X-RAY STUDY IN METALLURGY

Ulrich Dehlinger, Stuttgart

Introduction

The discovery of the interference of X-rays, caused by the spatial lattice of the crystal, makes it possible to trace the geometrical structure of solid bodies down to the smallest constituent particles: atoms and electrons. Therefore, as early as 1912, metallography was faced with the task of establishing, with the aid of X-rays, the structural plans of individual metals and alloys, and then of investigating the physical and chemical properties inherent in a definite structural plan.

This task forms part of the general problem of crystal bonding, i.e., the problem of how and with what strength a large number of given atoms must be arranged at low temperature. Purely theoretically this question has so far been solved only in a few of the simplest cases, for example, in the case of homopolar bonding—the hydrogen molecule[^10]—and in the ideal case of heteropolar bonding in ions with a spherically symmetric, closed shell of noble gases and with opposite charges.[^11] Proceeding from the theory of the hydrogen molecule, Slater[^12] obtained results that are important also for metallic bonding. For the understanding of many properties of a metal, although not for the solution of the problem of bonding itself, the task—though a particular one—is of essential importance: to calculate the motion of electrons around the given positions of atomic nuclei for a given crystalline structure.[^12]

Thus the problem of metallic bonding is still predominantly at the stage of experimental work. Only the investigation of a large number of structures of metals and alloys and of the typical properties inherent in them can advance us further in this field.

Study of States That Are in Thermodynamic Equilibrium

Interference of X-rays cannot be used directly to construct a microscope that would make it possible to see individual atoms. With its aid one can establish only those differences in the distribution of masses in a lattice that repeat periodically (the so-called space lattice), and this with the greater accuracy the smaller the period of the lattice is relative to the wavelength of the X-rays employed and the more often the periods are repeated. But a strictly periodic distribution of masses in a lattice is in more stable equilibrium than any deviation from periodicity, and according to Nernst’s theorem every thermodynamically equilibrium state at very low temperatures also tends toward this mechanically most stable state. Thus, if we investigate a body that is in complete thermodynamic equilibrium at sufficiently low temperatures, we may take into account a strictly periodic distribution of masses and then reliably determine the positions of individual atoms also by X-ray methods. Possible deviations from strict periodicity at higher temperatures, such as must be expected, for example, in mixed crystals if they are in thermodynamic equilibrium, can be calculated statistically and then, in most cases, verified by X-ray methods. The distribution of masses in an equilibrium crystal established in this way is called its structure.

In the case of bodies in which internal thermodynamic equilibrium does not yet fully exist (this includes all polycrystalline bodies), one must always take into account that, for example, at grain boundaries there exist groupings of atoms that do not repeat periodically and therefore are inaccessible to X-ray investigation, and that these may be the cause of properties especially characteristic of the given state.

Structure and Properties of Elementary Metals

Survey of the structures of the elements. Table 1 gives the crystalline structures of the elements of the periodic system that have been investigated up to the present time. The structures of chief interest to us, those not connected with the formation of molecules or layers (the so-called coordination lattices), when we are dealing with only one element, can be unambiguously characterized by indicating

TABLE

3 Li cub. 8
11 Na cub. 8
19 K cub. 8	20 Ca cub. 12	21 Sc ?	22 Ti hex. 12	23 V cub. 8	24 Cr cub. 8	· cub.	25 Mn cub.	tetr. 12	26 Fe cub. 8	cub. 12
37 Rb cub. 8	38 Sr cub. 12	39 Y ?	40 Zr hex. 12	41 Nb cub. 8	42 Mo cub. 8	43 Ma ?	43 Ma ?	43 Ma ?	44 Ru hex. 12	44 Ru hex. 12
55 Cs cub. 8	56 Ba cub. 8	58 Ce hex. 12	58 Ce cub. 12	72 Hf hex. 12	73 Ta cub. 8	74 W cub. 8	75 Re hex. 12	75 Re hex. 12	76 Os hex. 12	76 Os hex. 12
87	88 Ra ?	89 Ac	89 Ac	90 Th cub. 12	91 Pa	92 U ?

SYMMETRY OF THE UNIT CELL AND COORDINATION NUMBER.
In these lattices an atom always has several equidistant neighbors. Their number is called the coordination number; moreover, no distinction is made between atoms equidistant as a consequence of the symmetry of the space group and those equidistant “by chance,” i.e. as a consequence of particular values of the parameter. Accordingly, in Table 1 we use the following notation: cub. 4 denotes the known diamond structure, cub. 8 denotes the body-centered structure, cub. 12 denotes the structure of a cube with centered faces, which, as is known, represents a close packing of spheres; hex. 12 denotes the known hexagonal lattice with axial ratio \(c/a = 1.633\), which in this case represents the hexagonal close packing of spheres (in fact the measured,

				4 Be hex. 12	5 B ?	6 C hex. 3 (4) cub. 4	7 N rhomb. ?	8 O rhomb. ?	9 F	10 Ne
				12 Mg hex. 12	13 Al cub. 12	14 Si cub. 4	15 P rhomb. 3 (6)	16 S rhomb.	17 Cl	18 A cub. 12
27 Co hex. 12 cub. 12	28 Ni cub. 12	29 Cu cub. 12	30 Zn hex. 6 (12)	31 Ga tetr. 1 (6)	32 Ge cub. 4	33 As rhomb. 3 (6)	34 Se hex. 2 (6)	35 Br ?	36 Kr
45 Rh cub. 12	46 Pd cub. 12	47 Ag cub. 12	48 Cd hex. 6 (12)	49 In tetr. 4 (12)	50 Sn cub. 4 tetr. 6	51 Sb rhomb. 3 (6)	52 Te hex. 2 (6)	53 J rhomb. 1	54 X
77 Ir cub. 12	78 Pt cub. 12	79 Au cub. 12	80 Hg hex. 6	81 Te hex. 12 cub. 12	82 Pb cub. 12	83 Bi rhomb. 3 (6)	84 Po ?	85	86 Em

values of \(c/a\) lie between 1.58 and 1.63). In contrast to this, in the following lattices the formation of layers takes place: the hex. \(6(12)\) lattice has the same arrangement of atoms as hex. 12, but the axial ratio is \(c/a = 1.89\); therefore here only 6 atoms in one layer are separated by exactly equal distances, while the remaining 6 atoms have a somewhat greater distance. Thus the number in parentheses denotes only an approximate coordination number. Between the two structures just named there is a sharp difference, since an axial ratio lying in the interval between 1.63 and 1.88 has not been found even among metallic alloys2. Analogous to this is the structure of indium. An even more sharply expressed formation of layers is revealed by the structure of graphite, hex. 3 (4), and also by the structure of thallium. In re-

in lattices designated rhomb. 3 (6), and hex. 2 (1), one can note, from a geometrical point of view, a tendency toward the formation of molecules; with a slight distortion of the lattice, two or three of the six atoms that originally uniformly surrounded one atom approach the latter, and thus groups of four or three atoms arise. Since there are various possibilities for the orientation of these groups, such a lattice can no longer be uniquely determined from the data indicated in the table.

The following elements possess two or more structures: cobalt and thallium²² exhibit hexagonal close packing at low temperatures; above 477 and 231°C they acquire a cubic lattice with centered faces; consequently, in this transformation the coordination number does not change. Cesium probably behaves in the same way. Iron has at low temperatures a body-centered cubic lattice (α and β), which at 906°C passes into a face-centered (γ) lattice of the same symmetry; however, above 1401°C the former form, (δ), is restored (cf. also p. 128). Manganese initially has an (α) cubic lattice with 58 atoms in the elementary cell, which have different coordination numbers between 8 and 16.¹⁹ At higher temperatures this lattice passes into another, likewise cubic, one with 20 atoms in the elementary cell (β); somewhat below the melting temperature a third lattice (γ) is stable,³⁰ which has a face-centered elementary cell distorted to weak tetragonal symmetry, with 4 atoms. The gray modification of tin possesses the structure of diamond; above 18°C

Fig. 1. Structure of magnesium (hex. 12) and graphite as examples of coordination and layered lattices.

it passes into white tin, whose tetragonal lattice arises from the diamond lattice in such a way that one of the cubic axes contracts until six atoms instead of four become equidistant from each atom. According to some observations (see, however, ^23,24) mercury at \(-80^\circ\) C likewise exhibits a similar transformation; likewise one should assume a further modification with hex. 12 also for chromium. Of the transformations of sulfur, phosphorus, and selenium, in which nonmetallic molecular lattices arise, nothing more detailed is known. Likewise nothing is known about Be. In Zn, Cd, Al, and Ni transformations have also repeatedly been assumed. However, new X-ray investigations have shown that no change of lattice occurs anywhere. ^14,26,27 The disappearance of ferromagnetism at the Curie point (the so-called \(\alpha\)—\(\beta\) transformation) in iron, cobalt, and nickel is not accompanied, as Wefer has shown especially clearly, by any change of structure. K. Bach believes that he has succeeded in detecting radiographically in iron a discontinuous contraction of the lattice, of approximately \(0.2\%\); however, dilatometric measurements, more sensitive to such changes, have not revealed any discontinuous change in length.

Metallic Structures

Is the metallic character of a substance connected with definite structures? This question must be answered in the negative if metallic character is defined by metallic conductivity, i.e. electrical conductivity which decreases with increasing temperature approximately according to a linear law, or by the optical properties associated with it. For example, diamond is a perfect insulator, whereas silicon, possessing the same crystalline structure, has metallic conductivity, at least in the form of a single crystal. ^28 These peculiarities find a certain justification in the fact, theoretically established by Slater ^13, that only a small percentage of the electrons on which cohesion depends participates in metallic electrical conductivity.

If, however, those substances are called metals which not only conduct an electric current metallically but are also readily subject to deformation by slip (for more detail on slip, see p. 159 ff.), then already as a result of a rough measurement of strength one can divide the structures into metallic and nonmetallic. All elements with the structures cub. 4, rhomb. 3, and hex. 2, and also, of course, I, J, and S, are brittle, while all elements with the structures cub. 8, cub. 12, and hex. 12, and also with the structure

Mn undergo plastic deformation. The classification of gallium and solid mercury is doubtful. Numerous precise measurements of slip strength,²⁹ carried out on single crystals, have so far confirmed this subdivision: although the slip strength for individual substances possessing the above-mentioned metallic structures is very different—which in part may also be ascribed to impurities—nevertheless all these substances, especially at low temperatures, have a slip strength a whole order of magnitude smaller than substances with nonmetallic structures.

In some hexagonal and tetragonal structures a metallic character may be attributed to a definite direction. In the plane perpendicular to this direction there exists a sharply expressed possibility of slip, whereas in other directions the lattice proves to be brittle. Such substances, in the form of mono- and polycrystals, possess very different strengths. Thus, for example, in graphite the direction of the hexagonal axis bears a sharply expressed metallic character, for the plane perpendicular to this axis exhibits a very strong capacity for slip; in other directions the bonding conditions are apparently analogous to diamond, i.e. the bond here is homeopolar. A less strong preference for the hexagonal axis is found in structures hex. 6 (12); nothing is yet known about the properties of the indium structure tetr. 4 (12).

Whether this great capacity for slip, persisting down to very low temperatures, is the direct result of metallic bonding, as Jom-Rozery³⁰ assumes, or whether its occurrence is explained by an accidental circumstance due to the fact that metallic bonding proves more capable of forming weak points, according to Smekal,³¹ than other bonds, cannot yet be decided at present. The phenomena of work-hardening, which we shall discuss below and which are connected with features lying within the coherent lattice, apparently speak rather in favor of the first view. There is as yet no modern theory here. From the quantum-mechanical theory of Polanyi and Schmidt³² there still follows no distinction between metallic and homeopolar bonding such as results from the experimental facts indicated above.

Participation of free electrons in metallic bonding

A characteristic feature of metallic bonding appears especially clearly in the structure of α-manganese, a peculiar-

X-RAY RESEARCH IN METAL SCIENCE

whose features reappear in β-manganese and in the γ-phases of brass considered below, and so on. The cubic cell of this modification of manganese contains 58 atoms, which have neither simple positions in the lattice—that is, positions determined by symmetry elements—nor are they collected into groups whose centers of gravity would occupy such simple positions. Nevertheless, the symmetry of the entire arrangement is cubic, and, in addition, atoms halfway along the diagonal of this gigantic cell are in a body-centered arrangement. Similar cases are unknown among nonmetallic structures.^8 Where approximately the same number of atoms in a cell are not in simple positions—as, for example, in cyanite silicate—the symmetry of the entire arrangement is, at best, pseudo-cubic. Where, with a large cell, there is full cubic or hexagonal symmetry, neighboring atoms are arranged in groups whose centers of gravity occupy simple positions, as, for example, in cubic garnet or hexagonal quartz.

Other metallic structures likewise possess high symmetry. It should already be pointed out here that, through the addition of atoms such as C, N, H, which are capable of typical homeopolar bonding, this symmetry is very often, though only slightly, distorted.

In order to be able to explain all this, it must be assumed that, in the case of metallic bonding, a factor is involved which, within broad limits, does not depend on the position of the atom, but which is sensitive to changes in the symmetry of the lattice.

A large number of electrical and magnetic observations^12 has led to the conclusion that individual atoms lose one or several of their electrons, and that these electrons can move more or less freely among the various atoms; from the latter it follows that they may be the cause of the described features of metallic bonding. In analyzing the structure of those elements of the periodic system whose outer electron shell consists of 1 to 8 electrons (this so-called B subgroup in Table 1 is enclosed in a frame; the corresponding group A is formed by the transition metals listed on p. 137), Jom-Rozery^33 showed that, in the transition from homeopolar to metallic bonding that takes place here with decreasing atomic number, the bonding electrons suddenly become distributed over many atoms and, in this way, become freely mobile. As is known, in consequence of the Pauli principle, the electrons in an atom tend toward the formation of completed shells of noble gases, which in heteropolar bonding

achieved by the transfer of individual electrons from one atom to another. If, however, neighboring atoms together contain more or fewer electrons than is required, taking into account the uniform distribution of charges for the formation of such closed shells, then each pair of excess electrons of neighboring atoms enters into a homopolar bond, as in the hydrogen molecule. In the same way as excess electrons, vacancies in the shell of a noble gas may also be treated. Thus, in the case of iodine, one electron is lacking for completion of the xenon shell; therefore every two iodine atoms form a molecule between themselves, and these molecules are held together into a crystal by the so-called van der Waals forces. Bernal[^34], from magnetic facts, draws the conclusion that the latter forces in certain cases already have a metallic nature. In an analogous way there arise molecules of the structures of selenium or arsenic, containing 3 and, respectively, 4 atoms, as well as the fourfold coordination of the diamond structure. But already in this vertical series structures appear which reveal a tendency toward the coordination number 8 or 12, i.e. they reveal a higher coordination than that which would have to correspond to a homopolar bond at the valence determined by the given vertical series. With a further decrease in the number of outer electrons, only these, already non-homopolar, coordination numbers appear, and highly symmetric structures of metals are established. As can be seen, in these structures the outer electron no longer serves for the bond of two definite atoms, but in a certain sense effects a bond between all neighboring atoms—it becomes “free.”

