NEW INSTRUMENTS AND METHODS OF MEASUREMENT

CRYSTAL $\gamma$-SPECTROMETER

Yu. V. Kholnov

1. INTRODUCTION

The study of the $\gamma$-spectra of artificial and natural radioactive isotopes makes it possible to judge the spectra of the energy levels of the nucleus.

Comparison of the energies of $\gamma$-lines with the energies of other types of radiation ($\alpha$-, $\beta$-radiation) of some radioactive isotope leads to hypotheses concerning the decay scheme of the given isotope. The degree of reliability of assumptions of this kind depends on the accuracy with which the radiation energies are measured.

At present magnetic $\gamma$-spectrometers are very widely used; with their aid the energies of electrons produced by $\gamma$-radiation in interaction with matter are analyzed. The accuracy of determining the energies of $\gamma$-lines is then limited by the errors with which the absolute values of the magnetic-field strength are measured.

Interference spectrometry is free of these errors. It is extremely difficult to study $\gamma$-spectra by means of direct reflection of $\gamma$-rays from crystals if high accuracy is required. The difficulties arise first of all from the necessity of working with crystals having a small lattice constant and, in their structure, close to perfect ones. Such crystals, however, are not always available to the experimenter. In addition, in view of the rapid decrease of the reflection coefficient with decreasing wavelength of the radiation under investigation, it is necessary to have sources of high activity, and, in order not to reduce substantially the resolving power of the instrument, of high specific activity as well.

Because of numerous difficulties, for a long time there was no good crystal $\gamma$-spectrometer with the aid of

of which it would be possible to determine the energies of \(\gamma\)-lines so accurately that the measurement data could be used, for example, for calibrating numerous magnetic spectrometers.

The present review is devoted to a description of the design of a recently constructed interference \(\gamma\)-spectrometer, which is the best in accuracy for determining the energies of \(\gamma\)-lines. Experimental results obtained with the aid of this instrument are also presented here.

II. CRYSTAL SPECTROMETERS

The periodicity of the arrangement of atoms in a crystal leads to the fact that the rays reflected from them interfere and give intensity maxima at definite angles of reflection, the magnitude of which depends on the wavelength of the radiation and on the distance between the reflecting planes.

For radiation with a given wavelength \(\lambda\) and a distance between the reflecting planes equal to \(d\), there exists a definite series of reflection angles \(\varphi\) at which an intensity maximum is observed. Mathematically, the relation between these three quantities is expressed by the Wulff–Bragg equation

\[ n\lambda = 2d\sin\varphi, \tag{1} \]

where \(\lambda\) is the wavelength of the incident radiation, and \(n\) is the order of reflection.

This interference condition is the fundamental equation of all crystal spectrometers. To study radiation reflected from a known crystal, one determines the angle at which the maximum intensity of these rays is observed in any order \(n\). Then, from equation (1), the wavelength \(\lambda\) is found.

Crystal spectrometers were first used to study X-ray spectra. In connection with work in this wavelength region, all the basic methods of interference spectrometry with a crystal arose. By means of various methods employing reflection from a plane crystal (for example, the Bragg method with an oscillating plane crystal), not only X-rays but also \(\gamma\)-rays were studied already at the early stages in the development of interference spectrometry.

However, with the aid of a plane crystal only soft \(\gamma\)-rays were studied, and even then with low accuracy. The accuracy of the investigations was limited mainly by two reasons.

First, since the reflecting planes of any real crystal are not quite parallel (the crystal has a mosaic structure), there is a definite interval of angles in which rays of a given wavelength are reflected. Owing to this

γ-lines broaden and the accuracy of energy determination decreases.

This error becomes especially significant when the radiation wavelength is decreased, when the Bragg reflection angles become small.

Secondly, as the wavelength decreases, owing to a decrease in the reflection coefficient for the reflecting planes, the intensity of the reflected rays drops sharply.

From these limitations follow two requirements: a) in studying hard-ray spectra it is necessary to use the most perfect crystals; b) since the intensity of γ-radiation sources is limited, means must be sought for increasing the intensity of the reflected rays.

The second problem had already been solved to a considerable extent in the field of X-ray spectrometry. Focusing X-ray spectrometers with a bent crystal were created. With high resolving power—the same as that given by the best spectrometers with a flat crystal—focusing spectrometers give a gain in intensity of several tens of times.

There are two types of spectrometers with a bent crystal—the Johann spectrometer and the Cauchois spectrometer.

Fig. 1. Focusing according to Johann.

In Fig. 1 a diagram of the Johann spectrometer is given. The principal part of the instrument is a crystal bent along an arc of a circle of radius \(R\), drawn from point \(D\). The source of X-rays—an X-ray tube—is located on the segment \(AA'\) of a circle of radius

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

drawn from point \(O\).

The segment \(CD\), connecting the center of the bent crystal with point \(D\), serves as the diameter of this circle, so that the crystal is tangent to it at point \(C\).

Let the reflecting planes of the crystal be parallel to its bounding surfaces. Then it can be shown that a beam of rays of a definite wavelength emerging from the source \(AA'\) will meet, along the entire length of the crystal, reflecting planes at an angle satisfying condition (1), and will be focused on the symmetrically located segment \(BB'\) of the same “focusing circle.”

If some detector of X-rays is placed on the segment \(BB'\) (a photographic plate, a Geiger counter, an ionization

...camera, etc.), then one can study the intensity of the reflected rays as a function of the reflection angle \(\varphi\).

The angle \(\varphi\) can be varied by moving the source and, correspondingly, the detector along the focusing circle.

But since the angle \(\varphi\) and the wavelength \(\lambda\) are related to one another by relation (1), in this way one studies the dependence of the intensity of the reflected rays on the wavelength, i.e. the radiation spectrum.

It must be noted, however, that the Johann method does not give exact focusing. Thus, rays reflected from the points \(C\) and \(K'\) of the crystal will arrive, respectively, at the points \(B\) and \(B'\) of the focusing circle, owing to the fact that the crystal, at its extreme points \(K'\) and \(L'\), deviates from the focal circle.

This phenomenon is called geometrical aberration. Geometrical aberration makes the focusing of the instrument less sharp, broadening the lines under study and reducing its resolving power.

Fig. 2. Focusing according to Johansson.

Johansson proposed a method in which exact focusing of monochromatic rays is achieved. According to this method the crystal is first ground along a cylindrical surface of radius \(R\), as shown in Fig. 2 (position 1), and is then bent along the arc of a focal circle of radius \(r=\frac{1}{2}R\) (position 2). Then, if the source is located at point \(A\), rays with the wavelength corresponding to the given angle of reflection will be focused at the symmetric point \(B\) from the entire surface of the crystal. Because geometrical aberration is absent in this case, the crystal can be made large, which increases the intensity of the reflected rays. However, treatment of the crystal by the Johansson method is a very difficult task. Therefore the unmodified Johann method is most often used. A reflecting spectrometer of the Johann type is used for the study of soft radiation because of the high resolving power that it provides in this region.

For the study of hard X-ray radiation, a focusing spectrometer of the Cauchois type is used. According to the Cauchois method, the crystal is cut flat in such a way that its reflecting planes form a certain angle with the limiting surfaces...

nesses (in the case shown in Fig. 3, this angle is equal to \(90^\circ\)); then the crystal is bent along an arc of a circle of radius \(R\), drawn from the point \(\beta\). At this point the extensions of the reflecting planes intersect. The broad source \(A\) is placed on the convex side of the crystal. The beam of rays emerging from it penetrates the crystal. Some of the rays meeting the internal reflecting planes at an angle satisfying the Bragg condition will be selectively reflected. In this case the reflected rays of one wavelength are focused on the arc segment \(RR'\)—the real focus of the instrument, lying on the circumference \(CV\beta R\) of radius \(r=\frac{1}{2}R\), drawn from the point \(O\), the midpoint of the segment \(C\beta\).

Fig. 3. Focusing according to Cauchois.

The source may be placed in the position \(A'\). Then the rays will be reflected from the opposite sides of the same reflecting planes. Rays of a definite wavelength \(\lambda\), which were previously focused on the segment \(RR'\), are now focused in \(VV'\)—the virtual focus of the instrument. The segments \(RR'\) and \(VV'\) are situated symmetrically with respect to the line \(C\beta\). The arcs \(\beta R\) and \(\beta V\) are equal.

In studying X-ray spectra by the Cauchois method, a recorder is placed at the focus of the instrument, for which photographic film is usually used. On the photographic film bent along the focusing circle, a whole series of lines of a definite portion of the spectrum is photographed at once.

A Geiger counter may also be used as the recorder; in this case the counter is placed at the focus of the instrument, with a narrow slit placed in front of it. In studying the spectrum of rays, the counter together with the slit is moved along the focusing circle.

A Cauchois-type spectrometer, like Johann’s spectrometer, does not give exact focusing. The phenomenon of geometrical aberration is also inherent in this instrument. It consists in the fact that rays of one and the same wavelength (see Fig. 3), after being reflected from points \(C\) and, for example, \(D\) of the crystal, arrive at correspondingly different points of the focusing circle—\(R\) and \(R'\).

This phenomenon occurs because the crystal at its edge points \(D\) and \(D'\) is somewhat farther from the focal circle. If there were a perfect parallel displacement of the reflecting

planes onto the focal circle, the phenomenon of geometrical aberration would disappear.

If a perfect crystal with reflecting planes parallel throughout its entire length is chosen and bent so that the extensions of the reflecting planes intersect exactly at the point \(\beta\), the phenomenon of geometrical aberration will nevertheless occur. It is characteristic of the Cauchois method.

