The Electrical Nature of Molecular Forces in Crystals

V. R. Bursian.

Until comparatively recent times, the domain of intermolecular, or so-called “particular,” forces was very little accessible to our understanding. These forces determine the thermal and mechanical behavior of matter; they underlie the mechanism by applying which statistical methods (the hypothesis of thermal motion, modified where necessary by quantum theory) may be expected to provide a theoretical interpretation of these properties. The existence of such forces is beyond doubt; it is known that only in the very rarefied state of matter, in gases at low pressures, where the particles are on average far from one another, can they be neglected and, on the basis of statistical considerations, a satisfactory picture of the phenomena be obtained, i.e. a kinetic derivation of the equation of state of an “ideal” gas. It is also known that, in order to explain the deviations of real gases from the laws of an “ideal” gas, van der Waals had to assume the existence of two kinds of forces: attraction at large distances (“cohesive” forces) and repulsion at very small distances, the latter being taken into account by him summarily, in the form of a notion of the “volume” of rigid molecules. These formal assumptions were sufficient to illuminate, in many respects quantitatively and in any case quite satisfactorily qualitatively, an enormous range of phenomena in the gaseous and liquid states of matter. The indicated character of molecular forces undoubtedly appears also in the fundamental properties of the solid state: resistance to compression, too great to be explained by thermal motion alone—as in gases—testifies to a repulsion that manifests itself when particles approach one another, while the enormous cohesive forces testify to the appearance of attraction when an attempt is made to remove the particles from one another. The normal state of a solid body is evidently the result of the equilibrium of these two forces. However, the nature of these forces remained unknown, and it could be foreseen that, proceeding from the total effect of these forces—empirical equations of state, thermal and elastic properties

substances, their internal friction, and so forth, it is hardly possible to obtain an unambiguous answer as to their nature.

At the present time it has proved possible to approach this question from another side, namely, by attempting to construct theoretically the behavior of a substance on the basis of fundamental ideas about the nature of its particles, obtained in other fields of research.

Experimental investigations in recent years on the scattering of $\alpha$ particles and X-rays in passing through matter, and theoretical work on the structure of atoms and on the theory of series of spectral lines in the optical and X-ray regions of the spectrum, have given us a fairly definite conception of the general character of the structure of “particles”: according to Bohr’s theory these are complex dynamical systems consisting of positively charged nuclei and negative electrons, governed in their stationary state by electrostatic forces of mutual attraction and repulsion. The volumes of both nuclei and electrons are very small in comparison with the dimensions of the atomic system and with their mutual distances, which, together with the analogy between Coulomb’s laws and Newton’s law of universal gravitation, makes these models very similar to astronomical systems. The positive charge of the nucleus, equal to the sum of the charges of the electrons surrounding it (if the atom is in a neutral state), increases by one elementary charge ($e = 4.774 \times 10^{-10}$ absolute electrostatic units) in passing from one element of the periodic system to the next, beginning with hydrogen, which has one electron and one elementary positive charge in its nucleus. In nature, evidently, not the whole infinite multitude of conceivable kinds of motion of such systems is realized, but only a finite number of certain stationary dynamical configurations, corresponding to the number of elements known to us and possessing, under conditions accessible to us, quite definite properties; in this the theory of the atom sees a manifestation of quantum regularities. At present these regularities can be formulated quite definitely for simple cases; a necessary condition for this is the possibility of a complete solution of the mathematical problem of integrating the equations of motion in general form. It is known that for the hydrogen atom and for the ionized helium atom (nuclear charge $+2e$ and one electron), the application of quantum theory to the completely solved astronomical two-body problem has given results that agree strikingly with experiment, even in the smallest details. For more complex cases—the remaining atoms of the periodic system of elements—an exact theory is not yet accessible to us; however, the regularities observed in the X-ray spectra of all elements point to a very important circumstance, namely, that with the successive complication of systems by a large number of electrons, the arrangement of the inner electrons, closest to the nucleus, does not undergo any substantial change. Taking into account the periodicity of the properties of the elements, Kossel [1] expre-

...in the following way concerning the character of the structure of more complex atoms.

A neutral helium atom has two electrons; on the basis of its chemical inactivity one may conclude that they form a very stable configuration. The addition of one more electron and an increase of the nuclear charge to \(3e\) should lead us to a model of the lithium atom; it is natural to assume that the third electron does not enter into the configuration of the first two, but remains outside. A sufficiently close group consisting of the central charge \(+3e\) and two electrons, in a first approximation, represents as it were a nucleus with charge \(+e\), around which the outer electron revolves, i.e., a system very similar to hydrogen. Proceeding further along the first row of the periodic system, we shall add to the outer group one electron at a time until we reach the next noble gas, neon, in which this outer group will contain eight electrons, again forming a very stable combination. Continuing further, we shall build up a third, still more external group of electrons: sodium has one, magnesium two, and so on, while leaving unchanged both stable inner combinations of two and eight electrons. Let us also consider the element preceding neon—fluorine, which according to this scheme must have seven outer electrons. The chemical inactivity of the noble gases, attributed to the extraordinary stability of their outer electron configurations, gives us the key to understanding the simplest chemical compounds and the existence of electrolytic ions. Indeed, the idea naturally suggests itself that if a neutral atom of lithium or sodium with one outer electron and an extraordinarily stable configuration of inner electrons approaches an atom of fluorine, which lacks exactly one electron to form a similar (neon) configuration, then an exchange will occur, and the outer electron of the monovalent metal will enter the outer group of fluorine, completing there the formation of the neon configuration. In this process, of course, the central charges of the elements will remain uncompensated: the monovalent metal will receive a positive, and fluorine a negative, elementary charge; that is, two ions will be formed, whose electrostatic attraction will be capable, under certain conditions, of binding them into one chemical molecule of a salt. It is easy to see that, when this attraction is weakened by a surrounding medium with a large dielectric constant (water), the action of thermal motion can cause the molecule to split precisely into these two ions, i.e., in the manner required by our information about the electrolytic dissociation of aqueous solutions. These considerations are directly applicable to all halide compounds of the alkali metals; a single glance at Mendeleev’s table shows that next to any one of them there is, on the appropriate side, a noble gas of the zero group.

In the original sketch of Bohr–Kossel theory, the scheme of groupings of electrons in shells successively enveloping the atomic nucleus was proposed in the simplest form, namely as concentric rings, to which it was possible comparatively simply to apply the propositions of quantum theory in order to determine their radii. At present, confidence in the existence of such rings has been strongly shaken (among other things also as a result of the results of Born’s work, set forth below), whereas the general character of the considerations presented so clearly conveys the most essential aspects of the observed facts that it is hardly subject to substantial change.

