EXPERIMENTS WITH MONOCHROMATIC SLOW NEUTRONS

N. A. Vlasov

INTRODUCTION

Obtaining a monochromatic beam of neutrons of any energy is one of the most important problems of experimental nuclear physics. A solution of this problem would make it possible to study the cross sections for the interaction of neutrons of different energies with nuclei. Knowledge of these cross sections is essential, first, from the point of view of determining the fate of neutrons in systems with a self-developing chain fission process, and, second, for studying the spectra of excited states of nuclei manifested through resonance interaction.

The sources of neutrons are various nuclear transformations. Some of them can directly yield monochromatic neutrons. If, as the result of a nuclear transformation, one neutron is emitted, then its kinetic energy \(E_n\) is determined by the equality:

\[ E_n + E_z = Q + E_i, \]

where \(Q\) is the reaction energy, \(E_i\) is the kinetic energy of the particle bombarding the nucleus, and \(E_z\) is the kinetic energy of recoil of the final nucleus.

For different acts of one and the same transformation the quantity \(Q\) may have as many different values as the final nucleus has states immediately after emission of the neutron. If there is only one such state and, moreover, the bombarding particles have the same energy, then the sum \(E_n + E_z\) is the same for all acts of transformation, and the neutron energy \(E_n\) depends only on the direction of their emission relative to the direction of the bombarding particles. In a definite direction, therefore, monochromatic neutrons can be obtained if monochromatic bombarding particles and a thin target are used.

Examples of sources of monochromatic neutrons are the reactions \(\mathrm{D}(d,n)\mathrm{He}^3\), \(\mathrm{Li}^7(p,n)\mathrm{Be}^7\), as well as all \((\gamma,n)\) reactions with monochromatic \(\gamma\)-rays (impurities in the spectrum of \(\gamma\)-rays of soft components sometimes do not destroy the monochromaticity of the photoneutrons).

The most widespread sources, Be \((\alpha, n)\) and Be \((d, n)\), give a rather complex neutron spectrum, at best line-like.

Neutrons with energies of the order of \(100\ \mathrm{keV}\) and higher can be obtained directly from a source. To obtain neutrons of lower energy it is necessary to bombard the target with particles or \(\gamma\)-rays with an energy close to the threshold of the source reaction. In this case the yield becomes very low and the intensity of the source insufficient. Therefore the production of monochromatic neutrons with energies of the order of \(10\ \mathrm{keV}\) and below (we shall call such neutrons slow) proves to be a more difficult problem than the production of monochromatic fast neutrons.

Slow neutrons are usually obtained by slowing down fast neutrons in one medium or another (paraffin, water, graphite). Such a method of production inevitably gives neutrons with a continuous distribution in energy, i.e. with a continuous spectrum.

Methods for selecting from this continuous spectrum neutrons of a definite energy—though, it is true, not yet very high—have recently been developed quite well. A survey of these methods, and also of the results of investigations with monochromatic neutrons, constitutes the subject of the present article.

MECHANICAL SELECTOR

If neutrons of all velocities are emitted from a source in a short pulse, then after some time they will be distributed in space in such a way that the slow ones will be closer to the source and the fast ones farther away. The times taken by neutrons of different velocities to traverse a certain distance are different. This can be used to select neutrons of one velocity from a complex spectrum. Devices that carry out this selection of monochromatic neutrons from a continuous spectrum are called neutron velocity selectors. One of them is the mechanical selector, first realized in 1935 by Dunning, Pegram, Fink, Mitchell, and Segrè\(^1\) at Columbia University.

The use of a mechanical selector is based on the property of certain substances, for example cadmium, to absorb slow neutrons strongly. From such a substance one can make a screen impermeable to neutrons and, by opening it for short intervals of time, let neutrons out in portions. If such a screen is opened periodically, then for clarity it may be compared with a machine gun firing shot, from which the pellets fly out with different velocities. On the basis of this comparison we shall call the act of short-time emission of neutrons by the appropriately constructed source a shot. If, at a distance \(l\) from the first screen in the path of the neutrons, a second identical screen is placed, opening after a time \(t\), then only neutrons with

with velocities close to \(v=\dfrac{l}{t}\). Consequently, beyond the second screen there will be only neutrons of approximately the same energy,—approximately to the extent to which the duration of the shot \(\tau\) is less than the delay time \(t\) (the time between the moments of opening of the first and second screens). The scheme of the Dunning et al. selector is shown in Fig. 1.

Cadmium, deposited in sector-shaped layers on four duralumin disks, was used as the material for the absorbing screens.

Fig. 1. Scheme of the mechanical selector of Dunning et al.

Two disks were stationary, and the slits between the cadmium sectors in them selected a beam of neutrons. This beam could be passed or blocked by two other disks rotating on a common axis. The combination of two disks—one movable, the other stationary—thus formed a periodically opening screen. The second (exit) screen opened with a certain delay after the first, which was achieved by displacing the slit of the second movable disk by an angle \(\beta\) relative to the slit of the first. The duration of the shot is evidently determined by the angular width of the slit \(\alpha\) and by the rotational speed of the disk, while the delay is determined by the angular displacement of the slits of the rotating disks \(\beta\) and likewise by the rotational speed. The smaller the ratio \(\dfrac{\alpha}{\beta}\), the more monochromatic a neutron beam the selector picks out, and the better its resolving power. But increasing the resolving power leads to a decrease in the intensity of the selected beam. Indeed, the subsequent shot must occur no earlier than the closing of the slit of the exit screen; otherwise neutrons with a very short flight time will pass through it, in addition to those which passed during the preceding shot. Hence the angular displacement of two neighboring slits of the movable disk must be no less than \(\beta\). Therefore the ratio \(\dfrac{\alpha}{\beta}\) is the upper limit of that fraction of time during which the beam is open. If the source emits, in the selected direction, \(N\) neutrons per second, then the first screen transmits no more than \(N\dfrac{\alpha}{\beta}\). Of this number of neutrons that have passed the first screen, only a part will pass the second as well. If the neutron spectrum between the screens were such that equal numbers of neutrons reached the second screen per unit time, then the second screen would trans—

would seek the number of neutrons equal to \(N\left(\dfrac{\alpha}{\beta}\right)^2\). If one also takes into account the solid angle subtended by the neutron-beam selector, it turns out that the beam intensity at the detector is proportional to \(\left(\dfrac{\alpha}{\beta}\right)^3\), i.e., it decreases sharply as the resolving power is improved. For this reason the first mechanical selector, used with a weak Rn + Be neutron source, proved to be very imperfect. According to the published descriptions,\(^{1,4}\) it was constructed in two versions. In the first version the cadmium sectors on the disks had an angular width of \(3.7^\circ\), and the gaps between them \(3.5^\circ\). The distance between the movable disks was \(54\ \text{cm}\), and the distances between the movable and fixed disks \(5\ \text{mm}\). The disks could rotate at a speed of up to 5000 revolutions per minute. At this rotation speed, a turn through an angle of \(3.6^\circ\) was accomplished in the time:

\[ t=\frac{3.6\cdot 60}{360\cdot 5000}=1.2\cdot 10^{-4}\ \text{sec}=120\ \mu\text{sec}. \]

During this time the distance of \(54\ \text{cm}\) between the disks was traversed by a neutron having velocity

\[ v=\frac{54}{1.2\cdot 10^{-4}}=4.5\cdot 10^5\ \text{cm/sec} \]

or an energy of about \(0.1\ \text{eV}\).

With such wide slits, however, the selector had very poor resolving power and was transparent to neutrons of a very large range of velocities.

Because of the complexity of the spectrum of neutrons transmitted by the selector, interpretation of the measurement results is rather difficult and reduces only to rough qualitative conclusions.

In the second version of the selector, in order to improve the resolving power, the ratio of the slit width to the width of the cadmium sector was reduced. With a slit width of \(1^\circ\), the sector width was \(3^\circ\). To preserve sufficient beam intensity, however, it was necessary to bring the disks closer together, to a distance of \(25.7\ \text{cm}\). Protection against fast neutrons was also improved. But even with these parameters the resolving power of the selector was still very poor. Fig. 2 shows the transparency of this selector, calculated by Fink,\(^{4}\) as a function of neutron velocity. Although for neutrons with velocity

Fig. 2. Transparency of a mechanical selector for neutrons of different velocities \(v\).

\[ v_1=\frac{l}{t_1}, \]

where \(t_1\) is the time of rotation of the rotating system through an angle equal to the opening angle of the slit,

of the relative displacement of the movable disks, the selector has maximum permeability, but the width of the selector’s transmission band is very large. In addition to the main band, there also appear a secondary, tertiary, etc., which correspond to smaller velocities and multiple flight times.