The blurring of the focus as a result of geometrical aberration depends on the solid angle \(\alpha\) under which the crystal is seen from the point \(\beta\), and on the Bragg angle \(\varphi\). For small \(\varphi\) and \(\alpha \ll 1\), the broadening of the lines due to geometrical aberration is proportional to \(\alpha^2\):

\[ \frac{\Delta \lambda}{\lambda} \sim \alpha^2 . \]

Thus, in the case when the crystal occupies a very small part of the focal circumference, the broadening of the focus due to this kind of aberration is very small and, in practice, may be disregarded. Other causes have a more substantial influence on the sharpness of focusing: improper bending of the crystal and its mosaic structure. We shall discuss this in greater detail below.

To eliminate the geometrical aberration, the Cauchois method can be modified according to Johannson. However, as already mentioned, the manufacture of crystals by Johannson’s method is extremely difficult.

All methods employing bent crystals make it possible to focus, more or less accurately, monochromatic rays reflected simultaneously from the entire crystal. Owing to this, the intensity of the reflected rays increases considerably. In view of the small values of the reflection coefficients for hard X-ray and \(\gamma\)-radiation, it is desirable, in investigations of these types of radiation, to use focusing crystal spectrometers.

Such a focusing \(\gamma\)-spectrometer was built by a group of physicists at the California Institute of Technology, headed by DuMond. This work was begun as early as 1938, interrupted by the war, and then resumed.

They built a spectrometer of the Cauchois type\(^1\). However, the authors introduced into the design of the instrument an essential, fundamental change that increased the intensity of the reflected rays many times over. In the Cauchois spectrometer a broad source is placed on the convex side of the crystal (position \(A\) in Fig. 3); in this case part of the rays passing through the crystal is collected at the focus of the instrument \(R\).

In DuMond’s method, however, a small, very concentrated source is placed at the focus of the instrument \(R\) (Fig. 4). The radiation falls on the crystal within a certain solid angle determined by the working surface of the crystal. Encountering the reflecting planes

crystal at an angle satisfying condition (1), some of the rays will be reflected and will change their direction. The rays reflected will be precisely those wavelengths which would be focused at the given focus \(R\), if the source were arranged as in the Cauchois method.

In the DuMond method, rays falling from the focus \(R\) onto the crystal will meet the reflecting planes of the crystal at an angle that certainly satisfies Bragg’s condition.

The probability that a \(\gamma\)-quantum emitted by a particular nucleus will meet the reflecting planes of the crystal at an angle satisfying Bragg’s condition is determined by the solid angle under which the crystal is seen from the focus of the instrument.

Fig. 4. Schematic of the DuMond apparatus (not to scale).

This same probability for a source located on the outer side of the crystal is determined by a very small angle (of the order of several seconds), since for each quantum in this case there is only one direction in which it will meet a reflecting plane at the Bragg angle. If the activity of the sources in the first and second cases is the same, then the intensity of the reflected rays in the second case is incomparably greater. The authors believe that with such an arrangement of the source, in their geometry, a gain in intensity of approximately 700 times is obtained.

III. APPARATUS

a) Principal parts of the apparatus[^1]

The basic schematic of the apparatus is shown in Fig. 4. At the focus of the apparatus there is a source of \(\gamma\)-quanta. Part of the rays passing through the crystal is selectively reflected from it and is recorded by some detecting device (a Geiger counter, ionization chamber, scintillation counter, etc.).

To measure the spectrum of \(\gamma\)-rays it is necessary to vary the Bragg angle, which is achieved by moving the source along the arc of the focusing circle.

The most important part of the apparatus is the crystal, since it is chiefly this that determines the resolving power of the instrument. It is necessary to choose such a crystal whose reflecting planes are sufficiently parallel to one another. In addition,

CRYSTAL γ-SPECTROMETER

it must be easy to machine and, when bent, must not develop cracks or other defects.

In the spectrometer described, a quartz crystal was used. As reflecting planes, the planes \((310)\) were used, for which the lattice constant is

\[ d = 1177.6\ \text{X-units}.*) \]

A quartz plate, \(70 \times 50 \times 1\) mm in size, was polished with optical precision along surfaces perpendicular to the reflecting planes. In order to bend the crystal, special holders were made of a special steel (not subject to temperature and other influences), consisting of two blocks—one with a convex and the other with a concave cylindrical surface. Between these surfaces, by means of screws with springs, the crystal was clamped, assuming the shape of the holder (Fig. 5). It is clear from the figure that one surface of the crystal was pressed tightly against the convex plate of the holder. Therefore the radius of curvature of this cylindrical surface had to be constant with a high degree of accuracy over its entire length.

Fig. 5. Crystal holder (diagram).

This radius was equal to two meters. The quality of focusing of the beam depends on the accuracy with which the crystal is bent along a circle of the chosen radius. Therefore, for machining the cylindrical surface of the large-radius holder, a special mechanical method was developed². After long and painstaking experimental work, the authors succeeded in obtaining holders whose surfaces, in their qualities, satisfied the stated requirements.

Subsequently, the focusing properties of the bent crystal were tested by special experiments. To allow the radiation to pass, openings were cut in the steel holders. The dimensions and shape of the holder are shown in Fig. 5. The partitions between the openings were left in order to give the crystal the shape of the holder with the greatest possible accuracy. The total area of the openings was \(13\ \text{cm}^2\). The crystal, clamped in the holders, was installed as shown in Fig. 4.

*) \(1\) X-unit \(= 0.001\ \text{Å}\).

The source was located at point \(R\). Some of the rays penetrating the crystal were selectively reflected from the \((310)\) planes and were registered by a Geiger counter \((B)\). The remaining rays passed through without changing direction.

Since the Bragg angle for hard radiation is very small (\(34'\) for \(510\ \text{keV}\); \(4'\) for \(3\ \text{MeV}\)), the angle between the directions of the transmitted and reflected beams is also very small: \(\alpha = 2\varphi\). Therefore, under these conditions the two beams overlap, and, because the direct beam has an incomparably greater intensity, measurements in the hard-radiation region become impossible.

To separate the transmitted and reflected beams, one could increase the distance between the crystal and the counter. In that case, however, in order to avoid a loss of intensity it would be necessary to increase the dimensions of the counter, which would increase its background and, possibly, complicate its manufacture. Therefore, to separate the beams, a lead or tungsten collimator was placed between the crystal and the recorder; it absorbed the direct beam and transmitted the reflected beam. The design of the collimator is based on the following principle: the extensions of the rays reflected by the crystal intersect at point \(V\)—the virtual focus of the instrument (as if they originated from it). The collimator must be arranged so as to select precisely these directions. Along the rays emerging from point \(V\), long lead strips were placed, fastened in special grooves, so that a row of openings remained for the passage of the reflected \(\gamma\)-rays. The direct beam, however, was absorbed in these lead strips. The length of the collimator was \(80\ \text{cm}\). The number of lead strips in the first version of the instrument was six (see Fig. 4). It is obvious that such a system entails a loss of intensity of the reflected beam owing to its absorption in lead. In practice, in the first version this loss amounted to more than \(50\%\). However, the presence of such a collimator shifts the limit of applicability of the instrument toward higher \(\gamma\)-ray energies. If, in the absence of a collimator, the geometry of the instrument permits separation of the direct and reflected beams only for \(\gamma\)-ray energies not exceeding \(450\ \text{keV}\), then with the collimator this limit reaches \(1.5\ \text{MeV}\). For the study of still harder rays, further improvement of the collimator is necessary.

The quanta transmitted by the collimator are registered by a Geiger counter with enhanced sensitivity to \(\gamma\)-rays. In order for a quantum to be registered by the counter, it must produce in the substance of the counter wall a Compton or photoelectron, or form in it an electron–positron pair. An ordinary \(\gamma\)-counter consists of a metal cylinder along the axis of which a thin filament is stretched. A potential difference is created between the body of the counter and the filament. A \(\gamma\)-quantum, passing through ...

walls of the counter, creates electrons which ionize the gas located inside the counter; in this process an electrical pulse arises. The efficiency of this kind of γ-counter (the fraction of registered γ-quanta) is small.

Diamond constructed a new type of γ-counter with increased efficiency3. In longitudinal section the scheme of Diamond’s counter is shown in Fig. 6. Inside a copper tube of diameter 8 cm (A), lead disks (B) several tenths of a millimeter thick, serving as one of the counter electrodes, are arranged at equal distances from one another. In the center the disks have round holes (V), through which passes a steel rod (G). Sleeves (D) are mounted on this rod, each of which is located exactly between the disks. Thin tungsten whiskers with glass balls at the end (E) are soldered onto the sleeves. This whole system of the rod with sleeves serves as the second electrode of the counter.

Fig. 6. Counter.

The lead disks were coated with a thin layer of silver in order to retain the α-particles from radioactive impurities in the lead. A counter of this type consists, as it were, of several cells. The number of such cells was changed in the course of operation. The counter was filled with a self-quenching mixture. The plateau of a counter consisting of four cells was approximately 200 volts. Such a good plateau is necessary when measuring with a single counter.

The multicell counter was positioned so that the γ-rays emerging from the collimator were directed along its axis, passing through the lead disks. As the number of such disks is increased, the probability of formation of electrons by a quantum increases, and with it the efficiency of the counter. However, as the dimensions of the counter increase, its working region—the plateau—decreases and the background increases.

