Analysis of Crystals and Atomic Forces

W. H. Bragg and W. L. Bragg1

The unit of crystalline structure may be called that smallest portion of a crystal which, when repeated without change in character or orientation as many times as is necessary to fill a certain space, gives an ordinary crystal. This smallest portion contains within itself all kinds of atoms of the given crystal, all types of bonds, and all the properties possessed by the crystal as a whole. In investigating the physical properties of a crystal, we study the physical properties of the unit of crystalline structure. If it proves possible to find the number and arrangement of the atoms within such a unit, then one may hope to reveal also the role of each atom in producing the characteristic properties of the unit. The methods of analysis by X-rays provide a simple means of finding the number of atoms in the unit of structure and gradually shed more and more light on the question of their arrangement.

This character of the development of the study of the crystal has its analogies in the history of physics and chemistry. Some of the properties of a substance as a whole are already present in the individual atoms or molecules of which the substance is built. Such properties are manifested chiefly in the gaseous or liquid state of the substance, when the atoms and molecules are not bound by crystalline form. An important step forward was the discovery of the structure of the smallest quantity, i.e. of the molecule or even the atom, in which the properties just mentioned are already fully manifested. Thereafter the role of each atom in the molecule is investigated. The deeper this analysis goes, the more possible becomes the reverse synthetic process and attempts to build substances with any desired properties.

To confine oneself to the study of the properties of a substance in the gaseous and liquid states is in fact very restrictive. We circumvent this limitation by investigating with X-rays the unit of crystalline structure, in which all the properties of the solid state are manifested.

A crystal possesses elasticity, thermal expansion, thermal conductivity and electrical conductivity, dielectric properties, optical activity; moreover, all these properties may be not only scalar, but also vectorial. As a simple example let us take quartz, in which many crystalline properties—and very remarkable ones at that—are found to be due to the unit of its crystalline structure. We now know that the unit of the structure of quartz consists of three molecules of \(\mathrm{SiO_2}\), and we have advanced considerably in understanding the relative arrangement of the nine atoms of this unit. The ability to rotate the plane of polarization of light, if it is taken as an example of a physical property, is a property of the individual crystalline unit, just as it is of the whole crystal; it follows from the arrangement of the atoms within the unit. It is not a property of the molecule, still less of the individual atoms. We must consider this phenomenon and other properties of the crystal in dependence on the structure of the crystalline unit, and ultimately on the forces manifested by each atom—in other words, on the structure of the atom. The crystalline unit rivals the atom and the molecule in its usefulness, and very possibly also in its importance.

Of course, one of the chief aims of science is to discover the mode of action of atomic forces; the investigation of the unit of crystalline structure gives exceptionally important help in this. Before turning to an account of the information obtained in this way, it is necessary to look over what has been learned by other methods. In this connection, a brief outline must be given here of the structure of the atom in the form in which it is established by other fields of investigation.

It is assumed that the atom consists of a nucleus surrounded by electrons. Each electron carries a negative charge of \(4.77 \cdot 10^{-10}\) electrostatic units. The nucleus is positively charged; its total charge is an integral multiple of the same electrical unit. Consequently, the positive charge of the nucleus will be \(Ne\), where \(e\) is the electronic charge with positive sign, and \(N\) is an integer called the “atomic number.” In an uncharged atom the nucleus is surrounded by \(N\) electrons, each with charge \(e\). The nucleus itself has a complex structure; its dimensions, however, are very small in comparison with the dimensions of the surrounding electronic system. The whole mass of the atom, with the exception of a negligible fraction belonging to the electrons, is associated with the nucleus.

With the exception of the mass of the atom and the properties determined by it, the other physical and chemical properties of the atom apparently depend only on the magnitude of the nuclear charge. For example, all atoms with a nuclear charge of \(76e\) are, both chemically and physically, atoms of lead, although the atomic weights of lead isotopes may range from 206 to 214. Each cell of the periodic system corresponds to an element with a definite nuclear charge. In hydrogen the nuclear charge—

...one unit, helium—two units, and so on to the end of the periodic system, to uranium with atomic number 92. In chemical transformations of atoms, the external electronic system surrounding the nucleus changes, but the nuclei remain unchanged, thus preserving the individuality of the atoms. Only in the case of radioactive transformations, where the charge of the nucleus changes as a result of the emission of \(\alpha\)- and \(\beta\)-particles, does the transformation of elements into one another occur.

Very little is known about the arrangement of electrons around the nucleus and about the nature of the forces that hold the electrons in their places, or orbits. Bohr’s theory, elaborated in detail by Sommerfeld, predicted with astonishing accuracy the character of spectra; recently Bohr has expanded his original theory, attempting to give an explanation of the groupings of elements in the periodic system. This theory, however, has not received an exact expression, except for the simplest atomic systems.

Yet even in the absence of a complete theory of the structure of atoms, it is possible to give a partial explanation of valence; this explanation is of considerable interest in connection with the structure of crystals.

From time to time various electronic hypotheses of chemical compounds and valence have appeared, and the discovery of the connection between the atomic number and the place occupied by an element in the periodic system has greatly strengthened these hypotheses. As an illustration, let us consider a series of consecutive elements of the periodic system, for example: sulfur, chlorine, argon, potassium, calcium. The atomic numbers, or the numbers of electrons surrounding the nucleus, for these five elements are respectively equal to 16, 17, 18, 19, and 20. The argon atom does not enter into any chemical compounds with other atoms, and its valence must be equal to zero. We must suppose that the 18 electrons surrounding the argon nucleus are arranged in some manner in the form of an extremely stable system with a weak external field. In the next atom of this series, potassium, the nuclear charge consists of 19 units, and the neutral atom is surrounded by nineteen electrons. Potassium is a typical monovalent electropositive element. In solutions of potassium salts its atoms are found in the form of ions with one positive charge, whence it follows that one of the electrons has been lost. This difference in the properties of argon and potassium is explained by the fact that in potassium, as in argon, 18 electrons form a very stable system around the nucleus, but the 19th electron of potassium no longer enters into the composition of this stable system and is easily detached from the atom. From this point of view the potassium ion is extraordinarily similar to the argon atom, differing only in that the charge of its nucleus is \(19e\), while that of argon is \(18e\). Thus, upon the weak field characterizing the system of 18 electrons in the potassium ion, there is superimposed an electrostatic field corresponding to the charge \(+e\) of the whole atom. The tendency to lose an electron and to turn into a positive ion determines the chemi-

chemical properties of the potassium atom. In the calcium atom, the transition to an electronic configuration of the argon type occurs upon the removal of two electrons. On the other hand, chlorine has one electron fewer than argon, and in order to realize the argon configuration an additional electron must be captured by chlorine. In this case the charges of the electrons exceed the charge of the nucleus by one unit; therefore the chlorine ion is monovalent and electronegative. Similarly, sulfur is electronegative and divalent.

