Abstract
A paper read at the Faraday Society conference on “Crystal Structure and Chemical Composition.”
Full Text
THE PROBLEM OF THE METALLIC STATE1
J. Bernal, Cambridge
Of the four types of structure of a solid substance—ionic, homopolar, molecular, and metallic—the first two have received a sufficiently complete explanation on the basis of electronic and quantum theory. The third has not yet found an explanation, but it evidently depends simply on van der Waals forces; the metallic state, however, although physically it has been studied most fully, has so far received no satisfactory explanation.
It is difficult, first of all, to establish what precisely is meant by the term “metallic state.” In a broad sense this term is applied to a substance that transmits electricity by means of the transport of electrons. This property of “metallic” conductivity characterizes not only solid but also liquid metals, and it would be correct to apply the term “metallic” to all similar substances. A definition by means of conductivity is, however, merely a more rigorous way of expressing the directly observed fact that metals are almost opaque and possess a metallic luster, which can easily be recognized but not described exactly. If, however, one were to choose any other properties—mechanical, thermal, or chemical—then it would be possible to differentiate certain metallic
substances, and the entire metallic state would have to be divided into a whole series of subdivisions. Since, from both the theoretical and the experimental side, the electrical properties of metals had been studied best of all, the presence of subdivisions did not attract attention. In this article, on the contrary, an approach is taken from the side of other properties of metals and, in particular, from the side of the study of the structure of solid metals, for example metallic compounds and solid solutions of various kinds.
From the point of view of crystal chemistry, metallic substances occupy, as it were, an intermediate position between ionic and homopolar substances. The transition from metallic to ionic and from ionic to homopolar structure is almost continuous. In what follows it will be shown that there exists a large number of metallic compounds which by their nature are ionic, and whose metallic properties are of secondary importance. The connection with homopolar substances is still closer, because there exist elements and compounds in which the character of the bond between the ions can be, in one and the same crystal, both homopolar and metallic. At the same time there also exists a quite definite, purely metallic structure of solids and liquids, in which neither homopolar nor ionic forces participate. Examples of such a structure are furnished, in particular, by the metallic elements. On the basis of what has been said, the metallic state should receive the following subdivisions: the ionic-metallic, homopolar-metallic, and purely metallic states. The properties which serve to define the metallic state appear most distinctly in true metals; they possess the highest electrical and thermal conductivity and are the most opaque. In the transition from them to substances of the ionic and homopolar type, these properties become expressed more and more weakly. Along with them go other properties which likewise could characterize the metallic state; in particular, the tendency of the elements to form mixed liquid phases with a very extensive region of miscibility—
metallic nature of the components; toward the formation of solid solutions with almost any other element; and the tendency of compounds to form solid solutions with each of the components or with other compounds. But one of the most essential properties of the metallic state is revealed in the study of the structure of metallic crystals by X-rays: all metallic substances have a close-packed structure, with the densest arrangement of atoms occurring in purely metallic substances, while, in passing to substances with a homopolar or ionic structure, the arrangement becomes less dense.
At the present time the structure of most metallic elements has been determined. A large part of the metals has a very simple structure and possesses a cubic or hexagonal close-packed arrangement. A certain portion of metals has the structure of centered cubes, which also represents an arrangement close to close packing. The remaining substances, which possess a more complex structure, prove, as we shall see later, to be substances not of purely metallic nature. On the other hand, metallic compounds have been studied much less completely, although it is precisely with the aid of X-rays that it has been possible to establish their existence beyond doubt.
The laws governing the formation of metallic compounds differ so greatly from the laws of the rest of chemistry that it is not surprising that ordinary methods do not always make it possible to discover them. Only 56 of 3000 individual binary equilibrium metallic systems have been studied by X-rays, and only 8 ternary ones; for many of them the available information is far from complete.