The natural background of the counter is due to radiation from radioactive contaminations of the counter material and the objects surrounding it, as well as to the influence of cosmic radiation. To reduce the background, the γ-counter was surrounded by six other counters connected with it in an anticoincidence circuit, owing to which the background was reduced by a factor of two. In addition, the entire group of counters was surrounded by a thick layer of lead.

b) The instrument in operation1

To measure spectra, the source located at the focus of the instrument must move along the focusing circle. If, in doing so, the rigid system—the crystal, the point β—remained immobile,

and the focusing circle (see Fig. 4), then the point \(V\) would also have to move together with the focus of the instrument, and consequently the collimator and the counter as well. But the lead collimator and the counter surrounded by lead are a very heavy part of the instrument. It is therefore desirable to leave them stationary during measurements. From this follow the following requirements for the construction of the instrument.

The counter and the collimator must be stationary, and consequently the virtual focus of the instrument must also be stationary, since it determines the direction of the reflected rays and therefore the position of the collimator. The following may move: the source along the focusing circle, the crystal, and the focusing circle itself; moreover, the point \(\beta\), the crystal, and the focusing circle must be rigidly connected. These conditions are fulfilled if, when the rigid system is rotated about the point \(C\)—the center of the crystal (Figs. 4 and 7)—the distances \(V\beta\) and \(R\beta\) remain equal at all times, while the point \(V\) does not move. For this it is sufficient that, when the line \(C\beta\), together with the focal circle and the crystal, is rotated about the point \(C\) with angular velocity \(\omega\), the line \(CR'\) rotate with twice the velocity \((2\omega)\), and that the source \(R\) move along \(CR'\), remaining at all times on the focal circle. These displacements are carried out by means of the mechanism shown in Fig. 7.

Fig. 7. Illustration for section b); \(K\) — collimator, \(S\) — counter.
(The remaining designations are in the text.)

The slides \(T\) are fastened to an axis at the point \(V'\), located on the continuation of the line \(CV\), and can rotate about this point. Along the slides \(T\) run other slides \(Q\), which at the point \(B\) are pierced through by the continuation of the rod \(C\beta\), which performs longitudinal motions transverse to the slides \(Q\). The rod \(C\beta\) is immovably connected with the axis of the crystal \(C\) and rotates it. Along the slides \(Q\) there also runs a bar \(L\), with which the rod is hingedly connected

$CR'$, and along the latter the source $R$ can move, remaining all the time on the focal circle owing to the radius-rod $OR$, fixed at the point $O$ on the axis and hinged at $R$. The motion of the carriages $Q$ and $L$ is effected by means of two specially made, very precise longitudinal screws, firmly fixed on the carriages $Q$ and connected to one another by means of two equal gears, so that when, for example, the lower screw is rotated at a definite speed in one direction, the upper screw rotates at the same speed in the opposite direction. The lower screw runs in a threaded coupling which is rigidly connected with the lower carriages $T$ at the point $V'$, whereby, when the screw rotates, the carriages $Q$ together with the screw move relative to the carriages $T$. Along the upper screw there runs another coupling, fastened to the bar $L$. In view of the fact that the screws rotate at equal speed in opposite directions, both bars $Q$ and $L$ move in one direction, while the bar $L$ moves relative to the carriages $T$ twice as fast as $Q$ (to the speed of the bar $L$ is added the speed of the carriages $Q$, along which the bar $L$ moves).

Thus, when the screw moves, the carriages $Q$ move, rotating the rod $CB$, which is rigidly connected with the axis of the crystal, as a result of which the crystal also turns. At the same time the line $CR'$, connected with the carriages $L$, turns twice as fast as the line $CB$; the point $V$ remains stationary.

It should be noted that the path traversed by the carriages $L$ is proportional to the sine of the reflection angle, i.e. is proportional, by the condition $n\lambda = 2d \sin \varphi$, to the wavelength. The pitch of the screw is chosen so that one complete revolution of the screw corresponds to a change in wavelength of 1 X-unit. By the vernier it was readily possible to take readings with an accuracy of up to 0.001 X-unit (up to 0.000001 Å).

In Fig. 7 the instrument is shown in three positions. In position I the instrument is set for measuring the intensity of radiation of a definite wavelength $\lambda$, which is reflected at the angle $\varphi$. In position III the instrument measures the intensity of radiation at the same Bragg angle, but upon reflection from the opposite side of the reflecting planes. In what follows we shall distinguish the positions of the instrument I and III, calling the reflection in case I “reflection from the left,” and in case III “reflection from the right.” In case II the zero position of the instrument is given, when $\varphi = 0$, $\lambda = 0$; the points $\beta$, $V$ and $R$ then coincide, and the direct beam of rays is passed through the collimator and recorded by the counter.

In Fig. 8 a general view of the instrument is given. Here the upper rod corresponds to the line $CR'$. It rotates freely about the axis on which the crystal is situated.

The lower rod corresponds to the line $C\beta$ and is firmly connected with the axis of the crystal. The long collimator, located ...

placed between the counter and the crystal. Between the bars one can see the axis \(OR\), which holds the source on the focusing circle. The point \(O\) can be shifted somewhat on the lower bar by means of screws. This is necessary in the case when, for example, the reflecting planes of the crystal are not strictly perpendicular to its bounding planes and the point \(\beta\) is displaced. The source of \(\gamma\)-radiation, placed in a spherical lead holder, was secured at the focus of the instrument.

Fig. 8. General view of the instrument.

IV. EXPERIMENTAL RESULTS

1. Investigation and calibration of the instrument

The lower limit of the wavelengths accessible for study with the aid of the instrument is determined by that angle between the transmitted and reflected beams at which the collimator is still able to separate them. The wavelength of the softest radiation (the upper limit) that can be investigated with the aid of the instrument is approximately 500 X-units; thus, in its first version, the working range of the instrument extended from 7 to 500 X-units in wavelengths, from 20 kev to 1700 kev in energies, and from \(10'\) to \(13^\circ\) in angles \((\varphi)\).

Such a broad working region of the instrument made it possible, for the investigation of the focusing properties of the crystal and for the calibration of the instrument, to use well-studied X-radiation.

The accuracy of the focusing of the radiation by the crystal was investigated in the following manner. An X-ray tube with a tungsten anticathode was placed in position \(A\) (Fig. 3) with its convex side toward the crystal. The recording device was a photographic plate located on the focal circle. The dimensions of the focal spot of the tube were considerably smaller than the dimensions of the working surface of the crystal. Therefore, for a definite position of the anticathode of the tube, the rays of the tungsten \(K\)-series were focused by only a small part of the crystal. When the X-ray tube was moved to another position and

the neighboring region of the crystal was operating, the tungsten \(K\)-lines became somewhat blended. Thus the focusing of the tungsten \(K\)-spectrum was carried out with the aid of ten different regions of the crystal. The results of this investigation are presented in the diagram of Fig. 9.

The straight lines, denoted by numbers, connect the points of focusing of the line on the photographic plate with that region of the crystal from which these points on the plate were obtained. Along the horizontal axis are plotted distances along the focusing circle. In the direction perpendicular to the rays the scale is greatly enlarged. In the direction parallel to the rays the scale is natural. It is seen from the diagram that the smallest distance between the rays of the diagram is \(0.06\) mm. Consequently, all regions of the crystal focus rays of the given wavelength on a section of the focal circle not exceeding \(0.06\) mm. The corresponding blurring, expressed in wavelengths, is \(\Delta\lambda = 0.07\) X-units. This corresponds to turning the precision screw by approximately \(0.07\) of a full revolution. Such blurring of the focus occurs mainly because of nonuniform bending of the crystal (a variable radius of curvature), and also, perhaps, because of the mosaic structure of the bent crystal itself.

Labels in the figure: “focus width \(0.06\) mm”; “\(1\) mm”.

Fig. 9. Investigation of focusing sharpness.

As a result of the blurring of the focus, the lines investigated with the aid of the instrument are broadened. If the relative broadening of the lines \((\Delta\lambda/\lambda)\) due to inaccurate focusing of the rays by the crystal in the wavelength region \(\sim 200\) X-units (the tungsten \(K\)-series) is

\[ \frac{\Delta\lambda}{\lambda}=\frac{0.07}{200}=0.035\%, \]

then as the wavelength decreases this value increases. Thus, for annihilation \(\gamma\)-radiation,

\[ \left(\frac{\Delta\lambda}{\lambda}\right)_{\text{ann.}}=\frac{0.07}{24}\simeq 0.3\%, \]

and for lines with energy \(\sim 1\) MeV,

\[ \frac{\Delta\lambda}{\lambda}=0.6\%. \]

However, the measurement of wavelengths, as the authors believe, can be carried out more definitively, since the position of the center of the curve can be determined with an accuracy of up to 0.01 of its width.

For measurements in the region of energies exceeding 1 MeV, it was desirable to reduce as much as possible the blurring of the focus due to inaccurate focusing of the rays by the crystal. For this it was necessary to improve the crystal holders, which determine its shape and are responsible, in the main, for the sharpness of the focusing.

The quality of the focusing of the crystal was also checked by means of the silver \(K\)-series\(^2\).

X-rays from a tube with a tungsten anticathode, shielded by a layer of lead in order to exclude the direct incidence of the X-radiation of the tube on the photographic plate, fell on a silver screen. The fluorescing silver radiator served as the source. A photographic plate was placed in the focus of the instrument. In this case, in view of the large dimensions of the radiator, the entire surface of the crystal was operating. The photograph obtained of the silver \(K\)-series shows that the width of the lines does not exceed their natural width, obtained earlier by other authors. This conclusion is further confirmed by the following: when part of the working surface of the crystal was covered with lead, the lines did not become narrower. Work with the silver \(K\)-series is also of interest because it was carried out in the region of rays that are extremely soft for this instrument.

