STRUCTURE OF GLASS AND METHODS FOR ITS INVESTIGATION BY X-RAY STRUCTURAL ANALYSIS

A. I. Kitaigorodsky, Moscow

From the standpoint of the modern theory of matter, the following models of the structure of a solid body are possible:

1. A single crystal, i.e., a solid body whose mass is a periodic function of three coordinates.
2. A polycrystal, i.e., a conglomerate of small single crystals bound to one another by forces of atomic interaction.
3. An amorphous solid body, whose mass is not a periodic function of the coordinates (or, in modern terminology, has a period equal to infinity).

There are no other possibilities. The concept of a solid body is quite definite in the sense that, for it, we may assume the probability of exchange of positions between any two atoms to tend to zero (naturally, what has been said does not apply to a body whose energetic state is disturbed by some action of the external medium). By this definition we separate liquid and solid bodies, although, at the same time, we emphasize the possibility of considering a solid body as a limiting state of a liquid body. A crystallizing liquid is a polycrystal; a liquid supercooled to the glassy state is an amorphous body, as will be shown below.

In investigating the structure of a given glass, just as, incidentally, in investigating a crystalline substance, two fundamental questions must be resolved; these two questions constitute the “problem of the structure of glass.”

First, it is necessary to assign the given glass to one of the three groups indicated; second, to indicate the forces of bonding between the individual atoms. The solution of the second question gives us knowledge of the character of those chemical compounds whose molecules are present in the solid state as separate units. The problem of the structure of glass, posed in this way, is the same as the question of the structure of any other solid body.

It is obvious that, in resolving the first of the two questions we have posed, the first kind of solid body is excluded from consideration. Glass is not a single crystal. The evidence is so tri-

are trivial and may be omitted. Two possibilities remain: either glass is a polycrystalline substance, the crystallites of which are extremely small (since, either visually or with a microscope, the polycrystallinity of glass cannot be detected), or glass is an amorphous substance in the sense of the definition given above. In order to choose between these two hypotheses, the following experiment must be set up.

If, for simplicity, vitreous quartz is chosen as the object of investigation, then, adopting the hypothesis of the polycrystallinity of glass, we must assume that this object is a conglomerate of crystallites of SiO₂. Knowing the structure of a single crystal of SiO₂, it is easy to calculate the interference pattern that should be obtained if SiO₂ in polycrystalline form is subjected to the action of X-rays.

The X-ray photograph (in the case of exposure with monochromatic radiation) should consist of a series of concentric rings, the mean radius of which is determined by the structure of the single crystal. The sharpness of the rings, i.e. their width in this case, should be determined by the size of the crystallites, namely it should be equal to

\[ \frac{0.95\lambda}{l\cos\vartheta} + b, \]

where \(\lambda\) is the wavelength of the X-ray radiation, \(2\vartheta\) is the angle between the incident and scattered beam, and \(b\) is a constant.

The fact that all good glasses, photographed with monochromatic radiation, give 1–2 concentric diffuse rings on the X-ray photograph has long been known. This was first discovered by Debye and Scherrer around 1920. In the initial investigations the radii of the rings were not related to the chemical composition of the glass; the measurements were carried out imprecisely and contradicted one another. The first work whose experimental result was confirmed by subsequent investigations is that of Randall, Rooksby, and Cooper. The authors investigated several of the simplest glasses, including vitreous SiO₂. The Debye patterns obtained with Mo \(K_{\alpha}\)- and Cu \(K_{\alpha}\)-radiation made it possible to establish firmly the angle \(\vartheta\) corresponding to the maximum intensity of the first ring. On recalculation by Bragg’s formula, it turned out that the found \(\vartheta\) corresponds to an interplanar spacing equal to \(d = 4.33\) Å. In addition, a second ring, much weaker in intensity, was found, to which there corresponded \(d = 1.5\) Å.

On the basis of the indicated experiment the authors concluded that glass is a polycrystalline substance consisting of small crystallites of cristobalite, of size \(1.5 \cdot 10^{-7}\) cm (of the order of 20 mol). Hence a broader conclusion: the term “amorphous” or “vitreous” body differs from the term “crystalline” body only quantitatively, and not qualitatively.

