

From the theory of collisions of slow neutrons with hydrogen atoms, taking into account the chemical bond of the latter (\(F\), Sections 10 and 11), it follows that if the energy of slow neutrons passes from values greater than \(h\nu\) to smaller values, then the mean free path decreases and tends to a limiting value equal to

\[ \frac{\lambda_g}{4}. \]

This change in the mean free path is also connected with a different angular distribution of the neutrons after collisions; if \(W \gg h\nu\), then the angle between the direction of motion of the neutron before and after the collision is always acute, whereas for \(W \ll h\nu\) the angular distribution after collisions tends to become isotropic.

In agreement with this theoretical result, a quite noticeable decrease in the mean free path is indeed observed in passing from the nonthermal groups to the thermal one. For this latter group one may take

\[ \frac{W}{h\nu} \]

to be approximately \(0.1\); although this value is fairly small, thermal neutrons still cannot be regarded as neutrons with energy practically equal to zero. Thus it follows (\(F\), formula 102) that for

\[ \frac{W}{h\nu}=0, \]

the mean free path \(\lambda\) is equal to

\[ \frac{\lambda_g}{3.3}, \]

whereas in the limit, for \(W=0\), it would be

\[ \lambda=\frac{\lambda_g}{4}. \]

We may also estimate to what extent coherence is preserved between the directions of motion of a thermal neutron before and after collision. As a quantitative expression of the coherence we shall take \((\cos\theta)_{cp}\), i.e. the mean value of the cosine of the neutron scattering angle after collision; \((\cos\theta)_{cp}=1\) means complete coherence, whereas for isotropic scattering \((\cos\theta)_{cp}=0\). For neutrons with energy greater than \(1\ \text{V}\), we find

\[ (\cos\theta)_{cp}=\frac{2}{3}=0.67; \]

for thermal neutrons, assuming

\[ \frac{W}{h\nu}=\frac{1}{10}, \]

we obtain (\(F\), formula 103)

\[ (\cos\theta)_{cp}=0.067, \]

i.e. one tenth of the preceding value.

These results justify the approximations that were sometimes made in this work when treating the diffusion of thermal neutrons as isotropic diffusion. It must be noted, however, that in this way we introduced a small error in the sense that the “diffusion length” given by equation (12) came out too small, while the albedo given by equation (11) came out too large. It would be easy to calculate these corrections if ...

to know the value of \((\cos \theta)_{cp}\); however, for this we must know \(\dfrac{W}{h\nu}\), for which we can give only the order of magnitude.

In Sections 6–11 five different quantities were measured, which can be expressed as functions of only two of them, when use is made of the relations obtained from the theory of neutron diffusion.

These five quantities are the following: the mean free paths \(\lambda\) and \(\lambda_g\) of thermal neutrons (Section 8) and of neutrons of groups \(D\) and \(A\) (Section 7), the albedo \(\beta\) for thermal neutrons (Section 6); the “diffusion length” of thermal neutrons \(l\) (Section 7) and the length \(|x_0|\), pertaining to thermal neutrons and considered in Section 11, which is related to \(\lambda\) by the relation:

\[ x_0=\frac{\lambda}{\sqrt{3}} . \tag{29} \]

By means of (28), (11), (12), and (29), the five quantities mentioned above can be expressed as functions of \(\lambda_g\) and \(N\) (the number of free paths made on the average by a thermal neutron). In Table 18 we compare the measured values of these five quantities with the values calculated with the aid of the above-mentioned formulae, assuming that \(\lambda_g=1.0\ \mathrm{cm}\) and \(N=140\).

TABLE 18

Comparison of the measured and calculated values: \(\lambda\) and \(\lambda_g\)—mean free paths of thermal neutrons and of neutrons of groups \(D\) and \(A\), \(\beta\)—albedo for thermal neutrons, \(l\)—diffusion length, \(|x_0|=\dfrac{\lambda}{\sqrt{3}}\)

Quantities	measured	calculated
\(\lambda\)	0.3	0.30
\(\lambda_g\)	1.1	1.0
\(\beta\)	0.82	0.83
\(l\)	2.1	2.05
\(x_0\)	0.18	0.174

