MICRORADIOGRAPHY AND X-RAY MICROSCOPY

D. B. Gokhberg

INTRODUCTION

The desire to increase the resolving power of the microscope and to enhance the contrast of the images obtained has become ever stronger in recent years¹.

It is evident that there are two paths to increasing the resolving power of the microscope: reducing the wavelength and increasing the aperture².

However, in practice it is not possible to increase the resolution greatly simply by reducing the wavelength of the light used. Indeed, when modern, highly perfected glass optical systems of microscopes are employed, only violet or near-ultraviolet light can be used (with wavelengths only slightly below 3600 Å). Shorter-wavelength rays are absorbed too strongly by glass for it to be possible to work with them using an ordinary microscope objective and condenser. The wavelength of the light can be reduced by using special objectives for this purpose (quartz or catoptric). They are usually designed for wavelengths down to 2500 Å, since at shorter wavelengths the absorption of radiation in air becomes very pronounced. The quality of the images produced by these objectives is still lower than the quality of images obtained with glass microscope objectives. Therefore the use of ultraviolet rays does not lead to a substantial increase in resolving power. Nevertheless, the use of ultraviolet rays in microscopy is very advantageous, since, by making use of differences in the absorptive capacity of the objects under study for ultraviolet radiation of the corresponding wavelength, it becomes possible to distinguish many details that are indistinguishable in visible light.

In recent years E. M. Brumberg³–⁶ has developed a method that makes it possible to make full use of this by artificially converting “color” contrast in ultraviolet rays into color contrast in visible light.

Another path to increasing resolution—increasing the so-called numerical aperture—is associated with the difficulty that the value of the aperture angle \(\alpha\) is determined by the refractive indices \(n\) of the immersion liquid and the glass, and this latter cannot be greatly increased—the limit lies at about \(1.79^*)\).

For a further increase in numerical aperture, one should use the mineral garnet (\(n=1.8\)) or, still better, diamond (\(n=2.4\)). However, objectives of this kind have not been manufactured up to now, despite the fact that the use of a diamond frontal lens promises an increase in resolving power by almost \(1/3\). This is connected with difficulties in selecting a suitable immersion liquid.

The circumstances set forth have compelled a transition to radiations having an even shorter wavelength.

For this purpose, the use of electrons or protons was undertaken. The wavelength of an electron or proton is determined, as is known, by the relation

\[ \lambda=\frac{h}{p}, \]

where \(h=6.6\cdot 10^{-27}\ \text{erg}\cdot\text{sec}\) is the so-called quantum of action, and \(p\) is the momentum of the electron. For electrons with energies of the order of 50 kV the wavelength amounts to small fractions of an angstrom. It would seem that, by using such short-wave radiation, one could achieve a very great resolving power. However, electron optics is still in an embryonic state of its development. For electrons there still exist no achromatic lenses, and the defects of lenses here are relatively many orders of magnitude greater than in the optical case. Nevertheless, despite the imperfections of electron lenses, by using electrons it has still been possible to achieve a resolving power of the order of several tens (30–50) and even up to ten angstroms\(^{7,8,9,10}\).

This achievement cannot be overestimated. In comparison with the resolving power of optical systems in visible light (of the order of 1500 Å) or in the ultraviolet region of the spectrum (of the order of 1200–1000 Å), this signifies an almost hundredfold increase in resolving power.

Apparently, for proton microscopes, for which the wavelength is much smaller, the resolving power can be made appreciably greater than for electron microscopes, reaching 1–2 Å, but at present they are still in the development stage.

\(^*)\) Objectives and condensers having a very large aperture require monobromonaphthalene with \(n=1.657\) as the immersion liquid and are used comparatively rarely.

However, along with its enormous achievements, electron microscopy also has certain weak points.

First, by no means all microscopic objects permit their placement in a vacuum, and without this an electron microscope of the usual type cannot be used for their study. Such are, in particular, solutions and liquids, as well as many biological objects.

Furthermore, the thickness of an object accessible to investigation with an electron microscope must be very small—on the order of 500 Å.

This leads to the fact that some biological objects, in particular microtome sections, cannot be studied under an electron microscope. Nor is it possible to study polished sections of metals.

Working in reflected light in an electron microscope is possible only with great difficulty. As a consequence, the structure of homogeneous objects, such as metals, cannot be studied, or can be studied only to an insufficient degree. It is true that the replica method can be used, but it does not always give good results. All these difficulties remain also in proton microscopy, and some of them are even intensified.