This view was developed in detail by Kossel1, from whom we borrow the diagram in Fig. 1. Here the number of electrons in the atom

Fig. 1.

of each element is plotted along the ordinate axis, while the atomic number is plotted along the abscissa axis. In an uncharged atom these two numbers coincide; therefore the small circles representing neutral atoms lie on one straight line inclined at an angle of 45°. An atom becomes an ion by acquiring or losing electrons; in this case the number of electrons in the atom is indicated by a point. The diagram illustrates the following conclusion of Kossel: “Elements with a sharply polar character, situated near the inert gases, have, in the ionized state, exactly

the same number of electrons as in the corresponding inert gas.” At the level of each noble gas there is a horizontal row of points representing ionized atoms having the same number of electrons when they exhibit their full valency. This regularity is undoubtedly close to the inert gases, where a very stable system can be attained with only a small change in the total number of electrons. Deviations are observed in the middle of the longer periods, where these conditions are absent and where, one may suppose, there is a tendency toward some intermediate stable system.

These views on the nature of atomic structure can now be compared with the results of X-ray analysis. It immediately becomes clear that the probability of these views is strongly supported by new data on the structure of such polar compounds as, for example, potassium chloride. In a crystal of potassium chloride each potassium atom is symmetrically surrounded by six chlorine atoms, and each chlorine atom by six potassium atoms. The potassium atom does not form a molecule with any one of the chlorine atoms, but extends its attraction to several neighbors. This becomes understandable if the crystal is regarded as an aggregate of potassium and chlorine ions bearing opposite signs and held together by electrostatic forces; these forces correspond to the valence bond, which may be subdivided in any manner. The equality of the numbers of atoms of both elements in solid potassium chloride is not the result of the pairing of atoms into potassium chloride molecules, but is caused by the necessity for an equal number of oppositely charged ions in order to obtain an electrically neutral structure. The same is true also for more complex structures, for example calcite. In this case the calcium ions have a double positive charge, and the group \(\mathrm{CO_3}\) a double negative charge; these ions are arranged so as to form a structure very similar to that of potassium chloride.

A large number of crystals belong to the same type. In calcium fluoride we have an example of a crystal in which for each positive ion there are two negative ions. Each calcium ion with a double charge is surrounded by eight fluorine ions with unit charges, and each fluorine ion is surrounded by four calcium ions. The structure of fluorspar represents the most symmetrical way of realizing such an arrangement. In zinc sulfide, ions of one kind are surrounded by four ions of the opposite sign. We do not know the reason for the difference between this structure and the rock-salt type, but here too we see the same absence of any indication of grouping into molecules; in \(\mathrm{NH_4Cl}\) the \(\mathrm{NH_4}\) ion is surrounded by eight chlorine ions and conversely. The number of complex salts that have been fully analyzed is very small, but most of them probably correspond to one of these ionic groupings. The replacement of a simple

…of an atom by a complex ion may, to be sure, disrupt the crystalline…

Each carbon atom in this structure is surrounded by four other atoms. And here there is not simply an aggregate of molecules, whereas in polar compounds the crystal is an association of charged ions; here there are no ions, and the entire crystal must be regarded as a single molecule. The structure of diamond is the best example of a system all of whose bonds belong to the type that we have called “bonds by shared electrons”; the properties of diamond are characteristic of structures with bonds of this kind.

We have contrasted above two types of bonds that connect atoms into molecules and into crystals: polar and non-polar bonds. In such cases as the alkali-halide compounds, on the one hand, and the molecules of chlorine, oxygen, or nitrogen, on the other, the type of compound is expressed quite clearly. There are, however, compounds whose interpretation from this point of view is not so clear.

According to Kossel’s theory, in the molecule \(N_2O_5\) the outer electrons of two nitrogen atoms, 10 in number, must pass to the oxygen atoms in order to complete the octet groups in the \(L\)-shell; in this case each oxygen atom in its outer arrangement will resemble neon1. The molecule consists of two nitrogen structures with a positive charge \(5e\), connected by electrostatic forces with five oxygens, each with charge \(-2e\). Langmuir assumes that in this molecule too there are “shared electrons.” His model of the \(N_2O_5\) molecule may be represented as follows:

\[ \begin{array}{ccccc} & O & & O & \\ & | & & | & \\ O = N - O - N = O. \end{array} \]

Here a single bond represents a pair of electrons shared by two atoms; a double bond represents two pairs of “shared electrons.”

In many cases the distinction between these two interpretations is more formal than real. For example, Kossel represents the \(CCl_4\) molecule as shown in Fig. 2.

Fig. 2.

Here carbon has a fourfold positive charge, and the negatively charged chlorines are held by electrostatic attraction—

by attraction. In Langmuir’s model, the pair of electrons is common to carbon and to each chlorine atom, as a result of which carbon has an outer “shell” of 8 electrons, like the stable structure of argon. Apparently it is of little importance whether these electrons are regarded as part only of the chlorine structure taken separately, or as the outer shell of the carbon atom. There are cases, however, when these two models lead to different physical consequences. The group CO₃ in calcite may serve as an example. It is an ion with a double negative charge. Kossel believes that the three oxygen atoms have taken 4 \(L\)-electrons from the carbon and have each captured two additional electrons; thus three oxygens with double negative charges are grouped around carbon with a quadruple positive charge. According to Langmuir, the ion has the structure shown in Fig. 3.

Fig. 3.

In the structure of calcite, carbon is surrounded symmetrically by three oxygens. From nowhere does there follow a difference in the behavior of one oxygen atom and the two others, as would follow from Langmuir’s model; the actual crystalline structure is more in agreement with Kossel’s interpretation.

There exists one more type of interatomic forces, about which we have not yet spoken. Most organic substances consist of molecules whose atoms are bound by shared electrons. These organic substances give well-defined crystals, where the unit cell contains molecules, each of which is electrically neutral. These molecules must be bound to one another by certain residual forces. Such bonds depend neither on shared electrons nor on attractions between ions, although they may also be partly electrostatic, corresponding to the polar distribution of charge in the molecule.

Finally, there remains the question of interatomic forces in metals. These forces are possibly similar to the forces in polar compounds, with the electrons playing the role of negative ions.

