INTERATOMIC FORCES IN SOLID AND LIQUID BODIES

Ya. I. Frenkel

1. THE ENERGY OF ATTRACTIVE FORCES AND REPULSIVE FORCES IN METALS

The question of the nature of interatomic forces in solid bodies has more than once been subjected to theoretical investigation. A brief summary of the results obtained may be found in the book by Born and Mayer. These results, however, require certain additions of a fundamental character, to the consideration of which the present note is devoted.

It is customary to divide interatomic forces into two classes: attractive forces and repulsive forces. This division appears fully justified in the case of metallic bodies, where the attractive forces have a purely electrostatic character—the attraction of positive ions to the negatively charged gas formed by collectivized electrons, which fills, approximately uniformly, all the space between the ions—whereas the repulsive forces have a kinetic character, reducing to the pressure of the electron gas. Thus the energy of the attractive forces \((U)\) is potential, while the energy of the repulsive forces \((K)\) is kinetic. The state of equilibrium, characterized by the condition that the total energy \(W = U + K\) be a minimum, is not static but dynamic, just as in the case of individual atoms or molecules, in which the kinetic forces reduce to centrifugal forces overcoming the attraction of the electrons to the positive nuclei.

In the simplest case of the hydrogen atom, the potential energy reduces to \(-e^2/r\), where \(r\) is the radius of the electron orbit; as for the kinetic energy, taking into account the constancy of the angular momentum \(J\), it may be expressed in terms of the distance \(r\) by the formula \(K = \frac{1}{2} m \dot r^2 = \frac{J^2}{2mr^2}\). Thus,

\[ W = -\frac{e^2}{r} + \frac{J^2}{2mr^2}. \]

The minimum of this expression is determined by the formula

\[ \frac{e^2}{r^2} = \frac{J^2}{mr^3}, \]

i.e. it reduces to the relation \(K = -\frac{1}{2}U\), which follows from Clausius’ virial theorem and in the present case imme...

on average expresses the equality between the electrical force of attraction of the electron and the centrifugal force. Substituting for \(J\) the expression \(\dfrac{h}{2\pi}\), we obtain for \(r\) the Bohr radius of the normal hydrogen orbit.

In a completely analogous way the virial relation \(K = -\dfrac{1}{2}U\) can be applied to the description of the equilibrium of an arbitrarily complex system of electric charges, provided only that this equilibrium can be characterized by a single parameter, for example, by the distance \(R\) between neighboring positive ions in the case of metals, and if the velocity of the electrons \(v\) can be related to this distance by the de Broglie quantum relation \(mv = \dfrac{h}{2R}\) (since the minimum length of the de Broglie waves describing the motion of the electrons is approximately equal to \(2R\)).

The treatment set forth above can in principle also be applied to any other—nonmetallic—bodies. In this case, generally speaking, it is necessary in order to characterize the potential energy to introduce not one, but several parameters \((R_1, R_2,\ldots)\), and to express the kinetic energy of the electrons, by means of quantum conditions, as a function of these parameters. The conditions of equilibrium are then reduced to a minimum of the total energy as a function of \(R_1, R_2,\ldots\). This route to solving the problem of the structure of molecules and solids has not yet been tried, and I shall not dwell on it. As for the more general and correct route, which consists in solving the Schrödinger equation for the wave function describing the behavior of the electrons for a given arrangement of positive nuclei (characterized by the parameters \(R_1, R_2,\ldots\)) and in finding the minimum of the energy \(W\) as a function of these parameters, in view of the complexity of this problem it has hitherto been replaced by the problem of the motion of individual electrons, without direct allowance for the interaction between them, in a certain external field which was assumed to be “self-consistent.” In the case of crystalline solids this self-consistent field was incorrectly defined as a periodic field with the symmetry of the crystal itself*). This method, which has found fairly wide application not only to metals but also to dielectrics, must be regarded as very crude, even when the effects of electron exchange are taken into account, and we shall not touch upon it further.

2. COHESIVE AND REPULSIVE FORCES IN DIELECTRICS

The scheme presented above for solving the problem of the structure of metallic bodies, based on the idea of the collectivization of the outer

*) This definition would be correct only with respect to an outside electron introduced into the given crystal; when the “self-action” of the crystal’s own electrons is excluded, the periodicity of the field acting on each of them (from the positive ions and the remaining electrons) is destroyed.