Thus, atoms, molecules, and ions are systems of somehow grouped and moving electric charges. It is natural to suppose that externally, on neighboring particles, they also act as such—that is, to make the assumption of the electrical nature of interparticle forces. However, our knowledge of the spatial distribution of charges in particles, as has just been indicated, is not sufficiently complete for us to proceed to the solution of the problem of determining the equilibrium and stable configuration of an aggregate of such particles in general form. But at first it would be sufficiently convincing to show that the interparticle forces, calculated on the basis of this assumption from some model of the Bohr type, have the proper character indicated at the very beginning of the present article, and the correct order of magnitude. And for this it is necessary to have a ready conception of the actual arrangement of the particles constituting the given body.

And here it must be noted that the results of experimental investigations of recent years have given us quite exact and detailed information about the stable arrangement of atoms in a large number of bodies, namely in a whole series of crystals. Among the crystals possessing a thoroughly investigated and at the same time very simple structure are precisely the crystals of alkali-halide salts of the type \(NaCl\), \(KJ\), etc. The study of the interference of X-rays scattered by these crystals, by the methods of Bragg and Debye, has undoubtedly shown that the elements of their structure are not molecules and not neutral atoms, but the ions considered by us above, for example, \(Na^+\) and \(Cl^-\), arranged in a regular spatial cubic lattice, alternating with one another along rows parallel to three mutually perpendicular fourfold symmetry axes of this structure. If attention is paid to ions of one kind (for example, those indicated by black circles in Fig. 1), then it turns out that they are arranged in a lattice whose element is a cube with centered faces; the other ion is arranged in exactly the same lattice, but displaced relative to the first along the diagonal of the cube by half its length, so that both kinds of ions together occupy

all the nodes of a simple cubic lattice with the cube edge equal to half the preceding one. Let us note that, although in the drawing of the elementary cube there are shown 13 points of one kind and 14 points of the other kind, altogether 27 points, nevertheless, for the volume of our cube, containing 8 small cubes, there are 8 points, since it is obvious that in a simple cubic lattice there is one point per cube. Consequently, for the volume of the elementary cube \(\delta^3\), where \(\delta\) is its edge, there are 4 ions of each kind, for which one may take, for example, all those lying at the vertices of the lower left front small cube. Let us observe that, once the type of structure is known, the edge \(\delta\) of the elementary cube can be calculated in the following way: if \(\rho\) is the density of the crystal, \(\mu_+\) and \(\mu_-\) are the molecular weights of the ions, \(N = 6.06 \times 10^{23}\) is the number of molecules in a gram-molecule, then the volume of a cube containing 4 pairs of ions (4 molecules) will be 4 times greater than the volume per one molecule, i.e. the volume of a gram-molecule divided by \(N\). Hence, for the edge of this cube we obtain:

Fig. 1.

\[ \delta = \sqrt[3]{\frac{4(\mu_+ + \mu_-)}{N \cdot \rho}} = 1.87 \sqrt[3]{\frac{\mu_+ + \mu_-}{\rho}} \cdot 10^{-8} \qquad \ldots (1). \]

With these data Born and Landé [2] proceed to solve the problem with the following assumptions and restrictions. The distribution of electrons in the ions is assumed according to the original Bohr–Kossel scheme, in rings with definite numbers of electrons, following from Kossel’s scheme, and with radii calculated by applying to this case the quantum theory with account taken of the interaction of the rings with one another; namely, it is assumed that the first ring has 2 electrons and that each of them has an angular momentum equal to \(\dfrac{h}{2\pi}\) (where \(h\) is Planck’s constant), i.e. one quantum of angular momentum according to Bohr; the second ring has 8 electrons and 2 quanta of angular momentum for each, the third ring again 8 electrons and 2 quanta, the fourth—10 electrons and 2 quanta, the fifth—8 electrons and 2 quanta, and so on alternately. Sommerfeld’s supposition that the \(k^{\text{th}}\) ring has \(k\) quanta of angular momentum does not lead to results agreeing with experiment. Bohr also considered the question of orienting the rings not in one plane, but in mutually perpendicular ones, for which there are indications in Som-

merfeld. The coplanar arrangement adopted by him, with the sole exception of the fifth ring \(C_s\), which he oriented perpendicular to the plane of the others, gave the best results. It will be seen below that the results of subsequent work forced one to abandon the idea of rings altogether, so that these considerations are not of special interest.

In order to preserve the cubic symmetry of the lattice when systems resembling Saturnian rings are placed in it, all axes must be assigned special directions, namely along the diagonals of the small cube, mutually intersecting at its center (Fig. 2, where, for greater clarity, the drawing has been extended so as to include three small cubes in a row).

Fig. 2.

The aim of the painstaking part of the calculations, which will be discussed below, is to obtain an expression for the electrostatic mutual potential energy arising from the interaction of the ions. This expression will include, together with the parameters determining the structure of the ions, only the edge length of the elementary cube. Of course, there also exists the energy of the ions themselves, depending on their internal configuration, which, when \(\delta\) changes—for example under compression as a result of the action of neighboring ions—may change. For simplicity it is assumed that no such change in the internal configuration and orientation of the ions in the lattice occurs. Then the only variable (under hydrostatic compression) remains the quantity \(\delta\), and one may confine oneself to the investigation of the mutual potential energy. The equilibrium state of the crystal in the absence of external pressure (atmospheric pressure, in view of the small compressibility of solids, may of course be regarded as negligibly small) is determined by the minimum of this potential energy. If we denote it by \(\Phi(\delta)\), then the normal value of the lattice constant \(\delta_0\) is determined from the equation

\[ \Phi'(\delta_0)=0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (2). \]

This \(\delta_0\) is what should be compared with the numbers obtained from experiment, i.e. from equation (1).