Before beginning the measurements of the \(\gamma\)-spectra, it was necessary to calibrate the instrument. The large range of wavelengths covered by the instrument made it possible to compare the wavelengths of X-rays and \(\gamma\)-rays. For X-ray characteristic radiation, the wavelengths can be determined with great accuracy. Therefore it was natural to calibrate the instrument by some X-ray line. In the present case the rays of the tungsten \(K\)-series were used. These X-rays were chosen because of the availability and convenience of an X-ray tube with a tungsten anticathode, and also because their energy is sufficiently large. But the value of the energies of the lines of the tungsten \(K\)-series had not yet been known with sufficient accuracy. Therefore, in the same laboratory in which the \(\gamma\)-spectrometer had been constructed, special work\(^{4,14}\) was carried out to determine the exact value of the energies of the tungsten \(K\alpha_1\)-line (corresponding to the transition \(L_{III} \longrightarrow K\)) by means of a double-crystal spectrometer, the scheme of which is given in Fig. 10.

On the rotating tables (1 and 2) two plane quartz crystals are placed, the planes \((310)\), which in the present case were the reflecting planes, being perpendicular to the surfaces of the crystals. The support on which both tables stand can rotate about the axis of table 1. Rays from the X-ray tube, located

in point \(A\), pass through slit \(B\), and penetrate crystal 1. This crystal is set with respect to the beam at such an angle that the \((310)\) planes reflect rays of the wavelength of interest to us. The rays reflected by crystal 1 are selected by the second slit \(V\) and fall on crystal 2. Reflection (in the first order) from crystal 2 occurs only in two of its positions—parallel and antiparallel (shown by the dotted line), which corresponds to reflection “from the left” and “from the right” from the planes of crystal 2.

The counter \(G\), which could be rotated about the axis of crystal 2, registered the reflected x-ray quanta at various positions of the crystals.

Fig. 10. Double-crystal spectrometer.

Crystals 1 and 2 could be rotated through \(180^\circ\). The angle between the direction of the beam and crystal 2 could be measured with very high accuracy.

To determine the exact value of \(\lambda_{K_{\alpha_1}}\) of tungsten, x-ray tubes with tungsten and molybdenum anticathodes were placed in turn in front of slit \(B\) of the double-crystal spectrometer. The Bragg angles for the \(K_{\alpha_1}\) lines of tungsten and \(K_{\alpha_1}\) of molybdenum were measured with the greatest possible accuracy and compared with one another. From the very well known wavelength of molybdenum \(K_{\alpha_1}\) \((\lambda_{\mathrm{Mo}K_{\alpha_1}} = 707.831\ \text{X-units})\) and by determining the ratio of the indicated angles, they found the wavelength \(\lambda_{K_{\alpha_1}}\) of tungsten, which they obtained as equal to

\[ \lambda_{\mathrm{W}K_{\alpha_1}} = \frac{\sin \varphi_{\mathrm{W}}}{\sin \varphi_{\mathrm{Mo}}} \lambda_{\mathrm{Mo}} = 208.575 \pm 0.008\ \text{X-units}. \]

Then the x-ray tube with the tungsten anticathode was placed in the focus of the calibrated spectrometer, and a narrow slit selected a beam of x-rays \(^{4}\).

The rays selectively reflected from the crystal were recorded by a counter with four cells.

For the \(K_{\alpha_1}\) line the value obtained on the \(\gamma\)-spectrometer was
\[ \lambda_{K_{\alpha_1}} = 208.623 \pm 0.008 \ \text{X-units}, \]
i.e. somewhat larger than on the double-crystal spectrometer. The ratio
\[ \alpha=\frac{208.623}{208.575}=1.00023 \]
is introduced as the calibration factor of the instrument.

Fig. 11. Tungsten \(K_{\alpha}\)-doublet.

The true value of \(\lambda\) was found by dividing the wavelength value obtained on the instrument by \(\alpha\).

Subsequently, not only the wavelength of the line \(WK_{\alpha_1}\) was measured, but also that of the second line of the \(K_{\alpha}\)-doublet \((K_{\alpha_2})\).

Fig. 12. Tungsten \(K_{\beta}\)-, \(K_{\gamma}\)-, \(K_{\delta}\)-lines.

In Figs. 11 and 12, where the abscissa gives the wavelength values in X-units, and the ordinate gives the number of pulses in the counter in 5 and 10 min, it is seen that the \(K_{\alpha}\)-doublet is completely resolved. The \(K_{\alpha_2}\) line corresponds

transition \(L_{11}\longrightarrow K\). The distance between the \(K_{\alpha_1}\) and \(K_{\alpha_2}\) lines (in wavelengths) was found to be \(4.812 \pm 0.007\) X-units. This value agrees with the value calculated theoretically from the difference of the energies of the \(L_2\) and \(L_3\) levels. The \(K_\beta\) doublet is also completely resolved (\(K_{\beta_1}\) corresponds to the transition \(M_{111}\longrightarrow K\), \(K_{\beta_2}\) to the transition \(M_{11}\longrightarrow K\)).

The separation is \(0.805 \pm 0.001\) X-units. Fig. 12 gives the incompletely resolved \(K_\gamma\) doublet (\(K_{\gamma_1}\)-line corresponds to the transition \(N_{111}\longrightarrow K\), \(K_{\gamma_2}\) to the transition \(N_{11}\longrightarrow K\)) and the unresolved \(K_\delta\) doublet (corresponds to the transitions \(O_{11}, O_{111}\longrightarrow K\)).

Below is a table containing the data obtained

Table I

Wavelengths of the tungsten \(K\)-spectrum (in X-units)

Author	\(\alpha_2\)	\(\alpha_1\)	\(\beta_2\)	\(\beta_1\)	\(\gamma_2\)	\(\gamma_1\)	\(\delta\)
Diamond . .	213.387 \(\pm 0.010\)	208.575*) \(\pm 0.008\)	184.772 \(\pm 0.020\)	183.967 \(\pm 0.020\)	179.212 \(\pm 0.020\)	179.038 \(\pm 0.020\)	178.052 \(\pm 0.020\)
Ingelstam	213.382	208.571	184.795	183.991	179.232	179.049	178.073
Gudjon . .	213.38	208.57	184.73	183.93	179.13	178.95	177.99
Siegbahn . .	213.52	208.85	184.36	184.36	179.40	179.40	—

*) Obtained on a double-crystal spectrometer.

in the study of the tungsten \(K\)-spectrum. For comparison, it also gives the data of other experimenters.

2. The 411-keV line of \(\mathrm{Au}^{198}\)

The first \(\gamma\)-line studied on this instrument was the \(\gamma\)-line of the radioactive isotope of gold \(\mathrm{Au}^{198}\) (411 keV)\(^{5}\).

The source was a gold plate of dimensions \(30\times5\times0.1\) mm, which was placed at the focus of the instrument and was turned with its end face (the side of 0.1 mm) toward the crystal. This plate had previously been irradiated with a powerful neutron flux in the Oak Ridge pile. As a result of the nuclear reaction \(\mathrm{Au}^{197}(n\gamma)\mathrm{Au}^{198}\), the radioactive isotope of gold \(\mathrm{Au}^{198}\) was obtained, with a half-life of 2.7 days. By the beginning of the measurements the activity of the source was equal to 1 curie. Such a high activity is necessary in these measurements because the reflection coefficient from quartz planes for 411-keV radiation is very small (0.3%).

To record the reflected rays, a counter with four cells was used, whose efficiency with respect to \(\gamma\)-quan-

there, with energy of the order of 0.5 MeV, amounted to 8%. The background of the counter in the absence of the source was 32 pulses per minute. Such a comparatively low background was obtained as a result of the satisfactory design of the counter itself, as well as its careful shielding.

The results of measurements with Au\({}^{198}\) are given in Fig. 13. Here the abscissa gives the wavelengths in X-units, and the ordinate gives the number of pulses in 5 minutes.

Fig. 13. 411-keV line of Au\({}^{198}\).

The measurements were carried out with reflection both “from the right” and “from the left” from the reflecting planes of the crystal. Altogether 4 series of measurements were performed, two of which are shown in the figure. From the curves it is evident that the abscissa values for the peak of the line obtained with reflection “from the left” and “from the right” are not identical. This indicates that point \(\beta\) of the instrument (see Fig. 4) is somewhat displaced. In this and in all subsequent cases the half-sum of the readings of the measurements “from the left” and “from the right” is taken. In all 4 series of measurements the magnitude of the peak exceeded the background by approximately a factor of 4. The background here is produced chiefly by the scattering, by the lead plates of the collimator, of rays from the transmitted beam. This background, evidently, should increase as the Bragg angle decreases. The latter, for rays with energy 411 keV, is \(20'\). It should be emphasized that the width of the curves for the 411-keV line of Au\({}^{198}\),

obtained with this instrument is sufficiently large (of the order of 0.125 X-units, or 0.4%). The natural width of the line is undoubtedly much smaller. The line is broadened by the geometrical factors of the instrument. These include, above all, the width (extension along the focal circle) of the source. The optimal source width for this instrument is the width of its focus, equal to 0.06 mm. A greater source width substantially broadens the line. However, creating a small but strong source is a very difficult task.

The second cause of line broadening is inaccurate focusing of the rays by the crystal. In the study of lines such as the 411 kev line of Au\(^{198}\), it is very convenient to analyze these geometrical factors, since the natural width of the line in this case may be taken to be zero.

If there were a broad source and the instrument had exact focusing, the experimentally obtained line would have the form of a rectangle whose width is equal to the width of the source. In the real case—the case of inaccurate focusing of the radiation by the crystal—the line is broadened still more. By analyzing the experimental shape of the line, one can determine the influence of inaccurate focusing and construct the corresponding curve. From the results of measurements of the 411 kev line of Au\(^{198}\), the authors construct the curve of the influence of inaccurate focusing and use it in studying the shape of an annihilation-radiation line that has a natural width.