As we see, the agreement is very good. From these values of \(\lambda\) and \(N\) we find the following values for thermal neutrons in paraffin: the effective cross section for elastic collision with hydrogen \(\sigma_e=43\cdot 10^{-24}\ \mathrm{cm}^2\), the effective cross section for absorption \(\sigma_C=0.31\cdot 10^{-24}\ \mathrm{cm}^2\), the mean lifetime \(\tau=1.7\cdot 10^{-4}\ \mathrm{sec}\). This last value is in very good agreement with the measurement of \(\tau\) made with a mechanical apparatus\({}^{29}\) and with the theoretical value obtained from the theory (Sec-

section 12) and based on the assumption that the absorption of slow neutrons by protons is due to the emission of a γ-quantum by the magnetic dipole radiation,^[30] there it is shown that, irrespective of whether the state \(^{1}S\) of the deuteron (\(\pm 120\,000\ \mathrm{V}\)) is taken to be real or virtual, the theoretical value of \(\sigma\) will be equal to \(6.5 \cdot 10^{-4}\) or \(2.6 \cdot 10^{-4}\). The satisfactory agreement of this latter value with the experimental result seems to indicate that the \(^{1}S\) state is virtual.

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5. F. Rasetti, E. Segrè, G. A. Fink, J. R. Dunning and S. B. Pegram, Phys. Rev., 49, 104, 1936; Rend. Linc., 23, 343, 1936.

6. E. Amaldi, E. Fermi, Ric. Scient., VI—II, 344, 1935.

7. L. Szilard, Nature, 135, 950, 1935.

8. E. Fermi and E. Amaldi, Ric. Scient., VI—II, 443, 1935.

9. E. Fermi, “Sul moto dei neutroni nelle sostanze idrogenate,” Ric. Scient. VII—II, 13, 1936.

10. P. B. Moon and J. R. Tillman, Nature, 136, 66, 1935; B. Pontecorvo, Ric. Scient. VII—II, 145, 1935; E. Amaldi and E. Fermi, Ric. Scient. VI—II, 344, 1935.

11. E. Amaldi and E. Fermi, Ric. Scient. VI—II, 344, 1935; L. Szilard, Nature, 136, 950, 1935.

12. N. Bohr, Nature, 137, 344, 1936; G. Breit and E. Wigner, Phys. Rev., 49, 519, 1936.

13. O. R. Frisch, G. Hevesy and H. A. C. McKay, Nature, 137, 149, 1936.

14. E. Fermi and E. Amaldi, Ric. Scient. VI—II, 443, 1935; E. Segrè, Ric. Scient. VII—I, 389, 1936.

15. E. Fermi and E. Amaldi, Ric. Scient. VI—II, 443, 1935.

16. J. R. Tillman, Nature, 137, 107, 1936.

17. T. Bjerge and C. H. Westcott, Proc. Roy. Soc., A 150, 709, 1935; E. Fermi and E. Amaldi, Ric. Scient. VI—II, 443, 1935; VII—I, 56, 1936.

18. E. Amaldi and E. Fermi, Ric. Scient., VII—I, 393, 1936.

19. E. Amaldi, O. D’Agostino, E. Fermi, B. Pontecorvo, F. Rasetti and E. Segrè, Proc. Roy. Soc., A 149, 522, 1935.

20. O. R. Frisch and G. Placzek, Nature, 137, 357, 1936; D. F. Weekes, M. S. Livingston and H. A. Bethe, Phys. Rev., 49, 471, 1936.

1. H. H. Goldsmith and F. Rasetti, Phys. Rev., 50, 328, 1936, showed that in this calculation it is more correct to take as the thermal energy \(kT\) instead of \(\frac{3kT}{2}\).

2. E. Amaldi and E. Fermi, Ric. Scient., VII--1, 56, 1936.

3. Preiswerk and von Halban, C. R. 202, 840, 1936.
4. E. Amaldi and E. Fermi, Ric. Scient., VII--1, 310, 1936.
5. See, for example, R. Jaeckel, Z. Physik, 91, 493, 1934; F. A. Paneth and H. Loleit, Nature, 136, 950, 1935.
6. J. R. Dunning, G. B. Pegram, G. A. Fink, D. P. Mitchell and E. Segrè, Phys. Rev., 48, 704, 1935; Lincei Rend., 23, 340, 1936.
7. P. Preiswerk and H. von Halban, Nature, 136, 1027, 1935; F. Rasetti and G. Fink, Phys. Rev., 49, 642, 1936.
8. H. von Halban and P. Preiswerk, Nature, 137, 905, 1936.
9. E. Amaldi, O. D’Agostino, E. Fermi, B. Pontecorvo and E. Segrè, Ric. Scient., VII--1, 581, 1935.
10. A preliminary communication of this theory was given by E. Fermi, Phys. Rev., 48, 570, 1935.

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