Naturally, these mechanical selectors found only very limited application. With their aid direct confirmation was obtained that neutrons emerging from a thick paraffin layer have thermal velocities and that the spectrum in this velocity region is close to Maxwellian. A shift of the spectrum toward lower velocities upon cooling the paraffin was also discovered, and thereby direct proof was obtained of the existence of thermal equilibrium between the neutrons and the paraffin. However, an exact quantitative study of the spectrum of thermal neutrons, as well as of the law of neutron absorption by various elements, could not be carried out.

Fig. 3. Diagram of the selector of Fermi et al. Rotating drum in cross section. Neutron source—the Argonne boiler.

With the appearance of uranium boilers, which are intense sources of neutrons, the possibilities for applying a mechanical selector increased. In 1947, in the Argonne Laboratory of the University of Chicago, Fermi, Marshall, and Marshall^5 again employed a mechanical selector, whose construction is clear from Fig. 3. In the path of a narrow beam of thermal neutrons emerging through an opening in the wall of the boiler, there was placed a cylindrical drum assembled from parallel alternating layers of aluminum and cadmium, of thickness respectively 0.75 and 0.15 mm. The drum could rotate about an axis perpendicular to the plane of the drawing, at a speed of up to 15,000 revolutions per minute. For the mechanical strength required at such a high rotation speed, the Al and Cd layers were packed tightly inside a steel tube of diameter about 4 cm with wall thickness-

0.8 mm. Since Al and steel are practically transparent to neutrons, while cadmium absorbs them very strongly, the drum transmitted about 75/90 (the ratio of the thickness of the Al layers to the total thickness of Al + Cd) of the neutrons when the planes of the layers coincided with the direction of the beam, and completely interrupted the beam when rotated through an angle greater than \(2.7^\circ\). The magnitude of this angle was evidently determined, first, by the ratio of the thickness of the aluminum interlayer to the width (practically equal to the inner diameter of the tube), and, second, by the angular divergence of the beam, which amounted to \(3^\circ\) (\(=1.5^\circ \times 2\)). Thus, as the drum rotated, the beam was opened and closed twice during one revolution, i.e., it pulsated with a frequency equal to twice the rotation frequency of the drum.

The ratio of the duration of the burst to the interval between neighboring bursts, which limits the resolving power of the selector, in this case was approximately \(1/30\), i.e., it was considerably more favorable than in the first mechanical selectors.

If a second identical drum is placed at some distance from the first drum in the path of the beam and rotated at the same speed, then such a system of two drums will obviously be a monochromator, since only neutrons of a definite velocity will pass through the second drum. In actuality, in the Fermi–Marshall selector, subsequently improved by Brill and Lichtenberger\(^6\), a second drum was not used; instead, neutrons of a definite velocity were selected by means of a detector with modulated sensitivity. The detector was brought into operating condition by a short pulse and could register neutrons only during this pulse. The pulse was supplied through an amplifier from a photoelement, onto which at a certain moment there fell the image of a lamp reflected by a small mirror on the axis of the drum. The method of modulating the detector was probably borrowed from the works described below.

If the detector was located at a distance \(l\) from the drum and opened after a time \(t\) following the burst, then neutrons with velocity \(v=\dfrac{l}{t}\) and velocities close to it were registered. By changing this delay time \(t\), it was possible to tune the selector to neutrons of different velocities. The Fermi–Marshall selector, as we see, is not purely mechanical, but represents a combination of a mechanical device with an electromagnetic relay. Its principal difference from Dunning’s mechanical selector consists in the fact that it does not create, as the latter does, a beam of monochromatic neutrons, but only selects them by means of the relay, although if a second drum were present it could also be a monochromator. Fermi et al. used their selector to study the spectra of thermal neutrons emerging from a boiler under different filtration conditions, and also to measure the capture cross section of neutrons by boron and other elements. The results of these measurements are discussed below.

The maximum energy of neutrons investigated by means of mechanical and semi-mechanical selectors is 0.2 eV. Extending investigations with a mechanical selector into the region of higher energies is practically hardly attainable. For this it would be necessary, first, to create shields impermeable to neutrons of higher velocities, and, second, to impart higher rotational speeds to the corresponding mechanical devices. Cadmium proves unsuitable as a material for shields absorbing neutrons with energies greater than 0.5 eV, since in this energy region the absorption cross section of neutrons by cadmium decreases sharply, reaching \(10 \cdot 10^{-24}\,\text{cm}^2\) at \(E_n = 1.5\) eV (\(E_n\) is the kinetic energy of the neutron). Selective neutron absorbers such as silver, rhodium, etc., could be used for the construction of a multilayer shield, but apparently it is impossible to construct such a shield that would satisfactorily absorb a continuous neutron spectrum over wide ranges. Boron may be regarded as the best absorber for this purpose, since it has a large capture cross section for slow neutrons over wide ranges. As experiments with selectors have shown, the boron cross section may be represented by the formula \(\sigma_n = 114 E_n^{-1/2} \cdot 10^{-24}\,\text{cm}^2\) (\(E_n\) in eV). With increasing neutron energy the cross section decreases, and for a given attenuation of the neutron beam an ever greater thickness of absorber is required. For example, to attenuate by a factor of 1000 a beam of neutrons with energy 10 eV, a layer of boron carbide \((\mathrm{B}_4\mathrm{C})\) approximately 1.6 cm thick is required. The construction of rapidly rotating shields containing a large amount of boron is a rather complicated technical problem. Meanwhile, more advanced methods have been developed in recent years for separating and studying neutrons with energies greater than 1 eV, fully replacing the mechanical selector also in the region of thermal neutrons. They are based on the use of modulation of artificial neutron sources, on the one hand, and modulation of detector sensitivity, on the other. In contrast to mechanical selectors, we shall call them selector-modulators.

BRIEF HISTORICAL REVIEW

After 1936, no reports appeared concerning further applications of the mechanical selector. The author is aware, however, that at Leningrad University Yu. A. Nemilov, during 1936–1937, constructed a mechanical selector more advanced than Dunning’s selector. Nemilov modulated the neutron beam by means of two disks rotating toward one another, and instead of a second pair of disks used an ionization chamber with a linear amplifier of modulated sensitivity, i.e., he employed an arrangement that subsequently appeared in the work of Fermi and Marshall. The resul—

...the results of this work proved better than those of Dunning and co-workers, but were not published, in view of the fact that before the work was completed a more accurate study by Alvarez with a selector-modulator appeared.

In 1938 the first work with a selector-modulator, carried out by Alvarez[^7] at the University of California, was published. This work set forth the basic principles of the apparatus and the results of the first practical implementation of the selector. Using the target of the 37-inch cyclotron as the neutron source, Alvarez modulated the intensity of the source by modulating the amplitude of the high-frequency accelerating voltage. In the course of developing the apparatus, various methods and various modulation frequencies were tested. In practice, modulation frequencies of 120 and 60 cycles were used.

The ionization chamber (at first lithium, then with BF$_3$), serving as the neutron detector, was placed at a distance of 8 m from the target. The sensitivity of the recording circuit was modulated by means of a device analogous to the ordinary coincidence circuit.

In this work thermal and still slower (“cold”) neutrons were separated, and qualitative agreement of absorption in boron with the law $\frac{1}{v}$ ($v$ is the neutron velocity) was demonstrated. It was shown that the intensity of “cold” neutrons increases when the paraffin around the source is cooled and, consequently, the existence of thermal equilibrium of neutrons with paraffin was confirmed. At the same time the background from fast neutrons, which had constituted a very large quantity (larger than the effect itself) in the experiments of Dunning et al., was practically eliminated (down to 0.3%).