In order to pass to the influence of the types of atomic bond on crystalline structure, we shall review crystals analyzed by X-rays. At the same time we shall consider, for these cases, what the distances are between the centers of atoms. We shall see that comparison of interatomic distances makes it possible to draw certain empirical generalizations concerning that “share” which falls to each atom in the interatomic distance. Although these conclusions are very approximate, nevertheless they provide great help in analysis. On the basis of purely geometrical considerations, one could assign to an atom very different positions in most crystalline structures. In reality, each atom has a certain volume proper to it, and other atoms cannot enter this region. We shall try—

ANALYSIS OF CRYSTALS AND ATOMIC FORCES

we try to determine the distances within which the atomic centers may approach one another.

In Fig. 4 the well-known curve of Lothar Meyer is reproduced, which so well illustrates the periodicity of atomic properties. The atomic volume of each element (in cm³ per gram-atom) is plotted against the atomic numbers. The alkali metals, with which each period begins, are located at the peaks of the curve. The subsequent elements occupy positions on the regular portion of the curve, whose minimum corresponds to the elements in the middle of the period; then the curve

Fig. 4.

rises to the next crest, occupied by the next alkali metal.

The information on crystal structure that we now possess compels us to draw this curve in a new way. The structures of a large number of elements are known,—most of them were studied by Hull by the powder method.

Fifteen of these elements—all ductile metals—crystallize in a cubic lattice with centered faces. This is one of the possible ways of packing a certain number of equal spheres in the smallest volume. The cubic close-packed structure is shown in Fig. 5; in it each sphere touches 12 neighboring spheres.

There is also another way of packing spheres into a dense structure. The spheres in the planes (111) of Fig. 6 are arranged so that their centers lie at the vertices of a pattern of equilateral triangles. Successive (111) layers are packed one behind another so that the spheres of one layer fit into the spaces between three spheres of the upper and lower

layer. In the structure of Fig. 5, a line drawn from the center of one sphere perpendicular to the plane (111) passes through the center of another sphere situated three layers away from the first. It is possible, however, to pack these layers in such a way that every second layer, rather than every third, is encountered in a translational motion perpendicular to the layers. The structure thus formed has hexagonal symmetry and is shown in Fig. 6. The degree of packing density is exactly the same in both cases. Five elements crystallize, in this way, with an axial ratio that agrees very closely with the ideal ratio for close packing. Zinc and cadmium have a similar structure, but with a larger axial ratio. Cobalt and cerium are of interest as examples of metals crystallizing with both a hexagonal and a cubic close-packed structure.

Fig. 5.

In lithium, sodium, vanadium, chromium, iron, molybdenum, tantalum, and tungsten the atomic centers are arranged in a body-centered cubic lattice. Iron is of interest in that its modifications \(\alpha\), \(\beta\), and \(\delta\) have a body-centered arrangement, whereas \(\gamma\)-iron at temperatures approximately between \(900\) and \(1700^\circ\mathrm{C}\) has a face-centered arrangement1.

Fig. 6.

Silicon and one of the modifications of tin crystallize with a structure similar to diamond. Antimony and bismuth have a more complex structure, which will be discussed below. Ami-

ANALYSIS OF CRYSTALS AND ATOMIC FORCES

Nov¹) analyzed mercury and found that it forms hexagonal crystals. The structure which Aminoff ascribes to mercury can be represented as a hexagonal modification of the atomic arrangement of the diamond lattice. The hexagonal, approximately close-packed lattices interpenetrate in such a way that each mercury atom is surrounded by four others.

If, instead of atomic volumes, one plots the distances of closest approach between atoms as a function of the atomic number, one obtains a curve having the same periodic character and even greater regularity than Lothar Meyer’s curve. In some cases the structure of the element is such that there are two different distances of closest approach between neighboring atoms. In such cases two ordinates are plotted (Fig. 7).

Fig. 7.

We shall now consider the large class of polar compounds in which the bonding forces are determined by electrostatic attraction. It is convenient to divide them into several groups with characteristic crystalline structures.

Type I. The ion has six neighbors. A characteristic example of such a structure is rock salt. The ions are distributed in two interpenetrating face-centered lattices in such a way that each point of a simple cubic lattice is occupied by an ion of one sign or the other. Such a structure is possessed by the compounds of lithium, sodium, potassium, and rubidium with fluorine, chlorine, bromine, and iodine, with the exception of RbF. Ammonium iodide has this structure at ordinary temperature, while the chlorides and bromides of ammonium have it at 250°²). Other examples of this structure are AgCl and AgBr³).

Many simple salts of divalent ions have the same structure. The compounds of magnesium, calcium, strontium, and barium with oxygen and sulfur probably all belong to the same type. This has been verified in the cases of MgO, CaO, CaS, and BaS. The selenides and tellurides of these metals also

¹) Aminoff. Geol. Fören. Förhand. 44, Jan. 1922.
²) Barlett and Langmuir. Journ. Am. Chem. Soc. 43, 85, 1921.
³) R. B. Wilsey. Phil. and Mag. 42, 262, 1921.

cubic; therefore it is very probable that here we have before us another complete series of simple cubic crystals.

Another example of divalent ions forming simple cubic crystals is provided by the compounds CdO and PbS.

The lattice dimensions, i.e. the lengths of the edge of the elementary face-centered cube in this case, in Ångström units, are as follows:

SALTS OF MONOVALENT IONS.

SALTS OF DIVALENT IONS.

Type II. The ion has eight neighbors. The structure of ammonium chloride is typical of this class. Ions of one sign are located at the vertices of a simple cubic lattice, and ions of the opposite sign at the centers of the cubes, so that each ion is surrounded by eight ions of the opposite sign.

It is known that ammonium chloride and bromide at room temperature, cesium chloride, bromide, and iodide, and thallium chloride crystallize in this manner. The following table gives the lengths of the cube edges for these cases.

¹) At a temperature of 250° C.
²) This figure has not been verified by X-ray analysis.

Type IIIa. The ion is surrounded by four neighbors. An example is zinc blende. The ions of each kind are arranged in a face-centered cubic lattice, but the lattices interpenetrate in such a way that each ion is surrounded by four ions of opposite sign. All the points occupied by ions give a structure similar to that of diamond. Wyckoff showed that CuCl, CuBr, and CuJ have crystals of this type. The lengths of the edges of the unit cube are as follows:

The symmetry of this structure is lower than that of the first two groups of crystals. The trigonal axes are polar and the crystals are hemihedral. It is interesting to note that the crystals of the first two groups cleave parallel to the planes of the cube, whereas the crystals ZnS, CuCl, CuBr, and CuJ cleave parallel to the planes (110).