Without calculations the authors find that wollastonite glass consists of crystals of pseudowollastonite and glass \(Na_2B_4O_7\), of the corresponding crystallites of the same composition.

We noted earlier that the hypothesis of polycrystallinity cannot be rejected without special experimental consideration. However, what experimental evidence must be given in order to accept this hypothesis? First of all, if glass is a conglomerate of cristobalite crystals, then the interference pattern of the glass must be calculated exactly from the known structure of cristobalite. However, the interplanar spacing of cristobalite closest to \(4.33\ \text{Å}\) is \(4.05\ \text{Å}\)—a difference of more than \(6\%\) (it must be borne in mind that in measuring \(d\) by the Debye method the measurement error lies in the third digit). Thus even here additional hypotheses are necessary in order to save the basic one. Therefore Randall is forced to admit the possibility of a dependence of the interplanar spacing on the number of atoms forming the crystal.

One could be reconciled to the first circumstance, only for lack of anything better. However, there remains the second, most essential objection. Numerous studies of the recrystallization of metals and alloys show us that, upon heating a polycrystalline specimen, the blurred diffuse rings of the X-ray diffraction pattern begin to narrow, while the mean radius of the ring remains unchanged. This phenomenon is naturally explained by the growth of crystals. If, however, glassy \(SiO_2\) is heated, then, while the width and diffuseness of the principal rings remain unchanged, sharp rings of cristobalite, displaced relative to the principal ones, are suddenly superimposed on the X-ray diffraction pattern. In addition to the objections mentioned above, an essential circumstance is also that, with crystallites of size \(1.5 \cdot 10^{-7}\) cm, the interference pattern from the glass would be strongly dependent on the previous history and origin of the material, which, however, is not observed. These experimental refutations thus make Randall’s hypothesis unacceptable.

Randall’s hypothesis found no supporters. The attitude toward it of all investigators working in the field of the physical chemistry and structure of glass may be characterized by a quotation from an article by G. W. Morey: “From the very limited similarity of the X-ray diffraction patterns of cristobalite and \(SiO_2\) glass, Randall asserts that ordinary glassy quartz contains \(80\%\) cristobalite. Probably none of the subsequent investigators will be so categorical.” Of the works that have appeared from 1930 to the present and are devoted to the problems of the structure of glass—and there are quite a few of these works—only one, published in 1936, defends Randall’s point of view\(^1\).

\(^1\) This is the work of Valenkov and Porai-Koshits, printed in Z. Krist. (95, 195, 1936). The point of view of these authors is supported by Shishakov, who investigated glassy \(SiO_2\) by electron diffraction. Randall himself abandoned his hypothesis of the structure of glass in later works.

The contradictions of the “crystallite” hypothesis make it possible, even by the method of exclusion, to recognize that glass is an amorphous body (in the sense of the definition given above). This proposition is supported, on the one hand, by the serious theoretical works of Zachariasen and in part Hagg concerning the atomic arrangement in glasses and, chiefly, by the experimental works of Warren and his collaborators, carried out very precisely and elegantly.

The theory of X-ray interference and experiment thus compel us to suppose that the arrangement of atoms in glasses possesses no symmetry. At the same time, this arrangement is not entirely disorderly either, since the distances between the centers of atoms have a minimum determined by the size of the atoms entering into the composition of the glass. Such an arrangement of atoms, according to the theory of X-ray interferences, gives a Debyegram consisting of diffuse rings, the radii of which are connected with certain most frequently occurring interatomic distances. In any case there is no reason to doubt that the general idea in this consequence of the theory and experiment of X-ray structural analysis is correct. All the characteristic properties of “glasses” (and, of course, of organic ones as well) are natural for substances with a disordered atomic arrangement (isotropy, absence of a melting temperature, conductivity, absence of a chemical formula). These natural propositions were made concrete by Zachariasen. This investigator, on the basis of very interesting arguments, arrives at a concrete form of a three-dimensional, aperiodic lattice, at the nodes of which the atoms of the glass are situated (the concept of a lattice here is quite specific).