Slater[^13], using a model in one dimension, showed theoretically how these metallic bonding electrons also determine the electrical and magnetic properties of metals.

As can be seen, on the basis of the points of view set forth, one can divide the structures of the elements into metallic and nonmetallic in the same way as was done on p. 104. On the basis of Bernal’s assumptions, such a strict division would be impossible.

Measurements of lattice energies occurring in transitions between three simple metallic structures, cub. 8, cub. 12, and hex. 12, were theoretically calculated by Hundfield[^35]. In addition, the thermodynamic equilibrium of these structures and their transformations is apparently influenced also by the form of the orbit of the metallic electrons or, from the point of view of wave mechanics, by the distribution of charges around the atomic nucleus; the number of metallic-

narrow electrons has no direct significance in this respect.

As was mentioned above, experiment leads to the conclusion that metallic electrons must be sensitive to changes in the symmetry of the crystal lattice. The relation between the form of the trajectories of individual electrons revolving around an atom and the symmetry of the lattice in which the atom takes part was investigated theoretically, from the standpoint of wave mechanics, by Bethe. According to his calculations, in cubic symmetry the orbits, generally speaking, are degenerate, i.e., there exist several orbits with the same energy but with different positions of the axes. If the symmetry is tetragonal, then the orbits situated perpendicular to the principal axis of the lattice differ from the orbits parallel to it, so that the degeneracy is partially removed. Dellinger[^36] extended these results to the case of metallic electrons; he assumed that, at least at moderate temperatures, this system too is degenerate in the manner described. But according to the fundamental laws of statistics, a double degeneracy, i.e., a doubling of the number of possible states of equal energies, corresponds to an increase of entropy by \(R \cdot \ln 2\), where \(R\) is the gas constant. Thus a direct connection is obtained between the entropy of the system of metallic electrons and the symmetry of the lattice, which, if Bethe’s numerical results are taken into account, may be formulated as follows: upon transition from tetragonal or hexagonal to cubic lattice symmetry, the entropy of the electrons increases by an amount lying between \(R \cdot \ln 1\) and \(R \cdot \ln 2\) per mole-electron.

An analogous increase in the degeneracy of the system of metallic electrons must be assumed if, instead of the symmetry of the lattice, the coordination number is increased. In this way we arrive at the following position: in passing from coordination number 8 to coordination number 12, the entropy of the electrons increases by an amount lying between \(R \cdot \ln 1\) and \(R \cdot \ln 1.5\) per mole-electron.

For cobalt, the specific heats were measured above and below the transformation point[^37] (Fig. 2). According to these measurements, both modifications—at low temperatures hex. 12, at high temperatures cub. 12—near the transformation point have the same specific heats. Consequently, within this temperature region the difference between their internal energies \(\Delta U\) and their entropies \(\Delta S\) does not depend on temperature. The difference of free energies will be:

\[ \Delta F = \Delta U - T \Delta S . \]

As is known, a transformation—provided there is no change in volume—proceeds in that direction in which it is...

therefore $\Delta F$ is negative. If it is assumed that the total difference of the entropies between the cubic and hexagonal lattices is equal to the difference of the entropies of the electrons, as is indicated by the aforementioned independence from temperature, then, according to the preceding statement, $\Delta S$ in the transition from the hexagonal

Fig. 2. Specific heats of cobalt according to Umino. At 470°—lattice transformation, at 1100°—magnetic transformation.

lattice to the cubic lattice is greater than zero and, with the aid of observations of the heat of transformation $\Delta U_0$, one may represent the course of the difference of the free energies, as is done in Fig. 3. As can be seen, the picture obtained is correct: at low temperature the hexagonal lattice has the smaller free energy and therefore is thermodynamically stable; above the transformation point the cubic lattice appears, the electronic system of which possesses the greater entropy. The same relations are also observed for cerium and thallium. If, from the heat of transformation and the transformation temperature according to Fig. 3, one calculates the difference of the entropies, the values obtained lie within the above-mentioned limits.

Fig. 3. Free energy of the electronic system of two modifications of a metal.

For iron one may in the same way, on the basis of the second of the propositions stated above, first of all represent the transformation of the $\beta$-modification into the $\gamma$-modification. However, in its further course the equality of the free energies can no longer be independent of temperature, since the electrons responsible for ferromagnetism, in con-

unlike the other electrons of the metal, possess a finite specific heat. However, on the basis of W. Pauli’s formulas,12 it is possible, by measuring the paramagnetic susceptibility of these electrons and the heats of transformation $\beta$-$\gamma$ and $\gamma$-$\delta$, to estimate the temperature dependence of the total, and hence also the free, energy (above the Curie temperature)—again proceeding from the assumption that the total entropy difference is caused by both electron systems. In Fig. 4 both curves are shown. As can be seen, the curves of the free energy of the space-centered lattice $F_{\alpha\delta}$ and of the lattice with centered faces $F_{\gamma}$ intersect, with very satisfactory accuracy, also at the second transformation point $\gamma$-$\delta$ ($A_4$), if the entropy difference $\Delta S$, increasing as a result of the change in coordination number, is chosen so that they intersect at the transformation point $\beta$-$\gamma$ ($A_3$).

Fig. 4. Total and free energy of centrally and face-centered iron (valid only above the Curie point).

The uniform interpretation of metallic transformations given here must, of course, be confirmed by further measurements of specific heats, up to high temperatures and at very low temperatures. As will be shown below, it is also very important for understanding processes in mixed crystals.

Atomic radii of metals. V. M. Goldschmidt39 systematically investigated the distances, measured by X-ray methods, between the centers of gravity of atoms in the crystal lattice and the specific volumes of elements in the solid state calculated from them. In developing W. L. Bragg’s work,38 he succeeded in showing that atomic distances in alloys are calculated from the atomic distances for the elements in such a way that to each element there is assigned a definite atomic radius; the sum of the atomic radii of two elements may be taken as equal to their atomic distance in mixed crystals and compounds, if definite corrections are introduced for changes in coordination number and if one restricts oneself to crystals with one and the same type of structure.

bonds (such crystals are empirically designated as commensurable crystals). The correction consists in taking into account the contraction upon transition to smaller coordination numbers, which for all metals, with slight variations, has the values given in Table 2.

TABLE 2

Change in coordination number	Compression
12 → 8	3%
12 → 6	4%
12 → 4	12%

In the transition from hexagonal to cubic close packing, no appreciable change in the atomic distance is observed.

These contractions are directly related to the above-mentioned decrease in the total energy under the corresponding transformations, especially in iron. The quantitative form of the bond has been determined experimentally for a polar bond,⁴¹ but not for metallic structures. The large contraction in the transition to coordination number 4 is an indication of a change in the entire character of the bond, as was also evident from the preceding discussion.

According to Vestrgren and Almin,⁴⁰ the contraction in the case of compounds of the iron and platinum metals with other metals (see p. 136), in particular with aluminum, is considerably greater.

In Table 3 are given atomic radii referred to coordination number 12 (at room temperature). The quantities enclosed in boxes were determined by Goldschmidt,³⁹˒¹⁵ chiefly by direct preparation of the corresponding alloy with coordination number 12; the remaining quantities were obtained from the shortest atomic distances⁸ with corrections on the basis of Table 2. For deviations from the hexagonal axial ratio \(c/a = 1.633\), the atomic radius \(r\) is calculated as the mean of the horizontal distance \(a\) and the vertical distance \(e\) by formula (39):

\[ r=\frac{a+e}{2}=\frac{a}{2}+\frac{1}{2}\sqrt{\frac{a^{2}}{3}+\frac{c^{2}}{4}}. \]

The regular course of the atomic radii in Table 3 (see also Fig. 14) represents a more exact expression of the well-known course of the specific volumes of the elements in the periodic system according to Lothar Meyer (for the theory see also ³⁹).

Solid solutions, superstructures, eutectics

If various metals are alloyed with metals or nonmetals, then upon solidification either crystals arise with approximately the same symmetry and coordination as correspond to the initial bodies, or an entirely new lattice is obtained, which we call the lattice of metallic compounds*. The phenomena considered below are found in both cases; compounds, in turn, form mixed crystals, superstructures, and eutectics. Experimental investigations of these questions have hitherto been confined for the most part to the first case, which is why in the following chapter we shall speak only of it.

Mixed crystals. All metallic structures possess the property of forming mixed crystals, i.e., of admitting foreign atoms into their lattice (the basic lattice) without destroying it or changing its symmetry and coordination number. As is known, on the basis of pycnometric determinations of density and X-ray determinations of the lattice constant, one can find the number of atoms in the elementary cell of the lattice. If this number is equal to the number of atoms in the cell of the basic lattice, then the foreign atoms occupy the places of individual atoms of the basic lattice (substitutional mixed crystal); if the number of the former is greater than that in the basic lattice, then the foreign atoms are located in the interstices of the basic lattice, which are for the most part determined crystallographically. The first case is encountered much more often; the second has so far been observed only in the case of inclusion in the metallic lattice of metalloids with small atomic radius, such as C, N, H (and also when metals and nonmetals are included in nickel-arsenide structures).

In both cases, when a foreign atom enters, a change occurs in the constant of the basic lattice. This change, according to Vegard, in first approximation is directly proportional to the number of included foreign atoms, and, in the case of substitution, furthermore to the difference between the atomic radii of the foreign and the basic atoms.

The formation from two metals of a continuous series of mixed crystals is possible only in the case of substitution. In this case all the atoms of the basic lattice can be gradually replaced—

* Consequently, we distinguish superstructural phases, whose spatial lattice is to a considerable degree still similar to the lattice of the surrounding mixed crystals, from true compounds.

TABLE 3
Atomic radii in angstroms

3 Li 1.56	4 Be 1.13
11 Na 1.92	12 Mg 1.65	13 Al 1.42
19 K 2.32	20 Ca 1.86	21 Sc ?	22 Ti 1.46	23 V 1.35	24 Cr 1.30	25 Mn 1.30	26 Fe 1.27	27 Co 1.257	28 Ni 1.244	29 Cu 1.276	30 Zn 1.304	31 Ga —	32 Ge 1.394
37 Rb 2.54	38 Sr 2.15	39 Y ?	40 Zr 1.61	41 Nb 1.41	42 Mo 1.40	43 Ma ?	44 Ru 1.322	45 Rh 1.342	46 Pd 1.370	47 Ag 1.442	48 Cd 1.531	49 In 1.569	50 Sn 1.582	51 Sb 1.614
55 Cs 2.74	56 Ba 2.24	58 Ce 1.82	72 Hf 1.585	73 Ta 1.461	74 W 1.408	75 Re 1.371	76 Os 1.336	77 Ir 1.522	78 Pt 1.380	79 Au 1.439	80 Hg 1.55	81 Tl 1.707	82 Pb 1.744	83 Bi 1.82
			90 Th 1.80

heavy metals

bind atoms of the second metal, so that at no concentration does a second phase arise. Within an accuracy of 1% Vegard’s law is also valid here. More precise measurements, which were carried out on such series of mixed crystals: Ag—Au, Cu—Ni, Mo—W, confirmed the exact applicability of this law only in the last case. In other systems, in the formation of mixed crystals there occurs a noticeable contraction of the lattice, which likewise is not explained by assuming a higher degree of dependence on concentration in Vegard’s law.^65 For example, in the case of the silver-gold mixed crystal,^43 shown in Fig. 5, the lattice constant, as a result of the contraction, becomes even lower than the values for both components.

Fig. 5. Lattice constants of the series of gold-silver solid solutions according to Sachs and Weerts.^43

Superstructures in solid solutions. To explain the sharp concentration limits found by him, at which chemical action ceases, Tammann^60 in 1919 put forward a hypothesis according to which, in solid solutions that by prolonged heat treatment are brought into a state of thermodynamic equilibrium, chemically different atoms are likewise located at structurally different, periodically recurring points of the lattice; moreover, when lattice points are replaced, they alternate with the same regularity as, for example, the atoms of chlorine and sodium in the rock-salt structure. This hypothesis can be tested by X-ray methods, since such a regular arrangement requires the appearance of new lines, the so-called superstructure lines, which must lie only between the lines of the principal lattice, slightly shifted as a result of the above-mentioned change in the lattice constant, whereas an irregular arrangement of different atoms over the nodes of the principal lattice, as Laue’s calculations^69 show, becomes noticeable only owing to a uniform strengthening of the background located between the Laue lines. Numerous investigations devoted to this question in some cases confirm Tammann’s hypothesis, while in others they contradict it.

On the basis of statistical thermodynamics one must expect the following: it may be assumed—and from the facts of the above-mentioned contraction this is experimentally confirmed—

tally,—that the transition of an irregular arrangement into a regular one, since it is not connected with a change in the state of the bonds, i.e. with a change in the electronic system, is always accompanied by a decrease in the total energy. Therefore, in such a case, at absolute zero an entirely regular distribution of the atoms in the given mixed crystal must be established, if the latter, generally speaking, is still stable and does not decompose into two phases. In the case of higher temperatures it should be taken into account that an irregular distribution possesses greater entropy than a regular one, and that the influence of this entropy on the magnitude of the free energy, which determines stability (p. 102), increases with increasing temperature in comparison with the total energy. Therefore, at high temperatures the arrangement will be the more irregular, the smaller the work expended in transferring an atom from a regular position to an irregular one (Fehlordnungsarbeit).

In Table 4, in the form of stoichiometric formulas, there are given those concentrations of solid solutions near which, as was observed chiefly by Johansson and Linde, such superstructures exist. Here, in addition to the general coordination number of the solid solution, the mutual coordination numbers of chemically different atoms in their regular arrangement are also given.

TABLE 4

System	Structure of the solid solution	Correct distribution — composition	Correct distribution — structure
Au-Cu	cub. 12	AuCu₃	cub. 12; 12,4
Au-Cu	cub. 12	AuCu	tetr. 12; 8,8
Pd-Cu	cub. 12	PdCu₃	cub. 12; 12,4
Pd-Cu	cub. 12	PdCu	cub. 8; 8,8
Pt-Cu	cub. 12	PtCu₃	cub. 12; 12,4
Pt-Cu	cub. 12	PtCu	cub. 12; 6,6
Pt-Cu	cub. 12	PtCu	rhomb. 12; 6,6
Mg-Cd	hex. 12	CdMg₃	hex. 12; 12,4
Mg-Cd	hex. 6 (12)	Cd₃Mg	hex. 6(12); 12,4
Ir-Os	1	IrOs	hex. 12; 6,6

The lines of the superstructure prove to be of such strength here as, for example, also corresponds to the entirely regular arrangement of the atoms, and only in the case of AuCu₃, on single crystals, which are more accessible to X-ray investigation, between 350 and 400° there is observed a slight weakening of the superstructure lines,¹⁰⁷ which indi-

evokes the existence of the above-mentioned disorder, predicted by theory, which increases with temperature in the distribution of atoms. In other cases (AuCu, MgCd₃, Mg₃Cd) this disorder could more easily be recognized from the increase in electrical resistance and specific volume.