Type IIIb. We have seen that there are two ways of packing equal spheres with the smallest possible volume, corresponding to cubic and hexagonal symmetry. In crystals of the zinc-blende type the metallic ions are arranged in a cubic close-packed lattice, and the ions of electronegative elements lie between four metallic ions. There is another type of crystal with a very similar structure, in which ions of both signs are arranged in a hexagonal close-packed lattice; these lattices interpenetrate in exactly the same way, so that each ion lies between four ions of opposite sign. ZnO and CdS belong to this type, and such, probably, is the hexagonal form of ZnS (wurtzite). Aminoff1 recently showed that AgJ at ordinary temperature has this structure.

The hexagonal close-packed structure has the axial ratio \(c:a = 1.633:1\). The axial ratios for ZnO, ZnS, CdS, and AgJ are as follows:

Aminoff also investigated the complex form of the mineral miersite \(4\mathrm{AgJ}\cdot\mathrm{CuJ}\). He showed that in this case the structure is the same as in CuJ, i.e. of the zinc-blende type. The structure of AgJ is especially interesting because above \(146^\circ\mathrm{C}\) the hexagonal modification changes into the cubic one. The hexagonal modification has the same structure as ZnO and CdS; therefore, if the structure of miersite is taken into account,

then it is natural to expect that the cubic modification will prove to be of the zinc-blende type. However, according to Aminoff, this is not so. The structure is considerably more complicated; the elementary cube contains at least two molecules.

Type IV. The structure of fluorspar, \(\mathrm{CaF_2}\), has already been considered. \(\mathrm{SrF_2}\) and \(\mathrm{BaF_2}\) crystallize in the same way. The lattice constants are as follows:

\[ \begin{array}{lr} \mathrm{CaF_2}\ . . . . . . . . . . . . . . & 5.49\\ \mathrm{SrF_2}\ . . . . . . . . . . . . . . & 5.77\\ \mathrm{BaF_2}\ . . . . . . . . . . . . . . & 6.20 \end{array} \]

A very large number of the simplest chemical compounds crystallize according to one of the indicated types. Examples of more complex crystals with molecules containing a large number of atoms will be given below.

For a number of compounds, for example the alkali halides, one can study the effect of replacing one atom by another. When chlorine replaces fluorine or bromine replaces chlorine, the lattice dimensions increase. In the following table the distances between atomic centers are compared for a series of halide compounds, and the differences of these quantities are given, expressing the change in distance when one halogen is replaced by another:

Replacement of one halogen by another causes almost the same increase in distance, whatever the metal with which the halogen is combined, with the exception of copper and silver. From this it is natural to conclude that the same also holds for the replacement of one metal by another. If this rule were strictly valid, then the interatomic distance for any of these compounds could be expressed as the sum of two constants \(R_A + R_B\), where \(R_A\) is a constant for each metal, and \(R_B\) for each halogen. The actual values of these constants remain indeterminate, since we know only their sum.

Some interatomic distances in the table were obtained by measurements with the aid of X-rays, others were derived from the molecular volumes of crystals. It is possible that many of the irregularities of the table will disappear when it becomes possible to analyze all crystalline structures. Recently Davey1 made accurate measurements of a series of alkali-halide compounds: NaF, NaCl, NaBr, KF, KCl, KBr, KJ, RbBr, CsCl, CsJ, and showed that the indicated law of additivity is fulfilled very accurately, at least for these compounds.

Apparently the same is true also in the case of compounds of divalent ions, although here only a few structures have as yet been studied. The replacement of oxygen by sulfur causes a characteristic increase in the interatomic distances:

Similar relations also exist for other series of crystals.

It is extremely difficult to summarize these relations and express them in a form that would be useful for analysis, partly because many cases do not fit into them, and partly because our information about crystalline structure is still very limited.

One way of expressing these relations is to assign to each atom a certain constant, which we shall call the “radius of combination” (“radius of combination”); it represents the share that the atom contributes to the interatomic distance upon combination. If these constants for two atoms—nearest neighbors—are \(R_A\) and \(R_B\), then the distance between them should be \(R_A + R_B\). If we consider only polar compounds, in which \(R_A\) always corresponds to the metal and \(R_B\) to the electronegative element, then the constants remain indeterminate, since any other pair \(R_A + x\), \(R_B - x\) will give the same thing. In order to fix \(R_A\) and \(R_B\), it is necessary to make use of some other principle. Perhaps the best way to obtain general relations for interatomic distances is to find a series of consistently applicable constants and then consider the exceptions.

Such an attempt was made by one of the authors2. It turned out that a series of constants could be found for polar compounds of monovalent and divalent metals. As was already said above, in this case the actual constants nevertheless remain indeterminate. In order to fix these constants they were chosen so as to represent the distances not only in polar compounds, but also

in those cases when the compound contains two electronegative elements. The results are presented in Fig. 8, where the “combination radii” are the ordinates (circles), plotted against the atomic numbers as abscissae.

It must be emphasized, as was done in the cited work, that these “combination radii” are only a set of empirical constants, whose purpose is their use in calculating the dimensions of molecules entering into the construction of a unit of the crystal structure. Only by making new assumptions can we regard these constants as determining, to a certain degree, the actual dimensions of the atomic structure. We know so little about the structure of the atom that the old term “atomic diameter” of an element has no unambiguous meaning. In the original cited work, the quantities which we here call “combination radii” were, for brevity, called atomic radii, but, to avoid misunderstanding, the name has been changed here.

Fig. 8.

With a few exceptions, which will be discussed below, the distance between two atoms of any polar compound agrees with the values obtained by adding the two corresponding ordinates of the graph in Fig. 8 to an accuracy of 0.1 Å. On the basis of polar compounds alone it is not yet possible to fix the absolute values of the constants—the latter were obtained from data on the approach of electronegative atoms; in this case the interatomic distances are small, since the structure is interwoven by means of “shared electrons.” In this way a series of mutually consistent constants was obtained which can be applied to compounds in which both types of bond already exist.

Data on the distance between the centers of electronegative atoms in compounds are very scanty. We know the distances between carbon and oxygen and between nitrogen and oxygen, respectively, in carbonates and in sodium nitrite. We know the distance between two sulfur atoms in FeS₂, when the S₂ group plays the role of a negative ion. Further, we know the crystal struc-

type of the few electronegative elements, such as carbon and silicon, where the atoms are bound by “shared electrons” and where it is permissible, in order to find the radius of combination, to divide the interatomic distance in half. In this way constants have been found for N, O, S. The distance between the centers of sodium and oxygen in NaNO₂ is almost identical with the distance between sodium and fluorine in NaF. Similarly, the distance between Ca and O in CaO is the same as between Ca and F in CaF₂. Therefore we must ascribe to fluorine the same constant as to oxygen. The values of the constants for the other halides are obtained on the basis of the increase in interatomic distances when one halide is substituted for another. In these ways the radii of combination in Fig. 8 have been obtained.