It is natural to expect that consideration of the character of the principal glass-forming oxides will make it possible to draw a number of conclusions concerning the atomic arrangement. Goldschmidt had already connected the ability of an oxide of the type \(A_mO_n\) to form glass with the ratio of the radii of the ions \(R_A : R_O\). From the fact that \(R_A : R_O\) in different glasses fluctuates between 0.2 and 0.4, Goldschmidt concludes that a tetrahedral arrangement of oxygen atoms around the central cation is a characteristic and necessary concomitant of an oxide’s ability to form glass. In this assumption, undoubtedly, there lies a grain of truth; however, BeO, for which \(\frac{R_A}{R_O} = 0.3\), does not form glass. Obviously, the root of the question of the glass-forming ability of oxides lies deeper. This other factor of glass formation, in the opinion of Zachariasen (and also of a large number of other investigators), must be a small difference in the values of the potential energy of the atoms in the glassy and crystalline oxide. The relation between the energies of the crystalline and glass-like modifications of an oxide also makes comprehensible the small rate—

The results of the studies of Valenkov and Porai-Koshits, as well as Shishakov, are due, in all probability, to an impurely conducted experiment and the arbitrariness of its explanation.

THE STRUCTURE OF GLASS AND METHODS OF STUDYING IT

the rate of transformation of the vitreous form into the crystalline one, i.e., explains the stability of the corresponding glasses and, consequently, also answers the question of the glass-forming ability of the oxide.

From this proceeds Zachariasen’s basic assumption that glasses are similar in their structure to the corresponding crystalline modifications. The crystal chemistry of silicates makes it possible to accept that in glass (as also in silicates) cations are surrounded by anionic polyhedra. Just as in crystals, the polyhedra, joining with one another, form a spatial lattice. Crystal chemistry teaches us that the energy of the ionic bond of a group of atoms is determined chiefly by the coordination number, and not by the way in which the anionic polyhedra are connected with one another. Therefore the difference between glass and crystal lies in the inequality among the angles between those bonds of atoms which are common to two polyhedra. Vitreous quartz, according to what has been said above, consists of SiO₄ tetrahedra, as does cristobalite. The distances between atoms within a given tetrahedron are the same as for the corresponding crystal. Each oxygen atom is bonded to two Si atoms (i.e., to two tetrahedra). This atomic arrangement possesses no symmetry (nor translational symmetry) only because the angle between the two O—Si bonds is different for all oxygen atoms. Thus the separate tetrahedra are joined with one another randomly and disorderly, although an individual region of the vitreous substance is similar to an individual region of the crystal.

In order that an oxide \(A_mO_n\) may form both a glass and a crystal, it is necessary, Zachariasen thinks, that the crystalline lattice possess an energy close to the potential energy of the three-dimensional aperiodic lattice, at the nodes of which are located the atoms of the oxide in the vitreous state. For this, in turn, the following requirements must be fulfilled: 1) the atom A must have a small coordination, 2) the oxygen atom must be bonded with no more than two atoms A, 3) the oxygen polyhedra must have only common vertices with one another (not edges and not faces), 4) a minimum of three angles of each polyhedron must belong simultaneously to other polyhedra as well (this condition is, properly speaking, the condition for the occurrence of a three-dimensional lattice).

If the smallest coordination number is taken to be 3, then, bearing in mind the remaining requirements as well, oxides of the type \(A_2O_3\)—with coordination number 3, \(A_2O_5\) and \(AO_2\) with coordination number 4—may be mentioned as glass-forming oxides. These conditions make it possible to assign to the glass-formers the following oxides: \(B_2O_3\), \(P_2O_3\), \(As_2O_3\), \(Sb_2O_3\), \(SiO_2\), \(GeO_2\), \(P_2O_5\), \(As_2O_5\), \(Sb_2O_5\), \(V_2O_5\), \(Nb_2O_5\), \(Ta_2O_5\), which agrees excellently with experiment. It is interesting to note that the glass-forming properties of \(Nb_2O_5\) and \(Ta_2O_5\) were found after the publication of Zachariasen’s work.

With regard to more complex glasses of formula \(A_mB_nO_l\), on the basis of the same reasoning, a whole series of interesting remarks can be made (B are glass-forming cations, and A are the other cations).

B may be surrounded by triangles of O (trigonal arrangement) or tetrahedra of O (tetrahedral arrangement). The number of B atoms per one oxygen atom can be calculated from the assumption of the three-dimensional continuity of the tetrahedral and trigonal distribution. In the first case \(0.33 < n < 0.50\), in the second, \(n = 0.67\). A calculation of \(n\) for all known glasses gives a surprisingly good confirmation of the assumption just stated. Only for a small number of glasses with a high percentage content of \(\mathrm{B_2O_3}\) is \(n\) somewhat greater than 0.5.