Further, it is not difficult to see that the second derivative of \(\Phi\) with respect to \(\delta\) at \(\delta=\delta_0\) will be related to the compressibility of the crystal, which makes it possible once more to check the theory. The calculation of the compressibility on the basis of the considerations set forth is as follows. On the one hand, we

we can, for values of \(\delta\) differing little from \(\delta_0\), write the expansion of the potential energy in a Taylor series

\[ \Phi(\delta)=\Phi(\delta_0)+(\delta-\delta_0)\Phi'(\delta_0) +\frac{(\delta-\delta_0)^2}{2}\Phi''(\delta_0)+\cdots . \tag{3} \]

Here, by virtue of (2), the linear term is equal to zero, and the change in potential energy when \(\delta\) is changed by the amount \((\delta-\delta_0)\) is approximately equal to

\[ \Phi(\delta)-\Phi(\delta_0)= \frac{(\delta-\delta_0)^2}{2}\Phi''(\delta_0) \ldots \ldots \ldots \ldots \tag{4} \]

On the other hand, the work \(R\) of the external forces in compressing the volume by \(\Delta V\) is calculated from the formula

\[ R=\int_{V_0}^{V-\Delta V} p\,dV \ldots \ldots \ldots \ldots \tag{5} \]

where, for small changes of the volume \(V\), the pressure \(p\) may be taken proportional to the compression, i.e.

\[ p=-\frac{1}{\chi}\,\frac{V-V_0}{V_0} \ldots \ldots \ldots \ldots \tag{6} \]

where \(\chi\) is the coefficient of compressibility. Substituting (6) into (5) and integrating, we obtain:

\[ R=\frac{1}{2\chi V_0}(\Delta V)^2 \ldots \ldots \ldots \ldots \tag{7} \]

If our crystal contains \(N'\) elementary cubes with edge \(\delta\), then \(V=N'\delta^3\), and for \(\Delta V\) one may write:

\[ \Delta V=3N'\delta_0^2(\delta-\delta_0) \ldots \ldots \ldots \ldots \tag{8} \]

Substituting all this into (7) and comparing with (4), since the work expended must be equal to the increment of the potential energy, we obtain

\[ \frac{(\delta-\delta_0)^2}{2}\Phi''(\delta_0) = \frac{9N\delta_0}{\chi}\cdot \frac{(\delta-\delta_0)^2}{2}, \]

or finally

\[ \chi=\frac{9N\delta_0}{\Phi''(\delta_0)} \ldots \ldots \ldots \ldots \tag{9} \]

If, by \(\varphi=\Phi/N'\), we denote the potential energy belonging to one elementary cube, containing 4 ions of each kind, then the nearest task will be: the calculation of \(\varphi(\delta)\), the solution of the equation

\[ \varphi'(\delta_0)=0 \ldots \ldots \ldots \ldots \tag{2'} \]

and the calculation of the quantity

\[ \chi=\frac{9\delta_0}{\varphi''(\delta_0)} \ldots \ldots \ldots \ldots \tag{9'} \]

It should be noted that all these calculations refer to the temperature of absolute zero, since thermal motion is not taken into account in the calculation. However, the weak dependence of the volume and compressibility on temperature makes extrapolation of the experimental data to absolute zero, when comparing them with the results of the theory in its present state, unnecessary.

For very large \(N'\), as is the case in reality, one may disregard the elementary cubes lying near the surface of the crystal, and calculate as though the crystal were infinitely large. In that case the potential energy \(\varphi_1\) of an ion of one kind, arising from the presence of all the remaining ions, and the corresponding quantity for ions of the other kind, \(\varphi_2\), will be the same everywhere, and the mutual potential energy of all the ions, the number of which is \(N''\) of each kind, will have the value:

\[ \Phi=\frac{1}{2}N''\varphi_1+\frac{1}{2}N''\varphi_2 \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (10). \]

Since in our elementary cube there are 4 ions of each kind, the number of cubes \(N'\) is \(\frac{N''}{4}\), and the quantity \(\varphi\) introduced by us will be equal to

\[ \varphi=\frac{\Phi}{N'}=2(\varphi_1+\varphi_2)\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (11). \]

The problem is thus reduced to the calculation of the potentials \(\varphi_1\) and \(\varphi_2\) of one ion of each kind in an infinite lattice.

Since the electrons on the rings rotate with a large angular velocity, for the calculation of the mean static force they may be replaced by an equivalent charge distributed with constant linear density along the circumference of the ring. The basis of the calculation is the expression for the mutual potential \(\psi\) of two such rings with charges \(E_1\) and \(E_2\), radii \(a_1\) and \(a_2\), with the distance between centers equal to \(r\), whose axes make the angles \(\vartheta_1\) and \(\vartheta_2\) with the line of centers and \(\varepsilon_{12}\) with each other (Fig. 3). Expansion in negative powers of \(r\) gives the series

Fig. 3.

\[ \psi=\frac{\psi_{-1}}{r}+\frac{\psi_{-3}}{r^3}+\frac{\psi_{-5}}{r^5}+\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . (12). \]

The coefficients \(\psi_{-1}, \psi_{-3}, \psi_{-5}\) have the following values:

\[ \begin{aligned} \psi_{-1} &= E_1E_2,\\ \psi_{-3} &= -\,E_1E_2\,\frac{1}{2}\left\{a_1^2 P_2(\cos\vartheta_1)+a_2^2 P_2(\cos\vartheta_2)\right\}\ . \ . \ . \ . \ . (13)\\ \psi_{-5} &= E_1E_2\frac{3}{8}\left\{a_1^4 P_4(\cos\vartheta_1)+a_2^4 P_4(\cos\vartheta_2)+2a_1^2a_2^2 Q_4(\vartheta_1\vartheta_2\varepsilon_{12})\right\}, \end{aligned} \]

where \(P\) and \(Q\) denote the following functions of the angles:

\[ \left. \begin{aligned} P_2(\cos \vartheta) &= \frac{1}{2}\left(3\cos^2\vartheta-1\right),\\ P_4(\cos \vartheta) &= \frac{1}{8}\left(35\cos^4\vartheta-30\cos^2\vartheta+3\right) \end{aligned} \right\} \qquad \ldots\ldots\ldots\ldots\ldots (14) \]

\[ Q_4(\vartheta_1\vartheta_2\varepsilon_{12}) = \frac{1}{4} \left[ 1-5\left(\cos^2\vartheta_1+\cos^2\vartheta_2\right) +35\cos^2\vartheta_1\cos^2\vartheta_2 +2\cos^2\varepsilon_{12} -20\cos\vartheta_1\cos\vartheta_2\cos\varepsilon_{12} \right]. \]

