Diffuse Scattering of X-Rays at Small Angles

E. A. Porai-Koshits

I. General Characterization of the Scattering of X-Rays at Small Angles

The scattering of X-rays at small angles, like any scattering of radiant energy, is caused by the inhomogeneity of the structure of the medium through which the radiation passes. For the wavelengths used in X-ray structural analysis, such inhomogeneity is not only the arrangement of atoms in a crystal lattice, but also the arrangement of atoms or molecules in liquids or in amorphous substances, and even the distribution of electrons within atoms in gaseous substances. When X-rays pass through a substance in any state of aggregation, one or another diffraction pattern is formed, and the task of X-ray structural analysis is to study the geometry of the body, i.e., to determine the spatial arrangement of the scattering centers from the experimentally observed diffraction pattern.

The amplitude of the waves of scattered X-radiation is inversely proportional to the mass of the scattering particle. Since intensity is proportional to the square of the amplitude, scattering is due almost entirely to the electrons of the scattering substance (the intensity of radiation scattered, for example, by a proton will be \(1846^2\) times weaker than the intensity of radiation scattered by an electron). Consequently, it will be more accurate to say that the diffraction pattern makes it possible to detect the spatial distribution of electron density.

When X-rays pass through monatomic gases, the resulting diffraction pattern makes it possible to determine the arrangement of electrons around the nucleus. Depending on the concentration of electrons around the center of the atom, they will scatter either independently, in which case there will be a simple addition of the intensities of the waves scattered by each electron separately, or, interacting, in which case the amplitudes will be added. In the first case

the scattering intensity by an atom will simply be \(Z\) times greater than the scattering intensity by one electron (\(Z\) is the number of electrons in the atom). In the second case the phases of the waves scattered by the different electrons will be almost identical, which will lead to an increase in the total intensity as a result of interference. Considering the limiting case, when all the electrons in the atom act as one negative charge equal to \(Ze\), with mass \(Zm_e\), where \(e\) is the charge of the electron and \(m_e\) its mass, it is easy to see that, since the expression for the intensity of a wave scattered by one classically radiating electron contains the fraction \(\frac{e^4}{m_e^2}\), the total intensity will be proportional to the square of \(Z\). For intermediate cases an atomic factor \(F\) is introduced, equal to the ratio of the amplitude of the wave scattered by the atom to the amplitude of the wave which, under the same conditions, would be scattered by one electron according to the classical laws. Consequently, the intensity \(I_a\) of the radiation scattered by the atom is \(F^2\) times greater than the intensity \(I_{\mathrm{el}}\) of the radiation scattered by an electron according to the classical laws:

\[ I_a = I_{\mathrm{el}} \cdot F^2, \]

where

\[ Z < F^2 < Z^2 . \]

The intensity, expressed in absolute electron units (i.e., with respect to \(I_{\mathrm{el}}\)), is a function only of \(\frac{\sin \varphi/2}{\lambda}\) (\(\varphi\) is the scattering angle), and whatever wavelength we choose, for a given system of electrons the dependence of the intensity on \(\frac{\sin \varphi/2}{\lambda}\) will be expressed by one and the same curve. This curve (the curve of the square of the atomic factor) reaches its greatest value at small \(\frac{\sin \varphi/2}{\lambda}\), gradually decreasing as this quantity increases. The same will also hold for a not too dense gas containing \(N\) independently scattering atoms; the intensity of the radiation scattered by it

\[ I_{\mathrm{gas}} = N F^2 \tag{1} \]

will decrease monotonically with increasing \(\frac{\sin \varphi/2}{\lambda}\).

Consequently, the scattering intensity of monochromatic X-rays by a monatomic gas at small angles increases continuously as the angle decreases, reaching its maximum value \(N F^2 = N Z^2\) in the direction of the primary beam (\(\varphi = 0\)).

In a molecular gas, despite the presence of intramolecular interference and the appearance, owing to this, of the first interference maxima, the general form of the intensity curve remains the same: at small angles the scattered radiation is most intense.

A completely different form is exhibited by the scattering curve at small angles for liquids, certain amorphous, glass-like, and crystalline bodies. In contrast to scattering by atomic and molecular gases, when the scattering angle is decreased the scattering intensity here does not tend toward a maximum value, but falls to zero. This is explained by the structural homogeneity of such bodies over distances large in comparison with interatomic ones: the coherence of the scattering causes its complete cancellation as a result of interference.

Such a division of all substances into two classes—those that do and those that do not give scattering of X-rays at small angles—is to a certain extent artificial. The development of experimental technique continually reveals in substances of the second class the presence of at least weak scattering at small angles, forcing them to be transferred into the first group. In nature there are very few substances possessing an ideally homogeneous submicroscopic structure. Real bodies almost always have one or another deviation from regular structure, and, as is becoming increasingly clear, these deviations determine the most important properties of solid substances, important not only from a scientific but also from a purely technical point of view.

The simplest case of a gradual transition between these two classes of substances is provided by gases under high pressure. It is evident that, with respect to the scattering of X-rays at small angles, they must occupy an intermediate position between rarefied gases and liquids. This case was theoretically developed as early as 1925 by Debye^1, who gave a formula taking into account the interaction of atoms (or molecules) approaching one another, owing to which the intensity of radiation scattered at small angles decreases continuously with increasing gas pressure. According to Debye, this intensity depends on the factor \(\Omega/V\), characterizing the saturation of space by atoms of the substance \(\left(\Omega = N \cdot \frac{4}{3}\pi (2\rho)^3\right.\)—the total volume of the “spheres of influence” of \(N\) atoms, \(V\)—the total volume occupied by the gas or liquid), as follows:

\[ I = NF^2\left\{1 - \frac{\Omega}{V}\Phi(sp)\right\}, \tag{2} \]

where \(s = 4\pi \dfrac{\sin \varphi/2}{\lambda}\), \(\rho\) is the radius of the atom, and the function \(\Phi(sp)\) in the case of spherical atoms is determined by the following expression:

\[ \Phi(sp) = \frac{3}{(sp)^3}\,[\sin(sp) - sp\cos(sp)]. \tag{2'} \]

\(\Phi(sp)\) tends to 1 when \(sp\) tends to 0 (i.e., for small angles

scattering and for large wavelengths), which explains the fall of the scattering curve for a liquid when the angle decreases, since for a real liquid the fraction \(\Omega/V\) is close to unity. For a gas, however, the fraction is close to zero, and expression (2) coincides with (1). With increasing gas density the fraction \(\Omega/V\) takes various values between 0 and 1, and the intensity of scattering at small angles gradually decreases from its maximum value to zero.

Somewhat later, Zernike and Prins\(^{2}\), and then again Debye\(^{3}\), expressed the intensity of scattering by a liquid or a dense gas through the function of the radial distribution of atoms \(u(r)\) around any atom chosen as the initial one. In the form given by Zernike and Prins, this expression, which has found wide application in X-ray studies of liquids, glasses, and amorphous substances, has the following form:

\[ I = NF^{2}\left\{1+\int_{0}^{\infty} 4\pi r^{2}\,[u(r)-u_{0}]\,\frac{\sin sr}{sr}\,dr\right\}; \tag{3} \]

here \(4\pi r^{2}u(r)\,dr\) is the number of atomic centers in a spherical layer of thickness \(dr\), situated at a distance \(r\) from the selected atom, and \(u_{0}\) is the mean atomic density of the substance. One may therefore say that the intensity of scattering by a liquid is related to the intensity of scattering by a gas as the expression standing in braces is related to unity. For large values of \(s\) (large scattering angles or small wavelengths), the integral in these braces rapidly tends to zero, and scattering by a liquid practically coincides with scattering by a gas. For the small angles that interest us, i.e., for small values of \(s\), the fraction \(\sin sr/sr\) tends to unity; it can be shown that in this case the intensity of scattering by a liquid is related to the intensity of scattering by a gas as the “compressibility” of the substance in these two states, since compressibility is characterized by the same factor \(\Omega/V\). Debye’s formula (2) is obtained from (3) if the diameter of the atom \(2\rho\) is taken as the lower limit of integration and \(u(r)\) is regarded as constant.

Let us note that the a priori assumptions concerning the structure of matter adopted by the authors in deriving formulas (2) and (3), such as, for example, the assumption that already at very small values of \(r\) the distribution function \(u(r)\) coincides with the mean atomic density of the substance \(u_{0}\), or the neglect of the possibility of azimuthal ordering of atoms (the assumption that atoms are distributed all the more randomly the more), led to the fact that both formulas agree only approximately with the experimental data, diverging from them, in particular, at large values of \(\Omega/V\). The case

SCATTERING OF X-RAYS AT SMALL ANGLES

\(\frac{\Omega}{V}=1\) corresponds not to a liquid, but only to a more or less dense gas; for a gas under high pressure, or for a liquid, this quantity exceeds unity, which according to (2) leads to physically meaningless negative values of the intensity scattered by these substances at small angles. Subsequently, various corrections were introduced into formula (3), amounting to the selection of such a distribution function \(u(r)\) as would more closely express the real distribution of atoms in a dense gas or in a liquid\(^4\). Sometimes these corrections had a semiempirical character, and in this way satisfactory agreement was obtained between the theoretical and experimental curves of the intensity of radiation scattered at small angles by various gases near the critical point (Fig. 1). The absence of a more rigorous theory of the scattering of X-rays by a liquid compels experimenters to use formula (3) even in those cases which do not correspond to the assumptions underlying its derivation, which often leads to incorrect generalizations. This also applies to the scattering of X-rays at small angles by densely packed systems of particles, the theory of which, as will be shown below, is closely connected with the theory of scattering by a liquid and therefore uses the same formula (3).