In order that the fundamental requirement of this theory be fulfilled—namely, that the difference between the potential energies of the crystalline and glassy states be small—it is necessary that the atoms A have a large radius and a small charge. This requirement is satisfied by K, Na, Ca, Si, Ba, Pb.

The first part of Zachariasen’s arguments, concerning the structure of glass, finds its most decisive confirmation in X-ray experiment. The results of the interpretation of X-ray photographs of glasses (mainly the work of Warren; see below) leave no room for even slight doubt that glass is an amorphous body (where by an amorphous body, as has repeatedly been emphasized, is meant a body whose mass distribution is not a periodic function of the coordinates and does not depend on time). Hypotheses contradicting the above, such as, for example, Randall’s crystallite hypothesis and others, do not withstand the criticism of modern experiment. Such a view of the structure of glass is at present accepted by the overwhelming majority of physicists, chemists, etc., working in the field of glass (Tammann, Morey, Weyl, Zachariasen, Hegg, and others).

However, the question of the causes of glass formation by a given compound and of the impossibility of the existence in the glassy state of some other compound is at present, to a large extent, unresolved.

Zachariasen’s very fruitful point of view nevertheless cannot in this respect be recognized as general, suitable for any glass-forming compounds.

Interesting considerations that make it possible to explain the cause of glass formation of some bodies not subject to Zachariasen’s theory may be found in Hegg’s recently published work.

This investigator quite rightly draws attention to the fact that Zachariasen’s reasoning is applicable only in the case when the oxygen atoms form a three-dimensional lattice. Naturally, Zachariasen’s theory will lead us to false results if it is applied, for example, to silicate glasses whose corresponding crystals are formed by chains or layers of oxygen polyhedra. In this case one cannot assume the existence of a three-dimensional aperiodic lattice if one considers that the coordination numbers of the atoms in the glass and in the crystal are identical. The opposite conclusion is a priori incorrect, for in this case the difference in the energies of arrangement of the atoms in the glass is too great

and the crystal. To allow the existence of chains and layers in glass would mean recognizing it as anisotropic. Thus the limited applicability of Zachariasen’s arguments is clear.

Without touching upon questions of structure proper, Hagg considers possible causes of glass formation. The author believes that, in the case of compounds that have a tendency to form glasses, there is no mobility of ions and there is a relatively strong interionic field in the melt. Naturally, these two factors, in their turn, are determined chiefly by the complexity of the structure or, more precisely, by its “island” character. When there is a large difference between the forces of bonding within a group (island) and between groups, the melt consists of fragments of the crystal lattice, which hinder crystallization. As an example Hagg cites the crystallization of $\mathrm{Ca(BO_2)_2}$. The melt of $\mathrm{Ca(BO_2)_2}$ consists of boron-oxygen chains and calcium ions. For the growth of a crystalline nucleus there is required either a very slow process of joining the fragments of chains to one another or the destruction of the chains, for which very great energy is needed. Naturally, in Hagg’s opinion, when a certain limit of mobility of ions and chains is passed, we obtain a solid glass. Hagg considers it possible to extend all the considerations given above to the case of any glass-former.

Hagg’s theory undoubtedly contains a number of very valuable indications that make it possible to look more deeply into the processes of glass formation; however, the present writer considers the author’s claims to the universality of his explanations to be insufficiently founded. First of all, Hagg’s explanations of the process of formation of vitreous quartz seem to us unconvincing. The assumption that the strength of the $\mathrm{Si—O}$ bonds is equal at low temperatures and, at the same time, that there are differences between them at the melting temperature has no serious grounds; to say that part of the original crystal lattice is contained in the melt and to conclude from this that crystallization is difficult in this case is, at the very least, unconvincing.

Thus, while agreeing with Hagg that the phenomena of polymerization, association, island formation, etc., play a major role in the processes of glass formation, one cannot regard his arguments as adequate to all the requirements of a theory of glass formation.