V. M. Goldschmidt1 measured the atomic radii of approximately all metals having 12- or 8-fold coordination, choosing pure metals when they exhibited such coordination, or simple metallic compounds with a suitable coordination number if for the pure—
of pure metals this did not prove to be the case. In structures with close packing, which characterize both metals and metallic mixtures, the coordination number is large—12 for cubic and hexagonal lattices with close packing, and 8 for a body-centered cubic lattice, whereas for various complex structures this number ranges from 16 to 6. At such high coordination numbers, small changes in atomic diameter in passing from one coordination to another are averaged out in determining the atomic volume, and this atomic volume is constant for each given element throughout the whole range of its metallic compounds and solid solutions. The approximate constancy of atomic volumes is one of the primary facts with which any theory of the metallic state must reckon; but before drawing the conclusions that follow from it, one should examine in more detail the properties of true metals and truly metallic compounds.
Typically metallic substances, both solid and liquid, are characterized above all by their high electrical conductivity, which has a sharply pronounced maximum both for pure metals and for alloys; in other words, the conductivity rapidly decreases under the influence of impurities. But perhaps the most characteristic of metals are their mechanical properties, which sharply distinguish metals from substances of ionic and homopolar structure. A crystal of a pure metal under pressure does not cleave along cleavage planes and does not fracture, but forms slip planes, along which layers of atoms move relative to one another, requiring the expenditure of only a small amount of energy. It is possible that, in the ideal case of an absolutely pure metal and at an infinitely small stress, there would be no expenditure of energy at all in shear. The metal would then behave, in fact, like a liquid, and a crystal of a pure metal would differ from a liquid only in the regular arrangement of its atoms. In reality, however, when a certain shear has occurred, hardening begins, which prevents further—
nearest to sliding. This hardening is in all probability caused by a disruption of the regularity of the crystal lattice. Apparently it is quite similar to the same kind of disruption caused by the presence of foreign atoms in a solid solution. (It is precisely to this capacity for hardening during cold working and during the formation of alloys that metals owe their technical importance.)
The distortion of the regularity of the lattice manifests itself in various ways: along with mechanical hardening, the electrical resistance always increases very strongly. The nature of the change that has occurred is revealed most clearly in X-ray analysis. In a strained or imperfectly pure metallic crystal, the reflection of a monochromatic ray from a crystal plane is obtained with indistinct outlines and, moreover, as the strain increases, it becomes more and more diffuse. This indicates that the atoms are no longer arranged in absolutely plane parallel layers, but that here a more or less disorderly displacement of some of the atoms occurs relative to the plane in which they had been situated.¹
Similar displacements, but periodic in time, are caused by the thermal vibrations of atoms; and indeed, when looking at an X-ray photograph of a metal on which the lines appear very diffuse, it is impossible, without further investigation, to decide whether the photograph pertains to a crystal that is insufficiently pure, or to a crystal that has been subjected to pressure or heated. It is possible that the phenomena which P. L. Kapitza discovered in strong magnetic fields are caused by a similar distortion of the crystal lattice.² One may suppose that solid solutions constitute unstable states,
¹ After this article had been written, a further confirmation of this idea was given in Prof. Mark’s report presented at this conference. In this report he proves that the intensity of reflections from a stretched crystal decreases very rapidly with increasing stretching.
² P. Kapitza, Proc. Roy. Soc. 123, 292 (1929).
and metastable, and that, given sufficient mobility of the atoms or sufficient time, all solid solutions would decompose into their constituent parts or would form regular superstructures. From this point of view, a solid solution represents, as it were, a kind of glass, only one of regular structure.
The criterion of the presence of lattice disorder not only makes it possible to distinguish pure metals from solid solutions, but also makes it possible to call into question the existence of true intermetallic compounds. The number of the latter must be very large, and only a negligible fraction of them has been investigated. For establishing definite intermetallic compounds, the methods of thermal and micrographic analysis have in general been insufficiently sensitive; but X-ray investigation, which reveals the existence of new structures, usually makes it possible to arrive at sufficiently convincing conclusions, provided, however, that the preliminary heat treatment of the material has been sufficiently thorough. Along with ascertaining the existence of a definite compound, the method of investigation by X-rays usually proves sufficiently sensitive to determine whether we have before us a chemical compound of constant composition, or whether it gives a series of solid solutions with one or another region of homogenization. This can be established from the change in the dimensions of the lattice, which can be measured with an accuracy up to one ten-thousandth.