In the same year, 1938, a note by Fertel, Gibbs, Moon, Thomson, and Wynn-Williams[^8] was published, reporting measurements of the velocity distribution of thermal neutrons by a somewhat different method. Here the neutron source was the target (of heavy ice) of a high-voltage tube operating at accelerating voltages of 200–250 kV. The target was bombarded by a beam of deuterons emitted in pulses of duration 500 μsec, repeated every 5000 μsec. The modulation was effected by means of a tuning fork and a photocell with a subsequent amplifier controlling the voltage across the discharge gap of the ion source (20 kV). An ionization chamber with BF$_3$ was located at a distance of 5.4 m from the target. The amplified pulses from the chamber were fed to an oscillograph and photographed on a rapidly moving film, on which marks were also made for the instants at which the source was switched on. Processing the film made it possible to establish the distribution of pulses in time and, consequently, the distribution of neutrons by velocities. With not very great accuracy this distribution proved to be close to the Maxwellian distribution corresponding to 15° C.

In 1940 this same group of authors published[^9] more detailed results of measurements of the distribution of neutrons by velocities and of absorption in boron and cadmium. With regard to absorption in boron, the results apparently proved to be erroneous, since a deviation of the absorption from the law $\dfrac{1}{v}$ was found, which was not later confirmed in more accurate experiments.

In 1941 the first paper by Beker and Becher[^10], carried out at Cornell University, was published. The selector constructed by them already contained all the basic features of the modern one. The neutron source here was the target of a small cyclotron. The yield of neutrons from the target was modulated by means of a thyratron circuit controlling the operating regime of the cyclotron ion source. The duration of the bursts was regulated and could reach a minimum value of 15 $\mu\sec$ at a repetition rate of 400 times per second. An ionization chamber with $\mathrm{BF}_3$ (the detector) was placed at distances of 1.5 and 3 m from the source. The sensitivity of the detector was also modulated by pulses with adjustable duration from 15 $\mu\sec$, while the delay could be varied from 0 to 2500 $\mu\sec = \dfrac{1}{400}\sec$. Whereas Alvarez’s selector made it possible to isolate only a group of thermal and slower neutrons, and the English group went in the direction of higher energies of the neutrons under study up to 0.1 eV, Beker and Becher studied neutrons in the energy interval from 0.006 to 3–4 eV. The result of this work was a rather accurate investigation of the law of neutron absorption in Cd, proof of its resonant character, establishment of the upper limit of the absorption band, and of the fact that the absorption law agrees with the well-known Breit–Wigner formula. In addition, resonant absorption in indium and rhodium was directly demonstrated, and the lifetime of neutrons in paraffin was also measured. The methodological significance of this work is very considerable. The authors chose the most effective path of investigation and gave a thorough analysis of the characteristics of the selector. The further development of work in this field proceeded along the path of improving the Beker–Becher selector.

By 1943 Reingold and Havens[^16] had built a selector of the same type at Columbia University. Owing to the fact that the intensity of the neutron source (a 37-inch cyclotron with an internal target) was considerably higher here than at Cornell University, it proved possible to increase the distance between the source and the detector to 5.4 m and, as a result of this and of several other improvements, to increase the resolving power considerably. Both selectors, the Cornell and the Columbia, were improved over time,[^11][^12][^16][^17][^18][^19] and according to the latest descriptions the Columbia University selector is the most advanced of all described in the literature. It makes it possible to investigate

properties of neutrons with energies from 0.001 to 100 eV and even higher. With it, the most accurate measurements of the velocity distribution of neutrons emitted by paraffin were carried out, and absorption in boron, lithium, cadmium, and many other elements was investigated. The results of these measurements we shall present and discuss below.

At approximately the same time as the Cornell University selector, in 1941 a selector was built at the University of Illinois. It was first described in papers by Havors, Manley, and Luebke \(^{13,14,15}\). Here the velocity distribution of neutrons was measured and the absorption law in boron was verified. The neutron source was the target of a high-voltage tube. Probably, after 1941 this selector was not used, since, naturally, it could not compete with selectors using a cyclotron—a more powerful neutron source. The last paper \(^{15}\), although published in 1946, is dated October 1941.

In addition to those listed, a Stanford University selector has also been described, using a small cyclotron. With the aid of this selector, Frayer \(^{36}\) investigated the dependence of the absorption and scattering of neutrons of various velocities in iron on the degree of magnetization of this iron. In its construction this selector differs substantially from the Becher and Becker selector.

In 1947 two notes were published by Sutton, McDaniel, Anderson, and Davatel \(^{21}\) on measurements of neutron absorption in boron and gold at the Los Alamos Laboratory of the University of California. The selector used here was built by McDaniel at Cornell University and transported to Los Alamos. A detailed description of it was not published in 1947, but it is similar to the latest models of the Cornell University selector, described in papers by McDaniel \(^{12}\) and Jones \(^{22}\).

During 1947 the operating selectors were apparently three: the Cornell, Columbia, and Los Alamos selectors. The others, less perfect, were probably not used. In addition to the works mentioned, Havens, Rainwater, and Rabi \(^{34}\), using the Columbia University selector, investigated the interaction of neutrons with electrons in molten lead, and established an upper limit for the cross section of neutron scattering by an electron.

FUNDAMENTALS OF THE DESIGN OF SELECTOR-MODULATORS

Source

The decisive feature of the neutron source for a selector is its intermittency in time, its pulsation. In connection with this, such continuous neutron sources as, for example, Ra + Be or a uranium boiler are unsuitable for a selector-modulator. In the case when, in order to obtain neutrons, artificially

accelerated charged particles; to modulate the intensity of a neutron beam it is sufficient to modulate the intensity of the beam of charged particles. If, in doing so, only the intensity changes with time, but not the energy of the particles, then the neutron beam will be modulated to the same degree as the beam of charged particles.

Most selectors and, in any case, the most advanced among them use as the neutron source a beryllium target of a cyclotron, bombarded by deuterons and surrounded by paraffin or water. Fast neutrons emerging from the target are slowed in the paraffin or water, and the surface of the moderator emits a practically continuous neutron spectrum. One of the surfaces of the moderator is also the immediate source of neutrons. The source is arranged in exactly the same way when high-voltage tubes are used instead of a cyclotron. In order that the surface of the moderator emit neutrons in short pulses, it is necessary to bombard the target with correspondingly short pulses, i.e. to modulate accordingly the beam of bombarding particles.

The methods of modulation may be quite varied. The first method, practically applied by Alvarez, consists in modulating the amplitude of the accelerating high-frequency voltage on the cyclotron dees. According to the focusing conditions of the ion beam in the cyclotron, the magnetic-field intensity must decrease from the center toward the periphery. As a result, the ions in the cyclotron move not synchronously with the change of the high-frequency accelerating field, but with a certain phase difference that changes with each revolution. The greater the number of revolutions of the ion, the greater the phase difference. But the number of revolutions is determined by the magnitude of the accelerating voltage. With a very small accelerating voltage, the ion beam may fall so far out of phase that it will no longer be accelerated, but retarded, and consequently will not pass beyond a certain distance from the center, i.e. will be scattered without reaching the target. In connection with this, and also for certain other reasons, the intensity of the ion beam on the cyclotron target depends rather sharply on the amplitude of the high-frequency accelerating voltage. Alvarez made use of this. By modulating the anode voltage on the last two stages of the high-frequency generator by applying an unsmoothed rectifier voltage, he obtained on the target an ion beam pulsing at a frequency of 120 or 60 times per second. The pulse duration, however, was rather large—25–30% of the cycle at a frequency of 60 cycles and 50–60% of the cycle at a frequency of 120 cycles.

A more convenient method, which subsequently became exceptionally widespread, proved to be modulation of the current in the ion source. This method was used in all subsequent work both on cyclotrons10, 11, 16–20, 22 and on high-voltage tubes8, 9, 13, 14, 15. Alvarez also tried to modulate the intensity of the ion source by means of grids controlling the current of electron emission from the filament.

source. But these attempts did not yield a satisfactory result. A more convenient and reliable method proved to be modulation of the anode voltage on an arc ion source. The circuit used for this purpose, controlling the arc voltage, was given in the work of Becker and Bacher in 1941,^10 and is shown in Fig. 4.

Two thyratrons, included in the alternating-current circuit of the mains frequency of 400 cycles, by selection and by changing the phase of the voltage on the grids, could be fired practically at any moment. One of the thyratrons was connected in series with the ion source and, at the moment of firing, supplied voltage and, consequently, controlled the moment of switching on the ion source. The second thyratron was connected in parallel and, igniting somewhat later than the first, short-circuited the voltage on the arc, thus controlling the moment of extinguishing the arc.

Fig. 4. Circuit for modulating the ion source of the cyclotron, used in 1941 by Becker and Bacher.