That ionic crystals do not form glasses, i.e., that the bonds in silicates are not ionic in character, and also that association processes play a major role in the processes of glass formation, was pointed out repeatedly by Weyl and many others.

Both from the standpoint of physicochemical considerations and from the standpoint of physical theories, the process by which glass-forming properties change as a function of the composition of the glass (for the simplest cases) is approximately the same. It is known that when $\mathrm{Na_2O}$ is added to $\mathrm{SiO_2}$, the viscosity of the melt falls, and in strongly basic glasses the capacity for glass formation decreases. Sodium metasilicate, under ordinary cooling, already does not yield a glass.

According to Weyl, an increase in the number of Na ions in the glass hinders asso-

…association of SiO$_2$ molecules, since each Na ion surrounds itself, as it were, with a shell of SiO$_2$ molecules. The ordered and directed action of the ion on the SiO$_2$ molecule explains the increase in the probability of formation of a crystallization center. Obviously, everything said above can also be interpreted in the spirit of Zachariasen’s theory. True, the latter, in contrast to Weyl’s theory, also takes into account an increase in the number of oxygen atoms, which changes the bond strengths.

It is known that Li lowers the viscosity of SiO$_2$ more than Na and K. From Weyl’s point of view this is connected with the strength of the corresponding ionic fields. The latter, in turn, are connected with the size of the ion and, consequently, with the interatomic distances in the corresponding silicates. Thus Weyl’s and Zachariasen’s explanations of the glass-formation processes of the simplest silicates are closely related.

We have already noted above that Zachariasen’s and Hagg’s arguments concerning the ability of substances to form glass are based on the assumption that the energy of the arrangement of atoms in glass is very close to the energy of the corresponding crystal lattice. However, in a number of cases this assumption is not confirmed by experiment. It is known that the heats of fusion of silicates are not small in comparison with those of other substances; moreover, it is known that many silicates forming glasses possess higher latent heats of fusion than other silicates that do not solidify into glass. For example, the latent heat of crystallization of orthoclase is equal to 70 cal/g, that of leucite 26 cal/g. However, orthoclase does not crystallize, while leucite does not form glass. A whole series of compounds that never form glasses have much smaller heats of fusion. For example, the latent heat of fusion of KNO$_3$ is equal to 25 cal/g.

Experimental verification of all the hypotheses cited above is at present developing in two directions. First, numerous investigations of the properties of glass-forming compounds have as their aim the search for and determination of the characteristic properties of a glass former. This problem, the solution of which would allow us to explain why a given substance is a glass former and another is not, is still far from being solved.

Much more fruitful are investigations of the structure of glass, namely by methods of X-ray structural analysis. Even now the great possibilities of this method are evident. A calculated, profile-measured X-ray diffraction pattern of glass (for the time being, this has been done only for the simplest glasses) makes it possible, with very great precision and quite unambiguously, to determine the mutual arrangement of atoms in glass, and consequently not only the nature of the structure of the glass, but also the nature of the existing chemical bonds between individual atoms (as well as the nature of the chemical compounds), and also the interatomic distances.

The beginning of this work was laid by Warren in 1934, i.e., only 4 years ago. In the first work the simplest glasses, of composition SiO$_2$, GeO$_2$, were investigated. The analysis of the X-ray interference pattern was carried out at first by the method of trials, namely:

a hypothesis was made concerning a definite atomic arrangement, the theoretical interference pattern was calculated, and the result was compared with experiment. If the mutual arrangement of the atoms in the scattering body is known, as well as the atomic factors, i.e., the scattering magnitude for a given atom as a function of the scattering angle, then the theoretical interference curve can be calculated by the formula:

\[ I=\sum_m\sum_n f_m f_n \frac{\sin\left(\dfrac{4\pi \sin\theta}{\lambda}\right)r_{mn}} {\dfrac{4\pi \sin\theta}{\lambda}r_{mn}}, \tag{1} \]

where \(f_m\) is the atomic factor, \(r_{mn}\) is the distance between atoms \(m\) and \(n\), \(\lambda\) is the wavelength of the X-ray radiation, and \(2\theta\) is the angle between the incident and scattered rays. There is no need to know exactly the mutual arrangement of an infinite number of all atoms with respect to one another, if it is assumed that the mutual arrangement around each given atom is the same within small limits. This assumption is entirely legitimate—it only emphasizes the fact that the interference effect is determined by a series of constant interatomic distances.