In solid solutions with an atomic ratio of 1 : 1, the regular distribution of atoms represents a lower symmetry; correspondingly, the form of the unit cell also always changes. Thus, for example, the AuCu cell is tetragonal, with an axial ratio of

\[ \frac{c}{a}=0.93. \]

With increasing temperature this ratio increases, together with the increase in electrical resistance, and approaches unity. It is noteworthy that in AuCu, even at a lower temperature, owing to strong deformation, the axial ratio may reach 0.98 without any noticeable change in the intensity of the superstructure lines. In PdCu, with a regular arrangement, a cubic body-centered lattice arises, i.e. the so-called cesium chloride structure appears. As is known, a cubic body-centered lattice may be regarded as a tetragonal lattice with centered faces, with a quite definite axial ratio

\[ \frac{c}{a}=\frac{1}{\sqrt{2}}=0.7. \]

Therefore the PdCu structure may be considered as a limiting case of the tetragonality of AuCu. On the other hand, it may also be assigned to the compounds considered in the next section of the present paper. In PtCu the distribution of atoms and the form of the cell have cubic symmetry; in the case of a content somewhat greater than 50 atomic percent Pd, a superstructure of rhombohedral symmetry is also found, and a cell in which the angle between the axes is \(90^\circ 54'\), so that the deviation from the cubic system is very small. A more accurate state diagram for this system has not yet been established. In regular atomic groupings with an axial ratio of 1 : 3, the symmetry of the lattice, as compared with solid solutions, does not change. In all superstructures the density is several per mille greater than in the solid solution.⁶³

Already the first measurements of hardness and conductivity, which were carried out by Kurnakov on the gold–copper system and which make it possible to draw conclusions about the regular distribution of atoms in solid solutions, also show that this distribution, at quite definite temperatures, passes over into a distribution of another

sort, probably into a disordered solid solution. Numerous further investigations of conductivity and specific volumes fully confirm this; this is seen, for example, from Fig. 6, taken from the work of Grube and collaborators⁶³ and showing the course of the change in resistance upon very slow heating of an AuCu alloy with 50 atomic percent Au; similar curves are obtained for AuCu₃, CdMg₃, and Cd₃Mg. This diagram was checked by X-ray methods by Dehlinger and Graf¹⁰⁶ on single crystals quenched from different temperatures, which are especially convenient for observations. In this, as the investigations of Gorsky have already shown, the following is found: below the sharp jump of the resistance curve, lines of superstructure appear which, in their strength, approximately correspond to a fully regular distribution. The slight increase of resistance above 300° corresponds to a slow approach of the tetragonal axial ratio to unity. All these phenomena undoubtedly occur as a result of the above-mentioned, and theoretically easily explained, increase in the irregularity of the atomic distribution with increasing temperature. However, the very sharp jump in resistance that occurs at 425° remains unexplained; by X-ray methods it corresponds to the complete disappearance of the superstructure lines. Above this jump the resistance curve runs completely linearly; by X-ray methods no trace of a regular atomic distribution is any longer noticeable here. The symmetry is perfect, cubic.

Fig. 6. Resistance of AuCu in thermodynamic equilibrium (after Grube).

The complications that arise upon rapid cooling from high temperatures and upon heating specimens that are not yet in complete thermodynamic equilibrium⁶² can perhaps be explained by studying the sharpness of the superstructure lines,¹⁰⁶ as will be done below (see p. 148 ff.).

Now the question arises: will that, in any case, almost completely irregular atomic distribution which is detected by X-ray methods above the jump be in thermodynamic equilibrium? The following experimental facts speak in favor of this opinion:

a) the linear course of the resistance curve as a function of temperature is in this region always reproducible without any heat treatment, whereas in the regions where the ordered arrangement is stable, further changes in resistance are always noticeable even after many hours of annealing; b) the rate at which atoms not yet in equilibrium rearrange into the ordered distribution can be measured by means of measuring the intensities of the superstructure lines without any further assumptions. It increases very rapidly in the temperature region below the jump as one approaches it, only then, at the jump, suddenly to fall to zero. Consequently, below the jump the rate of diffusion of atoms in a uniform distribution has a normal temperature course; but above the jump, where it ought, owing to the higher temperature, to have been still greater, it is equal to zero.

On the basis of the quite understandable assumption that an isothermal phenomenon tends toward thermodynamic equilibrium with the greater speed the farther it is from the state of equilibrium, it follows from both experiments that the disordered distribution above the jump is already in equilibrium, so that it no longer needs any change.

I — ordered arrangement.
II — disordered arrangement.
III — disordered arrangement with increased electronic entropy.

Vertical axis: Free energy.
Horizontal axis: Absolute temperature.
\(\Delta S\)

Fig. 7. Free energy of the solid solution and of the ordered atomic distribution (without taking into account that part which is due to lattice vibrations).

Indications of the causes of this sudden jump of the atomic distribution are provided by the X-ray studies, described in more detail below, of the unstable intermediate state arising upon cooling AuCu above the jump. \(^{100}\) According to these studies, the tetragonal symmetry and the ratio of the axes \(c/a\) of the basic cell are already fully formed, while only a small part of the atoms is correctly arranged (Fig. 20). The primary phenomenon at the jump will apparently be a change in the symmetry of the lattice, after which the ordered or disordered arrangement of the atoms already follows. However, according to the ideas already set forth above, the transition from tetragonal to cubic symmetry has as its consequence an increase in the electronic entropy, which, as is seen from Fig. 7, is the reason that the free energy of the cubic solid solution with disordered atomic distribu—

...above a certain definite temperature, namely the jump temperature, becomes less than the energy of any tetragonal lattice with a more or less ordered atomic distribution (the sign of the change in the total energy at the jump proves to be precisely the one adopted in Fig. 7, since it is determined experimentally by observing the expansion of the lattice upon transition to cubic symmetry; therefore one cannot in any way dispense with the assumption of a change in entropy).

Essential for this explanation of the experimental facts is the assumption that the change in the shape and electronic symmetry of the entire coherent lattice must already be caused by only a small number of atoms correctly arranged with tetragonal symmetry. It is easy to see that, if a more or less tetragonal atomic distribution in a cubic lattice were thermodynamically more stable than a completely random distribution, then, on the basis of mechanical considerations, it would have to be excluded. The immediate experimental confirmation of this assumption is the existence of the intermediate state AuCu, illustrated in Fig. 20.

Even if a correct distribution of atoms without a change in the symmetry of the solid-solution lattice is possible, as is the case for a ratio of 1:3 in the mixture, then upon its occurrence the state of the electrons must change. In the case of an irregular distribution, chemically different atoms coordinate with one another indiscriminately (the first coordination number in Table 4), whereas in a correct distribution one must assume coordination only between like atoms. Therefore, in the transition from an irregular distribution to a correct one, the coordination number, and according to the propositions stated above consequently also the electronic entropy, decreases, so that everything discussed above can be directly transferred to this case as well.

Properties of the irregular distribution of atoms in solid solutions. On the basis of what was set forth above, it should be accepted that, in the mixed crystals listed in Table 4, at temperatures above the resistance jump (Fig. 6), the irregular distribution of chemically different atoms at the lattice sites is in thermodynamic equilibrium. In other systems of solid solutions, such as, for example, Ag-Au-Cu, Ni, Mo-W, where in various studies of casts and of single crystals no superstructure lines of Au-Ag have been detected (there was also found the increase in intensity characteristic of an irregular arrangement...

with the lines), one can expect this equilibrium in such systems with a high degree of probability. Here, however, at still lower temperatures, one must reckon with a transition to a regular arrangement, or with the decomposition of the solid solution into two phases, or with both, as seems probable in the Au–Pt and Ag–Pt systems. The special properties acquired by solid solutions as a result of the irregular atomic distribution present in them can be determined experimentally by comparing their numerical values above and below the resistance jump, or they can be studied from their variation in the region where the irregularity increases.

In the case of electrical resistance this is done as shown in the curve of Fig. 6, which for low temperatures was supplemented by Zeeman’s measurements.^66 Thus one can see that in AuCu, and also in AuCu₃, the resistance, beginning with the temperature of the jump and down to the very lowest temperatures, decreases with temperature considerably more steeply than in the case of pure metals, whereas the temperature coefficient of disordered mixed crystals above the jump is approximately equal to the temperature coefficient of pure metals. It follows from this that the disordered arrangement of atoms entails an additional resistance almost independent of temperature (Matthiessen’s rule), which below the jump decreases with temperature because, in this case, as was said above, the irregularity of the atomic distribution also decreases. How this additional resistance depends on the degree of this disorder is best shown by measurements of the course of the resistance depicted in Fig. 8, where a complete series of solid solutions is presented.^68 At low concentrations the resistance has a linear course with concentration, but at high concentrations there is a downward deviation from this straight line. According to Nordheim’s calculations,^79 from the standpoint of wave mechanics, this means the following: regularly, i.e. periodically, arranged foreign atoms do not entail any additional resistance, since the motion of the conducting electrons in the lattice, understood as wave motion, can enter into a certain kind of resonance with the period of the lattice,

Fig. 8. Course of the resistance of the gold–silver solid solution.^68

...resonance. With each deviation from periodicity the electron waves will be scattered incorrectly and will be lost for conduction, which manifests itself as resistance. But the disturbance of periodicity is least when a few foreign atoms are included; if a large number of such atoms is included, then with a known probability the next new foreign atom falls into such a position that it, together with several other foreign atoms, will form a periodic lattice, though with a larger period. Consequently, a new atom included in this way will produce no additional resistance, and this explains the deviation from the straight line observed in Fig. 8.

Fig. 9. Course of the shear stress necessary for the onset of slip in single-crystal gold-silver according to Sachs and Weerts.

The change in resistance, according to the data of Sachs and Weerts,\(^{14}\) in the state of correct arrangement is likewise considerably smaller than in the solid solution, and moreover it approaches those values which characterize the pure metals. If one represents graphically the course of the shearing force necessary for the onset of slip of solid solutions of gold-silver as a function of the concentration of these metals,\(^{14}\) then we obtain a picture (Fig. 9) strikingly similar to what we had for the resistance (Fig. 8). As in the case of the increase of resistance, the increase in strength (Verfestigung) of single crystals is apparently due mainly to irregularity in the distribution of atoms, whereas the force necessary for the continuation of slip probably depends to an even greater degree on the special properties of the atoms.\(^{14}\)

To explain the phenomena of aging (Vergütung) in the Au–Cu system, Dehlinger put forward the hypothesis that inhomogeneous, but not spherically symmetric, shear stresses arising as a result of some disturbances in the lattice cause strengthening of the metallic lattice. This hypothesis also makes it possible to explain subsequent facts. As can be seen from Fig. 10, a, the correct arrangement of different atoms produces, in the extreme case, stresses of spherical symmetry; this may be a consequence of the different radii of the individual atoms, but no inhomogeneous shear stresses. However, each deviation from periodicity, as shown by Fig. 10, b,

causes an elliptical state of stress, which in the individual stresses (indicated by arrows in Fig. 10, b) entails considerable nonuniform shear stresses. On the basis of geometrical considerations—namely, that at higher concentrations of foreign atoms their further inclusion entails smaller deviations from periodicity than at low concentrations, as is also the case with electrical resistance—the course of the experimental curve shown in Fig. 9 is quite explicable.

Multiphase systems. Most systems consisting of two or more components, upon solidifying from the liquid state, separate not only one phase (crystal lattice), but also a whole series of other phases. The greatest number possible under phase equilibrium is determined by the phase rule: for example, when two components are fused, at most two solid or liquid phases can coexist simultaneously, and only at quite definite temperatures and concentrations can their number be three. The region of temperatures and concentrations in which one or two definite phases can exist is represented by the well-known phase diagram.

a — lattice plane 111 of AuCu₃ with an ordered atomic arrangement; b — small distortions of the ordered arrangement due to the replacement of atoms of pairs.

Fig. 10.

In connection with this there arises the problem of determining, by the X-ray method, the boundaries of these regions of existence. More precisely, since the boundaries for liquid phases are readily determined by thermal methods, here the question is mainly of transformations in the solid state, and also of the temperatures and concentrations at which a mixed crystal can no longer take up any foreign atoms, and instead there appears

ULRICH DEHLINGER

some second phase. In this case one may speak of two methods of solving this problem.

a) First, the intensity of Debye–Scherrer lines in a multiphase region is investigated. The temperature and concentration of the system, while the latter is maintained in thermodynamic equilibrium, are varied until the intensity of the lines of one phase becomes equal to zero. For the corresponding temperatures and concentrations it would be possible to establish one boundary line of the phase diagram. Such a method is inapplicable, however, because the ratio of the intensities of the lines of two phases mixed with one another, as a result of the so-called extinction, will depend, as it turns out, not only on the ratio of the quantities of these phases, but also to a considerable degree on the sizes of their grains. Therefore determinations of the relative amounts of lattices from the intensities of the lines will be possible only for very small grain sizes.70,71

Fig. 11. Dependence of the lattice constant on the silver content in the Cu–Ag system.

If, by suitable treatment, it were possible to bring metal alloys as well into such a state, then by the radiographic method one could carry out an analysis not only of the elements, but also of the phases as such.

b) Secondly, as was mentioned above, when a foreign atom is incorporated, the lattice constant of the solid solution changes. However, in those regions where two phases can exist (with two components and constant pressure), the content of foreign atoms in the individual phases remains constant. If, therefore, the lattice constant of one phase is observed as a function of the total concentration of the system at some definite temperature, then it changes so long as we are in the single-phase region; when the separation of a second phase begins, it remains constant as the concentration is changed. Fig. 11 shows such a change in the syste-

…of Ag–Cu from the copper side. It is easy to see that the limiting concentration can be determined with great accuracy from the points of intersection of the two curves. By gradual cooling to various temperatures one can establish the course of this limiting concentration as a function of temperature. Fig. 12 shows the phase diagram corresponding to this case.

Part of the curves shown in Figs. 11 and 12 was obtained on polycrystalline material by Ageev, Tanzen, and Zaks.⁴⁹ The solubility curve found by them agrees fairly well with the curves obtained by other methods. X-ray investigations were repeated by Vist⁵⁰ on single crystals. As can be seen from the figures, he found that these single crystals change their lattice constant with increasing content of dissolved silver considerably less rapidly than polycrystalline copper, and that, especially at low temperatures, they are capable of dissolving a significantly larger quantity of silver than this copper. If the same single crystals are deformed, then the values of the lattice constant found for the polycrystalline material are reproduced again. By means of special experiments it was shown that in both cases thermodynamic equilibrium was attained (if one disregards the fact that polycrystalline material, in principle, should always recrystallize further, i.e. it can never be regarded as being in complete equilibrium) and that impurities do not exert any noticeable influence. As will be discussed later, there is a considerable difference between a single crystal and a polycrystal also in the kinetics of phase precipitation. These experiments, as well as the experiment with AuCu mentioned on p. 117, seem to reveal yet another typical feature of the metallic bond. Its significance can perhaps be explained only with the aid of wave mechanics.⁷⁴

Fig. 12. Diagram constructed from Fig. 11, supplemented with heterogeneous boundary lines.

Influence of foreign atoms on the transformations of iron. The question of how incorporated foreign atoms change the thermodynamic region of stability

both modifications of iron in a whole series of studies was elucidated chiefly by Wever.^72 In doing so it was found that, for alloys rich in iron, there exist two principal types of constitution diagrams. In the first case (the example of Fig. 13), beginning with a certain concentration of the mixture, the face-centered $\gamma$-lattice is no longer stable at any temperature; only the body-centered $\alpha$-$\beta$ lattice remains, which passes directly into the so-called $\delta$-modification, indicating the identity of both of these modifications.