If, for the moment, the quantity \(\delta\) is taken as the unit of length, then the coordinates of the ions may conventionally be denoted by the symbol \(\begin{pmatrix} ijk \\ lmn \end{pmatrix}\), where \(l, m, n\) are integers denoting the coordinates of the left front lower vertex of each cube (they are marked in Fig. 2), while \(ijk\) denote the coordinates of a point relative to this vertex and take, in each cube, the same values as those indicated in Fig. 1 for points assigned to the basic cube \(0,0,0\). The direction of the axis of the rings of each ion is given by the values of the cosines of their angles with the coordinate axes:

\[ \frac{(-1)^{2i}}{\sqrt{3}}, \qquad \frac{(-1)^{-2j}}{\sqrt{3}}, \qquad \frac{(-1)^{2k}}{\sqrt{3}} \qquad \ldots\ldots (15). \]

The distance between the ion \(\begin{pmatrix} 000 \\ 000 \end{pmatrix}\) and \(\begin{pmatrix} ijk \\ lmn \end{pmatrix}\):

\[ r^{\,ijk}_{\,lmn} = \delta \sqrt{(l+i)^2+(m+j)^2+(n+k)^2} = \delta\cdot \rho \ldots (16) \]

and the cosines of the angles of this direction with the coordinate axes:

\[ \frac{l+i}{\rho}, \qquad \frac{m+j}{\rho}, \qquad \frac{n+k}{\rho} \qquad \ldots\ldots (17). \]

On the basis of these formulas, the necessary values of the angles \(\vartheta_1\), \(\vartheta_2\), \(\varepsilon_{12}\) are calculated; when substituted into (14), they give the value for \(\phi\), the potential of some ring of the ion \(\begin{pmatrix} 000 \\ 000 \end{pmatrix}\) with radius \(a_1\), in the field of the ring of the ion \(\begin{pmatrix} ijk \\ lmn \end{pmatrix}\) with radius \(a_2\). It remains to carry out the summation, which is performed by the authors in the following order: first, summation is made over all ions of one kind which have the ring \(a_2\) and are located at one and the same distance; in doing so it turns out that the third-order term vanishes, so that only the first- and fifth-order terms remain. The same is done with ions of the other kind (for some of them, as is not difficult to see, the sum \(i+j+k\) is an integer, for others a fraction); let them have a ring with radius \(a'_2\). Then summation over the different values of the distance is carried out. Denoting this summation

with the sign \(\sum\), we shall obtain, for the potentials arising from the rings \(a_2\) and \(a'_2\) at the ring \(a_1\), expressions of the form:

\[ \psi(a_1a_2)=\frac{E_1E_2}{\delta}\sum\frac{C^{ijk}_{lmn}}{\rho} +\frac{E_1E_2}{\delta^5}\sum\frac{f(a_1,a_2)}{\rho^5}\,C^{ijk}_{lmn} \]

\[ \psi(a_1a'_2)=\frac{E_1E'_2}{\delta}\sum\frac{C^{ijk}_{lmn}}{\rho} +\frac{E_1E'_2}{\delta^5}\sum\frac{f'(a_1,a'_2)}{\rho^5}\,C^{ijk}_{lmn}, \]

where \(C^{ijk}_{lmn}\) denotes the number of ions having one and the same distance from the initial one. This number is in general equal to 48, but when the given point coincides with one or several planes of symmetry drawn through

\[ \begin{pmatrix} 0&0&0\\ 0&0&0 \end{pmatrix}, \]

this number is correspondingly decreased. The functions \(f\) and \(f'\) have the form

\[ -A\left(a_1^4+a_2^4\right)+Ba_1^2a_2^2, \]

where \(A\) and \(B\) are rather complicated sums. It then remains to sum the expressions obtained for all possible combinations of \(a_1\) (rings of the ion

\[ \begin{pmatrix} 0&0&0\\ 0&0&0 \end{pmatrix} \]

) and \(a_2\) (rings of ions of the same kind) and, correspondingly, \(a'_2\)—rings of ions of the other kind; moreover, in order to compute the potential of a nucleus in the field of rings and nuclei, and of rings in the field of nuclei of other ions, it is sufficient in the formulas to put the corresponding radii equal to zero. The summation of the terms of the fifth order, owing to the rapid convergence of \(1/\rho^5\), is carried out by direct calculation; the term of the first order is computed by special methods indicated by Madelung [3].

If we write \(E_{1,\mu}\) for the charges of the rings and nucleus of the ion

\[ \begin{pmatrix} 0&0&0\\ 0&0&0 \end{pmatrix}, \]

and \(E_{2,\nu}\), and respectively \(E'_{2,\nu}\), for the charges in the other ions, then, since the radii of the rings do not enter into the first term, it is easy to see that, for example, \(\sum_{\mu}\sum_{\nu} E_{1,\mu}E_{2,\nu}\) is represented in the form of the product of sums \(\sum_{\mu}E_{1,\mu}\sum_{\nu}E_{2,\nu}\), and since \(\sum_{\mu}E_{1,\mu}=\sum_{\nu}E_{2,\nu}=\pm e\), while \(\sum_{\nu}E'_{2,\nu}=\mp e\), the first-order term will be written

\[ \frac{e^2}{\delta}\sum\left(\pm\frac{1}{\rho}\right)\ldots\ldots\ldots\ldots\ldots\ldots (18) \]

where the sign \(+\) refers to ions of the same name, the sign \(-\) to ions of opposite name, and the sign \(\sum\) here denotes ordinary summation over all ions of the lattice. Since the shortest distance between two ions is \(\delta/2\), then, writing the expression for the first-order term in the form

\[ \frac{2e^2}{\delta}\sum\left(\pm\frac{1}{2\rho}\right)\ ;\ \ldots\ldots\ldots\ldots\ldots (19), \]

we find that \(\Sigma\) denotes the potential of an infinite cubic lattice, at whose nodes charges \(+1\) and \(-1\) are located alternately with an edge equal to unity. Madelung sums this expression in the following way: first the potential of an infinite row of alternating charges at one of them is computed; it is, obviously, equal to

\[ 2\left(-1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\ldots\right)=-2\lg 2=-1.386. \tag{20} \]