By changing the interval between the moments of ignition and extinction, it was possible to obtain a width of the ion-current pulse of the source variable over wide limits. In connection with the need to shorten the pulse duration as much as possible and to make it, as far as possible, rectangular, i.e. with very small rise and fall times (of the order of tenths of a microsecond), in later work the thyratron circuits were replaced by tube circuits. For example, in the works of Rainwater and Havens^16,^17, for modulating the ion source a unit of 12 6L6 tubes was used, and still later one of 12 6V6 tubes, connected in series in the anode circuit of the arc source.

Obtaining short, rapidly rising pulses presented certain difficulties, connected with the large magnitude of the arc current (up to 5 amperes) and with the need to block the capacitance of the arc circuit from the high-frequency voltage. The minimum pulse duration practically used in the later works reached 5 μsec. Already in the selector of Becker and Bacher in 1941, and in most subsequent ones, the pulse duration could be varied smoothly or in steps. Thus, for example, in the later works with the Columbia University selector^19, pulses of duration 5, 10, 20, 40, 80, 160 μsec were used, depending on the velocity range of the neutrons under investigation.

But the time of exit of neutrons from the front surface of the moderator, obviously, by no means coincides with the burning time of the arc of the ion

source, since a delay is inevitable. The first cause of this delay is the finiteness of the ion acceleration time in the cyclotron. This time can be estimated in the following way. Let the final energy of the deuterons be 8 MeV, which corresponds to the Columbia University cyclotron, and let the average accelerating voltage between the dees be 80 kV. Then the number of accelerations is equal to 100, and the acceleration time is 50 periods of the high-frequency generator. If the wavelength of the generator is 30 m and, consequently, the period is 0.1 μsec, then the acceleration time is 5 μsec. This estimate shows that the acceleration time is sufficiently large and must be taken into account. The same conclusion can be drawn by considering the length of the ion path in the cyclotron chamber. If the number of revolutions of the ion is 50–100, then the total path length is on the order of hundreds of meters. Although this path is traversed by an ion with high energy, the path length is so great that the acceleration time cannot be regarded as very small. However, it is impossible to calculate it because of the uncertainty in the magnitude of the accelerating voltage; therefore it is necessary either to mark the moment at which the deuterons strike the target, or to measure this delay directly. In practice the latter method is used, since the duration of ion acceleration is not the only cause of delay.

The second cause is the process of neutron slowing down. In fact, neutrons with an energy of the order of 1 eV do not emerge directly from the cyclotron target, but are formed as a result of multiple collisions of the primary neutrons with the nuclei of the moderator. It is not difficult to show that the slowing-down time of a neutron from an initial energy of 1 MeV to an energy of 0.1 eV in paraffin is about 4 μsec and, generally speaking, must be taken into account.

In addition to these causes, which pertain to the process of obtaining neutrons in the source, one more circumstance must be taken into account. The moment at which a neutron enters the detector in reality always differs from the moment at which the corresponding pulse is recorded, owing to the unavoidable delays of the pulse in the recording apparatus. This delay is connected with the fact that the pulses of the recording instrument have a finite rise time. Meanwhile, at some stage of amplification, clipping of the pulses from below is inevitably employed to remove noise, which also introduces a delay at least equal to the rise time of the leading edge of the pulse up to the clipped value. The magnitude of such a delay is determined by the design of the detector and amplifier. In the first experiments of Baker and Bacher in 1941, according to the authors’ estimate, it was equal to 25 μsec. Subsequently, however, both in this laboratory and at Columbia University, owing to improvements in the recording apparatus, this time was reduced to units or even fractions of a microsecond. In order to estimate the significance of these delay times, we give Table I, borrowed from the work of Rainwater and Havens[^16], in which various quantities are given that characterize

neutrons of different energies, including the slowing-down time from the initial energy of 7 MeV and the flight time over a distance of 1 m.

It should be borne in mind that in most experiments the distance between the source and the detector was several meters, for example 5.4, and later 6.2 m, in the selector of Columbia University. But even under this condition the delay is an appreciable quantity in comparison with the flight time of neutrons of not very small energies.

Table I

As early as 1941, Becker and Bacher used a direct experimental determination of the delay time. In subsequent work this practice became established, and the delay was taken into account with sufficiently high accuracy. The methods of measurement differ somewhat in different works. It is necessary to measure the intensity of the neutron source as a function of the time counted from the moment the ion source is opened. In order to exclude the flight time from the measurements, the detector is sometimes brought directly up to the source; in other cases a fast-neutron detector is placed at the same location where the slow-neutron detector usually stands. For fast neutrons the flight time is very small and practically does not distort the results of the measurements. A slow-neutron detector can also be placed near the source, but it must be shielded from thermal neutrons, which can exist for a long time in the body of the moderator. We shall discuss this in greater detail later, and for the moment shall confine ourselves to this remark.

If both the ion source and the detector are adjusted to the same and sufficiently small width of the opening pulse, then the measured intensity should be maximal when the delay is equal to the slowing-down time, and should rapidly fall to zero as the delay is increased or

decrease in delay. As an example, in Fig. 5 we present a curve obtained by Jones\(^{22}\) on the Cornell University selector under the following conditions. At a distance of 3 m from the source, instead of the usually used ionization chamber with \(BF_3\), a butane-filled chamber is placed, registering fast neutrons by recoil protons. The width of the opening pulse both at the ion source and at the detector is 5 μsec. In this case the water usually surrounding the source as a moderator is removed, and fast neutrons are registered,

Fig. 5. Fast-neutron delay curve with opening-pulse widths at the source and detector of 5 μsec.

coming directly from the target; consequently, the moderation time is not taken into account. The number of neutrons registered by the chamber is equal to zero when the source and detector are opened simultaneously; then, when the delay is introduced and increased, it at first grows, becomes maximal at a delay equal to 7.4 μsec and finally falls, forming a curve close in shape to a triangle. An ideal triangle should be obtained with a rectangular pulse both at the source and at the detector and in the absence of fluctuations in the delay times. The tails of Jones’s curve are explained by fluctuations in the acceleration time of ions in the cyclotron. The position of the maximum, evidently, determines the delay time, which, on the basis of this measurement, is subsequently taken into account as a correction in determining the time of flight; moreover, for different neutron energies a correction for the moderation time is also made by calculation (1.5 μsec for 1 eV).

Rainwater, Havens, Wu, and Dunning\(^{18}\) at Columbia University recorded an analogous curve, placing the usually used proportional counter filled with \(BF_3\) directly near the po-

of the paraffin surface serving as the source, and by surrounding the counter with a layer of boron carbide (\(\mathrm{B}_4\mathrm{C}\)), which shields it from long-lived thermal neutrons. The delay time obtained by them is equal to \(11\ \mu\mathrm{sec}\). Under these conditions it is no longer the primary fast neutrons that are registered, but probably neutrons with energies of the order of several eV and, consequently, some averaged slowing-down time is being taken into account.

The question of allowing for the delay of thermal neutrons is considerably more complicated. The point is that, even after slowing down, thermal neutrons may still remain for a long time within the body of the moderator before emerging through the front surface in the direction of the detector. The lifetime of neutrons in the moderator is determined, first, by its dimensions and shape and, second, by the capture probability. It is known that in large blocks of paraffin the mean lifetime of a neutron is of the order of \(200\ \mu\mathrm{sec}\). As is seen from Table I, this time is of the same order of magnitude as the flight time of the fast part of the thermal neutrons. It is clear that, as a result of this, the measurement results may be strongly distorted. An attempt to take these distortions into account by calculation was made by Baker and Bacher \(^{10}\). But, because of the complexity of the problem connected with allowing for the slowing-down process and the neutron spectrum, the calculations prove unreliable and only approximate. A practical way out of this difficulty is the artificial reduction of the lifetime to such limits that it becomes permissible from the point of view of the accuracy of the measurements. Obviously, to reduce the lifetime one must either decrease the dimensions (say, the thickness) of the moderator or introduce absorbers of thermal neutrons into it. Both methods have been used in different experiments.

Let us note, before considering the various methods of constructing the source, that with the aid of a selector the lifetime can be measured very well. For this purpose, obviously, it is necessary to place the thermal-neutron detector either inside the moderator or, at least, near its surface and to count the number of pulses as a function of the delay. Since the number of pulses is proportional (for a stable spectrum) to the number (density) of neutrons in the moderator, and this number will decrease exponentially with time after neutron production has ceased, the exponential curve thus obtained directly gives the mean lifetime as the time corresponding to a decrease by a factor of \(e\).