A natural desire therefore arises to construct an instrument having a resolving power if not greater than that of the electron microscope, then even smaller, but free of the indicated shortcomings, i.e., one that does not require placing the object in a vacuum and that permits the study of comparatively thick objects and, in particular, microtome sections and polished sections of metals.

For this purpose, a number of methods have been proposed, usually combined under the names of X-ray microscopy and micro-radiography (the term micro-radiography is also sometimes used[^1]–[^6]).

That it is precisely X-rays that are employed here is quite understandable. The wavelength of X-rays is sufficiently small (on the order of 1 Å) that, theoretically speaking, with their aid it would be possible to obtain a very high resolution. On the other hand, X-rays are weakly absorbed in air and are capable of penetrating thin layers of almost any substances. At the same time, the absorption characteristics of X-rays sometimes make it possible to obtain contrast in those cases where visible light or electron beams cannot give any contrast. Therefore, among the merits of X-ray microscopy one should also include the fact that with its aid it is possible to reveal details of the structure of the objects under study other than those revealed with the aid of optical or electron microscopes.

After these preliminary remarks we shall proceed to consider how microscopy in X-rays can be carried out, i.e., to a consideration of the various methods of X-ray microscopy proposed up to the present time.

D. B. GOTOBERIDZE

CONTACT MICRORADIOGRAPHY

Obtaining enlarged images in X-rays was tried long ago. As early as 1913, X-ray images of biological objects were obtained at small magnifications (17×).

Subsequently, a method of contact microradiography was developed by a number of authors11–16. In this method the object under study, in the form of a thin plate 0.05–0.15 mm thick (this may be a polished section or a slice), is transilluminated by a beam of X-rays. The image is recorded on a photographic plate placed in immediate proximity to the object (in close contact with it) (Fig. 1),

Fig. 1. Diagram for obtaining an image in contact X-ray microscopy by transmission.

and is then enlarged with the aid of an ordinary microscope, which in this case plays the role of a magnifier.

For photography, X-ray tubes with windows transparent to soft X-rays are used (beryllium windows, or those of getan glass). In general one tries to work with radiation of as long a wavelength as possible, since in this case the absorption of X-rays in the object increases and the contrast of the photograph is enhanced. And, in order to obtain significant magnifications, the image must be contrasty.

To obtain higher contrast, one may also make use of the selectivity of absorption. Let us recall that the X-ray absorption coefficient has the property, at certain values specific to each element, called absorption edges, of changing abruptly by several times. This makes it possible, when photographing an object in transmission consisting of several substances, to distinguish clearly elements that differ from one another by even one unit of atomic number, by using X-rays with a wavelength greater than the absorption edge of one of the elements under study and smaller than the absorption edge of the other. This gives the greatest contrast.

In practice, when working by the method of selective absorption, the characteristic radiation of the anode is usually used according to atomic

to a number smaller by one or two units of the atomic number of one of the substances being studied. This gives the greatest difference in the absorptions of the various constituent parts of the object.

Let us have two parts of one and the same object of equal thickness, but differing from one another in absorption coefficients \(\left(\dfrac{\tau_1}{\rho_1}\right.\) and \(\left.\dfrac{\tau_2}{\rho_2}\right)\). In this case the intensity of a beam of X-rays of some definite wavelength \(\lambda\), having passed through a layer of the first substance of thickness \(d\) and density \(\rho_1\), is determined by the expression

\[ I' = I_0 \exp\left(-\frac{\tau_1}{\rho_1}\rho_1 d\right), \]

and the intensity of the same beam of rays, having passed through a layer of the second substance of the same thickness \(d\), by the expression

\[ I'' = I_0 \exp\left(-\frac{\tau_2}{\rho_2}\rho_2 d\right). \]

Obtaining the greatest possible contrast on the radiograph corresponds to the greatest possible value of the ratio \(I' : I''\). The magnitude of this ratio is equal to

\[ I' : I'' = \exp\left[-d\left(\frac{\tau_1}{\rho_1}\rho_1 - \frac{\tau_2}{\rho_2}\rho_2\right)\right]. \]

Consequently, the condition for obtaining the most contrasty image is the choice of such radiation that the difference enclosed in brackets has the greatest possible value.

The possible magnitude of magnification (linear) depends on the type of photographic plates used and on the methods of their processing and, up to now, has not exceeded \(400\times\). Usually, however, one tries to work at smaller magnifications (\(50\)—\(60\times\)), since in this case one can use considerably more sensitive plates and reduce the exposure by tens, and sometimes hundreds, of times.