Then the potential of a parallel row on that same charge is computed and summed for all parallel rows of the same coordinate plane. For this part of the potential Madelung gives a very rapidly convergent double series:

\[ -8\left[\sum_n \frac{i\pi}{2} H_0^{(1)}(i\pi n) -\sum_n \frac{i\pi}{2} H_0^{(1)}(2i\pi n)+\ldots\right] = \]

\[ =-0.4225\ldots\ldots\ldots\ldots\ldots \tag{21}, \]

where \(n\) takes only odd positive values, and \(H_0^{(1)}\) denotes the zero-th Hankel function of the first kind. Finally, the potential produced by a parallel plane is computed, and summed over all planes. There is obtained likewise a rapidly convergent triple series:

\[ -16\left[ \sum_m\sum_n \frac{e^{-\pi\sqrt{m^2+n^2}}}{\sqrt{m^2+n^2}} - \sum_m\sum_n \frac{e^{-2\pi\sqrt{m^2+n^2}}}{\sqrt{m^2+n^2}} +\ldots \right. \]

\[ \left. \ldots \right] =-0.131\ldots\ldots\ldots\ldots\ldots \tag{22}, \]

where again \(m\) and \(n\) take only odd positive values. Adding (20), (21), and (22), multiplying by \(\dfrac{2e^2}{\delta}\) according to (19) and also by 4 according to (11), since the potentials for both sorts are equal, we obtain for the first-order term in the final expression for \(\varphi\), which we shall write in the form

\[ \varphi=\frac{a}{\delta}+\frac{\nu}{\delta^5}\ldots\ldots\ldots\ldots \tag{23} \]

\[ a=+13.94\,e^2\ldots\ldots\ldots\ldots \tag{24}. \]

The coefficient of the fifth-order term, after carrying out all the calculations, is the following expression

\[ \nu=2e^2\left\{ -28.7\cdot 2\left(\sum p_{1k}a_{1k}^4-\sum p_{2k}a_{2k}^4\right)+ \right. \]

\[ \left. +293.2\sum p_{1k}a_{1k}^2\sum p_{2k}a_{2k}^2 +37.4\left[ \left(\sum p_{1k}a_{1k}^2\right)^2+ \right.\right. \]

\[ \left.\left. +\left(\sum p_{2k}a_{2k}^2\right)^2 \right]\right\}\ldots\ldots\ldots\ldots \tag{25}. \]

where \(p\) denotes the number of electrons, \(a\) the radii of the rings, the subscript \(n\) the numbering of the rings, and 1 and 2 the kinds of ions. The following table gives, for various ions, the numbers of electrons and the relative radii \(a\) (the radius of the one-quantum circular orbit of the hydrogen electron, \(a_0=0.528\cdot 10^{-8}\,\mathrm{cm}\), is taken as unity).

With the aid of the data of this table one can, by (25), calculate the coefficient \(b\); it turns out to be positive for all the cases considered.

Writing the expression (23) obtained for \(\varphi\) in the general form with an undetermined exponent \(n\),

\[ \varphi=-\frac{a}{\delta}+\frac{b}{\delta^n}\ldots\ldots\ldots\ldots\ldots\ldots (26), \]

we can verify that it satisfies the general requirements set forth at the beginning: taking the derivative and setting it equal to zero, in accordance with \((2')\), we obtain for \(\delta_0\) the equation

\[ \varphi'(\delta_0)=\frac{a}{\delta_0^2}-\frac{nb}{\delta_0^{n+1}}=0\ldots\ldots\ldots\ldots\ldots (2''), \]

whence we obtain:

\[ b=\frac{a\delta_0^{\,n-1}}{n},\qquad \delta_0=\sqrt[n-1]{\frac{nb}{a}} \ldots\ldots\ldots\ldots\ldots (27), \]

and for the second derivative we obtain:

\[ \varphi''=-\frac{2a}{\delta_0^3}+\frac{n(n+1)b}{\delta_0^{n+2}} =\frac{a(n-1)}{\delta_0^3} \ldots\ldots\ldots\ldots\ldots (28), \]

thus for the compressibility from (9′) one obtains

\[ \chi=\frac{9\delta_o^4}{a(n-1)} \tag{9″} \]

Since \(n\) is a positive integer exponent greater than unity, (28) shows that at \(\delta_o\) the potential energy passes through a minimum; for \(\delta>\delta_o\) attraction predominates, for \(\delta<\delta_o\)—repulsion. One might have expected that Bohr’s model, placing a positive charge at the center and a negative one at the periphery of the atom, would give such a result; indeed, in the lattice the nearest neighbors are unlike ions, which therefore must attract one another, and this is expressed by the negative value of the first term; but when they are brought closer together, the peripheral parts, charged with the same sign and situated at a smaller distance than the centers of gravity of the total charge, must produce a rapidly increasing repulsion. Bohr’s calculation, which gave \(b>0\), confirms this expectation, at least for the model he adopted.

Substituting \(n=5\) in (27) and the numerical values of the coefficients \(a\) and \(b\), Born and Landé obtained very good agreement between the calculated and observed values of \(\delta\) (Fig. 4).

Fig. 4.

The verification of formula (9″) for the compressibility gave a not wholly satisfactory result: the \(\chi\)’s turned out to be, although of the correct order of magnitude, for all the crystals considered approximately twice as large as in experiment, whence it follows that such close agreement in the values of \(\delta\) should not be assigned special significance. Nevertheless, of course, fundamental importance must be attributed to the result obtained: the correct order of magnitude and the fairly close numerical values obtained in a twofold test of the theory by formulas (27) and (9″) on a large number of crystals of one type show very convincingly that the partial forces holding the ions of crystals can indeed be explained entirely by electrostatic forces between ions constructed according to Bohr’s model.

The next paper by Born and Landé [4] is devoted to the question of the discrepancy, found in the first paper, between the calculated and observed values of the compressibility, a discrepancy having not a random but a systematic character. Formula (9″) contains, apart from the number \(a\), which does not depend on the structure of the ions, and the lattice constant \(\delta_o\), which is given by experiment, only the exponent \(n\) of the negative power of \(\delta\) in the term expres-