At Cornell University the external target of the cyclotron was enclosed in a large block of paraffin of approximately rectangular shape, \(20 \times 40 \times 20\ \mathrm{cm}^3\), or in an identical block of water. Measurements \(^{11}\) with the aid of a selector showed that the mean lifetime of neutrons in such a block is \(123\ \mu\mathrm{sec}\). To reduce the lifetime the following method was used. The front face of the block was covered with a sheet of cadmium \(0.43\ \mathrm{g}/\mathrm{cm}^2\) thick, which absorbed all thermal neutrons but transmitted neutrons with energies above \(0.4\text{–}0.5\ \mathrm{eV}\). A layer of paraffin \(1.6\ \mathrm{cm}\) thick was placed over the cadmium, which

... which slowed neutrons passing through the cadmium, but could not retain them for long. For such an arrangement, resembling a layer cake, the mean lifetime turned out to be equal to 33 μsec. At Fermi’s suggestion, water with various concentrations of boron dissolved in it, but without cadmium and paraffin over the front surface, was tested\(^{11,12}\) as a moderator. With a water-tank capacity of 11 liters, measurements of the mean lifetime gave the following results:

in pure water . . . . . . . . . . . . . . . . . . . 107 μsec,
in a solution of 1 g \(B_2O_3\) per liter of water . . . . . . 70 μsec,
" " 7.3 g \(B_2O_3\) per liter of water . . . . . . . . 43 μsec,
" " 13.7 g \(Na_2B_4O_7 \cdot 10H_2O\) per liter of water . . 50 μsec.

At Columbia University, in order to increase the total neutron yield, an internal beryllium target was used (placed inside the chamber between the dees). A paraffin slab in a plywood box, placed directly next to the cyclotron chamber between the target and the detector, was used as the moderator. Measurements of the neutron lifetime were carried out\(^{16}\) for two paraffin thicknesses (including the plywood), and the following results were obtained:

at a thickness of 3.2 cm . . . . 45 μsec,
6.8 cm . . . . 100 μsec.

Although the yield of thermal neutrons was greatest at a paraffin thickness of 4.5 cm, a layer 2.6 cm thick was used. Later\(^{18}\) a thinner layer of paraffin was used, in which the lifetime was taken to be 30 μsec, but its thickness is not indicated.

With the lifetime reduced in this way, the correction was no longer very large and could be introduced approximately. For example, Rainwater and Havens simply subtract this time from the measured one in determining the time of flight and regard the resolving power of the selector in the thermal-neutron region as worsened by the corresponding amount, while in the intermediate region they interpolate the correction, which for fast neutrons reduces to the slowing-down time. At a distance between the source and the detector equal to 6.2 m, their correction for thermal neutrons is 4.5 μsec/m, i.e., of the order of one percent. This is already sufficient with the resolving power practically attainable.

Detector

Since the functions of modulating the sensitivity are carried out not in the detector itself, but in the amplifying apparatus, and the detector may be continuously in operating condition, no specific requirements are imposed on it. Any detector that records the moment of arrival of a neutron is suitable in the present case for measurements.

In most laboratories, including Cornell University, ionization chambers filled with BF$_3$ were used. At Columbia University all work was carried out with proportional counters, likewise filled with BF$_3$. To increase sensitivity to neutrons, both these and other instruments have recently been filled with gas enriched in the active isotope of boron, B$^{10}$.

From the standpoint of sensitivity, ionization chambers are probably more convenient, since they can operate at high gas pressures. Producing a higher pressure in proportional counters, however, leads to an increase in the operating voltage and to difficulties of stabilization. For example, for counters filled with BF$_3$, of diameter 5 cm and at a pressure of 50 cm Hg, an operating voltage of 3700 V was used, and, in order to get rid of leakage pulses over the moist surface, it was necessary to blow the counters with a continuous stream of dry air.

Increasing the sensitivity by increasing the working length of the instruments, which at first sight is quite possible when working with a beam, proves inadmissible, since it introduces an additional uncertainty into the neutron path length. It is clear that the working length of the counter must be taken into account in estimating the resolving power of the selector. In this connection the above-mentioned counters of 5 cm diameter had a length of only 10 cm.

Proportional counters, however, are more convenient than ionization chambers in that they require less complicated amplifying apparatus and, in particular, make it considerably simpler to obtain a steeply rising pulse, which is very important for combating the lag discussed above.

The sensitivity of counters* filled with BF$_3$ to neutrons of different energies is evidently determined by the dependence of the boron neutron-capture cross section on energy, since neutron registration is due to the process B$^{10}(n,\alpha)$Li$^7$, which follows capture and yields strongly ionizing $\alpha$-particles and Li$^7$. It is known, and has been established most accurately precisely by experiments with selectors, that the boron cross section is proportional to

\[ \frac{1}{v}\sim E^{-1/2} \]

($v$ and $E$ are respectively the velocity and kinetic energy of the neutron). Hence the sensitivity of the counters, in first approximation, is also proportional to $E^{-1/2}$. This first approximation is the better, the smaller the sensitivity, i.e. the less appreciable the attenuation of the beam in passing through the counter. With appreciable attenuation the sensitivity will be a more complicated function of energy. It is easy to see that this function has the form:

\[ \varepsilon(E)=1-e^{-n\sigma x}=1-e^{-\frac{n\sigma_0}{E^{1/2}}x}, \]

where $\varepsilon$ is the sensitivity, $n$ is the number of boron atoms per unit volume

* In what follows, for brevity, we shall speak of counters, also having chambers in mind.

counter, \(\sigma=\dfrac{\sigma_0}{E^{1/2}}\) is the capture cross section for a neutron with energy \(E\), \(\sigma_0\) is the cross section for a neutron of unit energy, \(x\) is the working length of the counter. It is clear that this circumstance must, generally speaking, be taken into account in measurements, since, for example, in counters with \(\mathrm{BF}_3\) at a pressure of \(50\ \mathrm{cm}\ \mathrm{Hg}\) and a length of \(10\ \mathrm{cm}\), if the boron is assumed to be one hundred percent enriched with the isotope \(\mathrm{B}^{10}\), the attenuation reaches 60% for average thermal neutrons, while the precisely calculated efficiency differs by 20% from that calculated without allowance for attenuation\(^*\).

For protection from scattered neutrons the detector was usually surrounded by large thicknesses of cadmium, boron, and paraffin, with an aperture only in the direction of the source. In addition to this shielding, immediately surrounding the counter, a collimating channel of the same substances was made. The adequacy of the collimator was checked by special experiments in which the beam was blocked by a thick layer of boron. In most experiments these measurements showed that the background from scattered neutrons was negligibly small, and that the collimator was quite reliable.

In the course of the measurements the counter was under voltage the whole time and registered neutrons irrespective of their time of flight. One of the amplifier channels registered all neutron pulses by means of scaling circuits and, consequently, measured the total neutron intensity integrated over all flight times.

For registering neutrons with a specified time of flight, the pulses from the same counter, after preliminary amplification and shaping, were fed to special channels, the number of which in the latest work at Cornell University reached 12, and at Columbia University—16. In addition, two counters were used simultaneously there, and in fact neutron registration was carried out at once over 32 channels, not counting the integrating ones.

Each channel was adjusted to its own definite delay and registered neutrons with a definite flight time. This selection of pulses according to the time at which they arrived in the channel was carried out with the aid of well-known coincidence circuits. All counter pulses that had passed preliminary amplification entered one branch of the coincidence circuit; into the other entered gating pulses adjusted to a definite delay corresponding to the specified energy. If the coincidence circuit did not receive a gating pulse, then the counter pulse also could not pass to the following stages of the circuit. Thus, after the coincidence cascade there followed already counter pulses selected in time. The actual design of the coincidence circuit varied somewhat in different works, but this is no longer essential.

\(^*\) Rainwater, Havens, Wu, and Dunning, who worked with such counters, do not mention taking this circumstance into account, nor do they indicate the degree of boron enrichment.

The use of a large number of channels, each of which ended with its own mechanical counter, naturally greatly shortened the measurement time, made it possible to obtain a high accuracy of the results, and to extend the measurements to a larger number of objects.