The question arises, however, as to what resolutions can be reached in this way. Obviously, in this case two different circumstances should be distinguished: first, the possibility of reproducing fine details on the photographic plate and, second, the possibility of examining and enlarging the image obtained on the photographic plate.

The first of these possibilities is completely determined by the grain size of the photographic plate used. The finer-grained the photographic plate used, the greater resolving power we obtain. At present, for this purpose, usually (at small magnifications) positive reproduction or diapositve fine-grained plates are used, and at large magnifications—special fine-grained plates. Usually, in doing this, they proceed as follows: during exposure the plate is heavily overexposed (by a factor of 30,

D. B. GOGOBERIDZE

compared with the usual exposure time), and then are greatly underdeveloped. This gives a very fine-grained and at the same time contrasty image, which cannot be obtained in any other way.

However, as a general rule, the greater the magnification one wishes to obtain, the finer-grained the plates with which one has to work, and the longer the exposures during photography become.

The resolving power of such a contact method cannot exceed the resolving power of an ordinary optical microscope. It is possible, however, that in the future techniques will be developed for the application of electron microscopy to this purpose.

This, however, by no means signifies that the method described has no future. On the contrary, this method is technically very simple, and the results obtained with its aid can in many cases serve as a very valuable supplement to ordinary microscopic and metallographic methodology. It should not be forgotten that, despite the small magnifications, this method already now makes it possible to obtain results, whereas many other methods of X-ray microscopy have not yet emerged from the stage of attempts, or even merely of ideas.

The principal advantages of this method, in comparison with the ordinary method of metallography, are the following:

1. A microradiograph makes it possible to reveal the structure of an alloy more fully, since it gives the distribution of the structural constituents throughout the entire volume of the specimen, and not only on its surface.

2. In a metallographic investigation of an alloy, the observed picture depends on the differential action of the etchant with which the polished section is treated on the surface of the specimen. If there is no difference in the action, then the structure is not revealed in the microphotograph. Meanwhile, in microradiography the contrast of the picture depends exclusively on the atomic weights of the components being studied. By using, for X-ray transmission, radiation of such a wavelength that it is selectively absorbed by one of the components of the alloy, the contrast of the image can be increased to a very high degree.

3. Finally, in microradiographs, micropores and microcracks are revealed much more distinctly than in ordinary microphotographs, which may be of undoubted practical interest in a number of cases.

In application to the study of metals by such a method, notable successes have already been achieved. It is possible to study the constituent parts of alloys, especially coarse-crystalline ones, and it is possible to distinguish from one another elements that differ by even only one atomic number. This method has found wide application in biology. With its aid one can, for example, study the deposition of one or another

of an inorganic substance in organic tissue. Thus, for example, with the aid of this method the problem of studying the distribution of iodine inclusions in the thyroid gland of patients with Graves’ disease and cretinism was solved. With its aid one can also conveniently study deposits of salts in sclerosis in the walls of vessels, etc.

In short, despite the small magnifications so far achieved by this method, its applications may be very

Fig. 2. X-ray microphotograph of steel with an admixture of lead.
Magnification 100×.

diverse. The simplicity of the technique will undoubtedly ensure for it still wider dissemination in the future.

In order to illustrate the resolving powers that can be achieved in the contact method of micro-roentgenography, we shall present some of our photographs (Figs. 2–4). Using the contact method of microphotography, we were able, in particular, to obtain very sharp contact images of fine copper meshes (Fig. 5). These images were considerably better than the optical contact photographs of the same meshes (obviously owing to diffraction in the optical case) and permitted magnification up to 600 times. Apparently, this is the greatest magnification that can be obtained by the contact method[^13].

Obtaining an X-ray and an optical contact image, or two X-ray images taken with the aid of rays

of different wavelengths, and simultaneously projecting them through two differently colored light filters by means of a special apparatus (as is done in the method of ultraviolet microscopy being developed by Brumberg for the optical case), it is possible to reveal such details of the structure that cannot be detected by other methods and that remain indistinguishable in each of the photographs taken separately.

Fig. 3. X-ray microphotograph of a cast aluminum alloy. Magnification 100 times.

Fig. 4. Contact X-ray microphotographs of mineral sections (photographs by the author).

In order to carry out such a comparison of two images—optical and X-ray—the author of the present article constructed a special instrument: a double microscope, representing a variant of E. M. Brumberg’s instrument.