Fig. 1. Experimental and theoretical curves of scattering by ether near the critical point. Around the experimental curves are given the temperature and the specific volume. On the left are indicated the positions of the zero on the ordinate axis corresponding to the different curves.

Up to this point we have considered the decrease in the intensity of scattering at small angles as a consequence of the emergence of uniformity in the arrangement of scattering centers through the creation of their, though disordered, sufficiently dense packing (transition from a gas to

E. A. Porai-Koshits

liquid). Obviously, if this question is approached from the other side, then, when inhomogeneities of corresponding dimensions are formed in the arrangement of atoms in the crystal lattice (if only by simply reducing crystallites to colloidal dimensions), the intensity of X-ray scattering at small angles will gradually increase from zero to some value characterizing the degree of this inhomogeneity. Such a phenomenon was observed, for example, in the study of various coals5–7 (Fig. 2), as well as of other substances possessing one or another fluctuation of electron density of submicroscopic dimensions. From the magnitude of the angles at which this scattering was observed, it could be concluded that the dimensions of such regions, differing in electron density, range from 20 to 1000 Å.

Fig. 2. Scattering of X-rays at small angles by nonactivated (A) and activated (B) coal.

All these observations led to the general conclusion that any scattering at small angles indicates one or another inhomogeneity of the structure of the substance, the presence of density fluctuations at distances of 20–1000 Å. Following this, a problem immediately arose belonging to the field of X-ray structural analysis: the determination of the sizes and shapes of submicroscopic regions of inhomogeneity from the diffraction pattern arising in the immediate vicinity of the primary X-ray beam, i.e., at small angles. Attempts in this direction were made by Krishnamurti8, Mark9, Hendricks10, Warren11, and others.12–17 However, such a problem required the creation of some theory of this phenomenon and a convenient mathematical apparatus for the quantitative interpretation of the experimentally obtained scattering curves. Such an interpretation was given for the case of “loose” systems of particles by Guinier18, who proposed a theory based on scattering by individual particles. The more complicated case of a system of densely packed particles was considered by Kratky19–20, Vineyard21–22, Hosemann23, and others24 only in recent years. Attempts of this kind are continuing at the present time almost without interruption—almost every physical journal of the current year contains an article devoted to diffuse scattering at small angles. And although there is still no more or less general theory of this phenomenon, the experimental results obtained have been successfully interpreted on the basis of the available theoretical works.

II. FUNDAMENTALS OF THE THEORY OF DIFFUSE SCATTERING OF X-RAYS AT SMALL ANGLES

X-ray structural analysis of crystalline bodies is based on the theory of scattering given by Laue. As for the discrete scattering of X-rays at small angles, it fits entirely within the framework of this theory. Indeed, according to the Bragg–Wulff law

\[ n\lambda = 2d \sin \vartheta \tag{4} \]

for the wavelengths used in X-ray structural analysis (\(\lambda\) of the order of \(1—2\ \text{Å}\)) and for ordinary lattice periods (\(d\) about \(3\ \text{Å}\)), diffraction maxima appear at scattering angles \(\varphi = 20^\circ\) to about \(30^\circ\), or \(0.5\) radian. If, at the same wavelength, the periods increase to \(20—1000\ \text{Å}\), then the discrete scattering in the form of sharp lines or spots shifts into the region of small angles, from \(5^\circ\) to \(7'\), respectively. In this way, various large molecules (sterins, globular proteins, etc.), fibrous substances (chrysotile asbestos, polyamide fibers, keratin, collagen, muscle fibers, myosin, etc.), high polymers, viruses, and so on have been investigated.^25—27 The largest periods were found in the adductor muscle of a shellfish (\(d = 725\ \text{Å},\ \varphi \approx 5.5'\)) and in the tail tendon of a kangaroo (\(d = 642\ \text{Å},\ \varphi \approx 6'\)).^27

Diffuse scattering of X-rays at small angles cannot be explained on the basis of Laue’s theory. This phenomenon, expressed in the appearance near the primary beam of a blurred spot or ring, has an obvious analogy with the scattering of light by droplets of fog or by powdered lycopodium, when from the diameter of the circle (halo) formed and from the scattering angle one can approximately determine the sizes of the scattering particles (provided they are monodisperse). In both cases the particle diameter is much greater than the wavelength of the scattered radiation, and the optical reciprocity theorem is applicable to the scattering of X-rays, according to which two mutually complementary systems of screens give one and the same diffraction pattern (with the exception of the illumination intensity at the point \(\varphi = 0\)). Consequently, diffraction patterns near the primary beam obtained when X-rays pass through an assembly of randomly arranged particles will not differ in any way from the patterns obtained in scattering by a homogeneous porous substance whose pores are located in the same places where the particles were located in the first sample, and have the same shape and dimensions. Moreover, owing to the penetrating power of X-rays, the same role will be played by particles surrounded by a homogeneous medium with a different electron density, with complete absence of pores.

For all these cases the scattering angle depends on the size and shape of the scattering regions, while the intensity of scattering depends on the difference between the electron densities in these regions and in the surrounding medium. Consequently, one may always assume that the scattering regions are surrounded by void, assigning to them an electron density equal to this difference.

It follows from this, first of all, that on the basis of X-ray data alone it is impossible to determine unambiguously whether the scattering is due to the presence, in the substance under investigation, of particles or pores; the choice can be made only by considering other physical properties (for example, density).

Let us note one more essential circumstance: the pattern of diffuse scattering at small angles is completely independent of the internal structure of the scattering regions and of the medium surrounding them, which may be either amorphous or crystalline; their internal structure affects only scattering at ordinary, not very small, angles, which obey relation (4).

Let us now consider scattering at small angles by a system of spherical particles of identical radius \(R\). In this case it is most convenient to start from the theory of scattering by gases and liquids. Indeed, fixing in space the positions of the atoms of a gas or liquid and imagining that their sizes have increased to the size of submicroscopic particles (with diameter from 20 to 1000 Å), we obtain two kinds of solids—“a solid of the gas-like type” and “a solid of the liquid-like type.” The first has a loose structure (the distances between particles are much greater than the size of the particles themselves), and its diffraction pattern resembles the diffraction pattern of gas-like substances, shifted into the region of small angles. The second is a system of closely packed particles; its diffraction pattern is close to the diffraction pattern of a liquid, and it too is shifted toward small angles. A schematic representation of the different kinds of

Fig. 3. Scattering of X-rays: \(a\)—by a crystal, \(b\)—by a liquid, \(c\)—by a gas, \(d\)—by a solid of the liquid-like type, \(e\)—by a solid of the gas-like type.

scattering \(^{7}\) is shown in Fig. 3. The primary beam is shown in each case by a vertical arrow.

When the theory of scattering by gases and liquids is applied to scattering by particles of submicroscopic dimensions, the basic formulas change only very slightly. Owing to the smallness of the angles,

\[ \sin \vartheta=\sin \varphi/2\simeq \frac{\varphi}{2} \quad \text{and} \quad s=4\pi\frac{\sin \varphi/2}{\lambda}\simeq 2\pi\frac{\varphi}{\lambda}=k. \tag{5} \]

The atomic factor \(F\) is replaced by the product of the number of electrons in each particle \(n\) and the function of the size and shape of the particles, which for spherical particles was determined by Debye (see equation \((2')\)):

\[ F \longrightarrow n\cdot \Phi(kR). \]

It is now easy to write the basic formulas relating the intensity of X-rays scattered at small angles to the radius of the spherical regions of inhomogeneity (particles or pores) that cause this scattering. Indeed, for a “loose” system of such regions, for example for “solid bodies of the gaseous type,” when the waves scattered by individual particles are not coherent and the intensities are simply added, instead of (1) we obtain

\[ I_{\infty}=Nn^{2}[\Phi(kR)]^{2}, \tag{6} \]

where \(N\) is the number of particles (pores) participating in the scattering.

When the regions of disorder approach one another to distances comparable with their sizes, a densely packed system is obtained (for example, a “solid body of the liquid type”), in which the interference between the waves scattered by individual particles must be taken into account; then equations (2) and (3) are applicable, taking the following form:

\[ I=Nn^{2}[\Phi(kR)]^{2}\left\{1-\frac{\Omega}{V}\Phi(kR)\right\}, \tag{7} \]

\[ I=Nn^{2}[\Phi(kR)]^{2}\left\{1+\int_{0}^{\infty}4\pi r^{2}[u(r)-u_{0}]\,dr\right\}. \tag{8} \]

(For small angles \(\dfrac{\sin sr}{sr}=1\).) The intensity is everywhere expressed in absolute electronic units, i.e. relative to the intensity scattered by one classical radiating electron.

If the particles are surrounded by a material medium having electron density \(\rho_{0}\), then in the last three formulas, instead of the number of electrons in one particle, there enters the difference of electron densities

particles and medium, \(\rho-\rho_0\). Since the electron density enters everywhere squared, it is evident that the term \((\rho-\rho_0)^2\) will also be positive in those cases where the particles have not an excess but a deficiency of electron density relative to the surrounding medium.