Fig. 13. Constitution diagram of the iron–silicon system with a narrow $\gamma$ region (from the table of Landolt–Börnstein).

Fig. 14. Types of iron alloys in relation to atomic radii, according to Wever.^72

In the second case (the example of Fig. 16, and also the well-known iron–carbon diagram), on the contrary, with increasing concentration of the mixture the temperature of the $\beta$-$\gamma$ transformation shifts to lower values, and that of the $\gamma$-$\delta$ transformation to higher values, so that the $\gamma$ region expands.

In Fig. 14, in the form of a single curve of atomic radii corresponding to Table 3, are presented the types of constitution diagrams which each of the elements forms with iron. It is easy to see that the lattice type of the alloying element plays no role here; for example, face-centered aluminum reduces the region of existence of the face-centered $\gamma$ phase, whereas a whole series of other face-centered metals exert the opposite influence. Conversely, the connection with the atom-

elements with radii, namely: elements lying near the minima of the curve of atomic radii expand the region of existence of the \(\gamma\)-phase; with an increase in the atomic radius, the stability of the \(\alpha\)-lattice increases; and with its further increase the elements become completely insoluble. Silver and cadmium are exceptions in this series. Quantitative investigations further show that the \(\beta\)-\(\gamma\) transformations are suppressed when the expansion of the lattice due to the inclusion of foreign atoms reaches \(3\%\).

In pores (Gitterlücken) only carbon and nitrogen atoms are included in the iron lattice—all other elements replace iron atoms. In accordance with what has been said above, this must be attributed to the especially small radii of both elements, but at the same time it must also be recognized that this has no direct influence on the above-mentioned relationships.

Mutual miscibility of metals (metals of the 1st kind). On the question of which pairs of metals form with one another continuous (uninterrupted) series of solid solutions, two-phase systems or a new lattice and compounds of another kind, the first investigations were carried out by Bernal.^34 The results of his experiments may be summarized as follows.

The groups of metal mixtures can be determined in the following way: to the first group must belong all, and only, those metals which, at least with some other metal, form a continuous series of mixed crystals, and the regular arrangements indicated in Table 4 must be interpreted as solid solutions. Since a continuous series of crystals can be formed only between two metals of identical structure, it follows directly from this definition that all metals of one group, at least in one of its modifications, must possess one and the same structure. Thus one may present, first of all, the group of face-centered metals shown in Table 5. In the table, for all investigated pairs of this group, the character of their phase diagrams is given; \(m\) denotes a continuous series of solid solutions, \(e\) (eutectic) a two-phase region with slight solubility on both sides (for example, as in Fig. 12), \(e_l\), \(fl\)—a continuous mixture in the liquid state, \(fl\)—immiscibility in the solid and liquid states; the superscript \(r\) indicates that the corresponding phase diagram has been studied by X-ray methods; the remaining data are borrowed from Landolt–Börnstein’s tables.

From the table below it is evident that no pair

of metals of the group, when alloyed, does not form a lattice of another kind (the compounds observed microscopically in the Ag–Mn and Au–Mn systems seem very improbable, and the hexagonal ε-phase in Fe–Mn is probably not stable)*. Where there is no continuous series of solid solutions, a two-phase system with the lattices of the pure metals is formed.

The sequence of the elements in Table 5 has been chosen in such a way that adjacent elements always mix with one another.

TABLE 5.

Mixture group I

	Ag	Au	Cu	γ=Mn	Ni	β=Co	γ=Fe	Pt	Pd	Ir?
Au	mᵣ
Cu	eᵣ	mᵣ
γ=Mn	fl?	e?	mᵣ
Ni	fl	m	mᵣ	m
β=Co		e	eᵣ	m	m
γ=Fe	fl	e	e, fl	mᵣ	mᵣ	mᵣ
Pt	eᵣ	mᵣ	mᵣ				m
Pd	mᵣ	mᵣ	mᵣ		m			m?
Ir?								m
Rh?								m

Mixture group II

	α=Fe	V
V	mᵣ
Cr	mᵣ

Mixture group III

	Mo
W	mᵣ

with one another. As to the reasons for the miscibility or immiscibility of the various metals belonging to group I, as yet absolutely nothing is known. In contrast to heteropolar crystals, atomic radii apparently play no role here.

From among the body-centered metals, two further mixture groups may be composed here from Table 5. Bernal assigns all the metals contained in Table 5 to one class, namely to metals of the 1st kind, and

* The phases recently observed by Wever and Ellinghaus in the Fe–V and Fe–Cr systems at 50 atomic percent (reported in K. W. Inst. f. Eisenf., Debye 12, 317, 1930 and 13, 143, 1931), judging by the nature of their phase diagrams, are superstructures of body-centered solid solutions.

all the other metals—to the metals of the 2nd kind (however, see below; alloys between metals of groups I, II, and III always form true compounds).

Apart from those listed in Table 5, and also the Bi–Sb system, no continuous series of solid solutions are known, so that no further groups of mixtures can be established. The following pairs of metals, as X-ray investigations show, form only a two-phase system with the original lattice, without compounds, and therefore reveal a certain kinship of their components: Mg–Cd, Cd–Hg, Zn–Ae, on the one hand, and Pb–Tl, Pb–Sn, Pb–Sb, Sn–Bi, on the other. From a technical point of view it would also be very important to extend these regularities to ternary systems, especially if, for the β- and γ-phases described below, groups of mixtures could likewise be compiled. However, this requires a very large amount of new experimental material.

Metallic Compounds

In the case of alloying two metals that belong to different groups, and also in the case of the remaining binary systems not considered in the preceding section, at certain concentrations there appear (insofar as this has already been studied) new crystalline lattices that differ from the lattices of the components and that, since they still have a metallic character, must be regarded as lattices of metallic compounds. In contrast to the superstructures already described,^106 as the latest investigations show, they have as yet been studied rather unsystematically and are considerably more brittle than pure metals. If some exceptions are not counted, the specific volume of these metallic compounds is only slightly smaller than that of pure metals and solid solutions;^40 contraction of the lattice as a result of the appearance of new valencies characteristic of the given compound also for the most part does not occur. The metallic compounds described here differ from normal chemical compounds especially in their ability to contain their components in excess, so that for the most part they do not have a definite stoichiometric composition.* Therefore, in what follows, wherever this is known, the region of homogeneity is always given, i.e. the compositional limits between which the type

* Almost all compounds of metallic character with a large homogeneity region can readily be distinguished from compounds of nonmetallic character with a very small homogeneity region.

the lattice of the compound is thermodynamically stable. In subdividing these compounds we follow Ewald and Hermann.^8 As metalloids are designated, above all, those elements which possess the homopolar structures characterized above. However, sulfur, and partly tin and aluminum, have to be assigned a variable character in compounds; for example, they are sometimes, as in Tables 6 and 7, metals, and sometimes, as in Table 10, metalloids.

Compounds Ag, Cu, Au with metals of the 2nd group. All the alloys of copper, silver, and gold with the metals Be, Li, Mg, Zn, Cd, Al, Sn (except Cu-Mg) investigated up to now, as metallographic investigations show—which, in the X-ray part, were undertaken chiefly by Westgren and Framen, and also by Goldschmidt—have a state diagram which is very similar to the state diagram for Cu-Zn [(Fig. 15) according to Bauer and Hansen] in the following respects:

Fig. 15. State diagram copper-zinc according to Bauer and Hansen [from Sauerwald (Sauerwald, Metallkunde, S. 414)].

on the side of Cu, Ag, and Au there is a large region of solid solutions of the 1st kind (substitutional solid solutions). Further, after a two-phase region, new phases appear, first of all—the stable β-phase, for the most part at high temperatures, with a space-centered cubic lattice, in which both components may be arranged in regular order (as, for example, in AgZn, where then a lattice of the cesium-chloride type arises) or else are distributed irregularly, and which can accept excess atoms of both components by substitution. Apparently, the further new phase β′ differs little from β; it has not yet been sufficiently investigated by X-rays. With still greater content of the metals of the 2nd group, which, generally speaking, is of no technical application, the γ phase appears, the lattice of which is

similar to the lattice of $\alpha$-manganese considered above. Its cubic fundamental cell contains 52 atoms, whose arrangement can be described as follows: 27 simple space-centered cells of the $\beta$-phase together form one cubic cell with a tripled lattice constant and contain 54 atoms. Two of them are removed, while the remaining ones are only very slightly displaced from their positions; moreover, even under such conditions the whole structure possesses a uniform distribution of masses while retaining cubic symmetry, and, of course, none of the atoms falls at those points which are determined by the intersection of symmetry elements. Chemically unlike atoms are very often distributed in this lattice in a regular manner. To describe this regularity one would have to take, for example, in $\mathrm{Cu}_5$, $\mathrm{Zn}_8$ eight of the above-mentioned large cells. This phase also has a large region of homogeneity. With a further increase in the concentration of metals of the 2nd kind, a new phase $(\varepsilon)$ appears, which has a lattice with dense hexagonal packing of spheres; here the atoms often appear to be arranged, probably, in a definite order.⁹⁷ Between the $\zeta$- and $\varepsilon$-phases two further phases often appear, the first of which, $\gamma'$, apparently has a lattice similar to that of the $\gamma$-phase, while the second, $\delta$, has a simple lattice which, for example, in the Cu–Al system is a tetragonal space-centered one with an approximate composition $\mathrm{Cu}_3\mathrm{Al}$. At a still higher content of metals of the 2nd kind, the compounds have the character of compounds with nonmetals (they are listed in Table 10); all of them have only a very small region of homogeneity. The metals of the 2nd kind themselves have only slight solubility for Ag, Au, and Cu (phase $\eta$). Table 6 gives the $\beta$-, $\gamma$-, and $\varepsilon$-phases found up to the present time; it gives the now known regions of homogeneity, expressed in atomic percent of metals of the 2nd kind, and also the mean stoichiometric composition. The latter is determined exactly only when the atoms are arranged regularly; in other cases, within the limits of the homogeneous region, the composition may be determined on the basis of the following rule, found by Hume-Rothery: if it is assumed that all participating atoms tend to include the majority of their electrons in the filled shell (which, according to spectroscopic observations, is the ten-electron shell appearing next), then for Cu, Ag, and Au one outer electron per atom remains in excess; for Mg, Zn, Cd, and Hg, two; for Sn, four; and for Sb, five outer, so-called valence, electrons per atom remain. Independently of the valence of the individual atoms, in the alloys considered here the structure of a given phase is determined by the ratio of the number of all valence electrons present to

TABLE 6

β-phases, cubic body-centered	β-phases, cubic body-centered	γ-phases, with cubic close packing	γ-phases, with cubic close packing	ε-phases, hexagonal close packing	ε-phases, hexagonal close packing
Composition	Homogeneity region in atomic %	Composition	Homogeneity region in atomic %	Composition	Homogeneity region in atomic %
Cu–Zn	46–49	Cu₅Zn₈	61–67	CuZn₃	78–80
Cu₃Al	(22–30)	Cu₉Cd₅	?	Cu₃Sn	25
Cu₅Sn	(15–17)	Cu₉Al₄	31–41?	Cu₃Sb	10–20
CuBe	47–49	Cu₃₁Sn₈	20.5	CuBe₃	?
AgMg	?	Cu₅Hg₈	30–33	Cu₃Te	?
AgZn	49–54	Ag₅Zn₈	60–64	AgZn₃	70–80
AgCd	49–51	Ag₅Cd₈	61–65	AgCd₃	69–83
AgLi	48–52	Ag₅Hg₈	?	Ag₅Al₃	27–43
AuZn	38–55	Ag₅Zn₈	65–69	Ag₃In	?
β′-phases (the same as β-Mn)	β′-phases (the same as β-Mn)	Au₅Cd₈	?	Ag₃Sn	?
Ag₃Al	25			Ag₅Sb	10–40 (–25)
				AuZn₃	81–89
				AuCd₃	?
				AuAl₃	?
				Au₄Hg	25

The percentage content is everywhere given for the metals of the 2nd group. The numbers standing in parentheses give the homogeneity region of the high-temperature phase that is unstable at room temperature. In Cu–Ag no ε-phase appears. With hexagonal close packing, Cu–Mg forms two complex compounds, Cu₂Mg and CuMg₂, with very small homogeneity regions.

to the total number of atoms. At a ratio of 3:2 a body-centered cubic β-phase is obtained; at 21:13, a cubic γ-phase; and at 7:4, a hexagonal ε-phase. As Table 6 shows, the compositions required by the rule—apart from the compositions for CuHg—lie, generally speaking, in the measured homogeneity region or very close to it, so that this rule is confirmed by a large amount of experimental material. There is as yet no theoretical explanation for this.

Compounds of the iron and platinum metals with metals of the 2nd group. According to Ekman⁷⁵, in all alloys of the platinum and iron metals with metals of the 2nd group investigated up to now, which are listed in Table 7, the β and γ phases described above are likewise formed. If, for the iron and platinum metals, the number of electrons is taken as equal to zero, then here too the rule concerning the ratio of the number of electrons to the number of atoms proves to be valid.

In this connection it should be said that the alloys Ag, Cu, and Au

with the iron and platinum metals, which fall into group 1 of alloys, form a well-known transition, which was especially well clarified by Forth[^67] on the basis of magnetic studies. Here, in the series of solid solutions Ag-Au, one must take into account the presence of one free electron, but in the Au-Pd series, at a low content of Pd atoms, the latter are apparently neutral, i.e., without giving a free electron to the Ag lattice, and only at high additions of Pd does the donation of one free electron on this Pd take place. The same can be said of Cu-Pd; the superstructure PdCu here may be understood as a compound in which the ratio of the number of electrons to the number of atoms is 1:2.

TABLE 7

β phases with a cubic body-centered lattice	β′ phases similar to β-Mn	γ phases with cubic close packing
CoAl NiAl MnAl FeAl (Cu,Mn)Al PtCu	CoZn	Fe₅Zn₂₁ Co₅Zn₂₁ Ni₅Zn₂₁ Rh₅Zn₂₁ Pd₅Zn₂₁ Pt₅Zn₂₁ Ni₅Zn₂₁

The remaining phases of these alloys have as yet been little investigated.

Compounds between other metals. Of the multitude of possible systems here, only very few have been investigated.

Part of the alloys between the various groups of alloys I, II, III, such as, for example, Ni-Cr, Co-Cr, and Ni-W, form far-extending solid solutions on both sides. In both of the latter cases, apparently, complex compounds appear, the structure of which has not yet been fully investigated. The systems Fe-W and Fe-Mo possess only slight mutual miscibility and also form complex compounds that have not yet been fully studied.

Of the compounds between metals of the 2nd kind, only the structures Mg₂Sn and Mg₂Pb, as well as MgZn₂, are known. The first two, like Mg₂Si, possess a cubic lattice of the fluorite type; the last has a hexagonal lattice with an axial ratio corresponding to close spherical packing, where, however, the atoms occupy non-simple positions. All three lattices are unknown for other metallic structures.