The electrical properties of intermetallic compounds apparently indicate that the compounds must be regarded as true metals, sharply differing from solid solutions. Especially eloquent in favor of this is the fact that the intermetallic compound Cu₃As, which Kapitza investigated, behaves in a strong magnetic field like a pure metal. The mechanical properties of pure intermetallic compounds have not yet been sufficiently studied. A truly metallic compound may be defined as a substance forming slip planes with the same ease as a pure metal, and possessing
of the same high symmetry. We shall see that the greater part of intermetallic compounds do not have this character, but tend more or less toward the homopolar and ionic types.
A further characteristic property of true metals consists in the ease with which they accept atoms of other substances in various proportions, forming solid solutions with them. The nature of the atoms in this substitution is not essential. There are metals with which every element—except, perhaps, only the inert gases, oxygen, and the halogens—can form solid solutions. This property makes it possible to distinguish clearly between different types of metals, and I decided that it was worth considering, if only in the roughest way, the ability of some metals to act as solvents for others and their own solubility in other metals.
Taking as a basis the data contained in the International Critical Tables, and supplementing them with data from more recent works, I compiled a table of the mutual solubility of all metallic elements for which it proved possible to establish it. It immediately became clear that mutual solubility is not a reciprocal property. Some metals have the ability to dissolve large quantities of other elements; others are practically incapable of taking into themselves even a few atoms, except atoms of elements closely related to them. The data I have collected are too fragmentary to have real significance; they can only sometimes provide useful indications. For the dissolving ability I derived average values from the atomic solubility of several different metals in a given metal, leaving aside those that mix with one another in any proportion. In doing so, however, I did not take into account the factor of differences in atomic diameters, which undoubtedly plays a major role, nor also the probability that many elements dissolve in metals not as separate atoms but as molecules of a chemical compound. Nevertheless, the results are extremely remarkable. Metals,
having the greatest dissolving capacity belong to the transition metals of groups VII and VIII, and also to the noble metals. Apparently, although the data are too scanty to assert this definitely, Pd has the greatest dissolving capacity. All these metals, from other points of view as well, are typical metals, and it is noteworthy that they all possess a cubic structure with centered faces; if an element such as Fe has more than one structure, then it is precisely its $\gamma$-lattice, i.e. the cube with centered faces, that has the greatest dissolving capacity. The dissolving capacity of metals of other groups of the periodic system falls off very rapidly; for Zn, for example, it is only one tenth of the dissolving power of Cu, and for Al only one twentieth. With the solubility of these metals, however, the matter is otherwise. Zn and Cd are apparently the most soluble; Al and Sn follow directly after them. Typical metals stand very low in the table, but this should not diminish their rank, since a considerable number of metals that mix with one another in any proportion did not enter the table.
From the data presented one may apparently conclude that the dissolving capacity of metals is a function of their crystal lattice, just as are their electrical and mechanical properties; solubility, however, is a function of the atom of the metal.
With a purely metallic lattice, in accordance with the fact that even the smallest force can cause whole rows of atoms to shift into a neighboring position, a foreign atom (which need not necessarily be metallic) can be introduced at the cost of only a small distortion of the lattice. It is precisely the magnitude of the distortion caused by the introduction of the given atom that determines its solubility. Nonmetallic atoms, for example N and C, produce the greatest distortion and therefore have the least solubility. On the other hand, the dissolving power of a given type of crystal is determined by the lattice distortion permissible for it. However, if the atom belongs
belongs to the ionic or homopolar type, the formulation becomes much more difficult and is selective, being carried out by a method that will be discussed below. The existence of a phase of complete miscibility of two metals is encountered in two cases: first, for true metals, most of which mix perfectly with one another in the solid state, provided only that their atomic volumes do not differ too sharply from one another; second, intermixing is possible for two elements that are isomorphous and possess very similar atomic volumes. For example, Zn and Cd are very similar chemically, but do not mix, whereas Mg and Cd, which are farther apart but have approximately identical atomic volumes, do mix, although not completely, as X-ray investigation has shown.^1
The difference among metals in the strength of their dissolving ability, or in the size of the region of solid solutions formed by them, may have an essential bearing on the theory of the electrical conductivity of metals.