of the potential energy arising from repulsive forces. The model of ions constructed from plane rings inevitably leads, as the preceding calculations have shown, to the value \(n=5\), and to an incorrect result for the compressibility. In order, while retaining the general expression for the potential energy (26), to obtain the compressibility required by experiment, smaller by a factor of two, it is necessary to increase \(n\) also by approximately a factor of two, and this compels us to abandon the model of plane rings and to assume some other structure. It is not difficult to see that the exponent \(n\) depends on the degree of symmetry of the ion model. If the negative charges are distributed uniformly over concentric spheres around the nucleus, then, as is known from electrostatics, such models will act as point charges at any distance greater than their radius, i.e., we shall obtain only the first term of the expression \(\varphi\); the exponent \(n\) will turn out to be equal to infinity. The lower the degree of symmetry, the smaller the value that may be expected for the exponent of the first non-vanishing term for \(\varphi\). Since experiment requires a larger exponent than is given by the model with plane rings, it must be admitted that the arrangement of the electrons is not plane, but more symmetrical and spatial. Born’s calculations show that different orientations of the inner rings have little effect on the result of the calculations; what is of essential importance are the dimensions and configuration of the outer shell. Thus the desired result cannot be obtained by placing the rings in different planes; rather, the outer layer must be given a spatial configuration. But since the outer layer of fluorine and sodium is the inner one for chlorine and potassium, and the outer layers of the latter are the inner ones for bromine and rubidium, etc., it is clear that the interpretation given above of the results of comparing experiment with the first attempt at a theory forces us in general to abandon the ring scheme, and to choose spatial schemes with a higher symmetry. The construction of such schemes, given our inability to apply quantum theory to complex mechanical problems, appears conjectural, and we must adopt a more phenomenological point of view, namely, try to obtain information about the character of these configurations from experimental data. This is precisely what Born and Landé did: they retained the general expression (26) for the potential energy and the constant \(a\), independent of special assumptions about the structure of the ions, while leaving the quantity \(b\) and the exponent \(n\) undetermined. Taking the experimental data for \(\delta_0\) and \(\varkappa\), one can determine the exponent \(n\) from (9″), and then compute the quantity \(b\) from (27). The following table gives the calculated values for \(n\); they fluctuate around 8.76. The authors take \(n=9\) and compute the theoretical value for \(\varkappa\), indicated in the last column of the table. Salts of monovalent thallium crystallize in a lattice of the same structure as the alkali halide salts.

A comparison of the calculated and observed values of $\chi$ is presented in Fig. 5. The agreement for $n = 9$ is quite satisfactory. This large value of the exponent indicates a rather high degree of symmetry of the outer electron shell, and it was possible to try to choose such spatial configurations as would give this exponent.

The number of 8 electrons in this shell suggests a cubic arrangement, and as a first orienting attempt Born tried to carry out the calculation of the mutual energy of ions whose outer shell consists of electrons situated at the vertices of a cube, with axes parallel to the axes of the lattice. In doing so, since the calculation is preliminary in character, all the inner electrons together with the nucleus may be combined into one central charge $(8+1)e$ for the alkali metal and $(8-1)e$ for the halide. Denoting by $\psi_{m_1 m_2}$ the potential of a neutral cube (with charge $8e$ at the center and eight electrons at the vertices) of one ion in the field of the same neutral cube of another ion, by $\psi_{k_1 m_1}$ and $\psi_{k_2 m_1}$ the potential of one neutral cube in the field of the remaining charge $+e$ or $-e$ of the other ion, and by $\psi_{k_1 k_2}$ the potential of the remaining charge $+e$ in the field of the remaining

Fig. 5.

of charge \(-e\) of another ion, Born obtains the following expansions in negative powers of the distance between centers (\(\alpha,\beta,\gamma\) are the cosines of the angles of the line of centers with the axes of the cubes, \(a_1\) and \(a_2\) are the radii of the spheres circumscribed about the cubes):

\[ \psi_{k_1k_2}=-\frac{e^2}{R} \]

\[ \psi_{k_1w_2}+\psi_{k_1w_2} = \frac{14e^2}{9R^5}(a_1^4-a_2^4) \left\{5(\alpha^4+\beta^4+\gamma^4)-3\right\} + \]

\[ +\frac{M(a_1^6a_2^6)}{R^7} + \frac{N(a_1^8-a_2^8)}{R^9} +\cdots \]

\[ \psi_{k_1w_2} = \frac{14e^2a_1^4a_2^4}{9R^9}\,f(\alpha\beta\gamma) +\cdots \qquad\qquad\qquad\qquad (29), \]

whence it follows that for \(a_1=a_2\), or for values very close in magnitude, the first term of the expansion will indeed have the proper exponent 9.

Of course, such a static arrangement of the electrons is probably impossible, but one may suppose that the spatial motion of the group of 8 electrons is such that, on the average, their configuration preserves cubic symmetry. Following the idea given by Born, Lande, in a series of works [5], investigated the search for periodic orbits of a system of several electrons, with the requirement imposed that their configuration at every moment satisfy the symmetry elements of a tetrahedron or a cube. Subsequently Lande found more general solutions. We shall indicate one of the latter, which Lande and Madelung [6] consider the most probable. Four electrons move along four circles perpendicular to the four axes of threefold symmetry of the cube (the diagonals), and in such a way that their coordinates satisfy the relation imposed by the requirement of tetrahedral symmetry: if electron I has coordinates \(x, y, z\), then the coordinates of the others must be II: \(x, -y, -z\); III: \(-x, y, -z\); IV: \(-x, -y, z\). It proves possible to place another four electrons, following the first along the same circles at a distance of \(75^\circ\). Such a system of four circles (shown in stereographic projection with simultaneous positions of 8 electrons in Fig. 6, the electrons located on the front part of the sphere being denoted by crosses, and those on the back by small circles) does indeed possess cubic symmetry, and Lande showed that for it the expansion \(\psi_{w_1w_2}\) also begins with the 9th power of the distance between centers. The energy of such a configuration, calculated by applying

Fig. 6.

of the quantum theory to these circular motions proves to be considerably smaller than the energy of a plane ring of eight electrons, so that it must possess greater stability and be more probable than the Bohr–Kossel ring. It is also remarkable that the total angular momentum and the resultant magnetic moment are equal to zero, which removes the difficulties arising in Bohr’s model because of the very large magnetic moments of coplanar rings, not observed in reality.

To confirm the theory being developed and to obtain indications as to the direction in which it needs further improvement, Born [7] calculated several other quantities characteristic of the mechanical behavior of crystals. The calculations, which are quite complicated, were carried out on the basis of the methods set forth in his book Dynamik der Krystallgitter, and also in the article [8], where the method is specialized for the cubic system. The basic assumptions of the theory are also simplified somewhat: it is assumed that the law of interaction between two ions \(k\) and \(k'\), situated arbitrarily with respect to one another, leads to the potential energy

\[ \psi_{kk'}=\pm \frac{e^2}{r}+\frac{b_{kk'}}{r^n}\ldots\ldots\ldots\ldots\ldots\ldots (30), \]

where the minus sign refers to the case \(k\ne k'\), and the plus sign to the case \(k=k'\), while the indices \(k\) and \(k'\) may take the values 1 or 2, corresponding to the two kinds of ions. The simplification consists in taking \(b_{kk'}\) to be constant, which reduces the force of interaction of the ions to a simple central force