Compounds between metals and metalloids. In his systematic investigations Hegg indicates that compounds of H, B, C, and N with metals in which the tenfold shell is not completed (the so-called transition metals with atomic numbers 21—28, 30—40, 57—78, 89—92) still possess a metallic character, which is expressed chiefly in their metallic conductivity, and partly also in superconductivity.¹⁰¹

The structures of these compounds listed in Table 9 are especially simple when the ratio between the atomic radii of the metalloid (X) and the metal (M) is sufficiently small. Here the metalloids are assigned such atomic radii (Table 8) as are exact only at 50 atomic percent; at lower concentrations of the metalloid the atomic radii have somewhat smaller values.

TABLE 8

B	0.97 Å	N	0.71
C	0.77	H	0.46

If the ratio of the radius of the metalloid atom to the radius of the metal atom is less than 0.59, then the coordination number of the metal atom will be 12 or 8, the same as in the case of pure metals, and the metal atoms form lattices of the type cub. 12, hex. 12, cub. 8, or, finally, a lattice not encountered in pure metals, hex. 8, i.e. a simple hexagonal lattice with axial ratio \(c/a = 1\). In certain interstices of this lattice the metalloid atoms are situated, for the most part singly, and partly in pairs. This arrangement is denoted in Table 9 by a second coordination number, which shows how many metal atoms a metalloid atom has as neighbors. It is characteristic of these phases that not all possible interstices of the lattice are filled by metalloid atoms; however, all metalloid atoms filling the interstices of the lattice have one and the same coordination number. Owing to such an arrangement, as was already indicated above (p. 105), a slight distortion of the high symmetry of the original metallic lattice often occurs. This distortion, in the case of a simple arrangement, amounts at most to 3.9%, and in the case of a paired arrangement reaches 20% (in the latter case the cubic lattice becomes tetragonal with ratio \(c/a > 1\) when this pair is arranged parallel to the tetragonal axis, and with ratio \(c/a < 1\) when it is arranged perpendicular to this axis).

As it turns out, the metalloid atoms are arranged according to a principle which, according to Goldschmidt,¹⁰⁰ is pri-

less so also to polar salts, namely: they are arranged with such a coordination number that contact is obtained between them and the surrounding metal atoms. Geometrical considerations show that, for example, in order for a metalloid to have a coordination number of 6, the ratio of the atomic radii must be at least 0.41. If the metalloid atoms were smaller, then free spaces would remain between them and the closely packed metal atoms. Since this apparently is not the case, then, as Table 9 shows, at an atomic-radius ratio smaller than 0.41 the coordination number will always be 4. What determines the type of metallic lattice is still unknown.

According to Hagg, in all the systems belonging here only those four compositions appear which are given in Table 9. In this case \(M_4X\) and \(M_2X\) have a broad, while \(MX\) and \(MX_2\) always have a narrow, region of homogeneity. However, some other compositions are also known. Thus, for example, although the \(\gamma'\) phase in the Fe–N diagram shown in Fig. 17 and studied radiographically by Eisenhut and Kaupp has the exact composition \(Fe_4N\), as corresponds to Table 9 (\(\alpha\) and \(\beta\) represent the corresponding modifications of iron), the \(\varepsilon\) phase in iron has the composition \(Fe_2N\). Above 11% N, Hagg found still another, more distant, \(\zeta\) phase of composition \(Fe_2N\), whose lattice differs only very little from \(\varepsilon\); therefore he combines the \(\zeta\) and \(\varepsilon\) phases into one homogeneous region of mean composition \(Fe_2N\). The phase appearing in Mn–N also has a lower nitrogen content than would seem necessary from Table 9. Thus, on the basis of the compositions given in this table, only summary rules can be derived.

Fig. 16. Equilibrium diagram of iron–nitrogen according to Eisenhut and Kaupp.\(^{83}\)

If, in the group of alloys considered above, the ratio of the atomic radii exceeds 0.59, then metalloids can enter the interstices of the lattice only in small concentrations, and sometimes, as was indicated above, a new lattice is formed; thus, for example, in the ternary Fe–Mn–C system there exists a hexagonal close packing with a carbon content up to 8 atomic percent. At a higher content of metalloid, its atoms no longer enter the lattice, but replace metal atoms, and considerably more complicated structures arise.

System	Ratio of atomic radii	Phases: \(M,X\) — lattice type	Phases: \(M,X\) — distortion	Phases: \(M\) — lattice type
Zr-H	0.29	Cub. 12.4		Hex. 12.4
Ta-H	0.32	—		" 12.4
Ti-H	0.32	—		" 12.4
Pd-H	0.34	—		Cub. 12.4
La-C	from 0.42 to 0.43	?		?
Ce-C	from 0.42 to 0.43	?		?
Pr-C	from 0.42 to 0.43	?		?
Nd-C	from 0.42 to 0.43	?		?
Th-N	0.43	?		?
Zr-N	0.43	?		?
Sc-N	0.47	?		?
U-C	0.48	?		?
Zr-C	0.48	?		?
Nb-C	0.39	?		?
Ti-N	0.49	?		?
W-N	0.51	—		Cub. 12.6
Mo-N	0.52	—		" 12.6
V-N	0.53	?		?
Mn-N	0.53	?		?
Nb-C	0.53	?		?
Ti-C	0.53	?		?
Ta-C	0.53	?		Hex. 12.6
Mn-C	0.55	Cub. 12.6	Tetr. \(<1\)	" 12.6
W-C	0.55	—		" 12.6
Cr-C	0.56	—		" 12.6
Mo-C	0.56	—		" 12.6
Fe-N	0.56	Cub. 12.6		" 12.6
V-C	0.56	—		" 12.6

A dash denotes that no phase of such composition exists; the change in atomic radii is considered in the text.

Of these, the structures of cementite \(Fe_3C\), as well as \(FeB\), \(Fe_2B\), and \(Ni_2B\), are definitely known.

If metalloids with a large atomic radius, such as those listed above, are alloyed with transition metals, then, according to Table 10, at 50 atomic percent there arises almost exclusively the so-called nickel-arsenide structure, which is type hex. 8.6 with complete replacement of all positions for metalloid atoms. The strong fluctuations of the axial ratio occurring according to Table 10 are a sign of manifesting homeopolar forces, which originate from the more or less closely packed atoms of the metalloid. 9) More complex, partly with molecular bonding, are constructed the compou-

TABLE 9

Composition: X — Distortion	MX — Lattice type	MX — Distortion	MX — Lattice type	MX — Distortion
	Cub. 12,4 ” 8,4 12,4 ?	Rhomb.	Cub. 12 R ? Cub. 12,4 ?	Tetr. < 1
	?		12 R	Tetr. > 1
	Cub. 12,6 ” 12,6 ? Cub. 12,6 ” 12,6 ” 12,6 ?		12 R ? ? Cub. 12 ? ? ? ? ? ? ? ?	Tetr. < 1 Tetr. > 1
Tetr. 1	Hex. 8,6 Cub. 12,6 ” 12,6 Cub. 12,6 ” 12,6 ” 12,6 ” 12,6		? ? ? ? ? ? ?
\(c/a < 1.63\) ” \(> 1.63\) ” \(< 1.63\) ” \(< 1.63\) Rhomb.	Hex. 8,6 Cub. 12,6 ? ? Cub. 12,6	\(c/a < 1\) Tetr. < 1 Tetr. \(c/a < 1\)	? ? ? ? ?

— paired arrangement of metalloid atoms. A system with a high ratio FeSi. At a higher metalloid content the pyrite-type cubic lattice often appears; more rarely there appears the marcasite-type lattice, which is not found at all in equilibrium, in which two metalloid atoms are joined into molecules, as well as the cadmium iodide type lattice, which forms sharply expressed layers.

In alloys of non-transition metals with metalloids of all kinds, certain simple structures and phase diagrams with a large region of homogeneity appear; in these, as was noted above, the metalloids still have a metallic character. In addition to those already named in Table 6, these include the hexagonal packing of composition \(\mathrm{Pb_2Bi}\) and the body-centered cubic structure

Tl₂Sb₂. The remaining compounds scarcely have a more metallic character. The structures known among them with 50 atomic percent are given in Table 10.

TABLE 10

Nickel-arsenide type \(c/a\) 2.8; 0.6

Compound	Value	Compound	Value	Compound	Value	Compound	Value
AuSn	1.23	CoS	1.52	FeS	1.69	NiS	1.55
CuSn	1.21	CoSb	1.34	FeSb	1.25	NiSb	1.31
PdSb	1.37	CoSe	1.47	FeSe	1.63	NiSe	1.40
PdTe	1.37	CoTe	1.38	FeTe	1.26	NiTe	1.36
PtSn	1.32	MnAs	1.52	FeAs (rhomb.)		NiBi	1.32
PtTe	1.32	MnSb	1.40			NiAs	1.39
PtSb	1.32					NiSn	1.30

Zinc-blende type, cub. 12; 4.4

AlAs	CuS	InSb	ZnS
AlP	CdSe	HgS	ZnSb
AlSb	CuTe	HgSe	ZnTe
BeS	GaAs	HgTe
BeSe	GaP	SnSb
BeTe	GaSb	CSi

Rock-salt type, cub. 12; 6.6

BaS	CaS	MgS	MnS	PbS	SrS	SnTe
BaSe	CaSe	MgSe	MnSe	PbSe	SrSe	SnSb
BaTe	CaTe			PbTe	SrTe

Wurtzite type, hex. 12; 4.4

ZnS	CdS	CdSe	MgTe

Compounds with halogens and the oxides of all metals are completely nonmetallic; as is known, they generally have heteropolar bonds.

A wholly peculiar type of bond is formed by a solution of hydrogen in palladium at low temperatures. Here the hydrogen atoms apparently adjoin the separate palladium atoms so closely that a formation arises which, in the lattice, behaves like a silver atom and has the same number of electrons and nuclear charge. As shown by the investigations of Kröger and Zakolevsky on hydrogen-containing solid solutions Ag—Pd, this pseudoatom can substitute for a silver atom in the lattice, with no change in the lattice constant taking place.

STATES NOT IN THERMODYNAMIC EQUILIBRIUM

In the investigations reported below, the discussion concerns not only the structure of the lattice, but also the constitution of the entire mass of the metal. By this we mean the orientation, the size of the grain, and distortions of the lattice of individual crystals.

The orientation of the crystallographic axes of a crystal lattice can be studied by means of photographs according to the Laue method and the rotating-crystal method.[^5] The statistics of the orientations of very many individual grains is also called the texture of a metal. If, in doing this, only one direction has to be recorded, one speaks of a complete fiber structure. The latter can already be determined with the aid of appropriately obtained Debye–Scherrer photographs. The grain size is measured for the most part microscopically. Lattice distortions, if they extend over a region encompassing more than \(0.5\,\mu\), can be recognized from the asterism of Laue photographs;5 if within this region they are strongly inhomogeneous, a broadening of the Debye lines is obtained.6 If they exist only in small regions of the lattice, while the remaining part of the lattice is undamaged, they can be detected only by decreasing the intensity of the lines and increasing the background.[^142],[^143]

Engineering uses metals exclusively in such states as are thermodynamically more or less unstable. These states always have a higher strength than the equilibrium states. By a greater or lesser approach to equilibrium, these strength properties of a given mass of metal can be subjected to very subtle changes. This is the meaning of most technical processes of change of shape, cooling, and hardening.

A. Intermediate states of metallic transformations

1. Martensite. The face-centered cubic lattice of \(\gamma\)-iron, stable above \(906^\circ\mathrm{C}\), can in thermodynamic equilibrium dissolve up to \(1.7\%\) carbon (austenite), whereas body-centered \(\alpha\)-iron can take up at most only \(0.04\%\) carbon (ferrite). The transformation of \(\gamma\)-iron into \(\alpha\)-iron proceeds in an entirely pure metal so rapidly that it cannot be quenched.[^104] However, if carbon-containing iron is cooled in water, a new state can be obtained, martensite, which differs from the equilibrium states by its microscopic structure and its great hardness, and upon tempering slowly passes into ferrite plus a eutectic of ferrite and \(\mathrm{Fe}_3\mathrm{C}\) (pearlite). In martensite the iron atoms possess a body-centered, i.e. a lattice similar to that of \(\alpha\)-iron, which still contains all the carbon in solution. Owing to this, the cubic cell of \(\alpha\)-iron is distorted into a tetragonal form, and the tetragonal—

…the axial ratio \(c/a\) increases with increasing carbon content linearly up to the value 1.068 at 1.4% C. Precise measurements of the lattice constants \(a\) and \(c\) according to Öman\({}^{102}\) are shown in Fig. 17. On the basis of these values and simultaneous density measurements, he concludes that, in contrast to \(\gamma\)-iron, the carbon atoms in martensite are not located in the pores of the lattice, but replace iron atoms in pairs, one atom at a time (similarly to what occurs in the carbide \(\mathrm{ThC}_2\), Table 9). On quenching nitrogen-containing \(\gamma\)-iron, a tetragonal phase entirely similar to martensite is likewise observed.\({}^{83}\)

Fig. 17. Lattice constants of tetragonal martensite according to Öman.

In studying tempered austenite crystals by means of rotating photographs, Kurdjumov and Sachs\({}^{103}\) showed how the face-centered \(\gamma\)-lattice passes into the tetragonal body-centered lattice of martensite and then into the cubic body-centered lattice of ferrite. These photographs (Fig. 18) show that the newly arising crystals of martensite and ferrite possess quite definite orientations with respect to the crystallographic axes of the original single—

Fig. 18, a. Rotating photograph of a quenched austenite single crystal (according to Kurdjumov and Sachs).

Fig. 18, b. The same after tempering for 30 min at \(650^\circ\).

the austenite crystal, namely a single austenite crystal, first splits into a large number of very small martensite grains, each of which is oriented in the directions shown in stereographic projections in Fig. 19. The orientations of the ferrite grains arising from them give the same picture. According to these figures, all the martensite and ferrite crystallites place their plane \((011)\) parallel to the plane \((111)\) of austenite, and their direction \([111]\) parallel to the direction \([101]\) of austenite. This determination is not unique. Since there exist 24 crystallographically equivalent positions that satisfy it, the fragments of the splitting austenite crystal occupy 24 different orientations.

The orientation described may arise in such a way that all the atoms of the austenite lattice begin to move along the crystallographic direction \([211]\) (the slip direction), and moreover to such an extent that the lattice planes \((111)\) lying in the slip direction slide, like cards, relative to one another. The magnitude of the “slip” of a given plane relative to the plane below is the same for all the lattice planes and is precisely determined with the help of the symmetry of the new lattice that arises. Through this first slip, from the face-centered lattice there arises a distorted body-centered lattice. By means of a simultaneous second slip, which proceeds in the plane \((211)\) of the body-centered lattice in the direction \([111]\), this distortion may

Fig. 19. Stereographic projection of normals to faces for martensite with respect to the axes of the primary austenite crystal (A 100 and A 010 denote two directions of cube edges of austenite).¹⁰³

be eliminated (slidings can be added geometrically). If both slidings stop at a certain definite point, a tetragonal lattice is obtained; if they go still farther, a cubic ferrite lattice arises.

Thus the martensite lattice appears in type as an orientation of an intermediate stage of the sliding process. The detection of this intermediate stage may be regarded as proof that, during the transformation of iron, the atoms perform the slidings described above, which in the same way also proceed during deformation.