Figure 4 gives the dependence of

\[ \frac{I}{N n^2} \]

on \(kR\) for various values of \(\frac{\Omega}{V}\). With increasing packing density of the particles, an interference maximum appears at \(kR=2.5\). In addition, the first of a series of secondary maxima is visible at \(kR=5.8\), which is approximately 100 times weaker than the scattering intensity at the angle \(\varphi=0\); the following maxima are still weaker. The presence of secondary maxima does not depend on the packing density; they occur even in cases of a loose system \(\left(\frac{\Omega}{V}=0\right)\) and are connected with the form of the function \(\Phi\), i.e., with the size and shape of the particles. To distinguish them from the interference maximum they are also called “form maxima.” Usually these maxima are not observed on experimental curves. There are two main causes leading to such smoothing of the scattering curves: the nonuniformity of the particles in size and shape, and the imperfection of the collimation of the primary beam.

Fig. 4. Influence of the packing density of particles on the intensity curve.

The second cause, associated with purely experimental difficulties, will be considered below. We shall now consider the possibility of taking into account the nonuniformity of particles in size and shape.

Let us first restrict ourselves to the case of spherical particles of various radii forming a loose packing. Introduce the particle size-distribution function, \(N(R)\), such that \(N(R)\,dR\) is the total number of particles having radii between \(R\) and \(R+dR\). Instead of expression (6) we obtain:

\[ I_{N(R)}=\int_0^\infty \frac{N(R)}{N} I_\infty\,dR = \rho^2 \int_0^\infty N(R) v_R^2 [\Phi(kR)]^2\,dR, \tag{9} \]

where \(v_R\) is the volume of a particle of radius \(R\), and \(\rho\) is the internal electron density of the particle. If one introduces the distribution function of the particles by their mass, then (9) takes the following form:

\[ I_{M(R)}=A\rho^2\int_0^\infty M(R)R^3[\Phi(kR)]^2\,dR, \tag{10} \]

where \(M(R)\,dR\) is the total mass of particles with radii from \(R\) to \(R+dR\), and \(A\) is a constant proportional to the total mass of the scattering specimen.

The limits of integration in expressions (9) and (10) are not defined. The upper limit can usually be established, but the entire region may be very large, and the distribution \(N(R)\) or \(M(R)\) remains unknown. One may adopt a definite type of distribution (Maxwellian, Gaussian, etc.), but this, generally speaking, may entail considerable errors owing to the neglect of a small number of particles having extreme values. Such a method of treatment was nevertheless applied by Schall and Ross \(^{28}\), who used an approximate value of the function of the size and shape of the particles, \(\Phi(kR)\), proposed by Guinier \(^{18}\) in accordance with the exponential form of the scattering curve,

\[ \Phi(kR)\simeq e^{-\frac{k^2R^2}{5}}. \tag{11} \]

Let us note that the exact function \(\Phi(kR)\), as we have already indicated, has a series of “shape maxima,” whereas function (11) does not have them, falling with angle according to an exponential curve. This neglect of secondary maxima can lead to errors only when using the scattering intensity curve at relatively large angles. However, expansion in a series shows that for \(kR \leq 1.5\text{--}2\) these errors do not exceed \(5\%\), and function (11) has found fairly wide use in the practical interpretation of roentgenograms.

Schall and Ross considered several types of distribution. In the case, for example, of a Maxwellian type of distribution, they substituted into (10) \(M(R)\) in the following form convenient for integration:

\[ M(R)=\frac{2}{\alpha^{\beta+1}\Gamma\left(\frac{\beta+1}{2}\right)}R^\beta e^{-\frac{R^2}{\alpha^2}}, \tag{12} \]

where \(\alpha\) and \(\beta\) are constants whose values must be brought into agreement with the experimental data. The symbol \(\Gamma\) denotes

\(\gamma\)-function. After substituting (11) and (12) into (10) we obtain:

\[ I_M(R)=4\rho^2\frac{2}{\alpha^{\beta+1}\Gamma\left(\frac{\beta+1}{2}\right)} \int_{0}^{\infty} R^{\beta+3} e^{-R^2\left(\frac{k^2}{5}+\frac{1}{\alpha^2}\right)}\,dR, \]

and after integration

\[ I_M(R)= \frac{2A\rho^2\Gamma\left(\frac{\beta+4}{2}\right)\alpha^3} {\Gamma\left(\frac{\beta+1}{2}\right)} \left[\frac{\alpha^2 k^2}{5}+1\right]^{-\frac{\beta+4}{2}} . \tag{13} \]

The requirement that the total mass be finite restricts the values of \(\beta\), which cannot be less than \(-1\).

Equation (13) gives the angular distribution of intensity if the distribution of the particles by size is expressed by (12). Consequently, if the parameters \(\alpha\) and \(\beta\) in (13) are chosen so that the curve \(I_M(R)\) coincides with the experimental one, then equation (12) will directly give the distribution law.

In an analogous way, Shull and Ross proceed in the case of the Gaussian distribution of particles by mass, which they seek in the following form:

\[ M(R)=\frac{2}{\sqrt{\pi}}\cdot\frac{\beta}{\alpha}\cdot\frac{1}{1+H(\beta)} e^{-\frac{\beta^2}{\alpha^2}(R-\alpha)^2}, \tag{14} \]

which, after substitution into (10) and integration, gives:

\[ I_M(R)=\frac{4\rho^2}{\sqrt{\pi}}\cdot \frac{\alpha^3}{\sqrt{\beta[1+H(\beta)]}}D(\beta,\alpha k); \tag{15} \]

\[ D(\beta,\alpha k)= \beta^{-\frac{13}{2}}\tau^4 \left\{(1+\tau^2)e^{-\beta^2} +\frac{\sqrt{\pi}}{2}\tau e^{\tau^2-\beta^2}(2\tau^2+3)[1+H(\tau)]\right\}, \]

where

\[ \tau^2=\frac{3\beta^4}{3\beta^2+\alpha^2 k^2} \quad\text{and}\quad H(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-u^2}\,du. \]

The practical application of these equations will be described below. It would be very convenient to invert the integral equation (10) with the aid of Fourier’s theorem, expressing the particle mass distribution function \(M(R)\) in terms of the experimentally found value of the intensity \(I_M(R)\). Such an attempt was made by Ross\(^{29}\), who, using the theory of Fourier integrals, Mellin’s theorem, and the theory of functions

Scattering of X-rays at Small Angles

Bessel, obtained the following inverse formula:

\[ \begin{aligned} M(R) &= \frac{4R}{9\pi A\rho^{2}}\, \frac{d}{dR}\int_{0}^{\infty} k^{3}I_{M(R)} \left[ \left(1-\frac{1}{k^{2}R^{2}}\right)\sin 2kR +\frac{2\cos 2kR}{kR} \right]\,dk = \\[4pt] &= \frac{8R}{9\pi A\rho^{2}} \int_{0}^{k_{0}} k^{4}I_{M(R)} \left[ \left(1-\frac{2}{k^{2}R^{2}}\right)\cos 2kR - \frac{2}{kR}\left(1-\frac{1}{2k^{2}R^{2}}\right)\sin 2kR \right]\,dk + \\[4pt] &\quad +\frac{4R}{9\pi A\rho^{2}}\, \frac{d}{dR}\int_{k_{0}}^{\infty} k^{3}I_{M(R)} \left[ \left(1-\frac{1}{k^{2}R^{2}}\right)\sin 2kR +\frac{2\cos 2kR}{kR} \right]\,dk . \end{aligned} \]

For sufficiently large values of \(k_{0}\), in the last integral one may asymptotically expand \(k^{3}I_{M(R)}\), and the integral can then be evaluated analytically. In many cases \(k_{0}\) can be chosen so that this integral may be neglected. This procedure is analogous to that encountered in the inversion of the scattering formula for liquids.

The laboriousness of the calculations proposed by Ross has led to the fact that no attempts at practical application of his formula have yet been made, and it has remained without experimental confirmation.

Turning now to scattering at small angles by a system of loosely packed particles of nonspherical shape, let us consider, first of all, the special case of particles of identical elongated form, whose long axes are arranged parallel to one another (fibrous materials), with the X-ray beam directed perpendicular to these axes (Fig. 5). It is easy to see that, in this case, the small-angle diffraction pattern will, as it were, depict the shape of a single particle rotated through \(90^\circ\). The intensity of the X-rays scattered along the equator will depend on the thickness of the particle \(2a\), and that of the rays scattered along the meridian on its length \(2l\). Using the function \(\Phi(kR)\) in exponential form (11) and restricting ourselves to the case of loose packing, we obtain the following approximate formulas for the intensity scattered along the equator and the meridian, relating these intensities to the dimensions of the particles:

\[ \left. \begin{aligned} I_{\mathrm{eq}} &\approx Nn^{2}e^{-\frac{k^{2}a^{2}}{5}},\\ I_{\mathrm{mer}} &\approx Nn^{2}e^{-\frac{k^{2}l^{2}}{5}}. \end{aligned} \right\} \tag{16} \]

Attempts to give more general and exact equations for particles of nonspherical shape have been made repeatedly. We shall dwell on these

attempts only very briefly, since their practical significance is not great. This circumstance, however, is explained not by an imperfection of the theory, but by the insufficient accuracy of the experimental scattering curves, which does not allow one to detect the weak effects predicted by the theory. One may therefore hope that, with the development of technique, these theoretical works will be able to assist in the interpretation of experimental results.