The latest experiments of Kapitsa have convincingly proved that the electrical resistance of metals is composed additively of two quantities, one of which decreases with temperature and rapidly vanishes in proportion to some high power of the temperature near absolute zero; the other term does not depend on temperature and remains constant for a given metal, being, however, a function of the degree of distortion of the crystal lattice, arising either from strain or from impurities. If all metals could be obtained in an absolutely pure form, then, according to Kapitsa, all of them would be superconductors. Combining this supposition with the fact of the different dissolving ability of metals, the distribution of the elements that are actually superconducting in the periodic system can be explained by means of a rather simple hypothesis. These elements—In, Sn, Ta, Hg, Tl, Pb—are located at the end of the periodic table, that is, they are ele-
^1 G. Natta. Annal. Chim. App. 18, 135, (1928).
heavy elements. At the same time, among the latter there occur remarkable exceptions, and precisely these elements that constitute the exception—chiefly the noble metals—possess, as we have already seen, a high solvent power. When a piece of superconducting metal is cooled, its resistance does not fall asymptotically to zero, but reaches a definite value; then, at a definite temperature characteristic of each metal, it suddenly falls to zero. (In the case of Hg, apparently, not only the thermal but also the residual resistance disappears in the superconducting state. If this phenomenon does not depend on stretching, the hypothesis set out above will have to be abandoned.) It is supposed that such a sudden fall, whose magnitude is a measure of the electrical resistance of the metal that arises from lattice distortion, is caused by the sudden freezing-out of impurities occurring at this temperature. So long as the mean thermal displacement of the atoms remains of the same order as the distortion produced by the presence of a foreign atom, the mixed crystal exists. But when, on cooling, this condition is violated, the lattice breaks and the foreign atom is expelled. Thus it is assumed that in the superconducting state the metal is stretched, but that its lattice is not absolutely disrupted. Cracks do not hinder conductivity, since, in view of the absence of resistance, the cross-section of the conductor plays no role. One may expect the rupture of the lattice to occur at higher temperatures in those cases where the amplitude of the thermal vibrations is smaller (that is, for heavier elements), and the distortion produced by the foreign atom is greater (for metals with low solvent power); we see that both these conditions are precisely fulfilled in actual superconducting metals.
The sharply defined temperature of lattice rupture evidently depends on the small number of quanta of thermal energy in this temperature region. Metals in which superconductivity has not been observed may turn out to be superconducting
THE PROBLEM OF THE METALLIC STATE
...at a threshold lying below the modern range of observations; it is also possible that the energy of half a quantum at absolute zero is sufficient in these cases to preserve the distorted lattice at all temperatures; in such a case the only way to make a metal superconducting is to obtain it in chemically and physically pure form. This view is partly confirmed by the latest experiments of de Haas1 on the superconductivity of metallic compounds, proving that superconductivity is a property of crystals, and not of atoms. Especially interesting is the case of the bismuth–gold eutectic, which is superconducting, whereas bismuth itself does not possess this property. It is possible, as Professor Goldschmidt informed me in this connection, that in the present case gold, with its high dissolving power, acts as a purifier upon bismuth.
The fact that for intermetallic compounds, as for example for $\mathrm{Ag}_3\mathrm{Sn}$, the sharp jump in temperature characteristic of a pure metal is not obtained apparently indicates the more complex nature of their thermal oscillations, due to the fact that such compounds no longer possess a simple crystal lattice. The present hypothesis on the nature of superconductivity is apparently accessible to experimental verification; but, of course, this will be sufficiently difficult, since for verification it will be necessary to detect definite changes in the reflection of X-rays or in the mechanical properties of the superconducting metal above and below its temperature boundary.