Synchronizing devices

To control the periodic shots of the source, which open the pulses of the amplifier channels, and their mutual displacement (delay), some master frequency is usually generated, with respect to which all elements of the circuit are correspondingly adjusted. In the first experiment of Bacher and Becker[^10] such a frequency was supplied in the form of the sinusoidal voltage of a generator at 400 cycles. All operating pulses were repeated at this frequency, and the delay was regulated by a phase shift.

Subsequently, at Cornell and at Columbia Universities another, more advanced, method of control was used. The master frequency was generated by a 100-kilocycle generator (50 kilocycles at Cornell University) with a quartz stabilizer. Frequency stabilization is necessary in order to preserve the constancy of the time scale. By dividing this frequency in several cascades of counting circuits and by using certain additional devices, it was possible to generate a rectangular pulse for the ion source and for all amplifier channels and to vary the duration of this pulse and the delay in steps of not less than 5 μsec (the half-period of the generator). The latest model of the Columbia University selector is a system with a rather complex control circuit. An idea of this circuit may be obtained from Fig. 6, which shows a diagram of the control blocks. At the upper left is shown a 100-kilocycle generator, from which pulses, after first being made rectangular, are fed, on the one hand, to the 5 μsec pulse generator, and, on the other, to ten successive 1:2 counting circuits. From various stages of this chain of counting circuits, signals are taken off by means of switches \(A, B, C, D\) to circuits that transmit the control pulses to the source and detector.

The pulses from the proportional counters, after passing through amplifiers and clippers, arrive, on the one hand, at integrating counters (through the corresponding counting circuits), and, on the other, at 16 channels which select pulses according to time. Each channel terminates in a 1:16 counting circuit and a mechanical counter. The switches and other control elements are brought out to a panel (the large rectangle on the right), where an oscilloscope screen is also mounted, permitting visual control of the operation of the circuit. The sweep of the oscilloscope is synchronized with the master frequency.

By means of the switches the selector is easily readjusted to any pulse width, any delay, and any period

repetition within the range from 5 μsec to 10240 μsec, each time being doubled on going to the next step.

In investigations of neutrons with a flight time (over a distance of 6.2 m) less than 640 μsec, a pulse-repetition cycle period of 1280 μsec was used. In order not to let slow neutrons from preceding cycles reach the detector, the beam was filtered through a thick layer of cadmium, which absorbed practically all neutrons with a flight time greater than 800 μsec. For flight times less than 1280 μsec, a cycle period of 5120 μsec was used.

Fig. 6. Block diagram of the control units of the Columbia University selector.

In this case the slow neutrons from preceding cycles were filtered out by a thin layer of mercury. For flight times from 1280 to 5120 μsec, the cycle period remained 5120 μsec, but the mercury filter was removed. For longer flight times, a cycle period of 10240 μsec was used. The width of the opening pulses at the source and on all detector channels was always set equal, namely 5, 10, and 20 μsec for a cycle period of 1280 μsec, and 20, 40, 80, and 160 μsec, depending on the flight time of the neutrons under study, for longer cycle periods. Adjacent channels were adjusted to adjacent delay intervals. For example, if the pulse on the first covered the delay time from 0 to 5 μsec, then on the second—from 5 to 10 μsec, etc., and on the sixteenth—from 155 to

160 μsec and, consequently, all sixteen channels simultaneously gave sixteen points on the curve corresponding to flight times from 0 to 160 μsec.

The latest model of the Cornell University selector was adjusted in an analogous manner for 12 measuring channels.

RESOLVING POWER OF SELECTORS

Owing to the finite width of the pulses opening the source and the detector, the selector registers neutrons that are not strictly monochromatic. In fact, if the width of the pulses at the source and detector is the same and equal to $\tau$, and their beginnings (or midpoints) are separated by an interval $t_e$, then among the registered neutrons there may be one which left the source just before the very moment of closing and arrived at the detector immediately after its opening. For such a neutron the actual flight time is $t_e-\tau$. Conversely, a neutron that left the source at the very beginning and arrived at the detector at the very end of the pulse will have a flight time $t_e+\tau$. It is obvious that neutrons with all intermediate flight times can also be registered. Consequently, a selector adjusted to a delay $t_e$ registers neutrons with all flight times contained in the interval $t_e-\tau<t<t_e+\tau$. It is easy to see that the probability of registration is maximal for neutrons with a flight time equal to the adjusted delay $t_e$ and, if the opening pulses are rectangular, decreases linearly in both directions, reaching zero at the edges of the interval, provided the detector is equally sensitive to all neutrons in this interval. In other words, the transmission of the selector as a function of the neutron flight time is represented by an isosceles triangle whose base is equal to $2\tau$, and the position of the vertex corresponds to the time $t_e$. This triangle, characterizing the resolving power, is usually shown on graphs depicting the results of measurements.

It is immediately evident that the resolving power of the selector is the better, the longer the neutron flight time, i.e. the smaller their energy. Conversely, in studying neutrons with a short flight time, when the width of the interval $2\tau$ becomes comparable with this flight time, the energy width of the interval becomes large and the resolving power poor. The dependence of the registered energy interval $\Delta E$ on the flight time or energy of the neutrons studied can be obtained directly by differentiating the formula

$$ E=\frac{mv^2}{2}=\frac{m}{2}\frac{l^2}{t^2}; $$

$$ \Delta E=-\frac{E^{3/2}}{\sqrt{\frac{ml^2}{8}}}\Delta t=-\frac{2E}{t}\Delta t. $$

From this it is seen that, for a given $\Delta t$, $\Delta E$ increases with energy as $E^{3/2}$, while the relative width $\dfrac{\Delta E}{E}$ increases as $E^{1/2}$. For a clear illustration of the resolving power we give Table II, in which the absolute and relative widths of the recorded interval are given on the assumption that $\Delta t=2\tau=10\ \mu\text{sec}$, which corresponds to the minimum practically used width of the modulating pulses, and $l=5\ \text{m}$.

Table II

From the table it is seen that in the region of thermal neutrons (with energy about $0.04\ \text{eV}$) $\dfrac{\Delta E}{E}$ is of the order of one percent, i.e. the resolving power is sufficiently high, and in practice it is quite possible to increase the pulse width in order to obtain a higher neutron intensity. But already at an energy of $10\ \text{eV}$, $\Delta E$ exceeds $1\ \text{eV}$. This means that, at energies of this order, sharp changes in intensity cannot be studied accurately. For example, the shape of resonance-absorption lines of neutrons, usually having a width of the order of $0.1\ \text{eV}$, can be determined sufficiently well at resonance energies of the order of fractions of an eV, but at higher energies it will be distorted.

Fig. 7. Transmission curve for $E_0=1\ \text{eV}$ and $\Gamma=0.2\ \text{eV}$ and its distortions due to finite resolving power.

The influence of resolving power on the shape of the curve under investigation was considered by Becker and Bacher in a 1941 paper. The results of their calculations, presented in the form of graphs, are given by us in Figs. 7, 8, and 9. Figs. 7 and 8 show the shape of the transmission curve $T(E)=e^{-\sigma n}$ of a sample resonantly absorbing neutrons with energy $1\ \text{eV}$ ($n$ is the number of atoms per unit surface of the sample). If...

the neutron capture cross section by the nuclei of the sample follows the Breit–Wigner resonance formula

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

where \(E_0\) is the resonance energy, \(\sigma_0\) the resonance cross section, and \(\Gamma\) the width of the level, and the attenuation of the beam occurs only through capture, then \(T(E)\) is represented by the curves marked \(0\) in Figs. 7 and 8. These curves can be obtained experimentally only with high resolving power. If the resolving power is not very good, then the experimental curves will have a distorted form, the distortion being the greater the worse the resolving power. The numerals \(1\) and \(2\) in the same figures denote the curves that should be obtained with a resolving power corresponding to the triangles marked by the same numerals \(1\) and \(2\).

Fig. 8. Transmission curve for \(E_0 = 1\ \mathrm{eV}\) and \(\Gamma = 0.4\ \mathrm{eV}\), and its distortion due to finite resolving power.

Along the axis of abscissas is plotted the neutron flight time in microseconds per meter, and the bases of the triangles are equal to twice the width of the modulating pulses, i.e. they represent the interval of the registered flight times. It is assumed that the pulse at the source is equal in width to the pulse at the detector.

Fig. 9. Distortion of exponential curves due to finite resolving power.