Fig. 20. Intermediate state in the transformation of the gold–copper solid solution into a tetragonal regular atomic distribution.

Nothing is as yet known about the motion of the carbon atoms during the transformation and about the causes of the increase in hardness in martensite.

Intermediate states in transformations of the atomic distribution in solid solutions.

If, in the course of several minutes, one cools from a point of the potential jump down to room temperature the solid solution in the Au–Cu system, stable above 425°C, with approximately 50 atomic percent Au and with an irregular distribution, then, according to Dehlinger and Traub,^106 there is established an intermediate state, particularly easy to observe radiographically on crystals, as follows (Fig. 20). The elementary cell in the entire lattice has a tetragonal form corresponding to the final state of equilibrium. On the other hand, the copper and gold atoms are regularly arranged only in part of the lattice cells, while in the remaining part of the coherent lattice—which is larger the greater the rate of cooling—the atoms, as in a single crystal, are arranged completely disorderly. This inhomogeneous disorder differs sharply from the homogeneous disorder that is in equilibrium (p. 117), which gives, although weakened, always sharp points of the superstructure. The inhomogeneity of the atomic distribution is detected with great certainty from photographs of a rotating crystal of the intermediate

states in which the points of the body-centered tetragonal lattice appear in full sharpness, while the superstructure points are strongly broadened and slightly weakened. This inhomogeneity is confirmed by the fact that the electrical resistance is greater than for the final state, and the superstructure lines are weaker. Only after further annealing do the superstructure lines appear in full sharpness and unweakened. In accordance with the rules mentioned on p. 119, at \(300^\circ\) several days are required for this, and at \(400^\circ\)—approximately one hour. At the same time the lattice constant and the tetragonal axial ratio do not change, within the limits of experimental error.

If the disordered solid solution is quenched in water, no transformation occurs. On heating, no intermediate state of a perfectly expressed regular arrangement is established, but, according to Table 5, the disordered distribution appears abruptly.

Here too, as in the case of martensite, the new lattice is oriented with respect to the axes of the old lattice, and here too, as there, the latter is split into single crystals. Here the tetragonal axes of the new lattice are parallel to the cubic axes of the old; but since the tetragonal axis of rotation of the new lattice can coincide with each of the three cubic axes of the old lattice, a splitting into three mutually perpendicular orientations results. Such an orientation may be regarded as proof that the tetragonal lattice arises as a result of the direct shortening of one of the three cubic faces of the crystal of the solid solution. With rapid cooling of perfectly undamaged single crystals, it is even possible to obtain x-ray patterns of the transitional intermediate forms of the unit cell along this path.

On cooling a solid solution containing about 25 atomic percent Au, an analogous intermediate state is also established. Since in this composition a regular cubic atomic distribution is possible, the symmetry of the lattice does not change; on the contrary, here too the lattice undergoes a slight uniform compression in all directions. Here the new lattice is likewise oriented with respect to the lattice of the solid solution. The old and new cubic axes are then parallel; naturally, splitting into different positions should not occur here.

The difference between the hardness of the intermediate state between AuCu and AuCu\(_3\) is very surprising. As is known for polycrystalline material, in the alloy with 50 atomic percent Au there can ...

a considerable refinement effect is obtained as a result of the following heat treatment: the solid solution is quenched from a temperature above 500° and then tempered for a short time at approximately 300°. It then becomes very hard. However, after annealing at this temperature for several hours it again becomes soft. As can be seen from comparison with everything stated above, and as is confirmed by direct X-ray investigations,¹⁰⁵ the hard state prepared in this way, the so-called refined state, is nothing other than an intermediate state, which only upon prolonged heating at a temperature below 425° passes into the soft, completely ordered state.

In contrast to this, the intermediate state of AuCu₃ is only slightly harder than the solid solution, i.e. the corresponding transformation does not produce refinement.

The suggestion that the hardness of the intermediate state of AuCu is due to the fact that, owing to splitting into three possible positions, very small grains arise can be rejected directly by means of the experiment with deformation mentioned on p. 117: if the grains arising because of the aforementioned splitting were very small, then, just as in the division of grains due to deformation, an axial ratio of 0.98 would again have been obtained. Consequently, the grains arising during splitting must be considerably larger than the slipping lamellae arising during deformation, and the cause of the increase in strength, which is considerably greater than in deformation, should not be sought in this.

Thus the increase in the resistance to transformation in the intermediate state of AuCu must be attributed to the stresses within the coherent lattice, acting as a result of the specific atomic distribution (Fig. 20). Strong stresses between the ordered and disordered parts of the lattice must exist both in AuCu and in AuCu₃; it is they, among other things, that force the disordered parts of the lattice to assume the lattice parameter of the ordered part. In the intermediate state of AuCu₃ these stresses apparently do not produce any increase in strength, since they evidently have spherical symmetry, i.e. tend to transform the larger cubic unit cell into a smaller one. Since externally applied pressures, as Schmidt and Polanyi have shown,¹⁰⁸ exert no influence on the resistance to deformation, it must be assumed that internal stresses of spherical symmetry also remain without influence. On the contrary, the stresses in the intermediate state of AuCu tend to transform the tetragonal lattice into a cubic one;

In this case shear stresses arise, whence one must conclude that the nonuniform shear stresses acting within the lattice have a hardening effect. As was mentioned above, this hypothesis is also justified in the case of ordinary disordered solid solutions; in the same way one can also explain hardening during deformation.

Since a modern theory of slip does not yet exist, further development of this hypothesis is still premature. According to the presently available, basically classical-mechanical views on the question of slip,³¹ nonuniform shear stresses in the lattice should not have a significant influence on resistance to shear; however, all these views explain only with very great difficulty the phenomena that have now been found experimentally.

General conclusions on the kinetics of transformations

Precise X-ray studies of martensite and of the gold–copper system make it possible to learn the basic mechanism of such transformations and of precipitation phenomena in solid metals. In both these cases the transformation is composed of two phenomena proceeding at different rates: first, there is a transformation of the form of the lattice, which, as may be concluded from an X-ray study of the orientation of the new lattice relative to the old one, consists of a possibly short displacement or slip of all atoms. This phenomenon, if it has become possible, apparently proceeds—at least practically—at an infinitely high rate. Naturally, therefore, no influence of temperature is to be expected. An example of a pure phenomenon of this kind is the $\gamma \to \beta$ transformation in completely pure iron, the rate of which was investigated by Wever.¹⁰⁴ According to Dehlinger’s views these phenomena occur by a quantum path, as a result of transformations in the system of metallic electrons. The second phenomenon, more or less independent of the first, is an exchange of places of individual atoms in the new or old lattice, in which each atom, at the cost of its own energy, must pass over a potential jump. It proves to be very similar to the normal diffusion of foreign atoms into a crystal lattice; moreover, its rate is so small that it is completed only after several hours or days, while being strongly dependent on temperature. An example of such a phenomenon is the exit of a carbon atom from the $\gamma$-lattice during the $\gamma \to \alpha$ transformation of carbon-containing iron. As was indicated above, it occurs only in the subsequent stage

of the transformation, namely after the martensite lattice has formed. Therefore the transformation of martensite into $\alpha$-iron, associated with this yield, proceeds considerably more slowly than the formation of the martensite itself.

In the transformation of solid solutions in the Au–Cu system, the possibility of a transformation of the lattice form is apparently connected with the fact that at least several pairs of gold–copper atoms of the coherent lattice attain, by a second-order process, a regular arrangement of definite symmetry (a theoretical quantum explanation of this was given by Dehlinger[^106]). Without such a condition an unordered solid solution would be almost wholly incapable of hardening, just as the $\gamma$-modification of pure iron is not retained on quenching. However, the transformation of the lattice form, at least in single crystals, proceeds much more rapidly than would be expected on the basis of the law of diffusion, because it occurs already when only a few pairs of atoms are arranged in order among many thousands of atoms of the whole lattice. Further investigations of this connection between transformations of the lattice form and changes in the positions of atoms in the lattice are evidently very important.

The following experiments show that the “range of action” of such ordered pairs of atoms in a coherent lattice may extend even farther than would be expected according to lattice theories developed up to now only for heteropolar lattices, so that it turns out that the finer details of the kinetics still depend in a not very clear way on the grain size. On cooling completely undamaged single crystals of a solid solution of composition AuCu,[^106] and also during the precipitation of silver from supersaturated single crystals of copper,[^50] individual transitional stages of the lattice transformation—for example, the tetragonal AuCu lattice with values of $c/a$ from 1 to 0.93—can be quenched and observed radiographically; here, on one specimen, the final and the initial form of the transformation lattice are never observed simultaneously. Conversely, if the crystals undergo even only slight deformations or become polycrystalline, then in both cases[^105] and [^49], during the entire transformation both lattices exist. The transformation proceeds in such a way that the intensity of the lines of the old lattice decreases and that of the new increases, while the constants of both lattices do not change appreciably. The course of the rate of transformation in both phenomena in single crystals is also different from that in polycrystalline material.

For every metallic transformation which consists of two phenomena proceeding at such different rates, as was described above, one should expect un-

stable intermediate states, in which one or another phenomenon is not yet at all, or only partially, completed. The only way of reliably eliminating the intermediate state would be the evaporation of the old lattice and the formation of a new one by condensation; here both processes would be directly connected with one another. Such ideas formed the basis of Tammann’s nucleation theory, the kinetic consequences of which, as is known, were excellently proved for the case of solidification of a molten metal. However, the observations described in the following sections show that, in any case, transformations in the solid state in metallic single crystals

Fig. 21, a and b. Course of hardness during the precipitation of copper from supersaturated solid solutions with silver according to Ageew, Hanson and Sachs. ^90 (Tempering temperatures 200° and 250°.)

do not proceed in accordance with this nucleation theory; as will follow from the next section, the development of the most recent theory of age-hardening (Vergütung) has also led, in other systems, to the admission of intermediate states, and consequently also to the admission of a transformation consisting of various phenomena.

Phenomena of age-hardening (aging). Many systems, such as, for example, Al-Cu-Si (duralumin), Ag-Cu, Cu-Be ^110 and others, were hardened by heat treatment in exactly the same way as the AuCu system described above. Fig. 21 shows the dependence of the change in hardness of solid solutions of copper containing silver, quenched from 770°, on time at various tempering temperatures. Here one should note in particular the repeated fall in hardness at long tempering times. This so-called aging is known to have great technical significance, for here, as in steel, and also

In many alloys of light metals, the specimen can be given its shape in the soft state and then, without further change of shape, its strength can be increased by aging. However, the atomic phenomena revealed in the aging processes mentioned here, in contrast to AuCu, have as yet been little studied. As is known, the principal condition for the hardenability of a system is the existence in the phase diagram of a transformation or precipitation line that is not parallel to the temperature axis (for example, as in Fig. 12). It is precisely then that, on quenching through this line, the transformation is first arrested and then, on subsequent tempering, proceeds slowly. And indeed we find, for example, in duralumin[^100] the X-ray lines of a new lattice arising upon precipitation, though only after prolonged tempering. According to the old hypothesis, with cautious tempering the new lattice should first be precipitated in colloidal form, as a result of which the slip planes of the old lattice become “riveted,” whence the increase in hardness. This hypothesis has never remained without objections, and all X-ray investigations[^111],[^112],[^113] consistently show that at the moment of greatest quenching no new lattice yet exists. At this stage, although some broadening of the X-ray lines does occur,[^110] which may be caused, for example, by small distortion,6 no change is yet observed in the constant of the old lattice, as should have resulted from a change in concentration in this lattice during precipitation.

Therefore, on the basis of Tammann’s magnetic investigations of CuFe,[^116] a new hypothesis was put forward, according to which, before the precipitation of a new phase within the coherent lattice, there must occur a regrouping of the chemically heterogeneous atoms contained in it, and it is at the expense of this that the increase in hardness takes place, in a manner still unknown. The possibility required by this hypothesis—previously considered unlikely—of a nonuniform distribution of atoms in a coherent lattice without a change in the lattice constant has been proved experimentally by direct X-ray investigations of the existence of the intermediate state of AuCu.[^106] As was indicated above, the increase in hardness may be connected with other phenomena. In addition, Hengstenberg and Wassermann[^114] found in duralumin in the state of greatest hardness some increase in the intensity of the lines and a diminution of the veil, which can also unquestionably be explained only on the basis of the assumption of an assemblage of atoms according to the new hypothesis.

On the contrary, the details, especially the temperature dependence—

can account for the phenomena of aging, which have not yet been explained; it is highly probable that the dependence, described in the preceding section, of the fine mechanism of transformation on grain size plays a significant role here.

Growth Textures

Casting textures. The simplest example of a casting texture is a single-crystal rod solidified in a crucible under the influence of one-sided cooling.¹¹⁷ Contrary to expectation, here too, as in the pulling of a single-crystal filament from the melt, according to Czochralski, the axis of the rod, in the direction in which the crystal grew, was in each experiment crystallographically oriented in a different way, and no orientation proved predominant. Here we are dealing with only a single nucleus, which is oriented in an arbitrary manner at the bottom of the crucible and which, with a proper choice of cooling conditions, can continue to grow.

According to Goetz and Graf, in some cases the orientation of this nucleus is apparently determined by the state of the melt. Thus, for example, the former orientation of a single-crystal rod of bismuth or copper, which was remelted without stirring or shaking, reappears after solidification with great sharpness, provided the melt was not heated very far beyond the melting temperature. A similar experiment is performed by foundrymen with many metals, for example with aluminum and zinc. If the melt was heated only slightly above the melting temperature, small grains appear; but if a certain temperature above the melting point was maintained for some time, very large grains are obtained, often because the grains present in the melt are destroyed as a result of overheating. In both cases, special experiments have shown that the walls of the crucible have no influence on these phenomena. This circumstance is very important from the point of view of understanding molten metals; the experiments described lead to the conclusion that, in the melt, crystal nuclei have a perfectly definite orientation, which is destroyed only at higher temperatures.

In polycrystalline cast specimens one can distinguish, generally speaking, three zones: immediately at the edge that first solidified there are small grains without a preferred orientation; then follows a broader zone with very long crystals, extending in a ray-like fashion from the outside inward. According to the investigations of Schmidt and Nix,¹²¹ these crystals have completely

a definite orientation: in face-centered and body-centered metals the direction \([100]\) lies parallel to the axes of the rays, and hence perpendicular to the wall of the specimen; in hexagonal metals this role is played by the direction \([001]\), in Sn by \([110]\), and in rhombohedral Bi by \([111]\). The other directions are not established, i.e., here one is dealing with a perfect fiber structure. The third zone again has small, disorderly oriented crystals.

The fact that, in contrast to growing single crystals, a definite orientation is established in polycrystalline metals is explained by Schmidt and Nix in the following way: in a polycrystalline specimen, among the multitude of nuclei arising in the first zone, there is a certain selection; only those of them can grow further for which the crystallographic direction of the greatest growth rate stands perpendicular to the walls of the specimen. All the remaining nuclei are suppressed by the former. In the growing of a single crystal, however, only one single nucleus arises, so that there can be no question of any selection.