Returning now again to the study of the properties of true metals, we shall see that in many properties—electrical, mechanical, and structural—the true metals form a very homogeneous group. If we wish to establish by what properties they differ from one another, we shall see that, of the whole mass of properties, the most characteristic is the atomic volume. The atomic volume
metals varies within wide limits, from 8 cubic angstroms for Be to 116 for Cs. The mechanical and thermal
Fig. 1
properties vary in close correspondence with the change in volumes. In Fig. 1 there is shown, first, the atomic density (the reciprocal of the atomic volume) of the elements
THE PROBLEM OF THE METALLIC STATE
three transition periods, and, for comparison with this curve, the curve of characteristic temperatures calculated from Lindemann’s formula from the melting points and compressibility has been drawn. It is evident that these curves run very close to one another over a great extent, from the alkali metals with which the group begins up to the Zn group; beyond this, irregularities begin, evidently depending on the presence of structures and binding forces no longer of a purely metallic character. The dependence of the properties on the curve of atomic densities and the very form of this curve—which clearly reflects the distribution of the electrons inside the atoms—plainly indicate that, for metals, the character of the bonds between atoms is sufficiently simple and homogeneous. Without yet defining these metallic bonds more closely, we shall now turn to the consideration of the homopolar and ionic variation.
It is clear that for the elements from Al to Cl, from Zn to Br, from Cd to I, and Hg, Tl, metallic properties gradually, step by step, pass over into non-metallic properties. This is also reflected in their crystalline structure. Close packing is no longer the rule. Pure Zn and Cd are usually assigned to the hexagonal system with close packing, but the axial ratio here is no longer 1.6 but 1.8; and from the discontinuity in the formation of alloys one can see that this is no longer a simple modification of the close-packed structure.^1 In Group III we have the special structures of Ga and In, and in Group IV the completely homopolar complex structure of Ge and gray tin. Group V is still more interesting: P has several complex molecular structures; As and Sb possess, in addition, a metallic form, as does Bi.
With this structure each atom has three near neighbors, while the next three are situated considerably farther away. Joining the nearest atoms by lines, we obtain an angle varying from \(97^\circ\) in As to \(93^\circ 50'\) in Bi, which departs more and more from the tetrahedral angle. With more distant
^1 For example, \(\varepsilon\)- and \(\eta\)-brass, Westgren and Phragmen, Phil. Mag., July, 311 (1925).
with atoms of the line joining the first atom, form considerably smaller angles of \(73^\circ 30'\) and \(81^\circ 40'\). The simplest assumption is that these structures do indeed represent layered crystal lattices, in which the nearest atoms are connected by homopolar bonds, and the more distant ones by metallic bonds. This point of view is confirmed by Kapitza’s1 work on the preparation and magnetic properties of single-crystal Bi. The considerable diamagnetism of Bi is readily explained on the basis of homopolar bonds, while the disappearance of diamagnetism under pressure and the simultaneous decrease in volume indicate that, in the liquid state, these bonds disappear. The peculiarities of bismuth crystals near the melting point under the action of stress, leading to a change of the directions \((111)\) into \((11\bar{1})\), are easily explained by a change in the direction of the homopolar bonds before their final rupture.2
The structures of group VI, metallic Se and Te, indicate that here the homopolar bonds should be regarded as linking atoms into helical chains parallel to the trigonal axes, while the bonds holding these chains together are again metallic.
Thus, among the elements there occur purely metallic structures, purely homopolar structures, such as Ge, and metallic-homopolar structures with ratios \(2:1\) and \(1:2\). These intermediate structures, since they are metallic, exhibit the electrical and mechanical properties of metals, but at the same time possess other properties as well, dependent on homopolar bonds. The most immediate of these properties is magnetic permeability. Many metals are diamagnetic because they contain diamagnetic atoms, but homopolar substances are diamagnetic to a far greater extent
and their diamagnetism, in contrast to true metals, rapidly decreases with temperature and disappears at the melting point, thereby proving that diamagnetism is characteristic of the crystal and not of the atom. It must be said that not only metallic elements such as Bi and Te possess diamagnetic properties of this type, but also a large number of intermetallic compounds. The study of the diamagnetic properties of alloys, given the vastness of this field, has so far been only fragmentary; nevertheless, from the work of Honda one can draw several convincing conclusions. I shall confine myself to considering one type of alloy, the so-called $\gamma$-structures, chiefly because this type is very well studied from the structural point of view.