The calculation showed the validity of the arrangement given in Table 1.

TABLE 1

The latter Si and O atoms included in the calculation are taken into account in the form of a spherical unit with atomic factor

\[ f=f_{\mathrm{Si}}+2f\, \frac{\sin 1.60\,\dfrac{4\pi\sin\theta}{\lambda}} {\dfrac{4\pi\sin\theta}{\lambda}}, \]

then, for each atom, the distribution for all atoms is continuous.

What, then, is the correct arrangement of atoms found in glassy quartz? From Table 1 we see that each Si atom is the center of a tetrahedron, at whose corners oxygen atoms are located at a distance of \(1.60\) Å. Each O atom has two silicon neighbors. Oxygen lies on the straight line joining two silicon atoms. Thus only two oxygen tetrahedra have a completely definite arrangement relative to one another. The arrangement of the remaining tetrahedra relative to the given one is completely disordered. As we see, Warren’s experiments brilliantly confirm Zachariasen’s theory as regards the structure of glass.

It is interesting to note that the results of this work make it possible at once to reject the hypotheses of association of \(\mathrm{SiO}_2\) molecules in glassy quartz and make it possible to answer the question of the chemical bond between atoms. Evidently it makes no sense to speak of the existence of a molecule in glassy quartz. Naturally, there are no grounds for expecting the same situations in all glasses (for in this respect the two existing theories, those of Zachariasen and Hugg, are contradictory).

The structure of amorphous \(\mathrm{SiO}_2\) and \(\mathrm{B}_2\mathrm{O}_3\) was determined recently by Warren and his coworkers once again, but now with the aid of Fourier analysis generalized to the case of noncrystalline bodies. By a direct calculation one can compute the amount of mass (in electron units) enclosed between two spheres of arbitrary radii, constructed about any center (any atom). The formula has the form:

\[ \sum_m K_m\,4\pi r^2(\rho_m-\rho_0) = \frac{2r}{\pi}\int_0^\infty sJ(s)\sin rs\,ds, \]

where \(\rho_m\) is the electron density of the \(m\)-th atom, \(K_m=\dfrac{f_m}{f_e}\), with \(f_m\) the atomic factor of the \(m\)-th atom, and

\[ f_e=\frac{\sum f_m}{\sum Z_m} \]

(\(Z_m\) is the atomic number of atom \(m\)), \(\rho_0\) is the mean number of electrons per unit volume, \(r\) is the distance from some atom, \(s=4\pi\dfrac{\sin\theta}{\lambda}\),

\[ J(s)=\frac{I-\sum f_m^2}{f_e^2}, \]

where \(I\) is the experimentally measured scattering intensity without change of wavelength, in electron units. The summation is carried out everywhere over all atoms of the molecule.

In this case the direct result of the experiment is the curve of the radial distribution of electron mass. The task of further analysis consists in identifying one or another peak of the curve with a definite atom.

The curve \(\sum K_m 4\pi r^2 \rho_m\), calculated from the X-ray intensity curve obtained from glassy \(B_2O_3\), is given in Fig. 1. The first peak on this curve corresponds to a distance of \(1.39\ \text{Å}\), the area of the peak being equal to 470 electrons. Knowing from studies of crystalline \(B_2O_3\) that the \(B—O\) distance is \(1.36\ \text{Å}\), we confidently take the first peak of the curve to be the sum of the masses of the boron atoms nearest to the given O atom. The number of oxygen atoms nearest to boron, or, conversely, the number of boron atoms nearest to oxygen, is calculated from the area of the peak and the effective numbers of electrons in the atoms. It turns out that the boron atom is surrounded by three oxygen atoms, and the oxygen atom is bonded to two boron atoms; in doing so we assume that the boron atoms

Fig. 1.                Fig. 2.

lie approximately opposite one another. The remaining peaks of the radial-distribution curve make it possible to determine the distances between neighboring boron atoms \((B—B = 2.78\ \text{Å})\) and between neighboring oxygen atoms \((O—O = 2.40\ \text{Å})\). All the distances measured in this way fit excellently into the scheme of an infinitely extended triangular network.

In Fig. 2, belonging to Zachariasen, a projection of the atomic arrangement of \(B_2O_3\) is shown.