Fig. 5. X-ray contact microphotograph of a copper mesh.
Magnification 150×.

METHODS OF ELECTRON MICRORADIOGRAPHY

The second method of contact microradiography, in the development of which A. K. Trapeznikov played an important role, is usually called the method of electron microradiography[^17-24]. In this method the specimen under study is brought into close contact with a fine-grain single-sided X-ray film. Then both the specimen and the film are irradiated (from the film side) with hard X-rays from tubes operating at voltages of about 200 kV and higher. To avoid the action of soft X-rays on the emulsion, it is necessary to use filters that absorb them (copper, with a thickness of not less than 2–3 mm, or lead). The hard X-rays produce a comparatively weak blackening of the emulsion, but they eject secondary electrons from the substance under study; these electrons, acting on the photographic plate, produce its blackening.

This method, by its very nature, apparently cannot yield such fine-grained radiographs as the purely X-ray method, since the electrons ejected by such hard X-rays have, although small, nevertheless not infinitesimally small, ranges in the emulsion. When viewed in a microscope at high magni-

...fine-grained films irradiated with X-rays of different hardness, we note that as the hardness of the rays is increased the graininess slowly increases. On account of this, the resolving power of the method under consideration will be slightly reduced, since one and the same fast electron may cause the phenomenon of decay in several grains of the emulsion (2–3 grains).

When it is necessary to obtain high magnifications, a Lippmann emulsion is sometimes used. The contrast of photographs obtained by this method, as some investigators believe, is considerably higher than that obtained by the ordinary method of X-ray microscopy.

Fig. 6. Scheme of the method of electron radiography.

The advantage of this method is the possibility of studying thick articles polished on only one side, and of using shorter exposures. To obtain satisfactory photographs it is necessary that the object fit tightly against the emulsion layer, since otherwise absorption and scattering of the X-ray photoelectrons in air occur.

It is also possible to work with another variant of the method of electron microradiography, namely: a specimen in the form of a thin film of constant thickness is placed between a source of X-ray photoelectrons (which is served by lead foil or a thin tungsten plate) and a photographic plate. The image on the plate is produced by electrons knocked out of the lead foil by a beam of X-rays and passing through the object being studied. To obtain an image, close contact between the lead foil, the film being studied, and the photographic plate is necessary, since electrons are strongly absorbed and scattered in air. In this variant of the method, what is in fact studied is the absorption of electrons in a very thin film, and in this form it is, properly speaking, already a special method of shadow electron microscopy at low magnification and with low resolution.

The first variant of the method under consideration seems to us more convenient. It may prove very useful as a method of microscopy at small magnifications, making it possible conveniently to investigate the structure of the surface of thick objects, or of such objects from which, for one reason or another, it is impossible to prepare a specimen sufficiently thin and transparent for use in contact microradiography.

PROJECTION X-RAY MICROSCOPY

The basic scheme of the projection method of X-ray microscopy8, 9 reduces to the fact that rays issuing from a point source pass through the object and give an enlarged shadow image of this object on a photographic plate. As is well known from ordinary optics, the resolving power (defined in this case by the width of the penumbral band) of the shadow method depends on the diameter of the radiation source. Thus the task consists in obtaining the smallest possible source of X-ray radiation. This is achieved by focusing a beam of electrons onto the surface of the anode of an X-ray tube of special design (Fig. 7). The extremely small diameter of the circle of convergence of the electrons (i.e., of the source of X-ray radiation) that can be attained at present is of the order of 50 Å. Thus, this method of X-ray microscopy can in principle provide almost the same resolution as an electron microscope. (It should be noted that in some variants of electron-microscopic methods a very similar principle is used.)

In reality, however, we encounter a number of difficulties. These difficulties are connected with the fact that the magnification in such a shadow method will be determined by the ratio of the distance from the focus of the tube to the object to the distance from the focus to the photographic plate. In practice it is difficult to make this ratio much greater than 100. With a distance from the object to the focus of 1 cm, this corresponds to a distance from the focus to the plate of 1 m. With any further increase of the distances, the intensity of the rays will fall too sharply.

Moreover, a source of very small size cannot give radiation of great intensity. It is usually assumed that an X-ray tube can withstand, in prolonged operation, a power of 200 W per 1 mm² of focal area. When the focal area is reduced, this specific (per unit of focal surface) permissible load increases sharply, but the total integral power falls sharply. Ardenne indicates as the possible current in such a tube \(10^{-9}\) A; however, it is possible that this figure is an underestimate.