\[ \psi'_{kk'}=\pm \frac{e^2}{r^2}+\frac{n b_{kk'}}{r^{n+1}}\ldots\ldots\ldots\ldots\ldots\ldots (31), \]

depending only on the distance, whereas it, of course, depends to a certain extent on the orientation of the ions (which have a definite configuration not of spherical symmetry) with respect to the line joining their centers, and moreover, even under the hypothesis of “rigid” ions made at the very beginning, in addition to the central force there must also appear, owing to the spatial distribution of the charges, moments of rotation. It is all the more interesting to trace how far such a simplification can correctly convey the observed phenomena. Summation of expression (30) over all ions of the lattice gives, for the potential energy of the elementary cube, an expression of the form (26)

\[ \varphi=-\frac{a}{\delta}+\frac{b}{\delta^n}, \]

where, by the method set forth above, \(n\) and \(b\) can be determined; for the calculation, however, of the quantities: the wavelength of the residual rays \(\lambda\) (corresponding to the frequency of oscillations of the lattice, when neighboring unlike ions oscil-

are in opposite phases) and the modulus of elasticity \(c_{11}\) (in Focht’s notation), it is necessary, in addition, to know the ratio

\[ \beta=\frac{b_{11}+b_{22}}{2b_{12}} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (32). \]

For the exponent \(n=9\), the value of \(\beta\) does not play a large role; Born takes it to be equal to unity for crystals of the type under consideration, to which the halide salts of monovalent thallium may also be assigned. The following table gives the observed and calculated quantities: the wavelength of the residual rays, the compressibility and modulus \(c_{11}\), and also the observed value of the product \(\chi c_{11}\), which according to this theory should come out equal to 2.28 for all crystals of the type under consideration.

The agreement may be recognized as quite satisfactory, and the remaining discrepancies may be ascribed to the inaccuracy of the basic assumptions of the theory.

Very characteristic results as regards the admissibility of the simplifications made were obtained in the study of the properties of the more complex crystal zinc blende \((ZnS)\), with two divalent ions, arranged in two cubic lattices with centered faces, displaced relative to one another by \(1/4\) of the diagonal of the elementary cube (Fig. 7) [9]. This crystal possesses polar axes and piezoelectric properties. The arrangement of the ionic planes perpendicular to the diagonal of the cube is shown alongside in Fig. 7; going upward, we meet pairs of planes always beginning with \(S\), and in the opposite direction we always meet first \(Zn\). Born’s theory makes it possible to calculate, under the same assumptions as in the preceding work, the piezoelectric constant \(e_{14}\). Formula \((9'')\), with the value \(a\) corresponding to this type of lattice and with ionic charge \(\pm 2e\),

\[ a=61.2e^2 \]

gives \(n\) very close to 5, which is also understandable owing to the lower symmetry of this lattice, evidently caused by the lower symmetry of the ions themselves. For this value of the exponent the quantity \(\beta\) (32) acquires a substantial value, and it has to be chosen so as to satisfy the experimental data as well as possible. In the following table the values for the calculated quantities are listed for the most suitable values of \(\beta\):

Here too the results may be regarded as satisfactory; the greatest discrepancy is shown by those quantities which, as is clear from the table, are most sensitive to small changes in the mechanism. Precisely for piezoelectricity one may expect the greatest inaccuracy of the theory of “central forces” and even of “rigid” ions. The electric moment arising upon deformation of the crystal is explained by the unequal displacement of the two lattices, i.e. by their relative displacement. Given the closeness of the mutual distances, because of the interaction of the ions there must also occur displacements of the charges within them themselves, which should cause an additional electric moment not taken into account by the theory. It is also noteworthy that for \(\beta\) one has to

Fig. 7.

take negative values, which indicates that the second term in (31) gives attraction for ions of the same name, and repulsion for ions of unlike name. This can be explained only by the fact that here the dependence of the force on the mutual orientation of the ions comes into play: because of their correct arrangement it may turn out that ions of the same name (at least the nearest ones) are situated precisely in those directions in which the second term of the force gives attraction.

The theory set forth gives an expression for the potential energy of an ionic lattice, i.e., for the work necessary in order to remove all ions from the lattice to an infinite distance from one another. Taken with the opposite sign, this quantity gives the heat of formation of the lattice from free ions. In connection with other thermochemical data this quantity may in turn serve to test the theory, while, on the other hand, thermochemical data may indicate shortcomings of the theory and the path toward their correction. A number of works [10] are devoted to the thermochemical applications of Born’s theory; in the conclusion of the present survey we cite some results most directly related to it.

Substituting in expression (26) the coefficient \(b\) from (27), we obtain for the potential energy, per 4 pairs of ions,

\[ \varphi=-\frac{a}{\delta_0}\cdot\frac{n-1}{n}\ldots\ldots\ldots\ldots (33). \]

Changing the sign, referring this quantity to one gram-molecule, substituting \(\delta_0\) from (1), and converting by multiplication by \(2.388\cdot10^{-11}\) into large calories, we obtain the heat of formation of a gram-molecule of solid salt from free ions:

\[ U=\frac{a(n-1)\,N^{\frac{4}{3}}\cdot 2.388\cdot 10^{-11}}{4\sqrt[3]{4}}\, \sqrt[3]{\frac{\rho}{\mu_{+}+\mu_{-}}}, \]

where all quantities for crystals of the type under consideration are known; the exponent \(n=9\) for all salts except lithium salts, whose ion, having only two electrons, cannot have cubic symmetry, so that for it \(n=5\), as also in the original theory. These numbers cannot be checked directly, but this can be done by indirect means. Namely, one can compute from the values of \(U\) the heat \(\Delta U\) of an imagined reaction, for example,

\[ NaF+KCl=NaCl+KF+\Delta U \]

where \(\Delta U=U_{NaF}+U_{KCl}-U_{NaCl}-U_{KF}\).