The exact value of the energy of the \(\gamma\)-line, obtained by averaging over all four series, proves to be

\[ E = 0.41118 \pm 0.00005 \ \text{Mev}. \]

Here the energy value is given with an accuracy of up to 0.01%.

This value is in agreement with the value \(0.408 \pm 0.004\) Mev for the Au\(^{198}\) line, obtained recently with the aid of a magnetic spectrometer by Levy and Growling. Here, however, the error in determining the energy is 1%.

3. Annihilation \(\gamma\)-radiation\(^{6}\)

For the calibration of magnetic spectrometers it is very important to have exact values of the energies of a number of \(\gamma\)-lines over a considerable range of energies. In the energy region of the order of 0.5 Mev, the radiation most often used for calibration purposes is annihilation radiation. Positrons emitted by a \(\beta^{+}\)-radioactive substance annihilate with the electrons of the substance, forming, in the overwhelming majority of cases, two \(\gamma\)-quanta. If the center of gravity of the electron–positron system does not possess noticeable kinetic energy at the moment of annihilation, then the law of conservation of energy and momentum will be satisfied only if, first, the quanta of annihilation radiation that are formed have equal ener-

gies exactly equal to \(m_0c^2\), where \(m_0\) is the rest mass of the electron, \(c\) is the speed of light; and, secondly, if the quanta fly apart at an angle of \(180^\circ\) relative to one another. If, however, the center of mass of the \(e^+ - e^-\) system possesses some kinetic energy, then the energy of the quanta will no longer be equal to \(m_0c^2\), and the angle of divergence between them will differ from \(180^\circ\).

If, before annihilating, the positrons completely lose their kinetic energy, which apparently does occur, then nevertheless the kinetic energy of the center of mass may be nonzero because of the velocities of the electrons.

In view of the fact that the quanta of annihilation radiation are emitted by a system moving with a definite velocity, there is a Doppler effect—an alteration of the wavelength of the radiation and, consequently, of its energy—observed by the experimenter in the laboratory coordinate system. For a given velocity of the center of mass of the \(e^+ - e^-\) system, this change in wavelength depends on the angle between the direction of the resulting \(\gamma\)-quantum and the direction of motion of this system (angle \(\alpha\)).

Since there is a probability that a \(\gamma\)-quantum will be emitted at any angle \(\alpha\), different changes in wavelength \(\Delta\lambda\) are observed. The annihilation line is broadened. The Doppler broadening of the line is the greater, the greater the velocity of the electrons with which the positrons annihilate.

In matter there exist several types of electrons possessing different velocities: conduction electrons (in a metal), and the electrons of the \(K\), \(L\), \(M\), etc. levels of the atom.

If positrons were capable of annihilating with all these electrons, the annihilation line would acquire a many-tiered structure—a number of broad wings (corresponding to the different types of electrons) would adjoin the narrow central part of the line. The probability of penetration of a positron into the inner shells of the atom is small, owing to its repulsion by the positively charged nucleus. Nevertheless, one should still expect that the annihilation line will not be very narrow even as a result of positron annihilation with conduction electrons (in a metal). Analysis of the shape of the annihilation curve makes it possible to answer the interesting question: with which electrons do the positrons annihilate? As the source of annihilation radiation, the radioactive isotope of copper \(\mathrm{Cu}^{64}\), obtained by bombarding ordinary copper with neutrons in a pile, was used.

Only \(20\%\) of the decays of \(\mathrm{Cu}^{64}\) occur with the emission of positrons. Only these decays lead to the formation of annihilation \(\gamma\)-rays. In view of this circumstance, and also because the reflection coefficient of the \((310)\) planes of quartz for radiation with energy \(510\ \text{keV}\) is very small \((0.2\%)\), a source of high activity is required for the investigation.

In the present work the activity of the source, which was a plate of radioactive copper measuring \(30 \times 10 \times 1\) mm, was 2.5 curies. The arrangement of the source in the instrument is shown in Fig. 14 (top view). In order to limit the extent of the source along the focal circle, a narrow slit (0.1 mm), located exactly on the focusing circle, was placed in front of it. The beam of \(\gamma\)-rays was selected by a lead collimator.

With such a wide source with a slit, a gain in intensity by a factor of 3 is obtained in comparison with the case where a source 0.1 mm thick is placed, but without a selecting slit. This increase in intensity can be explained as follows. The maximum energy of the positrons emitted by copper (\(\mathrm{Cu}^{64}\)) is approximately 0.66 MeV; the corresponding maximum range is about 0.3 mm. The positrons annihilate at the end of their range. Therefore, in the central region of the plate, positrons annihilate that come from all points of the plate separated from the center by the distance of the maximum range.

Fig. 14. Installation of the source (\(\mathrm{Cu}^{64}\)). (Scale not maintained.)
Labels in the figure: “Slit 0.1 mm,” “Source,” “Lead.”

The presence of such a wide source undoubtedly broadens the line and lowers the quality of the analysis of the curve shape. However, the accuracy of determining the peak energy does not suffer appreciably.

In the work a counter of the construction described above was used, consisting of nine cells (instead of 4). Its efficiency with respect to annihilation \(\gamma\)-rays was 30%; the counter plateau was 100 volts. To reduce the background introduced by cosmic rays, the counter was surrounded by several \(\gamma\)-counters connected with it in an anticoincidence circuit.

The results of the measurements are presented in Fig. 15, where the abscissa gives wavelengths in X-units, and the ordinate gives the number of pulses during 5 min. The maximum number of pulses counted in 5 min is about 3500. Three of the seven series of measurements made are shown here. Since the half-life of \(\mathrm{Cu}^{64}\) is 12.8 hours, the experimental curve had to be recalculated, taking into account the weakening of the source. The figure gives the curves already recalculated. The plus or minus sign before the abscissa indicates whether reflection occurs “to the right” or “to the left” of the crystal planes.

From the data of each series of measurements, the wavelength corresponding to the center of the annihilation curve was then determined. This determination was carried out in several ways. The first of them consisted in drawing, on both sides of the curve,

tangents to it in its steepest part up to their intersection, and the abscissa of the point of intersection was found.

By the second method, at half the height of the curve a horizontal line was drawn connecting its two branches, and at its center a perpendicular was erected. The point of intersection of the perpendicular

Fig. 15. 510-keV annihilation line.

with the abscissa axis gave the desired wavelength. Both of these methods of processing the results were applied both with preliminary subtraction of the background and without subtraction of the background.

All these data were then averaged. As a result, for the wavelength of the annihilation line the value \(\lambda = 0.024271 \pm 0.000010\ \text{Å}\) was obtained. The probable error, taken as the root-mean-square deviation from the mean, is \(0.01\%\). In the error quoted here it has been multiplied by 4, in order to allow for possible systematic errors of the instrument.

The value \(0.00001\ \text{Å}\) corresponds to a displacement of the carriage \(L\) (see Fig. 7) by \(0.01\ \text{mm}\). The value for the Compton wavelength, obtained by combining the well-known universal constants \(h\), \(m_0\), and \(c\),

\[ \lambda_c=\frac{h}{m_0c}=0.0242650 \pm 0.0000025\ \text{Å} \]

agrees well with this experimentally obtained value.

For the energy of the annihilation-radiation line the authors consider the most reliable value to be

\[ E=m_0c^2=0.51079\pm0.00006\ \text{MeV}. \]

Let us now dwell in greater detail on the second part of the analysis of the results of the work on annihilation radiation. In order to analyze the shape of the annihilation curve, the data of all series of measurements were summed into a single curve with allowance for decay. This curve—the sum of the experimental curves—is shown in Fig. 16.

Along the abscissa axis in the drawing are plotted, in X-units, the deviations from the center of the curve; along the ordinate axis, the intensity in arbitrary units.

Fig. 16. Sum annihilation curve.

If the natural width of the \(\gamma\)-line were equal to zero, the experimental curve would nevertheless have a certain width, determined by the geometry of the instrument. In the present case the broadening factors are the following: the finite width of the slit (\(0.1\) mm), penetration of the radiation through the edge of the lead slit (the source has a width of \(1\) mm, i.e. 10 times wider than the slit), and inaccurate focusing of rays by the curved crystal.

Fig. 17. Geometrical curve.

Fig. 18. Source curve.

Taking all these factors into account, for the given instrument one can theoretically construct a curve depicting the form of the line in the case where its natural width were equal to zero.

Such a curve, constructed by the authors, is given in Fig. 17 (the geometrical curve). In Fig. 18 a curve is given which takes into account the influence of the extension of the source over the focal circle.

The experimental curve (Fig. 16) is evidently obtained as a result of the superposition of the geometrical and the natural spectral curves. Comparing Figs. 16 and 17, we indeed find that the half-width (the width of the line at half its height) of the experimental curve is greater than the half-width of the geometrical curve. However, this excess is very small; therefore the natural and geometrical widths are comparable in magnitude, and here one cannot demand great accuracy in determining the form of the true annihilation curve.