Recently Goldstaub25 described an X-ray tube of this type, with a “point” focus, which he had constructed. The photographs he obtained were not of especially high quality, since the dimensions of the focus amounted—

took \(0.3\) mm. Goldschtaub gives the following approximate expression for the brightness of a “point” focus:

\[ B=\sigma\,\frac{r_0(t_1-t_0)}{r_1(r_0-r_1)}, \]

where \(B\) is the brightness of the focus (or the specific load proportional to it), \(\sigma\) is the thermal conductivity of the anticathode, \(t_0\) is the temperature of the cooling

Labels in the figure: focusing heated cathode; voltage to the lens; magnetic lens; object; short-focus electrostatic lens.

Fig. 7. X-ray tube with a “point” focus.

liquid, \(t_1\) is the temperature of the focus, \(r_0\) is the thickness of the anticathode walls from the focus to the cooling liquid, and \(r_1\) is the radius of the focal spot (which is regarded as part of a small sphere). We see from this expression that, indeed, \(B\) increases as \(r_1\) decreases, and for \(r_1=0\), \(B\) is equal to infinity.

Practically, this expression says that the power of an X-ray tube with a “point” focus will decrease not in proportion to the area of the focus, but in proportion to the reduction of its linear dimensions.

It must also be taken into account that the exposure time in photographing, as the specific power per unit area of the focal spot increases, grows not in proportion to the decrease in the integral intensity of the rays, but much more slowly, which constitutes a very favorable factor, though one difficult to allow for in advance.

Be that as it may, it is very probable that with such a method of photographing it will be possible to obtain an increase in resolution of approximately 10,000 times, and perhaps even more (100 times X-ray and 100 times photographic, by photographing on fine-grained plates and subsequent enlargement), which corresponds to a resolution of approximately up to 100 Å.

According to Ardenne’s idea, in this method a sealed glass X-ray tube of special design is used, while for focusing the electrons a magnetic lens, located outside the tube, and an electrostatic lens inside the tube are used. Because of this it is necessary to use a special electrical supply circuit, similar to that used in the electron microscope, but somewhat simpler. In the instrument there can be no parts that would have to be under vacuum, which undoubtedly greatly simplifies its construction.

On the other hand, it is possible to realize a tube of this kind also in the form of an evacuated construction, of a type somewhat similar to the electron microscope but simpler. We believe that precisely this type of X-ray microscope is the most promising for solving a number of problems.

Recently Cosslett and Nixon^26 constructed an X-ray tube of special design, in which the dimensions of the focal spot were reduced to 1 μ. With the aid of this tube these authors achieved an X-ray magnification of 95 times and attained a resolving power close to 1 μ.

X-RAY–ELECTRON MICROSCOPY

It is possible to realize a number of designs of X-ray–electron microscopes. In these instruments, under the action of a beam of X-rays, photoelectrons will be emitted from the surface under study, preferably a metallic one. These photoelectrons can be focused by an electron-optical system similar to the system of an electron microscope and can give a corresponding enlarged electron image of the surface. Thus, strictly speaking, we obtain an instrument that is an emission electron microscope of a special kind. In contrast to the ordinary electron-

... of an emission microscope in this instrument the emission will be produced not by thermionic emission, but by the X-ray photoelectric effect. The construction of such instruments is entirely possible at the existing level of technology, and on this path there are no fundamental difficulties. The only technical complication is that the velocities of the X-ray photoelectrons are rather large (several kilovolts) and are not the same for electrons ejected from different levels. Therefore it will be necessary to make a velocity filter for the photoelectrons.

DIFFRACTION X-RAY MICROSCOPY

In 1936 a work by the Czech scientist Bačkovský was published, in which a method was described for obtaining an X-ray photograph of the surface of a crystal by means of a spectrograph with Bragg focusing, with rocking of the crystal. Under these conditions Bačkovský succeeded in obtaining images on which, as the asymmetry of the instrument increased, the structure of the reflecting plane of the crystal became quite clearly visible.

In 1938 the author of the present article developed a method for studying the surface of a crystal in reflected X-ray light, using for this purpose Bragg–Vulff reflection in an asymmetric X-ray spectrograph with a stationary crystal ^27–29 (Fig. 8). As a result of using a stationary crystal and photographic plate it proved possible sharply to reduce the exposure (to 10–15 minutes) and at the same time to increase the sharpness of the image extraordinarily. At the same time, together with my collaborators E. E. Weinstein and M. N. Flerova, I developed a method for studying the structure of the surface of a bent crystal in Johann’s spectrograph ^30,31.