On the other hand, one can take from thermochemical data the heats of formation of these salts from the elements, i.e., from the solid metal and the gaseous halide according to the equation

\[ 2Na+F_{2}=2NaF+2Q_{NaF}\quad \text{and so on.} \]

The heat of the same reaction with these data is obtained as

\[ \Delta Q = Q_{NaF} + Q_{KCl} - Q_{NaCl} - Q_{KF} \]

Comparison of \(\Delta Q\) and \(\Delta U\) gives good results, but, unfortunately, not very convincing ones, since, owing to errors in \(Q\), the quantity \(\Delta Q\) is known with an error almost equal to its value. A table of theoretical values of \(U\) and experimental \(Q\) (which, of course, need not be equal to one another) is given below:

A more convincing test was carried out by Fajans, who observed that heats of solution are known much more accurately than heats of formation. Dissolving one gram-molecule of \(NaF\) and \(KCl\) one time, and \(NaCl\) and \(KF\) another time, in the same quantity of water leads to the very same final state—the dissociation equilibrium between four kinds of molecules and their ions; moreover, in very dilute solutions, the heat of solution of two salts simultaneously is equal to the sum of the heats of solution of each separately. Hence it is clear that the quantity calculated by Born,

\[ \Delta U = U_{NaF} + U_{KCl} - U_{NaCl} - U_{KF} \]

must be equal to

\[ \Delta L = L_{NaF} + L_{KCl} - L_{NaCl} - L_{KF}, \]

where \(L\) denotes the heats of solution of the salts. As an example we give the table:

Even more remarkable is another indirect confirmation, as well as results following from the following considerations. Let us denote the latent heat of vaporization of solid metallic sodium (which, as is known, is obtained as a monatomic gas) by \(D_{Na}\); the work of ionization of this vapor, expressed in large calories per gram-molecule, by \(I_{Na}\), and introduce the corresponding quantities for potassium. We shall denote the sum \(D+I\) by \(Z\). Next let us denote the work of dissociation of a molecule of gaseous chlorine by \(D_{Cl}\), and the work which must be performed in order to give to a neutral chlorine atom an electron removed from a metal atom, forming a negative ion, by \(E_{Cl}\); the sum \(D+E\) will be \(Z\) with the appropriate sign. Then the heat of formation of \(NaCl\) and \(KCl\) from the elements is obtained as equal to

\[ \begin{aligned} Q_{NaCl} &= -Z_{Na}-Z_{Cl}+U_{NaCl} \\ Q_{KCl} &= -Z_{K}-Z_{Cl}+U_{KCl} \end{aligned} \tag{34} \]

Taking the difference, we obtain

\[ \Delta Z_{Na,K}=Z_K-Z_{Na}=(Q_{NaCl}-Q_{KCl})-(U_{NaCl}-U_{KCl}). \]

On the right stand quantities that clearly depend on the anion \(Cl\), whereas the quantity on the left should not depend on the anion; we give a table showing that the combination of the experimental \(Q\) with the theoretical \(U\) in fact satisfies this requirement:

Obviously, \(\Delta Z\) is equal to \(\Delta D+\Delta I\). It turns out that \(D\) for the alkali metals are very close to one another, but are not known with particular accuracy, whereas for the works of ionization we have exact numbers from optical data on series of spectral lines: the number of oscillations \(\nu\) at the limit of a series of absorption lines is related to the work of ionization of the atom in the normal state by the quantum relation \(I=h\nu\). Since \(\Delta D\) is equal to only a few calories, comparison of the data:

shows that, in general outline, agreement between theory and experiment does occur.

As applied to halides, (34) makes it possible, from the quantities \(Q, U, Z_{\text{metal}}\), to determine \(Z_{\text{halide}}\), whence, from the known work of dissociation of the halide molecule, the quantity \(E\) can be found. The following table, taken from Fajans’s article, illustrates the entire calculation: on the left are written the equations of reactions with the halide \(X\) \((Cl, Br, J)\) and with the metal potassium; square brackets, as usual in thermochemistry, denote the solid state, parentheses the gaseous state, the symbol \(\Theta\) an electron, and the numbers on the right the heats of reaction (heat evolved positive). Of these, the first line follows from Born’s theory, the following ones from experiment, and the last—being a consequence of all the preceding—shows that attachment of an electron to a neutral halide atom occurs with evolution of heat and that \(E\) is negative. The halides have an “affinity” for the electron, as was already noted long ago and as follows quite naturally from the considerations set forth at the beginning of the present article on the tendency toward the formation of stable configurations of outer electrons.

Calculating according to this same scheme with other cations, we obtain numbers that agree satisfactorily with one another. From the subsequent literature on the development of consequences from Born’s theory, we shall point to the more detailed development of the theory in the work of Fajans and Herzfeld, in which the question of the volumes of ions is considered. The theory of cubic ions, as was shown above, gives the exponent \(9\), essential for all the calculations, only when the radii of the ions are equal, which is probably not the case; Fajans and Herzfeld showed that a much more detailed agreement with experiment can be achieved by taking into account

taking into account all the first terms of the expansion for $\psi_{k_2 x_i} + \frac{1}{2}\psi_{k_1 x_i}$ of formula (29). From a comparison of the experimental data with the results of the theory, it proves possible to obtain the radii of ions and considerably to improve the agreement of the calculated and observed elastic and thermochemical quantities.

The success of Born’s theory undoubtedly shows that the explanation of intermolecular forces by the electrical interactions of particles is entirely possible and promises good results.

[1] W. Kossel. Ann. d. Phys. 49, 229, 1916.
[2] M. Born und A. Landé, Ber. d. Preuss. Akad. d. Wiss. 1918, p. 1048; Verh. d. D. Phys. Ges. 20, 202, 1918. M. Born, Verh. d. D. Phys. Ges. 20, 224, 1918.
[3] E. Madelung, Phys. Zeitschr. 19, 524, 1918.
[4] M. Born und A. Landé, Verh. d. D. Phys. Ges. 20, 210, 1918; M. Born, Verh. d. D. Phys. Ges. 20, 230, 1918.
[5] A. Landé, Verh. d. D. Phys. Ges. 21, 2, 644, 653, 1919 Ztschr. f. Phys. 2, 83, 380, 1920.
[6] E. Madelung und A. Landé, Zeitschr. f. Phys. 2, 230, 1920.
[7] M. Born, Verh. d. D. Phys. Ges. 21, 533, 1919.
[8] M. Born, Phys. ZS. 19, 539, 1918.
[9] M. Born und E. Bormann, Verh. d. D Phys. Ges. 21, 733, 1919.
[10] M. Born, Verh. d. D. Phys. Ges. 21, 13, 619.
    K. Fajans, Verh. d. D. Phys. Ges. 21, 539, 549, 709, 714, 723. 1919.
    F. Haber, Verh. d. D. Phys. Ges. 21, 150, 1919.
    M. Born und E. Bormann, Zeitschr. f. Phys. 1, 250, 1920.
    K. Fajans und H. Grimm, Zeitschr. f. Phys. 2, 299, 1920.
    K. Fajans und K. F. Herzfeld, Zeitschr. f. Phys. 2, 309, 1920.