In Figs. (16) and (17) the Gaussian distribution curves of the form

\[ A_i(2\pi)^{-1/2}\sigma_i^{-1} e_i^{-\frac{x^2}{2\sigma_i^2}}, \tag{2} \]

are plotted by points, where \(\sigma_i\) is the half-width of the curve, \(A_i\) is a constant coefficient, and \(x\) is the coordinate, with the parameters \(\sigma_i\) and \(A_i\) chosen so that the Gaussian curves coincide as well as possible with the experimental and geometrical curves. The best such values are: for the experimental curve \(\sigma_{\mathrm{e}} = 0.15\) X-units and \(\sigma_{\mathrm{g}} = 0.115\) X-units—for the geometrical one. Let us suppose that the true annihilation curve is also a curve of the form (2) with \(\sigma_i = \sigma_{\mathrm{true}}\); then its half-width is determined from the relation

\[ \sigma_{\mathrm{true}}^2 = \sigma_{\mathrm{e}}^2 - \sigma_{\mathrm{g}}^2 \]

(\(\sigma_{\mathrm{true}}\) is the half-width of the natural annihilation curve). Substituting their values for \(\sigma_{\mathrm{e}}\) and \(\sigma_{\mathrm{g}}\), we have:

\[ \sigma_{\mathrm{true}} = 0.096\ \text{X-units}. \]

Such a broadening of the curve, under the condition that the positron is at rest at the time of annihilation, implies a kinetic energy of the electron equal to \(16\) eV. Only the conduction electrons of copper can possess such energies. Hence it follows that the principal mass of positrons apparently annihilates with conduction electrons.

The velocities of shell electrons are considerably higher. Thus, if for an energy of \(16\) keV \(\beta = \dfrac{v}{c} = 0.004\), then for the \(K\), \(L\), \(M\) shell electrons of the copper atom we have respectively (approximately) \(\beta_K = 0.20\), \(\beta_L = 0.060\), \(\beta_M = 0.017\).

If some of the positrons annihilated with shell electrons, then at the base of the annihilation curve there would be obtained cor—

stantial wings, whose width would far exceed the geometrical width of the instrument and therefore they could be studied well. However, the presence of a large background makes it difficult to analyze the base of the curve.

Knowing the background that was measured in the study of Au\(^{198}\), one can estimate the background expected for the given source under investigation and compare the results with the background for annihilation radiation averaged over all series of measurements.

Such a comparison shows that the background under investigation is higher than expected. But this excess lies within the limits of error.

Fig. 19. Diagram of the arrangement of N. A. Vlasov and B. S. Dzhelepov. (Scale not observed.)

Moreover, such an estimate is ambiguous, since the background varies strongly depending on the angle of reflection.

All this leads to the conclusion that in the present work no positive results were obtained in the study of the broad wings at the base of the curve.

The phenomenon of annihilation was also investigated by the Soviet scientists N. A. Vlasov and B. S. Dzhelepov\(^{7}\).

They carried out experiments to determine the angular distribution of annihilation \(\gamma\)-quanta.

It has already been said that if, at the moment of annihilation, the center of mass of the positron–electron system possessed no kinetic energy, then the two annihilation \(\gamma\)-quanta would fly apart in exactly opposite directions, as follows from the conservation laws. If, however, some kinetic energy is present, the angle between the directions of flight of the two quanta will differ from \(180^\circ\).

In Fig. 19 the arrangement of Vlasov and Dzhelepov is shown. A source of annihilation \(\gamma\)-rays (Cu\(^{64}\)) was placed at point \(A\). Two bundles of \(\gamma\)-counters, \(B\) and \(C\), connected in a coincidence circuit, are located on both sides of the source. The bundle \(C\) can rotate about an axis passing through the source; the bundle \(B\) is fixed immovably. The number of coincidences is recorded as a function of the angle \(\varphi\) through which the bundle \(C\) is turned from the straight line source–bundle \(B\). If the angle \(\varphi\) is plotted along the abscissa axis and the number of coincidences per unit time along the ordinate axis, a curve of the form shown in Fig. 20 is obtained.

The maximum number of coincidences occurs at \(\varphi = 0\).

The resolving power of the instrument constructed by the authors is so high that it enabled them to obtain the form of the curve,

very close to its natural form. From an analysis of its width there follows the unambiguous conclusion that, in the overwhelming majority of

Fig. 20. Angular distribution of annihilation γ-quanta (experiments of Vlasov and Dzhelepov). The triangle shows the resolving power of the apparatus.

cases annihilation occurs when the energy of the center of mass of the \(e^{+}—e^{-}\) system is less than \(10\) ev.

4. Decay scheme of \(J^{131}\)

Of the works on the investigation of \(\gamma\)-radiation in the energy region below \(0.5\) Mev, carried out with the \(\gamma\)-spectrometer described, it is necessary to dwell on the investigation of the \(\gamma\)-radiation of the radioactive iodine isotope \(J^{131}\)⁸.

Before this work there existed two hypothetical decay schemes of \(J^{131}\). In Fig. 21, on the left, is shown the scheme of Metzger and Deutsch⁹; on the right—the scheme of Owen, Moak, and Cook¹⁰. Both assumptions are based on measurements of \(\gamma\)-line energies by means of a magnetic \(\gamma\)-spectrometer. The accuracy of the energy determinations by the indicated authors was about \(1\%\). According to the first scheme the \(\gamma\)-lines of 284 kev and 80 kev must form a cascade; according to the second scheme there must exist two cascades, shown in the figure. If the first scheme is correct, then the sum of the energies of the 284- and 80-kev lines must be exactly equal to the energy of the 364-kev line.

If the second scheme is correct, the equality must hold

\[ E_{80} + E_{364} = E_{284} + E_{163}. \]

With the aid of a crystalline focusing spectrometer the energies of three \(\gamma\)-lines were measured.

As a result, the following energy values were obtained:

\[ E_{364}=364.18 \pm 0.1\ \text{keV}, \]

\[ E_{80}=80.133 \pm 0.005\ \text{keV}, \]

\[ E_{284}=284.13 \pm 0.1\ \text{keV}. \]

To an accuracy of up to \(0.02\%\), the equality

\[ E_{364}=E_{284}+E_{80}, \]

is satisfied, which speaks in favor of the Metzger and Deutsch scheme.

This elegant experiment is the first test of the Ritz combination principle as applied to nuclear levels. It also testifies to the perfection of the instrument itself.

[Diagram labels, left: \(53J^{131}\); \(600\ \text{keV}\ \beta\); \(315\ \text{keV}\ \beta\), \(15\%\); \(79\%\), \(364\ \text{keV}\); \(6\%\), \(284\ \text{keV}\); \(80\ \text{keV}\); \(638\ \text{keV}\); \(54Xe^{131}\); “Metzger, Deutsch.” Diagram labels, right: \(53J^{131}\); \(600\ \text{keV}\ \beta\); \(284\ \text{keV}\); \(163\ \text{keV}\); \(80\ \text{keV}\); \(364\ \text{keV}\); \(54Xe^{131}\); “Duyn, Moon and Cook.”]

Fig. 21. Two decay schemes of \(J^{131}\).

It would have been desirable also to measure the energy of the \(163\)-keV \(\gamma\)-line. However, the authors did not manage to study it because of the short half-life of the source.

If it were possible to show that the intensities of the lines \(I_{80}\) and \(I_{284}\) are equal, this too would testify that the two lines of 80 and 284 keV form a cascade.

In order to compare the intensities of these lines, it is necessary to introduce corrections into the intensities obtained from the experiment. These corrections must take into account the different efficiency of the counter with respect to \(\gamma\)-rays of different energies, as well as the different reflection coefficient of these lines by the planes of quartz.

After introducing the indicated corrections, it turned out that the intensities of the lines at 80 keV and 264 keV are of the same order.

5. The $\gamma$-lines of $\mathrm{Co}^{60}$

All previous measurements with this instrument were carried out with $\gamma$-rays whose energy lies in the region up to $0.5$ MeV.

However, there are many radioactive isotopes emitting $\gamma$-rays whose energies exceed $1$ MeV and even $2$ MeV.

In order to carry out measurements in this harder region, it was necessary to introduce substantial changes into the original version of the instrument[^11].

There are two principal difficulties in passing to the hard region. The first consists in the fact that, when the wavelength of the $\gamma$-radiation is decreased, the Bragg angle of reflection also decreases (for the same lattice constant $d$). This leads to the angle between the directions of the reflected and transmitted beams becoming very small, as a result of which these beams overlap one another.

The first collimator constructed by the authors made it possible to measure $\gamma$-spectra down to $\lambda = 7$ X-units.

At the present time a collimator of the same type as that described at the beginning of the review has been constructed, but somewhat modified. Instead of the six lead plates forming the collimator slits, 24 plates were used in the new version. Such a collimator made it possible to distinguish angles between the transmitted and reflected beams of the order of $8'$. This angle corresponds to a Bragg angle of $4'$ and to a limiting energy value equal to $3.0$ MeV.

The second difficulty consists in the fact that, when the wavelength of the $\gamma$-radiation is decreased, the reflection coefficient of the radiation from the crystal planes decreases strongly, as the square of its wavelength, and consequently the intensity of the reflected rays falls. This difficulty would be overcome if it were possible to create a strong $\gamma$-source. But the excessively small volume of the source, determined by the dimensions of the focus, requires the preparation of sources with an extraordinarily high specific activity, the greater the higher the quantum energy. And this is far from always possible. It is therefore necessary to find other ways of increasing the intensity of the reflected rays.

One may proceed by increasing the probability of scattering of the rays in the crystal, which is achieved by increasing its dimensions and by choosing other reflecting planes.

For hard $\gamma$-rays one can increase the thickness of the crystal without substantially attenuating them. The intensity of the reflected rays also increases in proportion to the thickness.

However, the thicker the crystal, the more difficult it is to bend it without various kinds of deformation.

An increase in the intensity of the reflected rays is achieved by increasing the working surface of the crystal—the dimensions of the crystal itself and of the windows in the steel holders.

In this case, however, the blurring of the lines obtained increases, owing to geometrical aberration.