Fig. 8. Diagram of the method of diffraction X-ray microscopy: п — plate, к — crystal, т — anticathode, 1щ and 2щ — slits.

In parallel with the development of the method for studying the structure of the surface of a crystal in monochromatic X-ray light, many

the development of methods of investigation in white X-ray light was being carried out \({}^{22-34}\).

These methods, in general, give results worse than methods of investigation in monochromatic light—less sharpness and contrast of the image—but they have the convenience that setting the surface of the crystal into a reflecting position is easier and faster with them, although the exposure itself is much longer.

Fig. 9. Diffraction X-ray microphotograph of the face of a cube of a mosaic crystal of rock salt. Magnification 7 times (author’s photograph).

If one takes a mosaic crystal*) of rock salt and makes a monochromatic beam of X-rays reflect from it, then we obtain the picture shown in Fig. 9 (distance from the plate to the crystal 3 cm, from the crystal to the focus of the tube—100 cm). This picture very accurately depicts the surface of the crystal. The agreement between the two pictures—the optical and the X-ray—is complete. Thus, this method makes it possible to depict with sufficient accuracy the defects present on the surface of a crystal and connected with inhomogeneities of its structure.

*) In speaking of mosaic structure, we have in mind the coarse mosaic structure, with block size of the order of millimeters, which we investigated in rock salt, and not the fine mosaic structure in the sense introduced by Darwin in constructing the theory of X-ray scattering. In an analogous sense this term is also used by Batschowski. This mosaicity, not the same in the various faces of a crystal, depends on disturbances during the growth of the crystal and on the artificial routes to its reproduction. The deformation blocks obtained when rock salt is compressed, especially at high temperature, are smaller and have smaller angles of rotation. An analogous mosaic structure can be obtained, for example, on zinc crystals (when single crystals are obtained by the Bridgman–Obreimov method, if crystallization is conducted too rapidly or from insufficiently pure materials).

Fig. 10. X-ray diffraction microphotograph of a smoothly polished face of a cube of a mosaic crystal of rock salt. Magnification 7 times (photograph by the author).

Fig. 11. X-ray diffraction microphotograph of a face of a rock-salt crystal deformed at high temperature. Magnification 5 times (photograph by the author).

Further, this method was applied by us to the study of the grinding and polishing of rock-salt crystals (Fig. 10), to the study of the deformation of this crystal (Fig. 11), and to the study of the question of the structure of quartz crystals. At the same time we made enlargements of the photographs obtained by us (reflexograms).

The American physicist C. Barrett^35, in 1945, applying our basic idea but without mentioning our work, developed a method of diffraction X-ray microscopy. In this method, too, a non-equilibrium X-ray spectrograph is used, together with an immobile crystalline specimen on which a beam of X-rays falls at the Bragg–Wulff angle. The plate, in the same way as was done by us, is placed as close as possible to the object under study

1—specimen, 2—photographic plate.

Fig. 12. Schematic of diffraction X-ray microscopy according to Barrett.

in order to achieve the greatest possible resolution. We give the scheme of Barrett’s method, borrowed by us from his article (Fig. 12).

Barrett considers that the resolving power of the X-ray-microscopic method in this variant is determined by the relation:

\[ W = \frac{sx}{l}, \]

where \(s\) is the slit width, \(x\) is the distance from the object to the plate, and \(l\) is the distance from the slit to the object. In this expression he repeats the relation \(\frac{x}{l}\) indicated by us and adds still another new factor, \(s\), which undoubtedly will be significant for a divergent incident beam of X-rays and will not be manifested for the parallel beam with which we worked. This expression does not take into account the divergence of the reflected beam of rays. Barrett used the method described for the study not only of single crystals or mosaics, but also for the study of polycrystals.

In 1952 Bond and Andrus^36 published an article entitled “Structural imperfections of quartz crystals.” This article gives a detailed account of the method for studying structural defects in crystal-

quartz by a method quite similar to ours, and photographs are presented, some of which are very similar to those that we published back in 1940.^27 References to our work are absent from this article as well.