Guinier\(^{18}\) was the first to consider scattering by densely packed particles of nonspherical shape. He introduced into the formulas, instead of the particle radius \(R\), the “radius of gyration of the particle” \(R_0\), characterizing not only its size but also its shape: the square of the radius of gyration \(R_0^2\) is equal to the mean square distance from each atom of the particle to its center.

Fig. 5. Schematic representation of scattering at small angles by oriented particles of elongated shape.

\[ R_0^2=\frac{\int x^2\,dv}{\int dv}, \]

where the integration is carried out over the entire volume of the particle. It may be said that \(R_0\) represents the radius of gyration in the mechanical sense, since \(mR_0^2\) is the moment of inertia of the particle with respect to its center of gravity, if \(m\) is its mass.

For a spherical particle of radius \(R\), for example,

\[ R_0=\sqrt{\frac{3}{5}}\,R=0.77R, \tag{17} \]

and for an ellipsoid of revolution with axes \(a\) and \(va\)

\[ R_0=\sqrt{\frac{2+v^2}{5}}\,a . \tag{18} \]

Having calculated the scattering by a single particle with radius of gyration \(R_0\) and then summed over all possible positions of the particles in space, Guinier obtained a cumbersome formula, which he replaced by an approximate exponential dependence between the intensity and the radius of gyration:

\[ I_{\infty}\simeq Nn^2 e^{-\frac{k^2R_0^2}{3}} . \tag{19} \]

The form function (11) is, obviously, a special case of the function entering into (19). The errors introduced by such an approximation were indicated above.

Fig. 6. Dependence \((\lg I_M(R),\ \lg k^2R_0^2)\) for particles of different shape.

Exact formulas relating the intensity to the radius of gyration of ellipsoidal particles in the case of various types of particle mass distribution (Maxwellian, etc.) were calculated by Ross and Schall\(^{30}\). For each type of distribution they gave series of standard curves for different values of the ratio of the axes \(v\), i.e. for different shapes of ellipsoids—from thin disks to long rods. The influence of the shape of particles of fixed mass (in the case of a Maxwellian distribution) on the scattering curves is shown in Fig. 6, where the abscissa is proportional to the radius of gyration (according to equation (18)). For a sphere \(v=1\), for disks \(v<1\), for rods \(v>1\).

Kratky\(^{20}\) gave formulas for particles not only of ellipsoidal shape, but also for other practically more important cases—cylindrical particles, particles in the form of rectangular plates or sheets (cellulose, proteins), in the form of ribbons, and, finally, in the form of infinite threads (fibrous substances). Using Debye’s theory of scattering by a molecular gas, Kratky calculated the scattering by a body of arbitrary shape as composed of spheres—up to an infinite row of spheres, i.e. a straight line. In doing so it was assumed that the electron density in such particles is distributed uniformly. In Fig. 7 an intensity curve is presented for particles consisting of three balls arranged in one row with radius equal to \(30\ \text{Å}\) (curve 1). Curve 2 was calculated for spherical particles of the same volume.

The ordinate scale is arbitrary. In the case of an infinite row of spheres the intensity is proportional to

\[ I \propto [\Phi(kR)]^2 \left[ 1 + 2\frac{\sin 2kR}{2kR} + 2\frac{\sin 4kR}{4kR} + \ldots \right]. \]

A more detailed survey of methods for the exact calculation of scattering curves for particles of various shapes is given in the cited work of Kratky, as well as in his earlier papers \(^{31-34}\).

The situation is considerably more complicated for closely packed particles of nonspherical form or of spherical form, but with different radii. One can point only to several theoretical papers by Kratky and his collaborators, the results of which were only partly confirmed in studies of small-angle scattering by fibers of natural ramie and regenerated cellulose (viscose) with various degrees of swelling. A detailed exposition of these works is given in Kratky’s most recent paper \(^{20}\). Here we shall confine ourselves to a concise statement of the principal results.

Fig. 7. 1 — scattering curve of three particles arranged in a row; 2 — scattering curve of one particle of the same volume.

Kratky considers in greatest detail the practically important case of thin-layer packings consisting of \(N\) layers of different thickness, situated at different distances from one another and limited in their total extent. Many fibrous substances have such a structure, for example protein fibers (cellulose, silk fibroin, keratin, etc.); moreover, the fibers of these substances consist of a series of packings, the number of which is inversely proportional to \(N\). As small-angle X-ray scattering has shown, the existence of “fringed” micelles, transverse covalent bonds, etc., does not affect, in the first approximation, the general picture of the structure of these substances formed from leaf-like micelles.

The author gives several methods for calculating scattering curves for such thin-layer packings. The results indicate, above all, the importance of taking external interference into account in addition to the scattering by a single layer (micelle). In addition, all calculation methods lead to scattering curves which the author considers convenient to divide into two parts: scattering “at small angles,” \(I_1\), in the form of a diffuse maximum situated at an angle that determines, according to the Bragg–Wulff law, the mean distance between the centers of two neighboring micelles; and scattering \(I_2\), the intensity of which rapidly increases as one approaches the primary beam; this scattering

the author calls “scattering at minimal angles.” Its intensity depends on the size of the entire packing of \(N\) layers as a whole, increasing as its extent decreases, whereas the angular dependence depends on the size of the individual layers. In Fig. 8 both types of scattering are given; the first of them is given for several degrees of spatial

Fig. 8. Scattering at small angles for various values of \(\dfrac{\Omega}{V}\) (curve \(I_1\)) and scattering at minimal angles for \(\dfrac{\Omega}{V}=\dfrac{1}{2}\) (curve \(I_2\)) (\(d\)—mean thickness of a layer).

saturation by layers, beginning with a loose packing \(\left(\dfrac{\Omega}{V}=0\right)\), and the second—for \(\dfrac{\Omega}{V}=\dfrac{1}{2}\). The total scattering at small angles is

\[ I = I_1 + \frac{1}{N} I_2, \tag{20} \]

where \(N\) is the number of layers in the packing. For \(N \to \infty\) (one unbounded system), scattering at minimal angles tends to zero, and \(I=I_1\).

Comparison of Fig. 8 with Fig. 4 shows that the curve of the total scattering (20) should not differ sharply from the scattering curve of a system of densely packed spherical particles of identical radius.

Next, scattering at small angles by a finite-size “bundle of rods” is considered. The results are quite analogous to the results obtained in considering the scattering by a thin-layer packing. The theoretical curves are confirmed in experiments with myosin fibers. Approximately the same result is obtained also when considering scattering by a “combination of spheres” up to

to their dense packing. Again the author finds it possible to separate scattering at small and at minimal angles.

In all cases Kratky notes the appearance, alongside scattering by individual particles of a given form, of a component caused by external interference and dependent on the packing density of the particles. Three factors affect the angular dependence of the scattering intensity belonging to this component:

1) The mutual impenetrability of the particles, which weakens the scattering intensity at small angles and makes possible the appearance of a series of weak maxima in the region of angles lying between the “minimal” angles and the angles corresponding to law (4).

2) The ordered arrangement of the particles, i.e., the presence of preferred distances between particles (analogous to the “quasicrystalline” structure of a liquid), which gives rise to maxima at angles corresponding to law (4). The appearance of these maxima is favored by uniformity of the particles in size and shape.

3) Finite size, which causes scattering at minimal angles. This effect was already considered by Debye, who believed, however, that it depends only on the total extent of one system (of layers, rods, spheres), whereas Kratky showed that only the intensity of scattering at minimal angles depends on it, while the angular distribution is connected with the dimensions of the individual particles.

Thus, if in a loosely packed system the shape of the scattering curve is directly connected with the size and shape of the individual particles, then the scattering curve of a densely packed system can provide information chiefly only about the arrangement of the particles and about their mean dimensions; the particle shape has little effect on the scattering curve. Indirectly, however, one can also judge the particle shape, since the latter undoubtedly influences the type of packing when the particles are packed sufficiently densely.

Below we shall consider examples of a concrete interpretation of experimental curves on the basis of the theoretical propositions set forth here. First we shall dwell on the characteristic features of the experimental technique now used to obtain as accurate as possible a curve of diffuse scattering at small angles, and indicate those paths for its further improvement which seem to us the most promising.

III. METHOD

To detect the diffraction pattern at the very smallest angles (usually obscured by the primary beam), one may proceed by maximally increasing the wavelengths used and by arranging a very fine diaphragm.

As the wavelength is increased, absorption increases; this requires a reduction in the thickness of the specimen, owing to which the intensity decreases

scattered by it, which in turn leads to very long exposures. The scattering of X-rays by air, which, as we have seen, is most intense precisely in the direction of the primary beam, makes it necessary to use vacuum chambers. At the same time, with very soft rays, absorption even in very thin beryllium chamber windows begins to play an important role. In addition, under all conditions strict monochromaticity of the radiation is necessary, since the presence of radiation with different wavelengths causes parasitic scattering to be superposed on the principal pattern, and the latter becomes strongly blurred. When very soft X-rays are used, monochromatization by reflection from a single crystal is difficult to carry out; monochromatization by means of filters, however, leads to a considerable increase in exposure because of the necessity of working at low voltages.