Comparison of curves \(0\), \(1\), and \(2\) shows that, at low resolving power, because of the distortion of the resonance curve it is impossible to determine either its true width or the value of the resonance cross section. The position of the resonance, however, is displaced only slightly and always toward lower energies. This is due to the asymmetry of the resonance curve. As a consequence, the curves of resonance absorption of neutrons with energies greater

1–5 eV depict not the true shape of the resonance absorption lines, but an experimentally distorted one. The actual width of the lines is, as a rule, smaller, and the resonance cross section larger (correspondingly, the true transmittance is smaller). In other words, the experimental curves give an upper limit for the width and a lower limit for the resonance cross section, while the true values may differ from these limits by tens or even hundreds of times.

Fig. 9 illustrates the influence of resolving power on the course of the descending part of the transmittance curve. The initial curve, marked 0, is a graphical representation of the functions \(A(2-e^{-\alpha t})\) and \(Be^{-\alpha t}\), joined at the value 0.5 by selection of the constants. The meanings of the numerals 1, 2, 3 and of the triangles are the same as on the preceding curves. Curve 1, coinciding with curve 0, corresponds to a width of the modulating pulse equal to the “decay constant” of the exponentials given, i.e. \(\tau=\frac{1}{\alpha}\); the others correspond, respectively, to double and triple pulse width. As was to be expected, deterioration of the resolving power leads to a decrease in the steepness of the descending (or ascending) portion of the curve.

In addition, Becker and Bacher indicate that resolving power does not affect the course of a simple exponential—the result is an exponential with the same “decay period.” Obviously, a straight line (inclined or horizontal) is also transmitted without distortion.

SPECTRA OF SLOW NEUTRONS

It is known that, in a sufficiently large volume of a substance weakly absorbing neutrons, thermal equilibrium is attained between the neutrons and the atoms of the substance, analogous to the thermal equilibrium between a solute and a solvent. But in the case of a “neutron solution” the thermal equilibrium is not ideal, and the distribution of neutron energies must differ from the Maxwellian distribution of the atoms of the “solvent.” This difference of the neutron spectrum from the Maxwellian one is due, first, to the presence of fast neutrons, from which thermal neutrons are formed as a result of multiple collisions; second, to the absorption process and, consequently, to the limited lifetime of the neutrons; third, to the finite dimensions of the substance; and, finally, fourth, to the inequality of the conditions of energy exchange with atoms for neutrons of different velocities.

The theoretical allowance for the distorting influence of these factors proves to be very complicated, and the calculation of the real neutron spectrum is practically impossible. Meanwhile, knowledge of this spectrum is of great theoretical and practical interest.

Before the appearance of selectors there was no possibility of investigating the spectra of slow neutrons. The experiments which revealed the temperature effect\(^{23}\) gave only qualitative proof of the presence of

neutrons with thermal velocities. And only with the aid of selectors was the actual velocity distribution of thermal and adjacent, faster neutrons found.

The method of studying the neutron spectrum with the aid of a selector is obvious. The number of pulses of the detector is measured as a function of the delay. If the resolving power is sufficient, then this number of pulses determines the number of neutrons as a function of the time of flight. This is, of course, the number of neutrons that have passed through the detector, i.e. the spectrum of neutrons in the beam. To pass from the number of pulses to the number of neutrons that have passed through the detector, one must take into account the dependence of the detector sensitivity on the velocity. Since the detector is an instrument filled with \(BF_3\), its sensitivity is proportional to \(\frac{1}{v}\) (without allowance for attenuation of the beam in the detector itself).

Consequently, if the experimentally observed spectrum of the number of pulses is represented by the function \(f(v)\), then the neutron spectrum in the beam will be \(v \cdot f(v)\). But the neutron flux through any surface, including any cross section of the detector, is represented by this same function \(vf\), if the spatial density of neutrons of different velocities is given by the function \(f\). This means that, owing to the specific properties of the detector (the sensitivity is proportional to \(\frac{1}{v}\)), the spectrum of the number of recorded pulses is the same as the spectrum of neutrons in the volume of the moderator, the surface of which is used as the neutron source. This conclusion, however, is valid only within such limits in which the mean depth from which the neutrons emerge through the surface, having undergone before this their last collision, is the same. This depth is, obviously, proportional to, and in the case of a direction normal to the surface simply equal to, the mean free path. If the moderator is paraffin, then, as is seen from Table I, the condition of constancy of the depth is not satisfied even for the whole region of thermal neutrons, and still less in the transition from thermal neutrons to faster ones, since the mean free path increases with energy. The data of Table I concerning the mean free paths, although not very accurate, are undoubtedly correct in the first digit after the decimal point, since the mean free path in paraffin is well known, on the one hand, for the main group of thermal neutrons, and on the other—for neutrons with energies above \(1\ \mathrm{eV}\). Obviously, the neutron spectrum in air beyond the surface of the paraffin, including also the neighborhood of the detector, must, in comparison with the spectrum in the depth of the paraffin, be enriched in those neutrons for which the mean free path and, consequently, the depth of emergence, are greater, i.e. in fast neutrons. This circumstance must also be taken into account in interpreting the experimentally obtained pulse spectrum.

Despite these considerations, which indicate an undoubted difference between the neutron spectrum and the Maxwellian one, comparison with

with a Maxwellian spectrum makes sense, first, for estimating the magnitude of distortions, and second, for determining the mean energy of thermal neutrons through the parameters of the Maxwellian distribution. In making these comparisons it is convenient to plot the Maxwell distribution as a function of the flight time, referred to the time interval, since in practice the number of registered pulses relates to an identical time interval determined by the given width of the opening pulses.

Fig. 10. Distribution of slow neutrons from a “thin” layer (2.6 cm of paraffin and 0.6 cm of vanadium). The solid line corresponds to a Maxwellian distribution at 430° K.

As a result of the corresponding transformation, Maxwell’s formula

\[ N(v)=v^{2} e^{-\frac{mv^{2}}{2kT}} \]

is transformed into:

\[ F(t)=Ct^{-4} e^{-\frac{ml^{2}}{2kTt^{2}}}. \]

Here \(C\) is a constant, \(l\) is the distance between the source and the detector, \(t\) is the flight time, and \(T\) is the absolute temperature, which does not necessarily coincide with the actual temperature of the moderator emitting the neutrons (in particular, paraffin), but can be chosen so that the latter formula agrees in the best way with the experimentally obtained distribution.

The first measurements of the neutron spectrum by means of a selector-modulator in the velocity range from \(10^{5}\) to \(10^{6}\) cm/sec, which corresponds to energies from 0.005 to 1 eV, were carried out by Manley, Haworth, and

Lubke[^14] at the University of Illinois. Their work was carried out in 1941, although it was published in 1946. The neutron source was the surface of a paraffin cube with a side of 14 cm, surrounding the target of a high-voltage tube in which, by the reaction \(D(d,n)\mathrm{He}^3\), neutrons with an initial energy of about 2.5 MeV were obtained. The measurements showed that the neutron distribution is close to Maxwellian,

Fig. 11. Distribution of slow neutrons from a “thick” layer (6.2 cm of paraffin and 0.6 cm of plywood). The solid line corresponds to a Maxwellian distribution at 390° K.

corresponding to a temperature of 400° K. But even in comparison with this distribution there is observed an excess of neutrons with velocities greater than \(3.5 \cdot 10^5\) cm/sec (energy 0.06 eV), understandable from the point of view stated above.

More accurate measurements were carried out by Rainwater and Havens[^16] at Columbia University in 1942. The neutron source here was the surface of a paraffin slab enclosed in a plywood box and placed near the chamber of a cyclotron. The internal thick beryllium target of the cyclotron was bombarded with deuterons of energy 8 MeV. Consequently, the primary neutrons entering the paraffin from the side of the target had a rather complex spectral composition, but in any case possessed energies of several MeV up to 12. The work gives neutron spectra corresponding to two different thicknesses of paraffin slabs. In Figs. 10 and 11 we reproduce the results of measurements published by the authors. In Fig. 10 the points show the experimentally obtained neutron spectrum for a paraffin thickness of 2.6 cm and plywood of 0.6 cm; the dashed curve was drawn through the experimental

points; the solid curve corresponds to a Maxwellian distribution at a temperature of \(430^\circ\mathrm{K}\). This temperature has been chosen as giving the best agreement of the Maxwellian curve with the experimental points.