(valence) electrons of individual atoms when they combine into a solid or liquid body, is wholly inapplicable to dielectrics, in which no such collectivization takes place.

Let us consider, for example, the simplest case of a solid (or liquid) dielectric formed by a liquefied or crystallized inert element—say, argon.

The collectivization of valence electrons in metallic atoms is due to the fact that in these atoms the valence electrons are bound relatively weakly and move (to use the language of Bohr’s theory) in orbits of large dimensions. Under such conditions, the approach of atoms, during the condensation of a metallic vapor, to distances comparable with these dimensions directly leads to the collectivization of the electrons; any further approach is prevented as a result of the growth of kinetic forces (the pressure of the electron gas).

In solid argon the orbits of the outer electrons remain small in comparison with the interatomic distances (i.e. the distances between the centers of the atoms), in consequence of which collectivization of these electrons (in the same sense as in metals) is impossible, or, more precisely, can be effected only by compulsion, namely by compressing the body with an extremely large external pressure (of the order of a million atmospheres). The question arises, then, what accounts for the circumstance that the approach of argon atoms during its condensation stops at such distances \(R\), which are large in comparison with the radius \((r)\) of their electron shells.

This question may be answered as follows. The attractive forces between argon atoms, under the condition \(R \gg r\), reduce to their mutual polarization, according to London’s theory, being determined by a potential energy of the form \(U = -\dfrac{C}{R^6}\), where \(C \simeq \dfrac{1}{2} h\nu_0 \alpha^2\) (\(\alpha\) is the polarizability coefficient of the atom, \(h\nu_0\) is the binding energy of one of the outer electrons, i.e. its ionization potential). These London attractive forces are small in comparison with those forces (of a chemical character) which are due to the sharing of electrons, in particular in metallic bodies.

As for the repulsive forces that prevent further approach of the argon atoms, they are due to the collectivization of the outer electrons—but not to that “effective” collectivization which leads to the emergence of a powerful “chemical” bond, but to an “ineffective” collectivization, which causes a monotonic increase of the repulsive forces upon approach.

As is known, one electron can bind two protons (in the ion \(\mathrm{H}_2^+\)), moving around them in a symmetric or “even” manner, as shown in Fig. 1 (which depicts the wave function of the electron at various points of the straight line joining the two nuclei). In this case the energy of the system \(W\), as a function of the distance \(R\) between the nuclei, is rep-

constituted by the curve of Fig. 2, which is characterized by a minimum at the point \(R_0\), corresponding to stable equilibrium.

However, in motion of the antisymmetric or odd type (Fig. 3), the energy of the system \(W\) increases monotonically, producing a continuously increasing repulsion. The latter reduces to electrostatic repulsion between like-charged nuclei; and partly to an increase in the kinetic energy of the electron, which in the limiting case \(R \to 0\) passes over to a two-quantum orbit of the helium nucleus (whereas in the case of Fig. 1 its motion reduces at \(R = 0\) to a one-quantum orbit).

Fig. 1.

The preceding results are not substantially changed when one electron is replaced by two, i.e. when the ion \(\mathrm{H}_2^+\) is replaced by the molecule \(\mathrm{H}_2\). In the case of two helium atoms the situation is complicated further, since one pair of electrons (with opposite spins) is shared in a symmetric manner, and the other pair in an antisymmetric manner; as a result the total energy changes approximately as shown in Fig. 4, which in fact excludes the possibility of formation of a stable molecule \(\mathrm{He}_2\), or of a more complex aggregate—a condensed body with strongly bound atoms.

Fig. 2.

Fig. 3.

All inert elements are characterized by the “closedness” of the outer layer of their electron shell, i.e. by the presence in it of an even number of electrons moving pairwise, with opposite spins, in orbits of identical quantum character. Therefore, when atoms of these elements approach one another, half of the electrons are shared in an even (symmetric) manner, and half in an odd (antisymmetric) manner, as a result of which repulsive forces arise, increasing monotonically as the distance decreases.