A comparison of microradiographs obtained by the method described with ordinary microphotographs showed the advantages of this method in the study of metals and crystals with respect to the considerably greater sharpness and contrast of the images obtained (at small magnifications, of the order of 250–300 times). In practice, however, although a magnification of approximately 40–50 times is achieved rather easily by using fine-grained positive plates and special development methods, magnifications of the order of 200–250 times require the use of special plates (for example, Lippmann plates) with very low sensitivity, as a result of which the exposure increases tens and hundreds of times.

Apparently, this method is especially promising in the study of various deformations and disturbances in single crystals and in polycrystals.

OBTAINING AN X-RAY IMAGE BY FOCUSING X-RAYS WITH CRYSTALS

All the methods of X-ray microscopy described above, except the diffraction method, did not use focusing of the radiation employed. Already in the diffraction method we have a peculiar focusing of X-rays due to their reflection from a crystal.

It is obvious, however, that reflection from a crystal can also be used in order to accomplish the focusing of converging beams of X-rays. As is known, bent crystals are used for this purpose. In this case bent crystals may be used both in transmission (Cauchois method) and in reflection (Johann and Gamosh methods). In this way Gamosh long ago obtained, in monochromatic light, a small magnification^37 (of the order of 10–15 times) of the focal spot of an X-ray tube. In doing so, Gamosh succeeded in obtaining an image of the structural components of the alloy from which the anode had been made. The images obtained were very indistinct and blurred. In any case, they could not be compared either with images obtained by us by the diffraction method, or with the Barret photographs, or with radiographs obtained by contact exposure followed by enlargement. It should be borne in mind, however, that Gamosh apparently worked with crystals whose surface was not sufficiently perfect and whose curvature was not sufficiently uniform (a nonconstant crystal radius due to various irregularities).

In a more recent work by Cauchois^38, photographs are presented obtained by a method similar to the Gamosh method, but in transmitted radiation (according to Cauchois). The images obtained were already somewhat better, although

and are inferior to those obtained by other methods. Let us note that, when working by her method, Kossel obtains a number of specific distortions, and she gives no indication of how to eliminate them. By using more perfect crystals and better methods of bending them, it will probably be possible to achieve a considerable improvement in the quality of focusing and, consequently, in the quality of the image in comparison

Fig. 13. Image of a grid obtained by the method of diffraction from a bent crystal of hard X-rays in transmission (after Kossel).

with that obtained by Gamosh. This method may become of especially great interest if, using a superpowerful X-ray tube (tens of kilowatts), the object under study is illuminated with a beam of hard (primary) X-ray light and then, with the aid of bent crystals, the secondary radiation emitted by the object under study is focused. Such a method is not applicable to all objects, since light elements (lighter than Ti) give X-rays that are too strongly absorbed by air for it to be possible to work with them. However, objects consisting of heavier elements can be studied successfully by this method.

When working not with secondary but with primary radiation, it is probably also possible to obtain good results, but in this case questions of contrast will be difficult. Indeed, the entire optical system of such a microscope (tube and mirrors) must be adapted to one definite wavelength. In investigating different objects, however, there may arise the need, in order to ensure proper contrast, to use rays of different wavelengths, i.e., there may arise the need for several optical systems fitted to different wavelengths, and several different tubes, or else a tube with several interchangeable anodes and interchangeable “optics.”

Recently Prof. B. Ya. Pines \(^{39}\) developed a method of double focusing of X-rays. In this method a crystal of double curvature is used. On the one hand, in the horizontal direction it is bent along a radius equal to twice the radius of the focusing circle, as in Johann’s method; on the other hand, in the vertical direction it is bent along a circumference of radius chosen so that the divergent beam on the focusing circle converges to a point. It can be shown that for this it is necessary that the radius of curvature of the crystal be equal to \(r\sin\theta\), where \(r\) is the radius of the focusing circle and \(\theta\) is the Bragg–Wulff angle. As a result one obtains a figure resembling part of the surface of a barrel cut off by a plane parallel to the axis.

Along this surface Pines bent rock-salt crystals at high temperature. As a result of such bending he obtained good intensity of the reflected monochromatic beam, which he used for purposes of structural analysis.

It is possible that the application of a similar focusing method will in the future make it possible to obtain a considerable magnification by X-ray means. However, up to now the image quality in the method of bending crystals according to Pines is still insufficiently high for the purposes of X-ray microscopy.

For the time being, a considerably more perfect image is obtained when the crystal is bent along a cylindrical surface.