From Honda’s measurements it follows that $\gamma$-brass, $\mathrm{Cu}_5\mathrm{Zn}_8$, and $\delta$-bronze $\mathrm{Cu}_{31}\mathrm{Sn}_8$, which have similar structures, are both strongly diamagnetic, whereas the basic metals composing them do not possess this property in the slightest degree.1
It occurred to me that this property must depend on the structure, and I immediately undertook the investigation of a whole series of specimens of the corresponding compound of copper and aluminum, $\mathrm{Cu}_9\mathrm{Al}_4$, kindly prepared for me by Dr. Stockdale and measured by Dr. Webster. This alloy also proved to be strongly diamagnetic over the whole range of homogenization, and the available data show that other mixtures of CuAl are still more diamagnetic. This is all the more interesting since Al itself is noticeably paramagnetic. Hence one may draw the entirely natural conclusion that in these structures there exist certain forces quite analogous to homopolar bonds, i.e. here there are electron orbits enclosing more than one atom. The physical properties of $\gamma$-structures, which are all similar to one another, confirm the same thing.
In contrast to pure metals they are extremely brittle and are distinguished by comparatively low electrical conductivity; but the hypothesis of homopolar bonds is supported still more strongly by their chemical composition. The $\gamma$-structures serve as the most—
A more remarkable example is provided by Hume-Rothery’s electronic laws, which were used to support the theory of complete ionization of atoms in the metallic state. The cell contains 52 atoms, and if we take 1 electron for Cu, 2 for Zn, and 4 for Sn, we see that for the compounds Cu₅Zn₈, Cu₉Al₄, Cu₃₁Sn₈ one obtains 84 electrons, that is, 21 electrons per 13 atoms. This rule seems very strange from the point of view of ordinary ionic chemistry, but becomes quite reasonable if one adopts the homopolar point of view. In a cell with several atoms and some number of polar bonds, it remains indeterminate from which particular atoms the structure receives its bonding electrons. The simplest example of this we see in diamond structures, where there are 8 electrons for two atoms, as is evident from the series GeSi, GdAs, ZnSe, and CuBr, although in the last case one of the atoms has one electron and the other 7. The correctness of this view is further confirmed by observation of the range of solid solutions of all three compounds. In γ-brass, replacement of one Cu atom by one Zn atom adds one electron to the cell; the corresponding substitution of one Al or one Sn adds 2 and 3 electrons. Hence, consequently, one may expect that it will be considerably easier to substitute Zn than Al, and Al easier than Sn; this is indeed observed: the solubility range of the solid solutions is respectively 8, 4, and < 0.5 atomic percent. On the other hand, replacing a Zn atom by a Cu atom would reduce the number of electrons by one, and by a correspondingly greater number in the case of Al and Sn.
If the stability of the structure depended on the number of homopolar bonds, this would be impossible to do, and indeed no solid solutions are formed on the copper side. This simple example is sufficient to show the important significance of homopolar bonds for a considerable number of intermetallic compounds; but, evidently, considerable experimental work still has to be carried out before it will be possible to establish which compounds are homopolar and which are not, and also to determine the structure of homopolar bonds.
In addition to truly metallic and metallic-homopolar compounds, there exists a whole series of compounds with metallic properties whose structure is already closer to ionic. A whole series of compounds of elements of the groups S, As, and Si with other metals have structures with small coordination numbers, usually 6, of the NaCl, NiAs, pyrite, and calcium chloride types. These compounds are compounds of metals and possess, generally speaking, metallic conductivity, but it is doubtful whether they could be classified as metallic substances. In their electrical properties they differ greatly from truly metallic compounds. Generally speaking, their electrical conductivity, instead of being greatest when the substances are taken in pure form and decreasing rapidly with the appearance of traces of impurities, shows precisely the opposite effect: the purer these substances, the greater their resistance; and almost certainly, if they could be obtained in absolutely pure form, they would prove to be transparent nonconductors. In general, the degree of approximation to the ionic state depends on the nature of the negative ion and, in the limiting case, as, for example, for \(\overset{++}{\mathrm{Mg}}_2\,\overset{4-}{\mathrm{Pb}}\), the polarizing effect must be so considerable that the entire ionic structure becomes very unstable. Such substances, if they could be prepared in pure form, would almost certainly prove to be extremely sensitive to all kinds of influences that disturb their structure, in particular electrical or light influences. A whole series of minerals, sulfides and sulfur compounds of antimony, apparently belong to this class. Their mechanical properties, in particular their tendency to split, also indicate their connection with ionic compounds.