In each period the constants approach a lower limit for the electronegative elements. These limiting values correspond on the graph to 0.65, 1.02, 1.17, 1.35 for the second, third, fourth, and fifth periods.

The crosses on the graph correspond to half the interatomic distance for crystals of the elements. For metals situated in the middle of the periods (for example: iron, cobalt, nickel, copper, zinc in the third period) it is evident that the distance of closest approach is somewhat greater than for the electronegative elements of the same period. On the other hand, the elements lithium, sodium, magnesium, calcium have a very large atomic volume and, correspondingly, a large interatomic distance. To illustrate this one may compare metallic calcium with its compounds. The atoms in the element calcium lie in a face-centered lattice, just as in CaF₂ and CaO. However, in CaF₂ the centers of the calcium atoms are closer to one another than in the metal itself, despite the presence also of fluorine ions; in CaO the calcium atoms are brought still closer together. For these elements the “radii of combination” in crystals of the elements apparently cannot be identified with their actual radii in compounds, and the crosses lie on the graph considerably above the circles.

In some cases we must assume that there exist at least two “radii of combination.” Silver, for example, in AgCl and AgBr must be assigned a constant equal to 1.72, almost the same as for sodium. Correspondingly, many salts of silver and sodium are isomorphous and have approximately the same molecular volume. On the other hand, in Ag₂O and AgJ1 silver must have a “radius of combination” of 1.42, corresponding to half the interatomic distance of metallic silver (1.43). In Fe₃O₄ and FeCO₃ the distance between the centers of Fe and O gives for \(R_{\mathrm{Fe}} = 1.35\). In FeS₂, \(R_{\mathrm{Fe}} = 1.25\), corresponding to \(2R_{\mathrm{Fe}} = 2.52\) for metallic iron.

In bismuth and antimony we have a case where, in the crystal of an element, there are two different interatomic distances.

For antimony^1) these distances are 2.87 and 3.37, for bismuth^2) 3.11 and 3.47. Here, apparently, the atoms are bound to three neighbors on one side by “shared electrons,” as in electronegative elements, and to three on the other side in the same way as atoms are joined in metals. The structure of bismuth is shown in Fig. 9.

The existence of two values of the “bonding radius” for these two elements also follows from the structure of their oxides.

Arsenolite \((\mathrm{As}_2\mathrm{O}_3)\), senarmontite \((\mathrm{Sb}_2\mathrm{O}_3)\), and cubic bismuth sesquioxide are isomorphous. The first two structures have been investigated by X-rays by Bozorth^1). The structures are of the diamond type, with the group \(\mathrm{As}_4\mathrm{O}_6\) or \(\mathrm{Sb}_4\mathrm{O}_6\) occupying the place of each carbon atom. The molecule is a regular tetrahedron, like the carbon atom. Just as in diamond, here too there are two orientations of the tetrahedron, one being the mirror reflection of the other in a plane parallel to a face of the cube. These conclusions follow from the experimental result that the plane \((lmn)\) gives a reflection only in the case when all \(lmn\) are even or odd; if one of the three \(l\), \(m\), or \(n\) is equal to zero, then the sum must

Fig. 9.

remain a multiple of four. A complete proof of the correctness of such an interpretation was given in the consideration of the crystal of basic beryllium acetate, which has the same structure^2). The \(\mathrm{Sb}_4\mathrm{O}_6\) molecule is, consequently, a regular tetrahedron; therefore the 4 Sb atoms must lie on the four lines joining the center of the tetrahedron with the vertices, and the six oxygen atoms—on the six perpendiculars from the center to the six sides. The distances of antimony and oxygen from the center are two parameters which are not determined by the nature of the structure, but may be estimated by comparison of the intensities of reflec-

^1) R. M. Bozorth. Journ. of the Amer. Chem. Soc., Jul. 1923, p. 1621.
^2) W. H. Bragg and G. T. Morgan. Proc. Roy. Soc. 104, 437. Oct. 1923.

...burning. Having thus found values for these two parameters, Bozorth concludes that the shortest distance between an antimony atom and oxygen within the same group or molecule, \(Sb_4O_6\), is equal to \(2.22\ \text{Å}\); if the two atoms belong to two neighboring groups, then the shortest distance is \(2.61\ \text{Å}\). If it is assumed that the contribution made by oxygen to the value of the distance between the centers of oxygen and antimony is \(0.67\ \text{Å}\), then the two corresponding contributions for the antimony atom are \(1.57\ \text{Å}\) and \(1.93\ \text{Å}\). These quantities are somewhat larger than the values found in crystals of antimony itself.

There exists a monoclinic variety of \(As_2O_3\), called claudetite, and an orthorhombic variety of \(Sb_2O_3\), valentinite. The unit cell of claudetite contains two groups \(As_2O_3\), and its dimensions are as follows:

\[ a = 5.3,\quad b = 6.5,\quad c = 4.45,\quad \beta = 94^\circ . \]

If one molecule is placed at the vertex of the cell, then the other is approximately at the center. It is possible that here, in the molecule, two arsenic atoms are joined together more strongly, as a result of which the symmetry of the crystal is lowered. In the arsenolite group \(As_4O_6\) the arsenic atoms form a regular tetrahedron; in the group \(As_2O_3\) two arsenic atoms may have an arrangement in the form of a dumbbell used for gymnastic exercises.

The work of Davey cited above may serve as an example of another point of view on these relations. We suppose that in such a crystal as, for example, potassium chloride, the atoms of potassium and chlorine occur as ions with a structure similar to argon, so that it may be thought that these ions have approximately the same dimensions. On this basis Davey calculated a series of constants, assigning to each ion one half of the interatomic distance. His results are as follows:

\[ \begin{aligned} R_{\mathrm{Na}} &= 1.25;\quad R_{\mathrm{F}} = 1.13;\\ R_{\mathrm{K}} &= R_{\mathrm{Cl}} = 1.56;\\ R_{\mathrm{Rb}} &= R_{\mathrm{Br}} = 1.73;\\ R_{\mathrm{Cs}} &= R_{\mathrm{J}} = 1.98. \end{aligned} \]

To ascribe equal “radii of combination” to these pairs of elements means to assert that each ion participates in equal measure in the interatomic distance. On the other hand, the large radius assigned to potassium and the small radius assigned to chlorine in our graph (Fig. 8) are not in agreement with such an equality. In reality, however, in both cases we are dealing with an additive law, but from two different points of view.

According to the principle of “radii of combination,” an atom surrounded by neighbors of one type, bonded to it in the same way, must be located...

to be at the same distance from them. In some cases, with the aid of this consideration alone, it is possible to approach the solution of a structure without knowing the actual distances between the centers.