Thus the above-mentioned direction of the crystal rays in polycrystalline specimens must coincide with the direction of the greatest growth rate of the corresponding metal. Since the latter is known precisely in the case of hexagonal crystals, this proves to be true.

Texture of electrolytically deposited precipitates. As was found by Glocker and Kaupp, in electrolytically deposited crystals a perfect fiber structure is also established (one crystallographic direction is fixed, the others arbitrary), the fiber axis being in the direction of the current lines. The crystallographic direction coinciding with the fiber axis depends on the composition of the electrolyte, but not on the material of the cathode. Here, too, the direction of greatest growth rate apparently is the privileged one.

Texture of precipitates arising from the vapor state. Thin metallic layers prepared by evaporation or cathodic sputtering are never amorphous, but always give good X-ray photographs. The face-centered metals Pt, Cu, Ni\(^{118,119}\), Co, as well as Bi\(^{120}\), studied up to the present time, have a perfect fiber structure with the \([111]\) direction of the fiber axis perpendicular to the surface. Whether in this case there is an influence of the material of this surface is as yet unknown.

Corrosion Structure

If a sample of a pure metal is acted upon by chemical agents, the structure of the surface does not change. Etching of the outer layers is even the only means of making the structure of the inner parts of the material accessible to X-ray or microscopic investigation without fear of altering them during treatment.

Something else occurs under chemical action on alloys that contain components with different capacities for resistance to solvents. The behavior of single crystals of the alloy AuCu with respect to liquid and gaseous solvents was investigated in this direction by Graf.^122 He finds that, under the action of aqua regia and nitric acid on the surface of the solid solution, a layer of completely pure gold is formed, which has the same orientation as the solid solution. Under the action of nitrogen oxides at high temperatures, an oriented layer is likewise formed, whose lattice constant directly at the free surface belongs to gold, but inward from it gradually approaches the lattice constant of the solid solution. Potassium cyanide etches the surface without change. The striking difference between liquid and gaseous solvents can be explained by the fact that the gold atoms remaining after the dissolution of the copper in the liquid become ionized and therefore acquire such great mobility that, by some as yet unknown path, they can form an entirely new lattice.

Under the action of a gas this possibility disappears; in this case the gold atoms can enter only the pores formed at the sites of the liberated copper atoms.

No differences in the behavior of the ordered distribution and the disordered atomic distribution of the quenched solid solution were found anywhere.

Thus Tammann’s explanation of the boundaries of resistance, based on the assumption of a regular atomic distribution, is no longer justified.

It follows from the calculations of Mazing and Borelius that even with an irregular distribution the noble atom also always stands in front of the base atom and protects this latter, if only there are more than 25% noble atoms. In solvents where, according to Graf, one may expect strong mobility of the noble atoms, this no longer holds. In this case at least 50% noble atoms are required, so that there can form

connected protective layer. Consequently, in these solvents the resistance boundary is shifted to 50 atomic percent, which is also in agreement with experiment.

Deformation Structure

The distortions of the lattice that appear under elastic deformation were first measured precisely by Sachs and Weerts.^124 Thus, by a very simple method it became possible to measure Poisson’s ratio.

Under residual (plastic) deformation, the crystalline structure changes only in a few cases [in AuCu (106), Cd-Mg (46), and Ag (50)]. On the contrary, the general structure changes substantially even under small deformations.

Single crystals. Single crystals plastically deformed in drawing or rolling change their orientation. As experiments with various initial orientations show, this can be explained on the assumption that the deformation consists of simple slip, which occurs in the manner described on p. 146.^121 Just as there, the slip planes and the directions of slip along which the motion of atoms occurs are established crystallographically (for a table of the slip elements found, see ^125).

In any case, only in one rare instance, in so-called twinning, is the magnitude by which individual planes slip also established; in this case the slip is precisely so large that the final position of the deformed region of the crystal can be obtained by reflection in a plane perpendicular to the direction of slip. From the point of view of deformation, the second type of slip is far more important, in which the magnitude of the displacement is not established and which therefore leads to considerably larger deformations. Moreover, not all parallel slip planes are displaced by the same amount; for large crystalline regions, which are called slip layers, this amount is zero and therefore becomes very large on the surfaces bounding these crystals, which become visible externally as slip ellipses. During slip the entire crystal rotates in such a way that the direction of slip approaches the direction of tension.

As is known, for slip to be determined one requires one quite definite component of the shear stress in the direction of slip,^126 whereas the components of the shear stress perpendicular to it [[unclear: text continues at bottom of page]]

whatever role they may play.¹³⁰ For further continuation of slip the stress has to be considerably increased, and there occurs what is called hardening.

If a single crystal is stretched in a tensile-testing machine, and for one reason or another only part of the entire length of the specimen is deformed, then the slip lamellae lying in the transition regions between the deformed part and the undeformed part must, owing to the curvature, assume a new orientation (slip with bending).¹³¹ Something similar was also observed by Burgers¹²⁶ in uniformly compressed crystals of aluminum; insofar as he succeeded in establishing, on the basis of quantitative measurements of the asterism figures on Laue photographs, here a small region of the lattice, probably a piece of each slip lamella, is bent in a definite direction, which differs by 10–20° from the normal orientation produced as a result of slip, and approaches the initial orientation, so that a cylindrical curvature arises with one axis lying in the slip plane and situated perpendicular to the active direction of slip.

According to Burgers this so-called scattering of orientation occurs because some of the atoms “catch” behind the slip plane, so that within a single slip plane, generally speaking, not all magnitudes of shear are the same.

If, as is the case for cubic crystal symmetry, there exist several crystallographically equivalent directions of slip, then, generally speaking, only one of them proves to be active, namely that on which the greatest component of the shear stress falls. As a result of the change of orientation during deformation, another direction gradually comes into a more favorable position; however, for it to enter into play, a somewhat greater stress is required than in the case of an undeformed crystal (latent hardening).

Polycrystalline metals. In the plastic deformation of a polycrystalline material, the orientation of all grains, according to Polanyi, just as in a single crystal, changes in such a way that certain crystallographic planes and directions, as the deformation increases, become more and more parallel to the planes and directions that are prescribed by the external conditions of deformation. In this way a small number of “ideal deformation positions” is obtained.¹²⁶ However, in reality there is a considerable scattering around them, so that a more exact descri-

The determination of textures is possible only with the aid of pole figures.¹²⁹ Very often, especially in wires drawn through a die, the texture is also nonuniform: the outer layers have a different structure, namely a weaker orientation¹²³ and greater lattice distortion,⁴⁸ than the inner ones.

The grain size changes considerably even as a result of a small plastic deformation. As radiographs show, the microscopically visible grains of the initial texture here break up into a large number of initially almost mutually parallel small grains, which should be understood as slip layers, such as are found in a single crystal. Their size has not yet been precisely measured; it probably lies beyond the resolving power of the microscope, but considerably above the value of \(0.1\,\mu\), at which a noticeable broadening of the Debye lines would have to appear.

Fig. 22. Stress and bending of slip plates of a grain during deformation under the influence of pressure directed perpendicular to the displacement of matter (\(A\) denotes the direction of pressure, \(D\) and \(R\)—the directions of the interaction forces of the slip plates).¹²⁷

Such broadening does indeed occur as a consequence of deformation; however, since in the metals Al and Zn it does not appear at all, and in other metals it appears only during rolling and drawing through dies, and not during free tension,¹²⁷ its cause should be considered the bending of the slip plates, which plays a secondary role in deformation and which occurs under the influence of a strong pressure perpendicular to the stress of motion of the substance (Fig. 22).

In general, polycrystalline deformation textures are obtained by the same slip mechanism as changes in the orientation of single crystals. Thus, for example, in rolled polycrystalline magnesium, as a result of simple slip, the hexagonal basal surface becomes perpendicular to the direction of pressure. In zinc twinning must also be taken into account. But in cubic metals, where there are many crystallographically equivalent slip directions, more subtle assumptions would have to be introduced. Polanyi, as well as Boas and Schmid¹²⁸, indicate that any desired change of shape could be obtained without nonuniform distortion or lattice curvature if, at the same time, with an appropriate velocity, at least

three directions of slip. These authors assume that in all grains the three slip directions which are under the highest external stress slip in this way. The orientation of the grain in which these three directions acquire equal stress is a stable final orientation appearing in the deformation texture.

In contrast to this, Wever and Schmid3 make the assumption that slip occurs simultaneously in only one direction, while the remaining deformation proceeds by bending of the slip lamellae, whose axis stands perpendicular to the slip direction and lies in the slip plane; moreover, this bending tends to rotate the active slip direction into the direction of the prescribed flow of the material (for example, in drawing wire, into the direction of its axis). If a second slip direction also falls into an equally favorable position, it too may become active, the crystal again straightening out. If this rotation again brings the first slip direction into action, then the position reached undergoes no further significant changes during subsequent deformation; this constitutes the stable orientation of the deformation texture. Thus, for example, as the stable position for drawn wires of face-centered cubic metals the direction \([111]\) is obtained, and for compression \([110]\), whereas the direction \([100]\) is labile, since as a result of a slight rotation during further slip it no longer returns.

Naturally, both of these assumptions should have served as the basis for various definite, though difficult to formulate, hypotheses concerning a hidden increase in the strength of individual slip directions and concerning the action of neighboring slip layers;4 it is not yet possible to choose between them. However, the experimental study of lattice distortions had already shown earlier that the latter conception is very often justified. Thus, for example, Burgers5 observed such orientation spreads as had been predicted by Wever and Schmid; likewise, on the basis of the broadening indicated above, one may also conclude that there is a considerable curvature of the slip lamellae. That this broadening occurs only under such deformations as are associated with pressure perpendicular to the direction of shear of the material seems especially important. It is precisely such a stress component that curves the slip lamellae shown in Fig. 22 in the manner proposed by Wever and Schmid,6 whereas in free tension the bending moment is so small that it

does not produce any more noticeable broadening. On the contrary, in polycrystalline aluminum there apparently act simultaneously several slip planes, so that the limiting case of the Boas–Schmid assumption is more suitable for it; it is possible that for this reason position \([100]\) is only weakly expressed. It thus appears that the difference between a single crystal stretched by means of a tensile-testing machine and a polycrystalline wire drawn through a die—the difference whose explanation was the purpose of all the works named here—arises rather from the specific influence of the dies than from the polycrystalline initial structure. Unfortunately, there are as yet no direct investigations on this question. Further material for studying these questions is provided by the recrystallization textures considered below.

The matters considered here have not only a purely geometrical significance; they provide the experimental basis for the still only slightly explained \(^{146}\) transfer of the strength properties of a single crystal to a polycrystalline structure.

Thus in every deformed polycrystalline metal (the converse will be a freely and, with due precautions, stretched single crystal) parts of the lattice with different orientations lie side by side. Without further assumptions it is impossible to understand why such an arrangement, at not especially high temperatures, can exist for a long (practically even infinite) time. The atoms at the boundaries of such a crystal lattice could, without any overcoming of a potential threshold—for example, in accordance with the ideas developed by Kossel \(^{135}\)—pass from one lattice to another. For this only small shifts would be needed, which were indicated above and which proceed at great speed independently of temperature. Therefore, as Tammann \(^{153}\) and Delinger \(^{123}\) indicated, one has to assume in the deformed formation some factor which, when the temperature-induced motions of the atom are not very strong, maintains the equilibrium of differently oriented lattices lying side by side. Van Liempt \(^{134}\) attributes this factor to changes in the electron shells of the atom that are not described more precisely by him. Tammann \(^{133}\) assumes that foreign substances—gases or solid impurities—are situated between the individual crystallites. In addition to Prandtl’s objections, Delinger describes a method of geometrical coupling (Verhakung) of lattices lying side by side (Fig. 23), which remain in a state of static equilibrium until the atoms come into too close a...

large temperature oscillations, but which disappears completely as soon as individual “caught” atoms, as a result of such a diffusion process, do not cross the potential threshold indicated in Fig. 23. Burgers5 draws attention to the fact that the lag he observed, described above, of individual pieces of the glide planes may lead to the formation of such linkages. As can be seen directly from Fig. 23, such linkage causes a strong shear stress in the lattice. However, according to the hypothesis set forth above (sections I, B₃ and II, A₂), in order to explain the phenomena of hardening and aging in mixed crystals, shear stresses act as a strengthening factor. Thus, by means of linkage it is also possible to explain the increase in strength that occurs during deformation.

It is very remarkable in this connection that not only an increase in temperature, but also numerous small bendings back and forth during fatigue testing can eliminate the stability of the deformed state of a specimen. If, for example, deformed copper or silver sheet is subjected to such a prolonged fatigue test, then always, immediately before fracture, the Debye lines, at first broadened as a result of deformation, become completely sharp and, in many cases, at temperatures considerably below the recrystallization temperature, new grains appear in the metal.[^137]

Fig. 23. Mechanically stable linkage of glide planes.

Recrystallization Textures

If a metal whose structure corresponds to the deformed state is subjected to gradual heating, then first there occurs the so-called recovery state (Erholung),[^138] i.e., the strengthening caused by deformation is partly reduced, and if as a result of deformation the Debye lines had been broadened, then under the indicated heating this broadening disappears (the so-called Arkell splitting of the K-doublet of X-ray radiation occurs).

In the general case, at higher temperatures (the recrystallization temperature), with further softening, the first new grains visible under the microscope are formed.

Only at very high degrees of deformation, and even then not in all metals (for example, in normal copper), do these two stages coincide with one another. New grains can orient themselves relative to the deformation texture in the following ways:

a) The deformation texture is preserved, since annealing is carried out only slightly above the recrystallization temperature. Only with a further increase in temperature do grains appear in all possible arrangements. This case occurs in rolled aluminum; in the case of technical material, a random orientation appears at a temperature only slightly higher than the recrystallization temperature, while in very pure material only at a temperature several hundred degrees higher.

b) One of the various orientations present in the deformation texture—namely, for the most part, only weakly represented—remains during recrystallization. All new grains grow precisely in this direction of orientation. This occurs in particular in cold-rolled copper, which before rolling had been subjected to hot rolling; in this case all grains grow in the position of the cubic orientation, only weakly represented in the deformation texture: (100) in the rolling plane and the direction 100 in the rolling direction. Something similar also takes place in the case of drawn copper wires. In rolled iron, of the three orientations of the deformation texture, two orientations with large scatter appear strongly, while the third remains without scatter in a weak form.

c) The new grains have a single orientation, always associated with large scatter, which has not yet occurred in the deformation texture, but which is in a geometrical relation to it. Thus, heavily rolled silver and α-brass recrystallize with orientations in which, as in the rolling texture, the direction (112) coincides with the rolling direction, but, in contrast to it, the plane (111) lies in the rolling plane. Consequently, this recrystallization texture formally arises from the rolling texture as the result of rotating the crystallites about the rolling direction as an axis through an angle of 30°. Drawn copper wire behaves in the same way. According to recent investigations, this orientation apparently is also preserved at high temperatures, so that in this case, just as in case “b,” it is completely impossible by means of annealing to obtain a technically isotropic material from heavily rolled sheet.