Fig. 11 shows analogous results for a paraffin plate thickness of \(6.2\ \mathrm{cm}\) and plywood of \(0.6\ \mathrm{cm}\). Here the experimental distribution is best represented by a Maxwellian at a temperature of \(390^\circ\mathrm{K}\). We again see a sharp difference between the real spectrum and the Maxwellian one in the region of short flight times, i.e., in the region of high velocities. Naturally, this difference is the greater, the smaller the thickness of the paraffin. But over a fairly broad range of thermal velocities the experimental spectrum agrees well with a Maxwellian corresponding to an elevated temperature. If, on this basis, one assigns to the thermal neutrons the corresponding temperature, then one may say that the neutron temperature for the described source configurations turns out to be \(30\text{–}40\%\) higher than the actual temperature of the paraffin. Meanwhile, in all previous neutron experiments analogous slowing-down conditions were used, and it was assumed that thermal neutrons (more precisely, neutrons absorbed by cadmium—\(C\)-neutrons) have a Maxwellian distribution corresponding to the temperature of the paraffin. In particular, the method for determining the energies of resonance-absorbed neutrons from their absorption cross section by boron was based on this assumption. The essence of this method is that the boron cross sections are measured experimentally, on the one hand for resonance neutrons, and on the other for thermal neutrons. Comparing these cross sections and bearing in mind that the boron cross section is proportional to \(\dfrac{1}{v}\sim E^{-\frac12}\), one can determine the energy of the resonance neutrons by the obvious formula:

\[ E_{\mathrm{res}}=\frac{\sigma_T^2}{\sigma_{\mathrm{res}}^2}E_T . \]

The energy of the thermal neutrons \(E_T\) entering into this formula was taken on the basis of the assumption of a Maxwellian distribution at the actual temperature. Owing to the fact that the boron cross section is not constant but proportional to \(E^{-\frac12}\), this energy is not equal to the mean thermal energy \(\dfrac{3}{2}kT\), but differs from it and depends on the thickness of the boron absorber used. For a very thin absorber \(E_T=\dfrac{\pi}{4}kT\). Bethe\(^{24}\) calculated the values of \(E_T\) for different absorber thicknesses, and these values were practically used in numerous experiments on the determination of \(E_{\mathrm{res}}\).

As a result of the difference between the neutron temperature and the temperature of the paraffin, all these measurements contain a systematic error of the order of \(30\%\), which, however, depends on the character and geometry of the source-

nique and cannot be precisely determined. However, at the present time, thanks to the successful use of selectors, the results of the earlier measurements of resonance energies have lost their practical significance, since more complete and accurate data have been obtained. The absorption method in boron itself, however, has not lost its importance and, on the contrary, now, when with the aid of selectors the law \(\frac{1}{v}\) has been well confirmed and the absolute values of the boron cross section have been determined to an accuracy of the third digit, it has acquired a firmer basis and may find application because of its simplicity.

One of the most recent examples of its use is the work of Fermi, Marshall, and Marshall\(^5\) on measuring the temperature of neutrons emitted by an argon uranium-graphite pile under various conditions of beam filtration. We have already noted that the neutron spectrum depends on many factors, including the quality of the moderator. Under certain conditions it may be so distorted in comparison with a Maxwellian one that the application to it of the concept of “temperature” requires special qualifications. Fermi et al. proceed in the following way. For an unknown neutron spectrum they measure the (mean) absorption cross section in boron. Knowing the boron cross section for neutrons with velocity \(2200\ \mathrm{m/sec}\) (energy \(kT\) at \(293^\circ\mathrm{K}\)) \(\sigma=703\cdot 10^{-24}\ \mathrm{cm}^2\) from their own measurements with a semi-mechanical selector, and assuming the law \(\frac{1}{v}\), and, in addition, using Bethe’s calculated values of the energy \(E_r\), they determine from the measured boron cross section for the unknown spectrum the temperature that would correspond to this spectrum if it were Maxwellian. Of course, such a definition of temperature does not correspond to its generally accepted meaning, but this effective temperature nevertheless characterizes a certain mean energy (velocity) of the neutron spectrum.

The results of measurements of this effective temperature for sources of various design are given in Table III.

As can be seen from the table, the effective temperature varies very strongly depending on the conditions under which the neutron beam is obtained. Let us briefly discuss these conditions and the nature of their influence on the neutron spectrum.

In the first case the source of neutrons is the surface of the graphite so-called thermal column, which is a very thick layer of graphite separating the interior of the pile, which is the immediate source of fast neutrons, from the place of observation. In this case what is essential is only that the large thickness of the graphite column is sufficient not to transmit the fast neutrons arising in the fission process; therefore near its outer surface there exist only slow neutrons in thermal equilibrium with the graphite and diffused from deeper layers. But the diffusion conditions are different for neutrons of different energies,

Table III

As the experiment of Anderson, Fermi, and Marshall\(^{25}\) showed, the cross section for the scattering of neutrons by graphite changes sharply from \(4.05 \cdot 10^{-24}\ \mathrm{cm}^2\) to \(0.70 \cdot 10^{-24}\ \mathrm{cm}^2\) when the neutron energy is decreased, the position of this jump in the cross section corresponding to a neutron wavelength of about \(7\ \text{\AA}\). The point is that neutrons having a wavelength comparable with the lattice constant of graphite, which is a polycrystalline substance, undergo Bragg reflection from the crystallites and therefore are strongly scattered. But the maximum Bragg wavelength for graphite is \(6.69\ \text{\AA}\), and neutrons with a greater wavelength do not undergo Bragg reflection and therefore are scattered by graphite much more weakly. This is the cause of the difference in diffusion conditions. Slow neutrons with a large wavelength, corresponding to the tail of the Maxwellian distribution for ordinary temperature, diffuse much better toward the surface of the graphite column; therefore the spectrum of neutrons emitted by this surface is enriched in slow neutrons, and the effective temperature \(198^\circ\mathrm{K}\) is considerably lower than the temperature of the graphite (about \(30^\circ\mathrm{C} = 303^\circ\mathrm{K}\)).

Case 2 differs from the preceding one only in that the beam has additionally been passed through \(3.7\ \mathrm{cm}\) of paraffin. Meanwhile the effective temperature increased to \(408^\circ\mathrm{K}\). This is explained by the fact that, for slow neutrons in paraffin, both the scattering cross section and the capture cross section are larger than for fast ones; therefore they are more quickly knocked out of the beam. In addition, forward scattering is less probable for slow neutrons.

In case 3, instead of paraffin, heavy water was placed in the path of the beam; it likewise raises the effective temperature of the neutrons from 198 to 288°K, since its scattering cross section is also larger for slow neutrons.

Case 4 illustrates the properties of graphite as a filter. The beam was passed through 22 cm of graphite, in which the fast neutrons were practically all scattered and left the beam, while the transmitted neutrons have a very low effective temperature—18°K.

Cases 5 and 6 correspond to the spectrum of neutrons obtained from the depths of a graphite thermal column. Deep cavities in the column form a sort of absolutely black body that emits an equilibrium spectrum. In case 5 the neutron temperature coincides with the temperature of the graphite. In case 6 it is somewhat lower, probably because the cavity was not deep enough.

The examples considered of studies of slow-neutron spectra show that thermal equilibrium between neutrons and a medium of sufficient thickness is indeed established, and that the neutron spectrum has a natural and distinct maximum in the region of energies of order $kT$. But both the position of the maximum and the entire shape of the spectrum are determined by the conditions of slowing down and by the configuration of the source. Paraffin as a moderator, as a rule, gives a neutron spectrum with an elevated effective temperature, since the slow part of the spectrum is strongly absorbed. Obviously, any absorber whose cross section increases for slow neutrons, for example according to the law $\frac{1}{v}$, will increase the effective temperature of the neutrons by impoverishing their spectrum in the region of small energies. Conversely, filtering neutrons through graphite, and in general through any polycrystalline scatterer, lowers the effective temperature owing to the preferential scattering of fast neutrons. This means, first, that the average energy of thermal neutrons is not a completely definite quantity equal to $\frac{3}{2}kT$, as was previously assumed for lack of sufficiently accurate experimental data. Second, this average energy of thermal neutrons turns out to be an experimentally controllable quantity. In this respect, Fermi’s method for obtaining “cold” neutrons is substantial and new.

We have not considered the results of spectral investigations using crystalline neutron monochromators. They are in agreement with what has been set forth and will be discussed in a special review.

(To be concluded in the next issue)