Suppose, for example, that it is required to determine the structure of rock salt on the assumption that each chlorine atom is surrounded by as many sodium atoms as the sodium is surrounded by chlorines; further, let the relations between neighbors of opposite signs always be the same, including the distances from center to center,—and let the structure be cubic. The number of possible solutions under such assumptions is limited to three. Each ion of one sign may have four neighbors of the opposite sign, as in zinc blende, or six, as in reality, or eight, as in ammonium chloride. A choice among the three possibilities may be obtained either by X-ray analysis, or on the basis of data on the actual distance between centers, in connection with data on the density of the crystal.

In exactly the same way, in calcium fluoride, each calcium is surrounded by twice as many fluorines as the number of calcium atoms surrounding each fluorine. By making assumptions similar to those which we used in the problem of rock salt, one can greatly limit the number of possible solutions. The final solution may be obtained on the same grounds as before.

In reality these two crystal structures were determined not in this way, but by a more direct application of X-ray analysis methods. The results were used to establish the principle of the “radius of combination,” but they themselves were obtained without the aid of this principle.

On the other hand, one may quite properly point to the determination of the structure of ice ¹) as an example of the use of this principle; subsequently the results were verified by X-ray analysis.

It is exceedingly interesting to compare the data on molecular dimensions obtained from the study of crystal structures with data from an entirely different source—the properties of gases. This problem has been posed in a number of recent works by Rankine (Rankine) ²). The theoretical expression for the viscosity of a gas, derived in the kinetic theory of gases, depends on the mean free path of the atom, or molecule, in the gas. The experimental data may be interpreted by making an assumption, for the purpose of mathematical treatment, about the sizes and shape of the bodies that rebound in collisions and that exhibit a certain attractive force when the distances between them are very small. Let us first take the simplest case of a monatomic gas, in which the atoms may be regarded as spheres of radius $\sigma$; let us suppose that there are no attractive forces and

¹) W. H. Bragg, Phys. Soc., London 34, p. 98.
²) Rankine, Proc. Roy. Soc. A 98, p. 360, 1921.

the mean free path is inversely proportional to the cross-sectional area of the atom, \(\pi\sigma^2\). The viscosity \(\eta\), the mean free path \(\lambda\), and the radius \(\sigma\) are connected by the following equations:

\[ \eta = 0.491 \cdot \rho \lambda \sqrt{\frac{3p}{\rho}} \]

\[ \lambda = \frac{1}{4\sqrt{2}\,N\pi\sigma^2} \]

Here \(p\) is the pressure of the gas, \(\rho\) its density, and \(N\) the number of atoms in \(1\ \mathrm{cm}^3\) of gas.

The attraction between atoms increases the number of collisions by bringing atoms to collision, whereas otherwise they might have passed by one another. One may use the correction proposed by Sutherland, which may be represented as follows:

\[ \sigma = \sigma_0 \left(1+\frac{S}{T}\right)^{\frac12}, \]

where \(\sigma_0\) is the true radius of the spheres and \(\sigma\) is the apparent radius calculated from the viscosity at temperature \(T\). The constant \(S\) can be calculated from the variation of viscosity with temperature. From these equations one can calculate the effective value \(\pi\sigma_0^2\) by measuring the viscosity \(\eta\) and its temperature coefficient.

In this case the quantity that is called the “radius” of the atom is equal to half the distance between the atomic centers in the case of closest approach during collision.

The values of \(\sigma_0\) calculated in this way are as follows:

In the case of molecules containing several atoms, the viscosity formula makes it possible to calculate not the atomic radius \(\sigma\) directly, but the quantity \(A\) (\(\pi\sigma^2\) in the case of a single atom), corresponding to the cross-section of the molecule in collisions. Rankin compares in the following way the area \(A\) for diatomic molecules of the halogen compounds with the area \(\pi\sigma^2\) for the inert gases.

The relation between chlorine and argon may serve as an example. Rankin chooses as a model of the chlorine molecule two spheres,

radii of which coincide with the value calculated for argon, but which slightly overlap one another at the place where they are joined to each other (Fig. 10).

The distance between the centers of these spheres is taken to be 2.05, in agreement with the data of crystal measurements, while the radii are equal to 1.43; consequently, the mutual penetration is considerable. In other words, Rankine assumes that each chlorine atom in the molecule resembles (with the exception of the region of the bond) an argon atom to such an extent that in collisions it acts as a sphere of the same size as an argon atom.

Fig. 10.

Further, he calculates the mean cross-sectional surface of such a molecule for all possible orientations of it in collisions. In this way \(Cl_2\) can be compared with \(Ar\), \(Br_2\) with \(Kr\), \(J_2\) with \(Xe\). There are no data for \(F_2\); therefore Rankine compares \(O_2\) with \(Ne\).

In the following table1 the values of the mean surface \(A\) of the model and the surface \(S\) of a diatomic molecule, inferred on the basis of viscosity measurements, are compared.

A surprising agreement is obtained. Without entering into a judgment as to the correctness of the physical interpretation of such agreement, one may point out that estimates of molecular dimensions from crystallographic measurements and from the kinetic theory of gases are in agreement.

Another interesting relation obtained by Rankine may be briefly set forth as follows:

\(CO_2\) and \(N_2O\) are regarded as three spheres lying on one straight line, with radius 1.17, corresponding to neon; the distance between centers is taken to be 1.30. The mean surface \(A\) of such a mo-

... is equal to \(0.895\cdot 10^{-15}\ \mathrm{cm}^2\). “\(S\)” for \(\mathrm{CO}_2\) and \(\mathrm{N}_2\mathrm{O}\) is respectively \(0.870\cdot 10^{-15}\) and \(0.867\cdot 10^{-15}\ \mathrm{cm}^2\).

The nitrogen molecule \(\mathrm{N}_2\) has a mean surface \(S=0.78\cdot 10^{-15}\ \mathrm{cm}^2\), which is very close to the corresponding value for krypton \((S=0.797\cdot 10^{-15}\ \mathrm{cm}^2)\). A similar coincidence of structural dimensions may be expected for the ions \(\mathrm{CN}^{-}\) and \(\mathrm{Br}^{-}\), just as for the neutral units \(\mathrm{N}_2\) and \(\mathrm{Kr}\). Experimental data speak in favor of this.^1) Two cyanogen ions combine, forming a molecule \(\mathrm{C}_2\mathrm{N}_2\), just as two bromine atoms form \(\mathrm{Br}_2\). The mean surfaces for these gases are as follows:

\[ \begin{aligned} \mathrm{C}_2\mathrm{N}_2\qquad & S=1.31\cdot 10^{-15}\ \mathrm{cm}^2,\\ \mathrm{Br}_2\qquad & S=1.28\cdot 10^{-15}\ \mathrm{cm}^2. \end{aligned} \]

Further, in crystals some cyanide compounds are isomorphous with the corresponding halide compounds. In particular, \(\mathrm{KBr}\) and \(\mathrm{KCN}\) have almost equal molecular volumes—43.1 and 42.8; although a complete X-ray analysis of \(\mathrm{KCN}\) is still lacking, the data obtained by Cooper^2) indicate that both compounds have the same structure.^3) Evidently, the \(\mathrm{CN}\) radical behaves in some respects like the simple bromine ion.