Fig. 9. Dependence of the scattering curve on the degree of monochromaticity of the radiation.

In Fig. 9 it is shown how the intensity curve changes as a function of \(k\) (on a logarithmic scale) in the case of using monochromatic (reflection from a NaCl single crystal), filtered, and unfiltered radiation\(^{28}\). In the case of using the latter, the error in determining the intensity reaches 50%, whereas with filtered radiation it is 20%. An error of 20% in measuring the intensity leads to a 25% error in determining the mean particle size. This is explained by the presence of the most intense short-wavelength part of the continuous spectrum, which partially passes through the filter, especially at increased voltage;

Fig. 10. Arrangement diagram of the apparatus.

The simplest arrangement of the source of X-rays, the slits, the specimen, and the film is given in Fig. 10[^35]–[^37]. The first two slits \(A_1\) and \(A_2\) cut out a narrow beam of X-rays incident on the specimen \(E\) and then on the part of the film \(n_1 n_2\). The slit \(A_3\), placed in front of the specimen, does not affect the primary beam and serves to catch the secondary X-rays scattered by the edges of slit \(A_2\). Both the primary beam and the secondary rays are absorbed by the metal plate \(M\). Thus, scattering at small angles by the specimen \(E\) is recorded by the X-ray film \(F\) outside the interval \(n'_1 n'_2\), so that the minimum scattering angle

\[ \varphi_{\min}=\frac{1}{2}\frac{n'_1 n'_2}{D} \]

will, evidently, be proportional to the size of the slits \(A_1\) and \(A_2\) and inversely proportional to the distance between them.

Fig. 11. Angle of maximum and minimum deviation in the case of a rectangular slit \(a \times b\).

Hosemann[^17] used windows of diameter \(0.0025\ \mathrm{mm}\) and obtained \(\varphi_{\min}=9.7'\); however, the exposure in this case, depending on the specimen, varied from 60 to 800 hours.

The influence of the shape of the slit on the diffraction pattern has been investigated repeatedly. Let us consider some point \(P\) of the diffraction pattern at small angles and determine how the scattering angle of the rays reaching this point changes depending on their path through rectangular slits of finite width \(a\) and height \(b\) (Fig. 11). The greatest scattering angle of X-rays incident at point \(p\) \((\varphi_{\max})\) is determined by the primary ray which has passed from one of the corners of the first slit (the one closest to point \(p\)) to the opposite corner of the second slit (Fig. 11, \(a\)); the smallest \((\varphi_{\min})\), by the ray which has passed from the edge of the first slit farthest from \(p\) to the opposite edge of the second.

As Yudovich[^23] has shown, the error introduced by the finite dimensions of rectangular slits and which may be called the “collimation error,” for a given slit area will be minimal for a quite definite slit width, by no means very narrow. In Fig. 12 the dependence of the collimation error is given

\(\Delta \varphi = \varphi_{\max} - \varphi_{\min}\) on the width of the slit for definite values of \(l, L\), and \(\varphi = \dfrac{r}{L}\), and for various slit areas \(A\). All the curves

Fig. 12. Dependence of the collimation error on the slit width.

have a minimum, which determines the optimal dimensions of the slit for the given distances.

Let us investigate the influence of the collimation error on the position and shape of the first of the series of secondary maxima (at \(kR = 5.8\)) in the graph of

Fig. 13. Dependence of the position and shape of the diffraction maximum on the wavelength and the shape of the slit.

Fig. 4. In Fig. 13, curve \(A\) depicts this maximum according to equation (8) for ideal experimental conditions, i.e. in the case of point diaphragms, while curves \(B, C\), and \(D\)—for the following slit dimensions

and values of the wavelength (the area of the slit remains unchanged):

\[ a = 0.25\ \mathrm{mm}, \]

Curve \(B\)

\[ \begin{aligned} b &= 0.80\ \mathrm{mm},\\ \lambda &= 8.32\ \text{\AA}\ (\mathrm{Al}), \end{aligned} \]

\[ a = 0.05\ \mathrm{mm}, \]

Curve \(C\)

\[ \begin{aligned} b &= 4.00\ \mathrm{mm},\\ \lambda &= 8.32\ \text{\AA}\ (\mathrm{Al}). \end{aligned} \]

\[ a = 0.25\ \mathrm{mm}, \]

Curve \(D\)

\[ \begin{aligned} b &= 0.80\ \mathrm{mm},\\ \lambda &= 1.54\ \text{\AA}\ (\mathrm{Cu}). \end{aligned} \]

As is seen from Fig. 13, curves \(B\) and \(D\) have been drawn for an almost optimal slit shape with an area equal to \(0.2\ \mathrm{mm}^2\), while curve \(C\) is for a narrow but long slit of the same area. The maximum on curve \(B\) (as compared with the maximum on curve \(A\)) is smoothed out and shifted slightly to the left, toward smaller \(kR\). The maximum on curve \(C\) is smoothed out still more and shifted; this is what should occur for larger \(\Delta \varphi\) (see Fig. 12); on curve \(D\) the secondary maximum disappears completely because of the harder radiation.

Fig. 14. Introducing a correction in the case of a narrow slit.

These curves clearly show with what caution one must approach the choice of radiation and slits in the case of rectangular cross section of the latter.

Let us give a method for recalculating the scattering curve obtained with rectangular slits in order to transform it into the ideal curve (i.e., for point diaphragms)\(^{38}\).

The diffraction pattern for a narrow slit is a long strip and is a superposition of an infinite number of diffraction patterns in the form of circles; the latter would be obtained with point diaphragms and therefore are the quantities sought. Let \(AA\) (Fig. 14) be the line of centers of such radial spots, and let \(BB\) be the narrow and long slit of the microphotometer, which moves in the direction \(ZZ\), remaining parallel to \(AA\). We observe the averaged intensity \(\Phi_1(z)\) as a function of the distance \(z\). The desired quantity is the radial intensity \(\Phi(r)\) of one elementary spot. The problem

SCATTERING OF X-RAYS AT SMALL ANGLES

is simplified by the change of variables \(F(z^2)=\Phi_1(z)\) and \(f(r^2)=\Phi(r)\). Since \(r^2=z^2+(U-V)^2\), then

\[ F(z^2)=\int_{-A}^{+A}\int_{-B}^{+B} f\{z^2+(U-|V|)^2\}\,dU\,dV, \]

where \(f\) is the required function. A solution is possible only for slits whose length is much greater than their width. In this case

\[ f(p)=-\frac{1}{2\pi B}\int_p^\infty F'(q)\,\frac{dq}{\sqrt{q-p}}, \]

where \(p=r^2\) and \(q=z^2\).

Using this formula, one can carry out the calculation without expressing the function \(F\) in analytic form. The experimental curve \(\Phi_1(z)\) is constructed as the curve \(F(q)\), and the differential curve \(F'(q)\) is found graphically. The values of the products \(F'(q)\) by \((q-p)^{-1/2}\) are computed for a series of arbitrarily chosen values \(p\) along the abscissa \(q\) and are plotted as curves. The areas under each of such curves, from \(p\) to \(\infty\), determine the ordinates of the sought curve \(f(p)\).

Fig. 15. Scattering at small angles by the outer layer of a strongly absorbing substance.

The effect of the profile of the slit edges on small-angle scattering was considered by Kratky \(^{20}\). Despite the fact that in the case of sharp slit edges a larger amount of substance takes part in the scattering (Fig. 15), sharp edges are preferable to blunt ones, for, as Kratky showed, the small-angle scattering by the outer layer of a blunt edge is considerably greater than the scattering by the outer layer of a sharp edge. Of course, the real circumstances are more complicated, since the vertex of the edge is always rounded, but the necessity of sharp edges is beyond doubt.

To shorten the exposure, Guinier \(^{18,39}\) used focusing monochromators in the form of bent crystals. Figure 16 shows the arrangement of his apparatus. Reflected from a bent single crystal (for example, quartz), the primary beam of X-rays is focused on the film in the form of a thin line. Slit \(A_2\) protects the specimen from parasitic rays scattered by the monochromator, without cutting off the working primary beam. The most convenient position of the specimen is midway between the crystal and the film. The trap for the primary beam is made not of lead, but of a less absorbing metal, calculated so that it transmits part of the energy of the primary beam and, in its intensity after the microphoto-

metering, it would be possible to make quantitative comparisons of different X-ray photographs (for example, for Cu \(K_\alpha\) rays a trap made of a copper plate \(0.2\) mm thick is convenient). The chamber with the specimen and the film—

Fig. 16. Arrangement of the apparatus when working with one bent single crystal.

—is vacuum-tight, with a window of aluminum foil \(0.01\) mm thick. The polished section serves as the cassette for the film. In the case of investigating a liquid, the latter is poured between two mica windows. The procedure for setting up is as follows: first the monochromator is installed; slit \(A_1\) regulates the divergence of the beam, which must not exceed \(20\)–\(30'\); in this case the opening of slit \(A_2\) is several hundredths of a millimeter. All this is checked photographically, first with slit \(A_2\) opened as wide as possible, then with a slit concerning the primary beam. Finally, the camera is installed, the trap being moved by rotating the camera about a vertical axis passing through the specimen. After putting on the polished section with the cassette, a control exposure without the specimen is made in order to make sure that there is no veil whatsoever.

Fig. 17. Arrangement of the apparatus when working with two bent single crystals; \(A_1\), \(A_2\)—bent single crystals, \(S\)—source of X-rays.