It is precisely the existence, in the region of intermetallic compounds, of three types—metallic, homopolar, and ionic—that has made the rules for the combination of metals so little understood. However, if ionic compounds could be distinguished by their electrical properties, and homo—
…polar into magnetic ones, the problem would become considerably simpler.
The laws of combination and structure of ionic compounds have at present been sufficiently established by the works of Goldschmidt¹ and Pauling;² homopolar compounds must obey the Hume-Rothery law, with various, more or less complex ratios of the number of electrons to the atoms; for purely metallic compounds the question still remains somewhat obscure, owing to the fact that the structure of these compounds has been determined only in a few cases. From theoretical considerations it seems probable that in this case the sole factor by which their composition and structure can be determined will be the atomic volumes. In the case of approximate equality of atomic volumes there is a tendency toward the formation, within wide limits, of solid solutions, although a regular structure is also encountered, as for example PdCu of the CsCl type. If the atomic volumes are different, such compounds are formed as allow a sufficiently simple cell of a high degree of symmetry, giving a minimal volume. If the difference is very great, a whole series of such compounds is formed. This is seen especially clearly in compounds of the alkaline-earth metals with large atomic volumes. There are, for example, series of compounds Cd, Cd₂Li, Cd₆Na and Cd₁₁K, for which the proportion of cadmium atoms keeps increasing, in order to balance the rapid increase of atomic volume, beginning with Li (22), Na (40) and up to K (74) ų. This tendency is encountered in all compounds of a similar kind, but it would be wrong to insist on the significance of volume before an X-ray investigation of these compounds has been carried out.
A general survey of the nature of the metallic state reveals those factors which it would be proper to call me—
¹ Cf. Goldschmidt’s article: “The Structure of Crystals and Chemical Composition.” Uspekhi fiz. nauk 9, 811. 1929. Ed.
² L. Pauling, J. Amer. Chem. Soc. 51, 1010 (1929).
metallic; separating them from one another, it is possible, from a purely empirical point of view, to give the principal characteristics of metallic bonds. In order to arrive at some explanation, it is necessary to assume special metallic bonds, distinct from homopolar, ionic, and molecular (van der Waals) bonds. Metallic bonds must possess the following basic properties:
1) They must act between identical atoms and at the same time between atoms completely different from them in structure, with the sole limitation that the greater part of such atoms must be metallic.
2) The bonds must be nondirectional; this is proved by their identity and by almost the same strength in the liquid state; they must be unsaturated, i.e., must always permit the possibility of the highest coordination numbers that can be realized stereometrically.
For some alloys this number reaches 16.
3) The forces of the bonds must vary inversely as some high power of the interatomic distance, which is confirmed by the extraordinary weakness of the bonds in alkali metals with a low melting point and very large atomic volumes, and by the high thermal energy of Ti and the small atomic volume of the Pt group.
4) In equilibrium with these attractive forces there must also be a repulsive force, which is a property of the atom, as is proved by the constancy of the atomic volume in the formation of alloys.
5) The bonds must permit the transition of electrons from one atom to another, in order to explain the electrical properties.
The extraordinary difference in the behavior of pure metals and compounds, on the one hand, and solid solutions, on the other, points to the electrical character of these bonds, depending on certain definite and regular interactions; moreover, in this respect, in contrast to the mechanical effect, the interaction is selective. Further, the electrical effect is not
of the energy function, as is evident from the extremely small magnitude of this effect for the most typical metals, such as Ni, Pd, and Pt.