It is difficult to achieve the desired results by increasing the working surface of the crystal also because, first, it is difficult to obtain a perfect crystal of large dimensions; second, as the crystal is enlarged, the dimensions of the counter also increase, as a result of which its background increases.

In the present work, in the second version of the instrument, the working area of the crystal was increased approximately twofold by enlarging the windows in the newly manufactured steel holders of the crystal.

One may also try to find other reflecting planes of quartz, or other crystals whose reflection coefficient for hard γ-rays would be much larger. But considerable difficulties arise here as well. The first of them is that the lattice constant of the reflecting planes must be as small as possible.

From the equation \(n\lambda = 2d \sin \varphi\) it is seen that when \(d\) is increased (for a given \(\lambda\)), the Bragg angle \((\varphi)\) becomes small, so that in the case of hard rays no collimator will be able to separate the direct and the transmitted beams. In addition, the crystal, while having large dimensions, must be close to the type of perfect crystals and must readily lend itself to mechanical treatment (grinding, bending) without significant deformation.

All these requirements greatly narrow the class of crystals suitable for the instrument.

The counting intensity of the reflected rays can be increased by improving the counter.

Not all γ-quanta already reflected by the crystal are registered by the counter. It is desirable, as far as possible, to increase the efficiency of the counter with respect to γ-radiation. This can be done by selecting the optimum thickness of the lead disks for the given radiation energy. The point is that increasing the thickness of the disks beyond a certain limit does not lead to an increase in the efficiency of the counter, since absorption of the electrons produced by the quanta begins to have an effect. When the thickness of the disks is decreased, the probability that electrons will be knocked out by the quanta decreases.

The efficiency of the counter can also be increased by increasing the number of lead disks per unit length of the counter. A counter with its efficiency increased in this way was used in the present work. The design and manufacture of such counters is a very painstaking and time-consuming task. For this instrument

a counter with a sufficiently low background is required; its plateau must be broad and flat.

Changing the collimator, and increasing the working area of the crystal and the efficiency of the counter, made it possible, with the aid of the instrument described, to study lines in the energy region exceeding 1 MeV.

Fig. 22. 1.1-MeV line of Co\(^{60}\).

Two lines of the radioactive isotope cobalt Co\(^{60}\) were investigated (1.1 MeV and 1.3 MeV); the activity of the source in this case was 50 millicuries.

Several series of measurements were carried out. As an example, Fig. 22 presents the results of some measurements. Along the abscissa axis the wavelengths are plotted in X-units, and along the ordinate axis—the number of pulses per 1000 sec.

Despite the high background, obtained as a consequence of the small Bragg angle \(\varphi\), the maxima are quite clearly visible. The vertical strokes at each experimental point indicate the statistical error (the square root of the number of counted pulses). As a result of averaging over all series of measurements, the following values for the line energies were obtained:

\[ E_1 = 1331.6\ \text{keV} \pm 1.0;\qquad \lambda = (9.308 \pm 0.005)\cdot 10^{-11}\ \text{cm}; \]

\[ E_2 = 1171.5\ \text{keV} \pm 1.0;\qquad \lambda = (10.580 \pm 0.005)\cdot 10^{-11}\ \text{cm}. \]

In the same work, the reflectivity of the quartz planes (310) for radiation at 1.1 MeV and 1.3 MeV was investigated. It was found that the areas of the lines with energies 1.1 and 1.3 MeV are approximately equal, and consequently the intensities of these lines are also equal, since the reflection coefficients for them do not differ very greatly from one another. If \(I\) is the total intensity of the source radiation, then

\[ I_{1.1}=I_{1.3}=\frac{I}{2}, \]

where \(I_{1.1}\) and \(I_{1.3}\) are the intensities of the source radiation with energies 1.1 MeV and 1.3 MeV, respectively. Then the reflectivity of the crystal, for example for radiation with energy 1.1 MeV, is determined as the ratio

\[ \alpha_{1.1}=\frac{I_{1.1\,\text{reflected}}}{I_{1.1}} =2\,\frac{I_{1.1\,\text{reflected}}}{I}. \]

\(I_{1.1\,\text{reflected}}\) is known from experiment. It remains to determine the total intensity \(I\). This was done as follows: the instrument was set to the zero position so that all radiation passing through the crystal was admitted by the collimator and entered the counter. Because of the high radiation intensity it was necessary to record an absorption curve. A lead filter was placed between the counter and the source, and the number of pulses per unit time was studied as a function of the filter thickness. The intensity of the radiation that had passed through a filter of thickness \(l\) is expressed by the formula \(I=I_0 e^{-\mu l}\), where \(I_0\) is the initial radiation intensity and \(\mu\) is the absorption coefficient of the substance.

If the filter thickness \(l\) is plotted along the abscissa axis, and the logarithm of the intensity along the ordinate axis, a straight line is obtained; extrapolating it to the value at \(l=0\) gave the required intensity \(I_0\).

As a result it was found that the ratios \(\dfrac{I_{1.1\,\text{reflected}}}{I_{1.1}}\) and \(\dfrac{I_{1.3\,\text{reflected}}}{I_{1.3}}\) have the following values:

\[ \alpha_{1.1}=4.97\cdot 10^{-4}\;(0.0497\%); \]

\[ \alpha_{1.3}=4.12\cdot 10^{-4}\;(0.0412\%). \]

6. Reflection coefficients[^13]

In the preceding section it was described how the reflectivity of the quartz planes (310) for \(\gamma\)-radiation of Co\(^{60}\) was determined.

Knowledge of the dependence of the reflecting power of a crystal on the wavelength of the radiation is extremely essential. It would make it possible to determine the intensities of the lines under study on the basis of experimental data.

On the other hand, comparison of the experimentally found dependence of the reflection coefficients on the wavelength with that theoretically predicted for various types of crystal would make it possible to draw conclusions about the structure of the crystal.

Dumond and co-workers carried out a series of measurements of the reflecting power of a quartz crystal of the plane (310) over a very wide range of wavelengths.

In exactly the same way as for the radiation of Co\({}^{60}\), experiments were performed for the \(\gamma\)-radiation of Au\({}^{198}\) (411 kev) \((\lambda = 30.09\ \text{X-units})\).

The reflecting power was also determined for the \(K_{\alpha_1}\)-lines of Th \((\lambda = 132.3\ \text{X-units})\), Au \((\lambda = 179.96\ \text{X-units})\), Ta \((\lambda = 214.88\ \text{X-units})\), and Sn \((\lambda = 489.57\ \text{X-units})\).

In these cases the source (located at the focus of the instrument) consisted of thin foils of the indicated elements, fluorescing under the action of x-rays from an x-ray tube with a tungsten anticathode (located at some distance).

The results of the measurements of the reflecting power of the crystal are given in Table II in the column headed \(\Gamma_\lambda\), where

\[ \Gamma_\lambda = \frac{I_{\theta\lambda}}{\gamma_\lambda I_0}, \tag{2} \]

where \(I_{\theta\lambda}\) is the intensity (number of pulses) at the peak of the experimentally obtained line when the source is in position \(R\) (Fig. 4). \(I_0\) is the total intensity of the radiation—the source is in the zero position (\(R\) and \(V\) coincide)—and \(\gamma_\lambda\) is the coefficient showing what fraction of all incident rays has wavelength equal to \(\lambda\).

In order to determine the value \(\Gamma_\lambda\), in addition to the experimentally determined \(I_{\theta\lambda}\) and \(I_0\), it is necessary to know \(\gamma_\lambda\). In the x-ray region the relative intensities of lines in the spectra of elements had previously been studied by other experimenters. Therefore \(\gamma_\lambda\) can be determined in this region as well.

In the region of \(\gamma\)-rays, however, to determine \(\gamma_\lambda\) it is necessary to know the decay scheme of the isotope under study.

For Au\({}^{198}\) we have only one \(\gamma\)-line \((E = 411\ \text{kev})\), and \(\gamma_\lambda = 1\). Co\({}^{60}\) gives two cascade lines \((E_1 = 1172\ \text{kev}\) and \(E_2 = 1332\ \text{kev})\); for them \(\gamma_{\lambda_1} = \gamma_{\lambda_2} = \dfrac{1}{2}\).

The value of the reflecting power obtained on the basis of the experimental data by formula (3) has the drawback that it depends substantially on the conditions of the experiment,

More precisely: the value \(I_{\theta \lambda}\)—the intensity at the maximum of the experimental curve—depends on the conditions of the experiment. This dependence is due to the influence, on the shape of the experimentally obtained line, of such “instrumental” factors as:

a) the width of the source (its extent along the focal circle),
b) inaccurate focusing of the radiation by the crystal,
c) the width of the line itself.

Depending on these factors, \(I_{\theta \lambda}\), and hence also \(\Gamma_{\lambda}\), may vary, as a result of which comparisons with any theory become impossible.

Therefore the so-called integral reflection coefficient is introduced, which does not depend on the indicated conditions and which can be calculated from the experimentally obtained value of \(\Gamma_{\lambda}\), taking these conditions into account. The integral reflection coefficient is defined as follows. Let an ideally parallel beam of strictly monochromatic rays be incident on a flat quartz crystal (Fig. 23, a), and, being reflected from the internal planes (310), perpendicular to the bounding surfaces, pass through the crystal. Let the intensity of the rays reflected at an angle \(\theta\) be determined by means of some recorder, with the angle \(\theta\) being variable. Plot (Fig. 23, b) the angle \(\theta\) along the abscissa axis, and along the ordinate axis the ratio

\[ \frac{N_{\theta}}{N_{0}} \]

of the intensity of the radiation reflected at the angle \(\theta\) to the intensity of the incident rays. At \(\theta=\theta_{0}\), satisfying exactly the Wulff–Bragg equation, the ratio \(\frac{N_{\theta}}{N_{0}}\) has its maximum value. For values of \(\theta\) adjacent to \(\theta_{0}\), the magnitude of this ratio rapidly decreases; however, the curve obtained, which is called the diffraction curve, has a quite definite (for the given crystal

Fig. 23. Diffraction curve.

and a definite wavelength of the incident radiation) a finite width.