In his last work, Guinier\(^{40}\), seeking to reduce the parasitic radiation still further, used an arrangement with two bent single crystals, the scheme of which is given in Fig. 17. The distance between the specimen and the film was \(100\) mm.

The collimation error in both of Guinier’s arrangements is large, since he uses, as it were, a narrow long slit.

Extremely interesting are attempts to use the advantages of ionization registration of X-rays. Diamond\(^{38}\), for example, proposed placing the specimen between two single crystals and, by rotating the second crystal, successively directing into a particle counter (or into an ionization chamber) all rays, both those scattered at small angles and the primary beam (Fig. 18). By subtracting from the obtained intensity curve the intensity curve of the primary beam, determined by the same method but without the specimen, one can follow the scattering down to very small angles (according to the data of Allison and Paratt for calcite crystals down to angles \(\varphi = 20''\)). For this purpose the slit of the counter must be sufficiently wide.

Fig. 18. Registration of scattering at small angles with the aid of two single crystals and a counter.

A similar arrangement was also proposed by other authors\(^{41—43}\). Quite recently Warren\(^{44}\) proposed the following method for measuring the radius of spherical equilibrium particles with the aid of two calcite single crystals and a counter with a wide slit. The relative intensities of the rays entering the counter are measured in three cases: 1) after reflection only from two single crystals, placed successively in parallel position (\(I_0\)); 2) with the specimen under investigation (in the form of a thin plate), placed between both single crystals (\(I_1\)), when only rays reflected from the second single crystal enter the counter, and, consequently, rays scattered by the specimen at small angles do not enter; 3) with the specimen placed after the second single crystal, directly in front of the counter (\(I_2\)), when the counter also registers the ray scattered at small angles. Obviously,

\[ I_1 = I_0 e^{-(\mu+\mu')m} \quad \text{and} \quad I_2 = I_0 e^{-\mu m}, \]

where \(\mu\) is the ordinary mass absorption coefficient, \(\mu'\) is the additional mass absorption coefficient due to scattering at small angles, and \(m\) is the scattering mass of the specimen in \(\mathrm{cm}^2\).

Taking logarithms, we readily find:

\[ \mu'=\frac{1}{m}\ln\frac{I_2}{I_1} = \mu\frac{\ln\left(\dfrac{I_2}{I_1}\right)}{\ln\left(\dfrac{I_0}{I_2}\right)}. \]

Warren further showed that this additional coefficient \(\mu'\) is proportional to the particle radius \(R\) and to their internal density \(\rho\):

\[ \mu' = 0.0108 \lambda^2 \rho R . \]

Thus, the particle radius \(R\) is readily determined from the three relative intensities \(I_0\), \(I_1\), and \(I_2\).

The last examples indicate that the use of two single crystals for the study of scattering at small angles is very promising. It should be noted that the idea of using, for purposes of structural analysis, a double spectrograph in combination with ionization registration of scattered X-rays, the second single crystal acting as a kind of diaphragm, was first proposed in the Soviet Union by Academician A. A. Lebedev as early as 1937, i.e., considerably earlier than by American scientists^45.

IV. INTERPRETATION OF X-RAY DIFFRACTION PATTERNS

A. Scattering by “loose” systems of particles

Let us first consider the simplest case of a loosely packed monodisperse powder (with particles of identical shape and size), i.e., the case of a “gas-type solid.” A typical example of such a body is a dilute solution of a substance with large molecules. It is obvious that the theoretical conditions adopted by Debye for a gas and underlying formulas (6), (16), and (19) will be satisfied in this case, and these formulas should be used for the interpretation of X-ray diffraction patterns.

Taking the logarithm of equation (19) and substituting the value \(k = 2\pi \frac{\varphi}{\lambda}\), we obtain:

\[ \lg I_\infty = \lg Nn^2 - \frac{k^2 R_0^2}{3}\lg e = \lg Nn^2 - 5.715 \frac{R_0^2}{\lambda^2}\varphi^2 . \tag{20} \]

Consequently, the dependence between the logarithm of the intensity and the square of the scattering angle will be expressed by a straight line, and the angular coefficient of this straight line \((\alpha)\) uniquely determines the radius of gyration of the particles \(R_0\):

\[ -\alpha = 5.715 \frac{R_0^2}{\lambda^2}, \quad \text{whence} \quad R_0 = 0.416 \lambda \sqrt{-\alpha}, \tag{21} \]

and in the case of using Cu \(K_\alpha\) rays \((\lambda = 1.539\ \text{\AA})\)

\[ R_0 = 0.644 \sqrt{-\alpha}. \]

Thus, in the case of a monodisperse powder, after microphotometry of the X-ray diffraction pattern and construction of the intensity curve, one should draw the graph \((\lg I_\infty, \varphi^2)\) or, still more simply, the graph \((\lg I_\infty, r^2)\), where \(r\) is the distance in millimeters on the film from the center of the primary

of the beam[^46] (in this case only the coefficient of \(\sqrt{-a}\) in expression (21) changes). From the slope of this straight line the radius of rotation of the particles is easily determined. For further interpretation it is necessary to bring in other data. For example, if \(R_0=20\ \text{Å}\), then this radius of rotation corresponds either to a spherical particle with radius \(26\ \text{Å}\), or to an ellipsoid of revolution of length \(84\ \text{Å}\) and diameter \(25\ \text{Å}\), or to a cylinder of diameter \(55\ \text{Å}\) and height \(14\ \text{Å}\), etc.

The most accurate results can be obtained in this way in the investigation of large molecules, owing to the sameness of their form. Knowledge of the molecular weight of these molecules makes it possible to obtain data on their shape.

Fig. 19. Determination of the radius of rotation of albumin molecules from the slope of the straight line \((\lg I,\varphi^2)\). \(a\)—intensity curve \((I(\varphi))\), \(b\)—curve \((\lg I,\varphi^2)\).

As an example one may cite the investigation, by the method of small-angle X-ray scattering, of dilute solutions of albumin[^39], hemoglobin[^47], and chymotrypsin (in acidified water)[^33].

From the slope of the straight line \((\lg I_\infty,\varphi^2)\) the radius of rotation of albumin was determined (Fig. 19), and was found to be \(R_0=20\ \text{Å}\). In the case of a spherical form of the albumin molecule, its radius would be \(26.2\ \text{Å}\), its volume \(74\,000\ \text{Å}^3\), and its molecular weight (from the density of albumin, equal to 1.3) \(58\,500\). This figure is higher than that obtained on the basis of other physicochemical data. Hence one may conclude that the assumption of a spherical form of the albumin molecule is incorrect. If the form of an elongated ellipsoid with an axial ratio \(w=2.4\) is adopted, then the molecular weights agree. In this example, in which the particle volume was known in advance from other data, small-angle scattering made it possible to determine its shape.

For the hemoglobin molecule, by the same method, a radius of rotation \(R_0=23\ \text{Å}\pm1\ \text{Å}\) was obtained. Perutz[^48], determining the form of the hemoglobin molecules, found it to be cylindrical, with diameter \(57\ \text{Å}\) and height \(34\ \text{Å}\), which corresponds to a radius of rotation exactly \(23\ \text{Å}\).

E. A. PORAI-KOSHITS

Determining the shape of a particle in the general case is very difficult. Theoretically, three methods may be used for this:

1) Continuing the straight line \((\lg I_\infty, \varphi^2)\) to its intersection with the ordinate axis \((\varphi^2 = 0)\), we find the intercept \(\lg N n^2\). Knowing the intensity of the primary beam from the scattering mass of the specimen, we determine the number of particles, \(N\), and then the number of electrons in each particle, \(n\). In this case the intensity \(I_\infty\) must be expressed in absolute electron units. The volume of the particle, determined from the number of electrons contained in it, and its radius of gyration make it possible to obtain information about its shape (the ratio of axes, etc.).

Fig. 20. Scattering curves for particles of ellipsoidal shape at different axial ratios.

2) One may consider, for example, the case of an ellipsoid of revolution and construct scattering curves \((I_\infty, \varphi)\) for one and the same radius of gyration \(R_0\) (obtained experimentally), but for different axial ratios \(v\). Coinciding at small angles, these curves diverge as \(\varphi\) increases. Prolate \((v > 1)\) and oblate \((v < 1)\) ellipsoids give different families of curves. By comparing their form with the experimental curve, one can in principle determine the shape of the particle. Unfortunately, however, the difference between the curves is very slight (Fig. 20), and such a procedure requires an unattainable accuracy of the experimental curves.

3) The third method, proposed by Kratky, was described above (see Fig. 7).

The simpler case is that of identically oriented elongated particles. For them formulas (16) are applicable, and the shape of the particles is determined comparatively easily. Such calculations have been carried out repeatedly in the study of highly oriented polymers (viscose, nylon fibers, strongly stretched rubber, cellulose acetate, etc.). Fankuchen and Mark studied in this way the process of drawing fibers.^25

The transition from a monodisperse powder to particles of different sizes immediately complicates the scattering curve at small angles and, consequently, makes its interpretation more difficult. Each particle size contributes its own scattering according to formula (19), its own straight line on the graph \((\lg I_\infty, \varphi^2)\), and all of them, when added together, form a curve convex toward the origin. Only at very small and very large angles can one expect rectilinear segments as a result of scattering by the largest and the smallest particles, respectively. If \(N_i\) particles containing \(n_i\) electrons have ra-

if the radius of gyration is \(R_i\), then the mean radius of gyration is determined by the formula

\[ R_m^2=\frac{\sum N_i n_i^2 R_i^2}{\sum N_i n_i^2}. \]

This shows that large values of \(R_i\) dominate, since larger values of \(n_i\) correspond to them.