The presence of ineffective collectivization, associated with the appearance of these repulsive forces, however, does not in the least prevent the simultaneous appearance of forces due to the mutual polarization of atoms. At large distances between them (more precisely, between their centers) these polarizational attractive forces prove greater than the repulsive forces caused by collectivization; at smaller distances, however, the latter gain the upper hand. The total energy \(W\), taking both effects into account, has the same (approximately) form as in Fig. 2, and corresponds to the formation of a stable system at some equilibrium interatomic distance \(R_0\).

Fig. 4.

3. SIMULTANEOUS REALIZATION OF MUTUAL POLARIZATION OF ATOMS AND INEFFECTIVE COLLECTIVIZATION OF ELECTRONS

The results set forth above contain nothing new: they are presented, in particular, in the book by Born and Mayer. However, in that book, as also in the original articles of a number of authors to which it refers, the physical side of the matter has not been clarified in a proper way. Namely, on the one hand, it is known that London polarizational (or “dispersion”) forces manifest themselves only at such interatomic distances \(R\) that are much greater than the “radius of atoms” \(r\), while, on the other hand, it is assumed that the collectivization of electrons—both effective and ineffective—can occur only in the case when \(R\) has the same order as \(r\), or is even smaller than the latter.

It is not difficult to see that this assumption does not correspond to reality. It would correspond to it only in the case where the electrons moved according to the laws of classical mechanics, excluding the possibility of their penetration through potential barriers into those regions in which the potential energy \(U\) is greater than the total \(W\), i.e., into which the electron could penetrate only with negative kinetic energy. Quantum mechanics, on the contrary, provides the possibility of such penetration—true, with a probability that decreases exponentially with increasing depth of penetration into the classically forbidden region. Thus, collectivization of an electron by two protons (or by any other nuclei), of the type of Fig. 1 or Fig. 2, is possible, in principle, at arbitrarily large distance \(R\) between them. The states usually described by the wave function

\[ \varphi_{+}=\frac{1}{\sqrt{2}}\,[\psi_a(r)+\psi_b(r)] \]

or

\[ \varphi_-=\frac{1}{\sqrt{2}}\,[\psi_a(r)-\psi_b(r)] \]

(where \(\psi_a(r)\) and \(\psi_b(r)\) are functions describing the motion of the electron about nucleus \(a\) or \(b\), in the absence of the other) must then be somewhat modified by taking into account the polarization experienced by one of the hydrogen atoms formed by the electron when it is localized on nucleus \(a\) or \(b\) under the influence of the electric field \(E=\dfrac{e}{R^2}\), created by the second nucleus. In other words, the functions \(\psi_a(r)\) and \(\psi_b(r)\) must be replaced by somewhat different functions \(\psi'_a(r)\) and \(\psi'_b(r)\), which take this polarization effect into account, i.e. the Stark effect in the field \(E\).

Analogous considerations apply to the case of two electrons. According to Heitler and London, the state corresponding to the effective symmetrization of both electrons is described by a symmetric function of the form

\[ \Phi_+(r_1,r_2)=\frac{1}{\sqrt{2}}[\psi_a(r_1)\psi_b(r_2)+\psi_a(r_2)\psi_b(r_1)], \]

whereas the antisymmetric function

\[ \Phi_-(r_1,r_2)=\frac{1}{\sqrt{2}}[\psi_a(r_1)\psi_b(r_2)-\psi_a(r_2)\psi_b(r_1)] \]

corresponds to ineffective symmetrization (i.e. such that the energy of the whole system increases monotonically with decreasing distance \(R\)). In order to take into account the influence of the mutual polarization of the atoms, it is necessary in these formulas to replace the functions \(\psi\) by functions \(\psi'\) that take this polarization into account. Under such conditions we could, in principle, alongside the ordinary hydrogen molecule, characterized by the function \(\Phi_+\) and a large binding energy, obtain a molecule of a second type, described by the antisymmetric function \(\Phi_-\), with a relatively very weak polarization bond, which corresponds to equilibrium between the London polarization forces and the repulsive forces caused by ineffective collectivization of the electrons. For such an antisymmetric molecule the equilibrium distance \(R_0\) would, of course, be considerably (several times) larger than for the “symmetric” one. Owing to the tendency of any system to pass (at low temperatures) into the state with the lowest possible energy, an antisymmetric hydrogen molecule would sooner or later have to pass into a symmetric one. In this case its resultant spin would have to change by 2 units, i.e. pass from the value 2 to the value 0 (if the spin of one electron is taken equal to unity). The probability of such a transition, which can occur only under the action of magnetic forces capable of changing the direction of the spin of one of the electrons (in particular, under the influence of magnetic inter-

...of the action of the latter), is relatively small—of the order of the probability of transition of helium atoms from the ortho state (with spin 2) to the para state (with spin 0).