A similar resemblance can be traced between the ammonium ion \(\mathrm{NH}_4\) and the molecule \(\mathrm{CH}_4\).^4) Ammonium gives salts isomorphous with salts of the alkali metals, with almost the same molecular volume as rubidium, which in the periodic system follows krypton. In accordance with this, the effective cross-section of \(\mathrm{CH}_4\) is very close to the cross-section of the krypton atom:

\[ \begin{aligned} \mathrm{CH}_4\qquad & S=0.772\cdot 10^{-15}\ \mathrm{cm}^2,\\ \mathrm{Kr}\qquad & A=0.757\cdot 10^{-15}\ \mathrm{cm}^2. \end{aligned} \]

Langmuir^5) drew attention to a remarkable series of cases of isomorphism which illustrates the role of ionic dimensions in determining the structure of crystals. One example of this is nitrates and carbonates. It had long been known that there is a practically complete coincidence of all crystallographic properties and molecular volumes of sodium nitrate and calcium carbonate. Crystals of sodium nitrate, deposited on a freshly cleaved surface of calcite, continue to grow parallel to it. X-ray analysis showed the identity of the structures. We explain such a coincidence of structures by the fact that the singly charged ion \(\mathrm{NO}_3\) has the same dimensions as the doubly charged ion \(\mathrm{CO}_3\), while the greater force associated with the double electric

1) Rankine. Proc. Roy. Soc. 99, 330, 1921.
2) P. A. Cooper. Nature, 1921, 107, 745.
3) Cf. R. M. Bozorth, Journ. Am. Chem. Soc. 44, February 1922.
4) Rankine. Trans. Far. Soc. 17, pt. 3, 1922.
5) Langmuir. Journ. Amer. Chem. Soc. 51, No. 10, October 1919.

with charge, brings the calcium ion to the group \(CO_3\) just as close as the smaller force of a singly charged ion brings the sodium ion to the group \(NO_3\). \(NaNO_3\), \(CaCO_3\), \(MgCO_3\), \(FeCO_3\), and, at high temperatures, \(KNO_3\), \(RbNO_3\), \(SrCO_3\), and \(BaCO_3\), form crystals of the calcite type. Similarly \(KNO_3\), \(CaCO_3\), \(SrCO_3\), \(BaCO_3\) are isomorphous with one another; the typical crystal of this class is the second form of calcium carbonate, pseudo-hexagonal aragonite. Langmuir also cites other cases of isomorphism found by Barker; only a small number of examples are given here. Just as calcium is extraordinarily similar to sodium with respect to the dimensions of the crystal structure, so strontium corresponds to potassium, and barium to rubidium. In accordance with this we have a series of similar crystalline properties in the case of such pairs: \(KClO_4 — SrSO_4\); \(RbMnO_4 — BaCrO_4\); \(RbClO_4 — BaSO_4\); \(KMnO_4 — SrCrO_4\), and for other more complex crystals. The fact of the parallel growth of certain crystals upon others is very interesting from the standpoint of molecular structure. Barker1 in 1907 studied the growth of crystals of alkali-halide compounds upon one another and came to the conclusion that these crystals must be divided into two groups with different structures. The crystals of group \(A\) have a relatively larger molecular volume than the crystals of group \(B\). Barker assumes that the members of each group have the same structure. Members of group \(A\) grow parallel upon one another; the same takes place within group \(B\); members of group \(A\) grow on crystals of group \(B\) quite irregularly. Barker’s table is given below; from it one can see how fully his conclusions have been confirmed by X-ray analysis.

The figures in this table denote the molecular volumes of the crystals:

As has already been said, \(AmCl\) and \(AmBr\) pass into group \(A\) at high temperatures.

We shall now add several remarks concerning other physical properties of the unit of crystalline structure.

The coefficient of linear thermal expansion of diamond is relatively small: at ordinary temperatures it has the value \(1.2 \cdot 10^{-6}\) per degree. The corresponding coefficient of expansion of gra—

phite, according to Fizeau, \(7.9 \cdot 10^{-6}\). But this considerable coefficient of graphite probably applies only to expansion in the axial direction. Backhurst found an increase of more than \(2\%\) in the spacing of the \((111)\) planes of graphite when the temperature was raised by \(900^\circ\mathrm{C}\); his results show that the expansion increases with temperature nonlinearly. On the basis of these results it may be concluded that the coefficient of expansion in the direction perpendicular to the axis is of the same order of magnitude as for diamond. This suggests that two atoms bound by “shared electrons” change their mutual distances with changes in temperature considerably less than in the case of bonds of another type. It is interesting to note that for bismuth and antimony the coefficient of expansion along the axis is much greater than in the perpendicular direction: in these substances there are atomic layers lying perpendicular to the axis; in them the atoms are probably bound to one another by “shared electrons,” but the bond between the layers rather resembles the bond between atoms in metals. Likewise in calcite, for which Benoist found a coefficient of expansion of \(25 \cdot 10^{-6}\) along the axis and a negative coefficient of \(-5.6 \cdot 10^{-6}\) in the perpendicular direction, the very strong bonds in the \(\mathrm{CO}_3\) group all lie in the \((111)\) planes. The structure of quartz is still not completely known, but one may suppose that the bonds by “shared electrons” in the crystal are, in general, parallel to the axis, since the coefficient of expansion along the axis is \(7.5 \cdot 10^{-6}\), and in the perpendicular direction \(13.7 \cdot 10^{-6}\).

There are still too few such observations, and they can serve only as a preliminary indication of the actual relations between the coefficients of expansion and the nature of the bonds.