In the case of a discrete series of sizes, the interpretation of radiographs can be approached in the following way. Continuing the rectilinear part of the curve \((\lg I_{\infty}, \varphi^2)\) at the largest angles until it intersects the ordinate axis, we find from its slope the smallest radius of gyration of the particles, \(R_1\). Subtracting the intensity (not the logarithm of the intensity) of the radiation scattered by particles with radius \(R_1\) from the whole curve, we obtain the scattering curve for all particles except particles of radius \(R_1\). To it we apply the same procedure, until as a result of the subtraction we no longer obtain a straight line, determining by its slope the radius of gyration of the largest particles. The intercepts cut off by the straight lines on the ordinate axis make it possible to determine the relative numbers of particles of each size. In the case of a large set of sizes, it is possible to construct point by point the distribution curve of particles by size.

Fig. 21. Processing of a scattering curve by the tangent method. Case of particles of two sizes.

This graphical method of processing curves was applied by the author of the present article in the study of microporous silicate materials. In Fig. 21 one of the curves obtained by the author is presented \((\lg I_{\infty}, r^2)\), from which it was possible to determine two radii of gyration, \(R_1=56\ \text{\AA}\) and \(R_2=115\ \text{\AA}\).

This same method was used by Iellinek, Solomon, and Fankuchen\(^{49}\) in interpreting radiographs of aluminosilica gel and amorphous

E. A. PORAI-KOSHITS

Table I

Ordinates	Ordinates	$\Delta \lg I$	$\Delta r^3$	Slope $(\Delta \lg I : \Delta r^3)$	$R$	$R^3$	$\dfrac{K}{R^3}\cdot 10^4$	Relative amount
$K_1$	0,61	0,484	630	0,000769	10	1 000	6,10	0,194
$K_2$	4,3	1,633	457	0,00358	22	10 600	4,06	0,129
$K_3$	39	2,591	240	0,0108	37	50 600	7,71	0,245
$K_4$	115	3,061	140	0,0219	54	157 000	7,32	0,232
$K_5$	140	3,146	107	0,0294	62	238 000	5,88	0,186
$K_6$	48	2,681	33	0,0812	103	1 092 000	0,44	0,014
						Total	31,51	1,000

coal. Fig. 22 and Table I illustrate their results for an alumosilica gel. These results were used for constructing the distribution curve by sizes and for calculating the area of the internal surface of the specimen.

Fig. 22. Processing of the scattering curve by the tangent method. Case of particles of six sizes.

It should be noted at once that such a graphical method for interpreting X-ray diagrams, first, requires an exact determination of the scattering curve for the largest angles and, second, may lead to considerable errors when large particles are present in the specimen, whose scattering is not captured because of its closeness to the primary beam: the observed scattering from a small number of fine particles may easily be attributed to the entire specimen. This is clearly seen from Table II, which illustrates how rapidly the intensity decreases with increasing particle radius. Therefore the limit of applicability of the graphical method should be carefully checked.

Distribution curves of spherical particles by size (or by their mass), in the case of various types of distribution, are expre-

Table II

	25	50	100	200	400
Particle radius (in Å)	25	50	100	200	400
Scattering angle at which the intensity decreases by half (in minutes)	63.0	31.5	15.8	7.8	3.9

are given by equations of the form (12) and (14)\(^{28}\). For their practical application it is necessary to fit the constants \(\alpha\) and \(\beta\) to the experimental data in such a way that the curves \(I_{M(R)}\), determined by equations (13) and (15), would coincide with the experimental scattering curves. This can be done by one of the following two graphical methods:

1) By superposing the experimental curve on one of a series of standard curves calculated from equation (13) or (15) for different values of \(\alpha\) and \(\beta\) (Figs. 23 and 24).

Fig. 23. Standard curves in the case of a Maxwellian particle distribution.

Fig. 24. Standard curves in the case of a Gaussian particle distribution.

The experimental curves in this case are plotted in the coordinates \((\lg I, \lg k^2)\), and the standard curves in the coordinates \((\lg I, \lg \alpha^2 k^2)\); \(\beta\) is determined from the standard curve, and \(\alpha\)—from the position of the \(\lg k^2\) axis of the experimental curve relative to the \(\lg \alpha^2 k^2\) axis of the standard curve, after which, with the aid of equation (12) or (14), the distribution curve is found. If none of the standard curves is suitable, the experimental curve may be fitted piecewise, each of its parts being required to agree well with one of the standard curves. The parameters are determined as before, and in constructing the total distribution curve one should take into account the relative intensities of the component curves. The whole procedure takes no more than 30 minutes.

2) By changing the experimental curve until it is transformed into a straight line. Equation (13) can be written in logarithmic form as

\[ \lg I_{M(R)}=\lg B-\frac{\beta+4}{2}\lg\left[k^2+\frac{3}{\alpha^2}\right], \quad \text{where } B=\text{const}. \]

It is obvious that the dependence of \(\lg I_{M(R)}\) on \(\lg\left[k^2+\frac{3}{\alpha^2}\right]\) is expressed by a straight line, from whose slope, equal to \(-\frac{\beta+4}{2}\), \(\beta\) can be determined. It is impossible to construct the graph directly, since \(\alpha\) is also an unknown parameter. Therefore one should construct a graph of the dependence of \(\lg I_{M(R)}\) simply on \(\lg k^2\), and then add to \(k^2\) various values of \(\frac{3}{\alpha^2}\) until a straight line is obtained. In Fig. 25 such a construction is given: at

\[ \frac{3}{\alpha^2}=220\cdot 10^{-4}\ \mathrm{rad}^2\ \text{\AA}^{-2} \]

the experimental points lie on a straight line. The found values of \(\alpha\) and \(\beta\) are then again substituted into (12). With this method of finding the parameters, it is sometimes also necessary to divide the experimental curve into parts and “straighten” them separately.

Fig. 25. Determination of the constants \(\alpha\) and \(\beta\) by straightening the experimental curve.

The mean radius of the particles, \(R_m\), is easily calculated from the parameters \(\alpha\) and \(\beta\). Its values are given in Table III. It is interesting to note that \(\frac{R_m}{\alpha}\) varies almost exactly linearly with \(\beta\) for \(\beta \gg 1\).

Table III

\(\beta\)	0	1	2	3	4	5
\(\dfrac{R_m}{\alpha}\)	0.227	0.693	1.183	1.617	2.176	2.674

Considering Figs. 23 and 24, one can see that all the standard curves have a rather flat section at very small angles, a curved section at intermediate angles, and a steep linear section at the largest angles. Since measurement of the absolute intensity is associated

with considerable difficulties, the comparison is based not on the absolute values of the ordinate, but on the general form of the curves. It is therefore essential to determine the experimental curve not only in the final (steep linear) part, but also in the region of angles that includes its bend.

Figure 26 gives the curve of the distribution of particles by mass for alumina gel, obtained by Shull and Ross\(^{28}\); the corresponding experimental intensity curve was given in Fig. 25. The mean particle size was found to be 36 Å, which agrees very well with the mean size of dispersed crystallites, determined from the broadening of diffraction lines in the region of ordinary angles, which was 38 Å. In the same way, distribution curves were obtained for two silica gels\(^{50}\), prepared by different methods, with additions of up to 20% Al\(_2\)O\(_3\). For the first gel, the mean particle diameter increased, as the Al\(_2\)O\(_3\) content increased, from 31.5 to 43.5 Å, and for the second from 58.3 to 65.7 Å. The calculated specific surface proved to be 30% greater than the surface determined by gas absorption, which the authors explain by the screening of part of the surface at the contacts of particles. The dependences of specific surface on the Al\(_2\)O\(_3\) content coincide in both cases.

Fig. 26. Particle distribution curve for silica gel.

These two examples, however, exhaust the application of the method of using standard curves. The curves are so close to one another in form that the choice between them is usually very difficult, not to mention the choice between different types of distributions and the errors introduced by neglecting the relatively small number of particles of extreme sizes. In general, other, more complex types of particle-size distributions are possible, which, moreover, may also differ in shape. All this requires extremely accurate experimental data, still unattainable at the present time, which, as we indicated above, also makes difficult the use of the standard curves proposed by the same authors for particles of nonspherical shape.

Thus, the method of interpreting small-angle diffraction patterns based on the selection of definite types of particle-size distribution is still of little effectiveness. Despite the great approximate nature of the method of graphical decomposition of the scattering curve into a series of curves, this method, when a certain caution is observed, is still the only one leading to satisfactory results in the study of a loose system of noninteracting particles.