When hydrogen atoms are replaced by helium atoms or by atoms of some other inert gas, we obtain an essentially different situation, since in this case states of the symmetric type cannot occur and, thus, only antisymmetric states are realized, in which the repulsive forces caused by the ineffective collectivization of electrons are balanced by the forces of polarization attraction at comparatively large distances, which corresponds to a comparatively weak van der Waals bond between the atoms.

4. INTERPARTICLE FORCES IN THE CASE OF COMPLEX MOLECULES

Ordinary hydrogen molecules are similar to helium atoms in the respect that they contain two electrons in one and the same stationary (quantized) state, with opposite spins. Therefore, when two similar molecules approach one another, the same thing occurs as when two helium atoms approach one another: namely, an ineffective collectivization of electrons, associated with the appearance of repulsive forces, and mutual polarization, leading to the appearance of van der Waals attractive forces. Liquid or solid hydrogen, formed upon condensation of the gaseous form, is therefore held together by the same forces as helium, which explains its similarity to the latter in the sense of low density, small latent heat of evaporation, and low critical temperature. Some authors have previously put forward the supposition of the possible existence of liquid or solid hydrogen in a metallic state (with a much greater density, a much higher critical temperature, and electrical conductivity). There are, however, no arguments in favor of such a supposition. Metallic hydrogen would probably have a much lower energy of formation than ordinary hydrogen, in which the principal part of the energy of formation corresponds to the energy of bonding of atoms in molecules, whereas the interaction of the latter with one another is very weak. Thus metallic hydrogen could be regarded, at best, as a metastable state of hydrogen, capable of passing, with the liberation of a large amount of energy, into the ordinary state. In the case of the majority of other substances existing in the solid state in the form of separate molecules, these molecules are bound to one another in approximately the same way as hydrogen molecules, i.e. by polarization forces, which are balanced by repulsive forces caused by ineffective collectivization of the outer electrons.

In some cases, however, owing to the antagonism between internal and external bonds, molecules, when joined together into a solid or liquid body, break up into atoms or radicals

with opposite electric charges, which give rise to a heteropolar or ionic bond between them. The prototype of such ionic compounds is rock salt.

In this case, the repulsive forces between positive and negative ions are again due to the ineffective generalization of the outer electrons.

In all these cases the energy of the repulsive forces, as a function of the interatomic distance, may be described approximately by an exponential function of the form \(Ae^{-aR}\) (or by its product with some polynomial), whereas the energy of the attractive (polarization) forces has the form \(-\dfrac{C}{R^6}\). The role of kinetic energy is thus played by a more complicated quantity, in which the kinetic and potential energies of the collectivized electrons are inseparably merged.

In the case of complex molecules, the conception of attractive and repulsive forces as central forces emanating from some point of the molecule—its center—or applied to this center, obviously becomes incorrect and must be replaced by the notion of a collection of a more or less considerable number of similar centers for one and the same molecule. These centers may be individual atoms forming the molecule, or else the bonds between neighboring atoms, formed by collectivized electrons, or, finally, both. The cohesive forces between such complex molecules may be reduced, as Langmuir showed in 1926, to a peculiar surface tension between molecules, if they are treated as miniature solid bodies, or, in the general case, between separate heterogeneous parts of identical molecules; the repulsive forces, which determine the “hardness” of molecules, i.e. the inability of individual links to approach one another beyond a certain limit, may still be associated with the ineffective collectivization of the outer electrons.

5.

We have seen above that, in the case of metals with their “effectively collectivized” electrons, the repulsive forces have a purely kinetic character, i.e. they are due to the pressure of the gas formed by these electrons. The circumstance that the kinetic energy of the electrons may be treated as the potential energy of the atoms, determined by the mutual distances of the latter, is connected with the fact that the electrons move much faster than the atoms and that, under given quantum conditions, which remain unchanged when the atoms are displaced, the kinetic energy of the electrons is determined uniquely by the arrangement of the atoms or, more precisely, of the atomic nuclei, which are then regarded as immobile.