In an entirely analogous manner, an analysis of \(SiO_2\) glass was carried out, which fully confirmed the investigation described above by the method of trial models.

Warren’s school did not limit itself to the study of glasses consist-

…consisting of one oxide. At the present time we have an analysis of the structure of the glass \(Na_2O—SiO_2\) (Warren and Loring) and of the glass \(PbO—SiO_2\) (Bär). These investigations were carried out, of course, by the trial method. The structure of sodium glasses was determined on the basis of roentgenograms obtained from seven glass specimens with an \(Na_2O\) content from zero to \(46\%\).

The guiding thread of the investigation was the idea that the number of neighbors of a given atom is the same as in the corresponding silicate in the crystalline state. If the molecular fraction of \(Na_2O\) in the glass is \(x[a\,SiO_2(1-x)]\), then the relative number of atoms \(Na\) is \(2x\), \(Si\) is \((1-x)\), and \(O\) is \((2-x)\).

In the absence of \(Na\) atoms, each \(O\) atom must be bonded to two \(Si\) atoms. In the presence of \(Na\) atoms one must assume that only part of the oxygen atoms is bonded to two \(Si\) atoms (denote them by \(O_2\)); the oxygen atoms \(O_1\) are bonded to one \(Si\) atom and one \(Na\) atom. The greater the percentage of \(Na_2O\) in the glass, the more the ratio of the number of oxygen atoms to the number of \(Si\) atoms differs from two, and consequently the greater becomes the number of \(O_1\) atoms. It is easy to show that the number of \(O_1\) atoms must be equal to the number of \(Na\) atoms1. Fig. 3 shows a preliminary scheme (in it only the number of neighbors is essential) of sodium glasses.

Fig. 3.

Fig. 4.

For carrying out the calculation by formula (1), it is necessary to count the number of neighbors for all types of atoms in the glass. For example, for a silicon atom: its nearest neighbors are 4 oxygen atoms at distance \(y_1\) [the corresponding term in formula (1) will be \(4f_{Si}f_O \dfrac{\sin y_1 s}{y_1 s}\)].

\[ 1O_1 + 2O_2 = 4Si, \]

(the number of \(Si—O\) bonds ending at oxygen and silicon atoms) after the simplest transformations we have:

\[ \alpha O + 2(1-\alpha)O = 4Si;\qquad 2-\alpha = 4\frac{Si}{O} = 4\frac{1-x}{2-x};\qquad \alpha=\frac{2x}{2-x}; \]

\[ \frac{O_1}{Na}=\frac{\alpha O}{Na}=\frac{2x}{2-x}\cdot\frac{2-x}{2x}=1. \]

The atoms \(O_2\) surrounding silicon are bonded to the second silicon, located at a distance \(y_2\) from the initial one. The number of these atoms is calculated as follows: the total number of Si—O bonds will be

\[ O_1 + 2O_2 = xO + 2(1 - x)O = (2 - x)O. \]

Thus the number of Si—O bonds falling on the atoms \(O_2\) is equal to

\[ \frac{2(1-x)O}{(2-x)O} = \frac{2-3x}{2-2x}. \]

Therefore, of the 4 O atoms surrounding the silicon atom,

\[ 4 - \frac{2-3x}{2-2x} \]

of them are also bonded to one Si atom. The number

\[ \frac{4-6x}{1-x} \]

will be the number of nearest silicon neighbors to a silicon atom [we obtain the term of the sum which in formula (1) is

\[ \frac{4-6x}{1-x} f_{\mathrm{Si},\mathrm{Si}} \frac{s' n_{y_2}s}{y_2 s} \].
]

Without argument it is clear that the number of oxygen atoms next in distance (at the distance \(y_3\)) will be equal to

\[ 3 \frac{4-6x}{1-x}. \]

One more group of nearest atoms can be taken into account; farther on, however, the interatomic distances become increasingly indefinite, beginning to depend on the orientation of the tetrahedral groups. We may, starting from a certain point, regard the distribution of matter around the initial atom as continuous and without maxima. It is known that this part of the sum (1) affects the values of \(I\) at small angles \(\theta\). Therefore, not wishing to take it into account, the authors did not use the initial part of the experimental interference curve, thereby obtaining the right to truncate the sum (1) at the 6th–7th term. After the working formula has been constructed, the painstaking work begins, consisting in the selection of the values \(y_1\), \(y_2\), etc., which correspond most closely to the experiment.