B. Scattering by Closely Packed Systems of Particles

When the particles of a disperse powder approach one another, interference arises between waves scattered by different particles; this leads to the appearance of an interference maximum at \(kR = 2.5\) (Fig. 4) and to its growth up to complete separation from the scattering at minimum angles. The diffraction pattern in this case degenerates into a diffuse ring, analogous to that which we observe at large angles in the scattering of X-rays by liquids or glasses (“solid bodies of the liquid type”). In this case formulas (7) and (8) are applicable. In addition, following Kratky \(^{20}\), one can divide the scattering curve into two parts—at small angles and at minimum angles—and from each extract information on the sizes of the particles; the diameter of the diffuse ring (scattering at small angles) must then correspond to the angle related by relation (4) to the mean distance between neighboring scattering centers, which, for close packing of spherical particles, determines approximately their mean diameter. Both of these methods of treating experimental data have been used by various authors, and we shall now briefly discuss their results.

Yudovich \(^{23}\) investigated samples of colloidal gold, the particle sizes of which had previously been measured with an electron microscope. Using soft aluminum radiation \((\lambda = 8.32 \text{ Å})\), he found on the small-angle scattering curve a series of maxima at \(k^2 = 0.85 \cdot 10^{-4},\ 4.2 \cdot 10^{-4},\ 11.5 \cdot 10^{-4}\), and \(20.5 \cdot 10^{-4}\ \text{radian}^2\text{ Å}^{-2}\). Comparing the positions of these maxima with their positions on the theoretical curve, constructed with the aid of relation (7), on which they are located at \(kR = 2.5;\ 5.8;\ 9.1;\ 12.3\ \text{radian}\cdot\text{Å}^{-1}\) (in Fig. 4 only the first interference maximum and the first form maximum are shown), Yudovich determined the following values of the particle radius:

\[ R = \frac{2.5}{\sqrt{0.85 \cdot 10^{-2}}} = 271 \text{ Å}, \]

\[ R = \frac{5.8}{\sqrt{4.2 \cdot 10^{-2}}} = 283 \text{ Å}, \]

\[ R = \frac{9.1}{\sqrt{11.5 \cdot 10^{-2}}} = 260 \text{ Å}, \]

\[ R = \frac{12.3}{\sqrt{20.5 \cdot 10^{-2}}} = 272 \text{ Å}. \]

Despite the good agreement of these quantities with one another, they all considerably exceed the value of the particle radius determined with the aid of the electron microscope and equal to \(232 \text{ Å} \pm 11\%\). The author explains this discrepancy by the displacement of all maxima toward smaller values of \(k^2\), and by their smoothing owing to the presence of collimat-

tional error discussed above (see Fig. 13), and also as a consequence of the inhomogeneity of the particles in size. Such a shift toward smaller scattering angles should give overestimated values for the particle size.

In determining the size of the same particles from the slope of the tangent drawn to the scattering curve at the minimum angles, the radius found was

\[ R = 248\,\text{\AA}. \]

Ginier\(^{51}\) studied hemoglobin in the red blood corpuscles of a horse. In contrast to the case of a dilute hemoglobin solution, the small-angle scattering curve had a distinct maximum near the angle of 0.025 radian (Fig. 27), which corresponds to an average distance between the centers of neighboring hemoglobin molecules of about \(62\,\text{\AA}\), and indicates their more or less close packing in the blood corpuscles. Since in crystalline hemoglobin the identity period varies, depending on the degree of hydration, from 36 to \(51.4\,\text{\AA}\), there is evidently an excess amount of water in the corpuscles, partially disrupting the order, and the arrangement of the hemoglobin molecules in them will be intermediate between the ordered arrangement in a solid crystal and the disordered arrangement in a dilute solution. Somewhat earlier, Perutz\(^{52}\) proposed a model according to which hemoglobin molecules in blood corpuscles are in contact and their diameter is \(75\,\text{\AA}\), which differs only slightly from the \(62\,\text{\AA}\) obtained by Ginier.

Fig. 27. Scattering curve of hemoglobin in the red blood corpuscles of a horse.

The scattering curve of a hemocyanin solution\(^{19,40}\), in contrast to hemoglobin, did not change at all with concentration and had a small horizontal plateau at the minimum angles and a sharp drop in intensity down to zero with increasing angle. The diffraction curve, consequently, reflects an unchanged arrangement of molecules inside certain particles, independent of the distance between the particles. The electron microscope revealed\(^{53}\) formations of 4–6 densely packed rods, whose diameter was approximately \(200\,\text{\AA}\). The plateau on the small-angle scattering curve corresponds to \(230\,\text{\AA}\).

Kratky\(^{20}\) investigated small-angle scattering by fibers of natural ramie and regenerated cellulose (viscose) with various

degree of swelling. Proceeding from his model of a thin-layer close packing and using the concept of scattering at small and minimum angles, the author determined the mean value of the distance, consisting of the thickness of the layer and the slit, for ramie (61 Å) and for viscose (80 Å), while the scattering by viscose at small angles was considerably more intense, and at minimum angles considerably weaker, than by ramie. The author explains the former by the large number of slits in regenerated cellulose (up to 15–20% of the total volume) as compared with ramie (2–3%), the slits, in accordance with Babinet’s principle, being regarded by him as scattering centers; and the latter by the regular growth of ramie fibers and the irregular growth of them in regenerated cellulose. The thickness of the layer and slit for ramie (61 Å) agrees well with measurements by other methods (for example, by linear expansion), whereas, if external interference is neglected, i.e., if the system is regarded as loose, an overestimated value is obtained—about 200 Å. Kratky convincingly explained the experimentally found dependence of the intensity of scattering at small angles on the degree of swelling of regenerated cellulose on the basis of his theoretical premises, again revealing the necessity of taking external interference into account.

A more detailed examination of Kratky’s works compels one to assert that, although in constructing the theory the author proceeded from special models of closely packed particles, allowing for various shapes, sizes, and distributions of particles, only the purely qualitative conclusions from his rather cumbersome formulas had practical significance. The quantitative results were obtained by him very approximately and relate to mean values of particle size. As before, here there is an obvious lag in the accuracy of the experimental data behind the subtle theoretical constructions. Apparently, in the complex region of closely packed particles of various shapes it is in general possible to obtain only very approximate, almost qualitative characteristics, which, however, may play an essential role in the study of the submicroscopic structure of substances, especially in combination with other physicochemical methods of investigation.

V. CONCLUSION

The great attention now being paid by physicists to the phenomenon of diffuse scattering of X-rays at small angles shows the importance of investigating objects that are too large for study by the usual methods of X-ray structural analysis and too small for microscopic study. Alongside the electron microscope, a new technique is being created for the study of such objects. In contrast to the electron microscope, its application to materials with high electrical resistance

does not cause any difficulties and does not require the sort of preparation of the specimen that might disturb the submicroscopic structure.

Very important macroscopic properties of many engineering materials depend on their submicroscopic granular structure. This is especially clearly seen in investigations of metals and their alloys, whose mechanical strength, electrical and magnetic properties, plasticity, etc., can sometimes change sharply under thermal or other treatment without any noticeable change in their atomic structure. Guinier^18,39, for example, investigated the supersaturation of a number of solid solutions during their heat treatment, which causes considerable changes in mechanical properties. X-ray photographs taken at ordinary angles revealed no changes whatsoever up to the precipitation of excess atoms of the dissolved metal. At the same time, X-ray photographs of Al—Cu and Cu—Be alloys taken at small angles showed that, before precipitation, copper and beryllium atoms gather into swarms, whose arrangement is determined by the orientation of the crystals of the alloy; in a single-crystal specimen of an Al alloy with 5% Cu, for example, the form of these swarms can be determined from the form of the small-angle diffraction pattern (Fig. 28): the swarms are flat formations situated in the planes of the crystal cube, and from the diffraction pattern one can follow the change in their dimensions (diameter and thickness) with temperature. In another case (Al—Ag, Al—Zn alloys), the swarms of atoms have no definite form or orientation, which is evident from the symmetric small-angle scattering diffraction pattern obtained in the form of a blurred ring; the latter indicates that the swarms are situated close to one another—there is a “local” assembly of atoms before their general precipitation. Likewise, obviously, one can observe the first traces of the so-called “fatigue of metals” before this phenomenon becomes noticeable under the microscope or affects the pattern of atomic arrangement.

Fig. 28. Scattering at small angles by Al alloys with 5% Cu.

Other fields of application of the new method are fibrous materials, where small-angle scattering can be used to study the size and shape of micelles or fibrils, and also to control the production of artificial fibers; asbestos-like materials, dielectrics and lubricants, in which the most important properties are determined by the spatial arrangement of atoms or molecules; the increasingly important field of high polymers; materials from which living organisms are built (muscle fibers); catalysts of all kinds, the internal surface of whi—

... depends on the dimensions and shape of submicroscopic pores (small-angle scattering is currently being introduced as a control method in the production of synthetic rubber and synthetic gasoline[^54]), various clays, whose molecular nature has been studied with great precision, but whose properties are determined by aggregates of individual molecules having colloidal dimensions; large molecules, discussed above, in particular giant protein molecules; microporous materials of every kind, etc.

This brief enumeration is quite sufficient to emphasize the important practical role that the application of the new technique may play in the most diverse scientific and industrial fields. In solving these varied problems connected with the submicroscopic structure of matter, the closest link between science and technology is realized. A detailed description of the experimental work and results obtained by the small-angle scattering method in all the fields just listed was not part of the author’s task. In view of the breadth and diversity of the material, such a task should be the subject of a separate article. The aim of the present review is merely to draw the attention of engineers and scientific workers to this new physical method, whose experimental difficulties are not great and whose application, alongside other methods, can bring unquestionable benefit.

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