The integral reflection coefficient is defined as the area of this diffraction curve.

The authors of the work being described, by analyzing all instrumental factors affecting the shape of the line, succeeded in expressing the integral reflection coefficient through the experimentally determined quantity \(\Gamma_\lambda\), both in the region of X-rays (the natural width of the lines is not equal to zero) and in the region of \(\gamma\)-rays (zero natural width). These calculations were carried out under the assumptions that:

1) the width of the source \((a_1)\) and the width of the focus \((a_2)\) are of the same order of magnitude and

\[ a_1 = a_2 \gg a, \]

where \(a\) is the half-width of the diffraction curve, and

2) the line shape is known.

In the last column of Table II are given the values obtained from the formulas relating \(R_\theta\) to the experimentally determined quantity \(\Gamma_\lambda\).

Table II

Line	\(\lambda\) (X-units)	\(E\) (MeV)	\(\Gamma_\lambda\)	\(R_\theta\) (radians)
Sn \(K_{\alpha_1}\)	489,57	0,0253	\(0,175 \mp 0,023\)	\((3,28 \mp 0,43)\cdot 10^{-5}\)
Ta \(K_{\alpha_1}\)	214,88	0,0506	\(0,050 \mp 0,002\)	\((6,63 \mp 0,27)\cdot 10^{-6}\)
Au \(K_{\alpha}\)	179,96	0,0688	\(0,044 \mp 0,004\)	\((5,52 \mp 0,50)\cdot 10^{-6}\)
Th \(K_{\alpha_1}\)	132,3	0,0936	\(0,017 \mp 0,004\)	\((1,99 \mp 0,47)\cdot 10^{-6}\)
Au\({}^{198}\)	30,09	0,411	\(0,0030 \mp 0,0001\)	\((1,53 \mp 0,05)\cdot 10^{-7}\)
Co\({}^{60}\)	10,578	1,172	\((4,97 \mp 0,50)\cdot 10^{-4}\)	\((2,54 \mp 0,25)\cdot 10^{-8}\)
Co\({}^{60}\)	9,308	1,332	\((4,12 \mp 0,50)\cdot 10^{-4}\)	\((2,11 \mp 0,25)\cdot 10^{-8}\)

The corresponding seven points are plotted in Fig. 24, where the wavelengths in X-units are laid off along the abscissa axis, and the integral reflection coefficients in radians along the ordinate axis (both on a logarithmic scale). Within the assumed error, the tangent of the angle of inclination of the resulting semi-experimental curve is equal to two.

Consequently, for a bent quartz crystal in DuMond’s spectrometer the integrated reflection coefficient is proportional to the square of the wavelength in the wavelength range from 9 to 500 X-units (from 25 to 1332 kev).

In Fig. 24 the dashed curve (No. 1) is constructed from the theoretical formula for a mosaic quartz crystal. Dashed curve No. 2 gives the dependence of \(R_\theta\) on \(\lambda\) for a perfect crystal \((R_\theta \sim \lambda)\). The experimental curve agrees well with theoretical curve No. 1. It follows from this that the bent crystal behaves as a mosaic crystal.

At the same time, experiments carried out by the authors to determine the integrated reflection coefficients for flat quartz crystals—for the same reflecting planes (310)—by means of a double-crystal spectrometer showed that in this case the behavior of the crystal is what may be expected from a crystal close to perfect.

Fig. 24. Integrated reflection coefficient for a bent quartz crystal.

In conclusion to the section, it is necessary once more to point out the extremely sharp decrease of the reflection coefficient with decreasing wavelength, which makes it difficult to increase the resolving power of the instrument.

V. CONCLUSION

In Table III the results obtained on DuMond’s crystal \(\gamma\)-spectrometer are compared with the results of other investigators.

The table convincingly shows that DuMond’s results far surpass all other results in accuracy.

DuMond and co-workers consider that in the region of 0.5 Mev they can freely measure the energy of \(\gamma\)-rays with an accuracy of up to 0.02–0.04%, and in the region of 1 Mev up to 0.04–0.06%. For the first time in the history of \(\gamma\)-spectrometry, measurement of the energies of \(\gamma\)-lines is being performed with such accuracy. Before DuMond’s spectrometer was constructed, the study of spectra by direct reflection of electromagnetic radiation from crystals was carried out mainly in the X-ray region, where very high accuracy was achieved in deter-

line energies. In the region of γ-radiation, the hardest rays that it had previously been possible to study by reflection from a crystal (a flat crystal was used) had an energy of 770 kev. The error in the determination of the energy was very large and amounted to no less than 2%.

Table III

No.	Isotope	$E_\gamma$ kev, DuMond	$E_\gamma$ kev, Other authors
1	Au$^{198}$	$411.18 \mp 0.02$	$408 \pm 4$
2	I$^{131}$	$80.133 \mp 0.005$
3		$284.13 \mp 0.1$
4		$364.18 \pm 0.1$
5	Co$^{60}$	$1171.5 \pm 1.0$	$1155 \pm 0.5\%^{12}$
6		$1331.6 \pm 1.0$	$1317 \pm 0.5\%^{12}$
7	annihil. radiat.	$510.79 \pm 0.06$

As was said, the accuracy of determining γ-radiation energy with the aid of magnetic spectrometers is limited by the accuracy with which the magnetic-field strength is measured.

The latter is measured with an error of no less than several tenths of a percent. With an error of the same order, the energy of γ-rays is also determined with the aid of the best modern magnetic γ-spectrometers. To determine absolute values of γ-line energies, the experimenter must construct, from a number of reference points, a calibration curve for his instrument. The accuracy of the results obtained by DuMond and his collaborators already makes it possible to use them for calibrating numerous magnetic spectrometers.

Magnetic spectrometers of various types are applicable in different energy regions. Thus, for example, γ-spectrometers using pairs are applicable only in the region of energies exceeding $E = 2m_0c^2 = 1022$ kev, since a quantum is capable of forming a pair in matter only if its energy exceeds this limit. It is therefore desirable to have very accurate values of the energies of certain widely known γ-lines over a very broad range of energies.

In connection with the task of extending the measurement range with a crystal spectrometer toward higher energies, there arises

question—how far in this direction one can go. In what region will the accuracy of energy measurements fall below the accuracy obtained with the best modern magnetic spectrometers?

This limiting accuracy is ultimately set by the imperfection of the working crystal itself—the nonparallelism of the reflecting planes, the mosaic structure. DuMond and his co-workers assert that, for their instrument, which uses the (310) planes of quartz for reflection, this limiting accuracy in determining the wavelength corresponds to an error of \(\pm 0.005\) X-units. This error is approximately constant for all wavelengths. Then, for example, for \(\lambda = 5\) X-units, which corresponds to an energy of \(2.500\) MeV, this error amounts to \(0.1\%\). Only the best modern magnetic spectrometers approach such accuracy, for example, the spectrometers of Korsunskii type with an inhomogeneous magnetic field (on the spectrometer of B. S. Dzhelepov, A. A. Bashilov, A. V. Zolotavin, and N. A. Anton’eva, see the first issues of Doklady Akademii nauk SSSR for 1950), which focus the electron beam very precisely.

However, going over to the study of the very hard region of rays entails all the difficulties already discussed and, in particular, a decrease in the intensity of the reflected rays.

In addition to those mentioned above, there exist at least two more ways leading to an increase in the counting intensity of the reflected rays.

Recently luminescent crystalline counters have been finding ever wider application; their efficiency with respect to \(\gamma\)-rays is many times greater than the efficiency of DuMond’s \(\gamma\)-counters.

An increase in the reflecting power of the crystal can be achieved by incorporating heavy atoms into its lattice, for example, tungsten atoms.

The realization of these two possibilities can significantly increase the luminosity of the instrument.

REFERENCES

1. J. W. M. Dumond, Rev. Sci. Ins. 18, 626 (1947).
2. J. W. M. Dumond, D. A. Lind, E. R. Cohen, Rev. Sci. Ins. 18, 617 (1947).
3. D. A. Lind, Rev. Sci. Ins. 20, 233 (1949).
4. B. B. Watson, W. J. West, D. A. Lind, J. W. M. Dumond, Phys. Rev. 75, 505 (1949).
5. J. W. M. Dumond, D. A. Lind, B. B. Watson, Phys. Rev. 73, 1392 (1948).
6. J. W. M. Dumond, D. A. Lind, B. B. Watson, Phys. Rev. 75, 1226 (1949).
7. N. A. Vlasov, B. S. Dzhelepov, DAN 70, 207 (1950).

References

1. D. A. Lind, J. R. Brown, J. W. M. Dumond, D. Klein, D. Muller, Phys. Rev. 75, 1544 (1949).
2. F. Metzger, M. Deutsch, Phys. Rev. 74, 1640 (1948).
3. Owen, Moe, Cook, Phys. Rev. 74, 1879 (1948).
4. D. A. Lind, J. R. Brown, J. W. M. Dumond, Phys. Rev. 76, 1838 (1949).
5. Jensen, Laslett, Pratt, Phys. Rev. 75, 459 (1949).
6. D. A. Lind, W. J. West, J. W. M. Dumond, Phys. Rev. 77, 475 (1950).
7. J. W. M. Dumond, D. Marlow, Rev. Sci. Ins. 8, 112 (1937).