In Fig. 4 we give the theoretical and experimental curve for one of the glasses, a table of interatomic distances, and a scheme of the atomic arrangement (Fig. 5).

Each silicon is enclosed in a tetrahedron of oxygen atoms. An oxygen atom either belongs simultaneously to one more tetrahedron or is bonded to a sodium atom. It is easy to see that the whole structure may be regarded as an infinitely extended network composed of Si atoms and \(O_2\) atoms, with the sodium atoms inserted into the various voids of the network so as to be in contact with as large as possible a number of \(O_1\) atoms. The structure found fully supports the theoretical work of Zachariasen.

It is very important that, even in the case of binary glass, X-ray analysis makes it possible to assert decisively the absence of any molecular groupings whatever in the glass. Moreover: the concept of a molecule loses its meaning for glass. Obviously, for silicate glasses this is unquestionably true, since in a separately existing molecule, for example \( \mathrm{Na_2O \cdot SiO_2} \), each Si atom would be surrounded by only three oxygen atoms. By calculating the energy of such a configuration, one can convince oneself of its instability.

Fig. 5.

Fig. 6.

We shall not dwell on the investigation of lead glasses. We shall give only the scheme of the atomic arrangement (Fig. 6) and the table of interatomic distances.

All conclusions drawn with respect to sodium glasses are also valid for lead glasses.

Conclusion

The short period of time during which attempts were made to establish the structure of glasses has proved very fruitful. At present one may regard as definitively settled the question of what kind of bodies glass belongs to, and also of what the amorphous state of matter is. The methods of X-ray structural analysis make it possible to begin and broadly develop the investigation of glasses more complex than oxides and the simplest silicates. The results of these investigations, by giving the immediate atomic arrangement in glass, make it possible to judge, from interatomic distances, the chemical bonding of atoms and, consequently, the nature of the compounds present in glass.

As for the second direction of work on the theory of glass, namely the theory of the nature of the glass-forming ability of oxides, very little has yet been done here, and from the existing theories

TABLE 2

TABLE 3

not one can be applied without reservations. However, here too we envisage a whole series of experiments, chiefly by methods of X-ray structural analysis. The writer of these lines has in mind, mainly: (a) investigations at different temperatures of the same glasses; (b) investigation of glasses obtained at different rates of cooling, which will undoubtedly help to solve this problem as well. It should be remembered that in this field literally nothing has been done.

Literature

Theory of the Structure of Glass

G. Tamman, The Glassy State, 1933.
G. Morey, Constitution of Glass, J. Am. Cer. Soc., 17, 315, 1934.
E. Berger, Contributions to Theory of Glass, J. Am. Cer. Soc., 15, 647, 1932.
W. Zachariasen, Atomic Arrangement in Glass, J. Am. Chem. Soc., 54, 3841, 1932.
W. Rosenhain, Structure and Constitution of Glass, Published by Soc. of Gl. Tech., Engl., 1927.
G. Hagg, Vitreous State, J. Chem. Phys., 3, 42, 1935.

X-ray Investigation of Structure

J. Randall, H. Rooksby and B. Cooper, Structure of Glasses, J. Soc., Gl. Techn., 14, 219, 1930.
B. Warren, X-Ray Diffraction of Vitreous Silica, Z. Krist., 86, 349, 1933.
B. Warren and A. Loring, X-Ray Diffraction Study of Structure of Soda-Silica Glass, J. Am. Cer. Soc., 18, 269, 1935.
B. Warren, X-Ray Determination of Structure of Glass, J. Am. Cer. Soc., 17, 249, 1934.
G. Bair, The Constitution of Lead Oxide-Silica Glasses, J. Am. Cer. Soc., 19, 339, 1936.
B. Warren and H. Krutter, Fourier Analysis of X-Ray Patterns of Vitreous SiO₂ and B₂O₃ (in preparation).

Scattering Formulas

Danilov, Theory of the Scattering of X-rays in Liquids, Uspekhi Fiz. Nauk, 14, 449, 1934.

1. Composing the evident equality: ↩