It does not seem impossible to illuminate these properties from the standpoint of wave mechanics. It is clear that the attractive forces must depend on the interacting terms and, furthermore, that these interacting terms must differ from the terms encountered in homopolar combinations; and this difference, quite possibly, is connected with the distinction between symmetric and antisymmetric electron rotation (spin). A certain hint as to what they are is given by Heisenberg’s theory of ferromagnetism. In the case of the molecule H$_2$, only symmetric terms give a stable combination; but for more complex atoms it is possible that antisymmetric terms can also give stable states. Without attempting to solve this problem—for even in the simplest case it would be extraordinarily difficult to write down even the most approximate wave equation for an entire crystal—we can nevertheless easily discern that the idea of interaction can explain the basic thermal and mechanical properties of pure metals. The form of the atomic-density curve is a measure of the strength of the mutual attraction as a function of the number of electrons lying beyond the last orbit (the electron shell) corresponding to the inert gas.
According to Hund’s principle of equivalence for incomplete and filled electron shells, this naturally leads to a minimum of the energy at the beginning and end of each period of 18 or 32 (i.e., before and after each inert gas), although at the end of the periods the effect is masked by homopolar bonds. Large atomic volumes are similarly explained by the actual disposition in space of the excess electrons; and, usually, the atomic volumes decrease when an 18-electron shell is reached. A filled 18-electron shell must possess, as was shown for the case of Pd, positive interacting terms; but the maximum of interaction is attained with an incomplete 18-electron shell—
point. One can also draw more tempting analogies between the mechanical and thermal properties of individual metals and their electronic state; for example, for manganese we have two very complex structures which, if one did not know that Mn is an element, could have been taken for alloy structures. In fact, the form $\beta\mathrm{Mn}$ is isomorphous with the compound $\mathrm{AlAg}_3$. This fact, and the sudden break in the curve of atomic volumes at Mn, may depend on the possibility of the existence of two principal terms with very similar energy, so that this element in reality constitutes an alloy of two kinds of manganese, in different quantum states. Bradley pointed out to me that the character of the allotropic changes of Mn confirms this point of view, since instead of a sharp transition point between $\alpha$ and $\beta\mathrm{Mn}$ there apparently exists, at different temperatures, an equilibrium of two forms replacing one another, in accordance with the temperature equilibrium between the two kinds of Mn atoms. In a similar way, the maximum on the curve of the characteristic temperature of Ti in the first 18-electron period, which is not repeated for Mo and W in the following periods, is very probably connected with the fact that here we are dealing with the filling of group III for the first case and of IV and V for the others.
The electrical properties present many difficulties. At first sight it may seem that the conception of a constant atomic diameter for the metallic atom, which must be adopted in connection with the study of the structures of metals and alloys, is inconsistent with the theory of free electrons required by the theory of electrical conductivity of Sommerfeld, Houston, and Bloch. According to Bloch’s most deeply developed theory, the conducting electrons are related to the crystal lattice as a whole, and not to an individual atom. It is possible, however, that the same electrons behave in this way in an electric field, but under conditions of static equilibrium may, with sufficient probability, be considered attached to individual atoms and ensure the preservation of the atomic volume. In any case, every theory of metals that aspires to success must
…of the metallic state must equally explain both the constancy of atomic volume and the electrical and magnetic properties.
I began this article with the intention of giving a general view of the nature of the metallic state. In the course of it I saw that the obscurity existing in this field depends in equal measure both on the absence of systematic experimental material and on the internal difficulties of the theory. An enormous number of physical measurements have been made on metals and alloys, but only a few of them measured anything beyond the accidental properties of individual specimens, and these observations naturally related chiefly to alloys and mixtures of metals having technical significance.
In order to understand the nature of the metallic state and to be able finally to test it scientifically, much more systematic and intensive work will be required. Above all it is necessary to carry out a structural, thermal, microphotographic, and radiographic survey of the whole region of binary phases and of a certain number of ternary ones. At the same time, guided by structural analogies, it would be necessary to carry out, on the principal types of compounds and solid solutions, changes of electrical, magnetic, and mechanical properties far more thoroughly than has been done up to now.
Without such a foundation of experimental facts, the quantum theory, from which we now expect the final explanation of the nature of the metallic state, will work in the dark and accumulate useless formulas at an enormous expenditure of labor. The present report, in which the author arranges long-known facts in a different order and combines them with rather bold assumptions, has as its aim to emphasize the existence of the general problem of the metallic state and the necessity, by common efforts, of advancing experimental observations for the solution of this problem.