The increase in the volume of solids when they are heated is connected, as is known, with the thermal motion of the atoms or, what is the same thing, with their kinetic energy. Since the potential energy (conditioned by the arrangement and motion of the electrons for a given arrangement of the nuclei) has the form shown in Fig. 2, the approach of atoms for a given energy of their oscillations turns out to be smaller than their separation—which, as is known, leads to thermal expansion. The latter may also be regarded as the result of a thermal pressure caused by the quasi-elastic oscillatory motion of the atoms, when account is taken of the dependence of the frequency \(\nu\) of these oscillations on the volume. At room temperatures this thermal pressure is expressed by the formula

\[ p=-3RT\,\frac{\partial \lg \nu}{\partial V}, \]

where \(T\) is the temperature and \(V\) the volume. In the general case \(3RT\) should be replaced by the mean value of the thermal energy (which at low temperatures reduces to \(AT^4\)).

Since the oscillatory motion of atoms, according to quantum mechanics, does not cease at the absolute zero of temperature, retaining an energy corresponding to the value \(\frac{1}{2}h\nu\) for oscillations of each frequency \(\nu\), the forces of interatomic repulsion in solids retain in part a kinetic character even at \(T=0\). In the case of most bodies, at not too low temperatures, this circumstance practically plays no role. However, in the case of such a substance as helium, it has an extremely important significance for the proper understanding of the properties of this substance both at \(T>0\) and at \(T=0\).

London showed in 1937 that the small density of helium (about 0.1 at \(T \simeq 0\)) is explained precisely by the residual pressure produced by its atoms and which can be calculated if one assumes that the helium atoms oscillate not harmonically, but as though the attractive forces between them were absent altogether. Under such conditions the motion of each atom may be represented as rectilinear and uniform within the limits of the “cell” which it occupies; this cell is bounded on all sides, as it were, by rods surrounding the atoms, from which it rebounds as from solid walls. If the radius of the helium atom is denoted by \(a\), and the radius of the spherical cell which it occupies by \(R\), then its center can move within a sphere of radius \(R-a\).

Its wave function in the normal state is then expressed by a formula of the form

\[ \psi(r)=\frac{A}{r}\sin\frac{\pi r}{R-a}, \]

and the kinetic energy in this state proves to be equal to

\[ K=\frac{\left(\frac{h}{\lambda}\right)^2}{2m} =\frac{1}{2m}\left[\frac{h}{2(R-a)}\right]^2, \]

where \(\lambda = 2(R-a)\) is the wavelength, and \(m\) is the mass. The radius of the cell is determined by the atomic volume \(v\) (i.e., the volume per atom) from the formula \(v=\frac{4\pi}{3}R^3\).

Starting from these considerations, it is not difficult to show that the most favorable arrangement of helium atoms at \(T=0\), from the standpoint of minimum energy, is not that which corresponds to close packing of spheres, i.e., to the presence around each atom of 12 neighbors in contact with it, with \(R=a\) (in this case the kinetic energy \(K\), according to the preceding formula, would become infinite), but rather one which corresponds to a two or even three times smaller number of neighbors (i.e., simple cubic or diamond-like), with the difference \(R-a\) proving comparable with \(a\). Thus, in this case the repulsive forces caused by the ineffective collectivization of the outer electrons lead to the emergence of additional, purely kinetic repulsive forces, caused by the oscillatory motion of the atoms themselves as hard spheres. Experimental study of the structure of liquid helium II near absolute zero by means of X-rays generally confirms London’s theoretical conclusions; however, the experimental results obtained speak rather in favor of a structure of the simple cubic lattice type than of a diamond-like structure corresponding to a cubic lattice of the tetrahedral type.

This question is at present not fully clarified, as is also the general theory of the properties of helium in the liquid state at very low temperatures. In any case, however, it is clear that kinetic repulsive forces not only of electronic but also of atomic origin (which determine the thermal pressure of atoms at \(T>0\)) must play a decisive role in the correct understanding of these properties.