CATOPTRIC FOCUSING OF X-RAYS

X-rays can be focused not only by reflection from a crystal, but also by total internal reflection. The latter, as is known, occurs for X-rays in passing from a less dense medium into a denser one, and the angle at which reflection takes place is small—of the order of \(\frac{1^\circ}{2}\). It is determined by the relation

\[ \frac{\cos\alpha}{\cos\beta}=n, \]

where \(n\) is the refractive index, \(\alpha\) is the glancing angle of incidence, and \(\beta\) is the glancing angle upon refraction. In order for total internal reflection to occur, it is necessary that \(\cos\beta=1\) (angle \(\beta=0\), i.e. that the refracted ray not enter the denser medium). In other words, for the cosine of the limiting glancing angle we obtain the expression:

\[ \cos\alpha=n. \]

Since \(n\) in the case of X-rays (in contrast to visible light) is less than unity and differs only slightly from it, it is convenient to represent it in the form

\[ n=1-\delta, \]

where \(\delta\) is a small positive quantity. As theory shows, the quantity \(\delta\), for not too large and small \(\lambda\), is approximately equal to

\[ \delta = 1.35 \cdot 10^{-6} \rho \lambda^{2}, \]

where \(\lambda\) is the wavelength in angstroms, and \(\rho\) is the density of the reflecting layer. Working in air, one can vary the wavelength of the radiation used in the range of approximately from \(2 \text{ Å}\) to \(0.5 \text{ Å}\) (proceeding from the need to obtain sufficiently large absorption in the object).

By coating the reflecting mirror with a layer of platinum or another heavy metal, one can vary \(\rho\) within rather wide limits (approximately from 0.5 to 21). However, even in the most favorable case \(\delta\) always remains very small.

This leads to the fact that the angles of total internal reflection are also small, and consequently it is necessary to work with mirrors having a large radius of curvature (small curvature and small aperture), a large focal length, and small magnification. Moreover, because of the smallness of the reflection angles it is not possible to use spherical mirrors, but cylindrical ones must be used. To achieve focusing under such conditions, it is necessary to use two (or three) cylindrical mirrors with mutually perpendicular axes. Such a system does not yet make it possible to obtain substantial magnifications. X-ray magnifications of the order of 10–15 have been achieved, which, in combination with photographic magnification, allowed Kirkpatrick\(^{40, 41}\) to obtain a magnification of the order of 300.

Kirkpatrick believes that, judging from the aperture angles of the cylindrical mirrors, the resolving power of this method can reach (theoretically) \(70 \text{ Å}\). To check the resolving power of his method, Kirkpatrick used a 350-mesh grating per linear inch, i.e. with cells of approximately about \(1/30\) mm in size. In doing so, using focusing in two crossed cylindrical lenses, he was able to obtain a quite satisfactory image. Judging from the photographs presented, the resolving power of such a method is already greater than \(0.02\) mm.

Ehrenberg\(^{42, 43}\), who studied the focusing of a beam of X-rays emitted from a point source and passing through a slit by means of a cylindrical glass mirror whose axis was parallel to the slit, was able to obtain a width of the focused line of the order of a micron. All this permits one to hope that in the future, by developing better shapes and combinations of shapes of reflecting surfaces, it will be possible to increase the resolution considerably in comparison with what has been achieved so far.

In particular, it may be that the use of a similar system of crossed mirrors in diffraction X-ray microscopy will prove very promising for the purpose of obtaining magnified diffraction patterns. In this, as it seems to us, lies one of the main areas of further applications of this method.

The scheme of the method used in this case may be as follows: behind the diffracting object—single-crystal or polycrystalline—there are arranged crossed cylindrical mirrors and, at their focus, a fine-grained photographic plate. Since the beam of X-rays reflected from the crystal has a very small angle of divergence (of the order of \(10 \div 20''\)), the diffracting object will play the role of an almost infinitely narrow slit, which will greatly increase the resolving power of the method.

DIOPTRIC FOCUSING OF X-RAYS

The use of dioptric systems for X-rays is hindered mainly by two circumstances: the smallness of the refractive indices and strong absorption.

Although in some cases the deflection of X-rays by prisms made of glass, diamond, aluminum, and other materials has been used, lenses for X-rays, and thus the entire method of dioptric focusing, will hardly find application. Indeed, for example, a biconvex lens with radii of curvature of \(10\) cm, made of a material with density \(2.5\), will have for X-rays of wavelength \(1.5 \mathring{\mathrm A}\) a focal length of \(6.6\) m. Reducing the radii of curvature to \(1\) mm will reduce the focal length only to \(6.6\) cm, while the aperture of such a lens will be negligible.

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