Undoubtedly the melting points are also connected with the nature of the bonds, and probably in an analogous manner. The very high melting temperature of carbon in all its forms may be explained by the strength of the “shared-electron” bonds that exist in both diamond and graphite; on the other hand, the very low melting temperature of organic substances is undoubtedly caused by the weakness of the intermolecular bonds. Very little is yet known about all this. The melting point of silicon, which resembles diamond in the nature of its bonds, is high—about \(1200^\circ\), but as yet there is no explanation, from the point of view of the nature of the bonds, for example, of the fact that the melting point of silicon is lower than that of tungsten or rhodium. Apparently one can say with certainty that the low melting temperature of sulfur is explained by the fact that its atoms are bound into molecules which, as in organic substances, are very weakly joined to one another. Indeed, it is impossible to devise for sulfur a structure that would satisfy the following conditions:

a) all atoms are identical;

b) the mass of each atom is \(32 \times 1.66 = 53.3\);

c) the distance between the centers of two neighboring atoms in all cases is equal to 2.05 (as follows on the basis of cases of bonding by means of “shared electrons”);

d) the density of the crystal is approximately only 2.

The lightest structure that could be realized from such atoms is the diamond structure, but it would lead to a density:

\[ 3.52 \times \frac{32}{12} \times \left(\frac{1.54}{2.05}\right)^3 = \text{approx. } 4. \]

Other properties of the crystalline unit are very closely connected with interatomic forces. The type of bond which we have called “bonds by shared electrons”—when the structures of two atoms merge, forming a complex unit—cannot be interpreted more closely until the structure of the atom has been studied.

However, in the case of polar compounds we stand on firmer ground; here it may be assumed that charged ions are bound mainly by electrostatic attraction, and the forces can be calculated. The works of Kossel, Born and Landé, Madelung, and Fajans have shown that these electrostatic forces can be quantitatively related to the forces manifested in chemical compounds.

Born and Landé1 discuss the case of alkali-halide compounds. Here ions of opposite signs are held in the crystal structure by electrical forces. In addition, there must exist a repulsive force which keeps the ions at a distance; it is balanced by the electrostatic attraction. The corresponding distances between ions are measured by us with X-rays. If it is assumed that the electrostatic forces obey Coulomb’s law, then one can calculate the potential energy lost when \(N\) positive and \(N\) negative ions, initially scattered in space, assemble into an ordered crystal structure. Madelung2 calculated that for a lattice of the NaCl type the required potential energy is equal to:

\[ N \cdot 1.74 \cdot \frac{e^2}{r} \ \text{ergs}. \]

This expression must be corrected, since there is also a repulsive force that appears when ions approach one another. We cannot calculate this force theoretically, since we do not know the structure of the atom, but its value can be found for positions of equilibrium, since here it is balanced by the electrostatic attraction. In addition, Born and Landé showed that the determination of the compressibility of a crystal

CRYSTAL ANALYSIS AND ATOMIC FORCES

makes it possible to find how this force changes with distance. They suppose that the potential energy released when ions combine into a crystal is expressed as follows:

\[ u = N\left(\frac{a}{r} - \frac{b}{r^n}\right), \]

where \(a = 1.74 \cdot e^2\), as above; the term \(\frac{b}{r^n}\) is determined by a repulsive force which rapidly decreases with distance; \(n\) is greater than unity.

From the condition of equilibrium \(\frac{\partial u}{\partial r} = 0\) it follows that:

\[ b = \frac{a r^{n-1}}{n} \]

and the expression for \(u\) takes the form:

\[ u = \frac{Na}{r}\left(1 - \frac{1}{n}\right). \]

Then Born and Landé calculated \(n\) from the compressibility. They showed that if \(k\) is the compressibility of the crystal, then

\[ k = 18 \frac{9 r^4}{(n - 1)} a. \]

For a number of alkali-halide compounds they find for \(n\) a value approximately equal to 9.

From the expression written above for the energy \(u\), we see that, under the assumption made regarding the repulsive force, the potential energy \(u\) is approximately equal to \(\frac{8}{9}\) of the value calculated for electrostatic attraction alone. The change introduced by the additional term is small, and although the process of calculating the additional term is not entirely exact, nevertheless the correction probably has the correct order of magnitude. Therefore the numbers calculated by Born and Landé for the total energy should be very close to the true values. The table gives the values of the energy, in large calories, released in the formation of a gram-molecule of crystal from dispersed ions:

W. H. Bragg and W. L. Bragg

These figures cannot be compared directly with the heats of formation of the corresponding compounds. If one starts from solid sodium and gaseous chlorine, then the total amount of energy liberated in the formation of NaCl will be the difference between the above-mentioned quantity \(u\) and the energy absorbed in the evaporation of sodium, the ionization of the dispersed atoms, the dissociation of the chlorine molecules, and their ionization. We know the latent heat of sodium, the energy required for the ionization of its atoms (measured by the ionization potential of the atom), and the energy required for the dissociation of chlorine molecules. Knowing from experiment the heat of formation of NaCl from sodium and chlorine, one can calculate the last unknown quantity, the energy of ionization of the chlorine atoms. It turns out that this quantity is negative, i.e. energy is liberated when a neutral chlorine atom absorbs an electron and becomes an ion.

It was found that upon ionization of one gram-atom the following energy is liberated:

\[ \begin{aligned} \mathrm{Cl} &= \mathrm{Cl}^{-} + 119\ \text{cal.}\\ \mathrm{Br} &= \mathrm{Br}^{-} + 84\ \text{cal.}\\ \mathrm{J} &= \mathrm{J}^{-} + 77\ \text{cal.} \end{aligned} \]

In other words, the value found for \(u\) can be checked directly against experiment.1

If one takes a gram-molecule of sodium chloride and potassium fluoride and dissolves the crystals in such an amount of water that dissociation is complete, then the final state will be the same as if we had started from equivalent amounts of sodium fluoride and potassium chloride. However, the sums of the heats of solution will be different in these two cases, since we are starting from different crystalline structures. The difference of the total heat of solution for NaCl and KF, on the one hand, and NaF and KCl, on the other, must be equal to the difference of the sums of the corresponding potential energies given in the last table. This is indeed fulfilled, as is seen from the following:

The energies are expressed in large calories per gram-molecule.

The agreement is not good in all cases; research in this field has not yet advanced very far. Nevertheless, there can be no doubt that here a large step forward has been made in the direction of a quantitative unification of physical and chemical forces.

We shall add only a few words as a summary of the preceding.

There exists a unit of crystalline structure that possesses all the properties of the crystal. It contains an amount of substance corresponding to a small number of molecules. The chemical molecule in it cannot be found, although it can be divided into similar groups of atoms, and this group, as for example in organic crystals, may already be very close to the chemical molecule. The task of research must be to reveal the relations between the structure of the unit and its properties.

The investigations described above should be regarded as examples of initial experiments in this direction. They are inevitably fragmentary and constitute only a preliminary excursion into a very extensive field of research.

1. K. Fajans, Verh. d. Deut. Phys. Ges. 21, 533, 1919. ↩↩↩↩↩↩↩↩↩↩↩

2. Madelung, Phys. ZS. 19, 524, 1918. ↩↩