As described in case “c,” the uniformly compressed crystallites studied by Burgers also recrystallize.

aluminum, whose deformation texture was described above. The orientation of the majority of the crystals coincides with the direction that follows from the theoretical orientation of deformation by means of the same rotations as were mentioned in the description of the position of the scattered parts of the lattice; the angle of this rotation is 20–60°. Consequently, here, during recrystallization, all grains grow precisely in those orientations which, in the deformation textures, were obtained from the most extreme scattered parts of the lattice.

Dillinger^132 and Burgers^126 attempted to explain the connection between recrystallization and deformation textures. Both assume that the orientations appearing during recrystallization must exist, in separate though very small regions of the lattice, already in the deformation texture, and that, consequently, the formation of entirely new nuclei, just as in the case of other transformations in the solid state, does not occur here. Further, both assume that the question of which of the orientations present in the deformation texture is the determining orientation of the new grains depends on distortions of the lattice present in the deformation texture. According to Dillinger, these are the least deformed regions of the lattice, to which the surrounding atoms attach themselves; for this, after the stabilizing cohesion has been removed, only short shifts or slips are required. In this way the behavior of drawn copper wire can be explained very simply. According to Wever and Schmidt, the curvature axis (Fältelungsaxe) of the slip lamellae lies perpendicular to the direction of the greatest flow of the material, i.e., to the axis of the wire. As Fig. 22c shows, precisely those regions of the lattice of these slip lamellae have the smallest radius of curvature, and hence the greatest distortion, which are most remote from the mean orientations of the slip lamellae. Consequently, according to the stated assumptions, as was indeed observed, precisely the mean orientation of the deformation textures should not appear, and the outermost scattered orientations will exclude all the other positions. In exactly the same way one could explain the recrystallization of a compressed aluminum crystal. The rolling texture, according to Wever and Schmidt, may be explained as the superposition of the compression texture and the texture of free drawing. As indicated above, under compression strong folding should be expected, since the direction of pressure is almost perpendicular to the direction of displacement of the material, whereas under free drawing only a very slight bending should appear. Therefore, during rolling the axis of the strongest folding will lie perpendicular to the direc-

the pressure, but parallel to the direction of rolling, and in accordance with this the new orientation will arise from the rolling texture by rotation about this direction, which is entirely consistent with the observations. To explain case “б,” further assumptions must be made, namely that the cube position, which appears weakly in the deformation texture, undergoes smaller distortions than the other orientations, which, of course, is connected with its instability. Then case “a” occurs, when no preferential folding exists.

Such a curvature, as Burgers has shown, exists in aluminum crystals, although it is so weak that it gives no broadening of the lines; but in the case of polycrystalline aluminum it is masked by irregular distortions originating from the grain boundaries, so that of polycrystalline material in general the following may be said: if, in the deformed state, there is broadening of the lines, then upon recrystallization case “б” or “в” occurs, and if there is no broadening, then case “a” occurs.^139

Burgers^134 assumes that the most strongly distorted parts of the lattice act as centers of recrystallization and that, for example, in rolled aluminum crystals the ends of the slip lamellae experience the greatest compression from the surrounding material; consequently, these regions of the lattice also determine the new orientation. It is not yet possible at present to decide the question of the correctness of both of these views.

On the question of the magnitude of the grain established during recrystallization, one may point to the communication of Altertum.^140

Conclusion

In the preceding exposition a whole series of properties of metallic structures has been reported which, apparently, are important from the standpoint of the metallic bond. These include their high, to a considerable extent independent of atomic positions, symmetry; their great capacity for deformation; and the widely developed “action at a distance” in changes of the lattice. The further task of physical metallurgy will be to investigate the causes of these properties and to compare them with the general theories of crystals. Only then will it be possible to bring the individual lattice measurements obtained with the aid of X-rays into quantitative relation with one another.

LITERATURE

General information on X-ray methods and structures.

1. Bragg, W. H. u. W. L., X-Ray and Crystal Structure. 4 Ed., London 1924.
2. Ewald, P. P., Kristalle und Röntgenstrahlen, Berlin 1923.
3. Jonson, A., Erg. exakt. Naturwiss., 1, 210, 1922.
4. Mark, H., Die Verwendung der Röntgenstrahlen in Chemie und Technik, Leipzig 1926.
5. Glocker, R., Materialprüfung mit Röntgenstrahlen, Berlin 1927.
6. Ott, H., Handbuch der Experimentalphysik, 7, 2, 1929.
7. Schleede, A. u. Schneider E., Röntgenspektroskopie und Strukturanalyse, Berlin 1928.
8. Ewald, P. P. u. Herrmann C., Strukturbericht der Z. Krist., 1926—1930.

The works analyzed there are not cited here in greater detail.

1. Neuburger, M. C., Röntgenographie der Metalle, Stuttgart 1928.

Works on the theory of the atom.

1. W. Heitler, Physik. Z., 31, 185, 1930.
   See also M. Born, Z. f. Physik, 64, 728, 1930.
2. Pauling, L., Z. f. Physik, 50, 1036, 1928; Z. Krist., 67, 377, 1928.
3. Darrow K., Elementare Einführung in die physikalische Statistik, Leipzig 1931.
4. Slater, J. C., Physic. Rev., 35, 509, 1930; 36, 57, 1930.

On Section I, A. X-ray investigations.

1. Simon, F. u. Volsen E., Z. f. physik. Chem., 1928, 133, 165 (alkaline earths Sr, Zn, Cd).
2. Goldshmidt, V. M., Naturwiss., 17, 134, 1929; Z. f. Physik. Chem., B. 3, 241, 1929 (Re).
3. Ebert, F. u. Hartmann H., Z. f. anorg. Chem., 179, 418, 1929 (Sr, Ba).
4. Jäger, F. M., Terpstra u. Westenbrink, Proc. Acad. Amsterdam, 29, 1193, 1926; Z. Krist., 66, 1928 (Ga).
5. Möisel, K., Z. f. allgem. Chem., 196, 237, 1930 (Nb).
6. Bradley, A. J., Proc. roy. Soc., 115, 456, 1927; Preston, G. D., Philosophic. Mag., 5, 1199, 1927, 1928 (α-u.β-Mn).
7. Öhman, E., Metallwirtschaft, 9, 825, 1930 (Mn).
8. Harros, Mack a. Blake, J. amer. chem. Soc., 1, 1583, 1928 (J).
9. Sekito, S., Z. Krist., 74, 189, 1930 (Tl).
10. Lark-Horovitz, K., Physic. Rev., 33, 121, 1929 (Hg).
11. Wolff, M., Z. f. Physik, 53, 72, 1929 (Hg).
12. Bach, C., Helvet. phys. Acta, 2, 95, 1930 (Fe).
13. Alichanow A. J., Z. f. Metallkunde, 21, 127, 1929 (Al).
14. Valentiner, S. u. Becker. G., Naturwiss., 17, 639, 1929 (Ni).

Other works.

1. Seemann, H. J., Physik. Z., 28, 765, 1927; 29, 94, 1928; Z. f. Physik, 61, 576, 1930.
2. Wassermann, G., Z. Krist., 75, 369, 1930; Landolt-Börnstein, 2. Erg.-Bd., 38—39.
3. Hume-Rothery, W., Philosophic. Mag., 4, 1017, 1927.
4. Smekal, A., Handb. d. physikal. u. techn. Mechanik, 4, 2, 1, 1931.
5. Polanyi, M. u. Schmidt E., Naturwiss., 17, 301, 1929; Polanyi M., Metallwirtschaft, 9, 553, 1930.
6. Hume-Rothery, W., Philosophic. Mag., 9, 65, 1930; ibid. 11, 640, 1931.

1. Bernal, A., Trans. Farad. Soc., 25, 367, 1929; Metallwirtschaft 9, 983, 1930.
2. Canfield, R. H., Physik. Rev. 25, 569, 1930.
3. Dehlinger, U., Z. f. Physik, 68, 535, 1931.
4. Masumoto, J., Sc. Rep. Tohoku Univ., 15, 449, 1926; Umino, S., ebenda, 16, 593, 1927.
5. Bragg, W. L., Philosophic. Mag., 40, 177, 1920.
6. Goldschmidt, V. M., Z. f. physik. Chem., Bd. 133, 397, 1928.
7. Westgren, A. u. Almin, A., ebenda, 5, 14, 1929.
8. Klemm, W., ebenda, 12, 1, 1931.

To Section I, B. X-ray investigations

1. van Arkel, A. E., Physica, 6, 64, 1926 (Mo, W).
2. Sachs, G. u. Weerts, J., Z. f. Physik, 60, 481, 1930 (Ag, Au).
3. Burgers, W. G., Z. Krist., 75, 155, 1930 (Cu, Ni).
4. Hengstenberg, J., Metallwirtschaft, 9, 463, 1930 (Ag, Au).
5. Dehlinger, U., Z. f. anorg. Chem., 194, 223, 1930 (Cd, Mg).
6. Johannson, C. H. u. Linde, J. O., Ann. d. Physik, 5, 762, 1930 (Au, Pt).
7. Johannson, C. H., ebenda, 4, 485, 1930 (Ag, Pt).
8. Ageew, N., Hansen, M. u. Sachs G., Z. f. Physik, 66, 350, 1930; 67, 293, 1930 (Ag, Cu).
9. Wiest, F., Stuttgarter Diss., 1931 (Ag, Cu).
10. Osawa, A., Sc. Reports Tohoku Univ., 19, 109, 1930 (Fe, Cu, Ni, Co).
11. Osawa, A. u. Oya S., ebenda, 18, 427, 1929 (Fe, V).
12. Osawa, ebenda, 19, 247, 1930 (Fe, Mn).
13. Öhmann, E., Z. f. physik. Chem., Bd. 8, 81, 1930 (Fe, Mn).
14. Sekito, S., Z. Krist., 72, 406, 1930 (Cu, Mn).
15. Persson, E., Z. f. physik. Chem., B. 9, 25, 1930 (Cu, Mn).
16. Solomon, D. u. Morris-Jones W., Philosophic. Mag. 10, 470, 1930 (Pb, Sb).

Other works

1. Vegard, L., Z. f. Physik, 5, 17, 1921.
2. Grimm, H. G. u. Herzfeld, K. F., ebenda 16, 79, 1923.
3. Tammann, G., Z. f. anorg. Chem., 107, 1, 1916.
4. Wagner, C. u. Schottky, W., Z. f. physik. Chem., Bd. 11, 163, 1930.
5. Borelius, G., Johannson, C. H. u. Linde, J. O., Ann. d. Physik., 86, 291, 1928.
6. Grube, G., Z. f. anorg. Chem., 1931.
7. Sachs, G. u. Weerts, J., Z. f. Physik, 67, 507, 1931; 62, 473, 1930.
8. Seemann, H. J. u. Vogt, E., Ann. d. Physik, 2, 976, 1929.
9. Seemann, Z. f. Physik, 62, 824, 1930.
10. Vogt, E., Z. f. Elektrochem., 1931.
11. Grüneisen, E., Handb. d. Physik, 13, 1, 1928.
12. v. Laue, M., Ann. d. Physik., 56, 497, 1918; 78, 167, 1925.
13. Rusterholz, A., Helvet. phys. Acta, 4, 68, 1931.
14. Schäfer, F., Z. f. Physik, 1931.
15. Wever, F., Arch. f. Eisenhüttenw., 2, 739, 1929; Erg. d. techn. Röntgenkunde, 2, 240, 1931.
16. Nordheim, L., Naturwiss., 16, 1046, 1928.
17. Delinger, U., Metallwirtsch., 9, 589, 1930.

To Section I, C. X-ray investigations

1. Eckman, W., Z. f. physik. Chem., Bl. 12, 57, 1931 (transition metals).
2. Hägg, G., ebenda 12, 33, 1931 (metalloids). In 75 and 76 there are

numerous literature data on compounds of transition metals with other metals and metalloids.

1. Solomon, D. and Morris-Jones, W., Philosophic. Mag., 11, 1930, 1930 (Ni, Bi, Sn, Bi).

2. v. Stackelberg, M., Z. f. physik. Chem., Bd. 9, 437, 1930 (carbides).

3. Katon, N., ibid., 6, 27, 1930 (Cu, Hg).

4. De Jong, W. F.—Willens H. W. Y., Z. f. anorg. Chem., 170, 241, 1928 (Ni, Sc, Co, Sn).

5. Pastorello, S., Gazz. chim. ital., 60, 493, 1930 (Lo, Ag).

6. Perconn, E., Z. f. Physik, 57, 115, 1929 (Mn, Cu, Al).

7. Eisenhut, O. and Kaupp, E., Z. f. Elektrochem., 36, 393, 1930 (Fe, N).

8. Hägg, G., Z. f. physik. Chem., Bd. 8, 455, 1930 (Fe, N).

9. Hendriks, St. b. and Kosting, P. R., Z. Krist., 74, 534, 1930 (Fe₃, C).

10. Hendricks, ibid. 74, 511, 1930 (Fe₂P, Fe₂N, Fe₃N, Fe, B).

11. Wever, F. and Möller, H., Z. Krist., 75, 362, 1930 (Fe, B).

12. Hägg, G., ibid., 71, 134, 1929 (Fe, As).

13. Hägg, Z. f. physik. Chem., Bd. 11, 15?, 1930 (Fe, B).

14. Osawa, A. and Oya M., Sc. Reports Tohoku Univ., 19, 95, 1930 (V, C).

15. Wever, F. and Haschimoto U., Mitt. K. W. Inst. f. Eisenf., 11, 293, 1929 (Co, Cr).

16. Westgren, A., Hägg G. and Erikson S., Z. f. physik. Chem., Bd. 4, 6, 1929 (Cu, Sb).

17. Howalls, E. V. and Morris-Jones W., Philosophic. Mag., 9, 993, 1930 (Cu, Sb).

18. Thomassen, L., Z. f. physik. Chem., Bd. 2, 349, 1929; 4, 277, 1929 (platinum metals with Sb, Te, Se, As).

19. Hägg, G. and Funke C., ibid., 6, 272, 1929 (Ni, Bi).

20. Kürber, F. and Haschimoto U., Z. f. anorg. Chem., 158, 1930 (Bi, Te).

21. Linde, J. O., Ann. d. Physik, 8, 124, 1931 [Cu, Sn. (?)].

22. Wever, F. and Müller A., Z. f. anorg. Chem., 192, 317, 1930 (Fe, B).

23. Westgren, A., Metallwirtschaft, 9, 919, 1930.

24. Goldschmidt, Geochem. Verteilungsgesetze d. Elemente, 7, 1926.

25. Meissner, W. and Frantz H., Naturwiss., 18, 418, 1930.

For Section II, A

1. Öhmann, E., Nature, 127, 270, 1931.

2. Kurdjumow G. and Sachs G., Z. f. Physik, 64, 325, 1930.

3. Wever, F. and Enge N., K. W. Inst. f. Eisenf., 12, 93, 1930.

4. Oshima, K. and Sachs G., Z. f. Physik, 63, 210, 1930.

5. Dehlinger, U. and Graf L., Z. f. Physik, 64, 359, 1930.

6. Dehlinger, ibid., 1931.

7. Polanyi, M. and Schmid E., ibid., 16, 336, 1923.

8. Schmid, E. and Wassermann G., Naturwiss., 14, 980, 1926; Metallwirtsch., 7, 1329, 1928.

9. Masing, G., Veröff. Simenskonz., 8, 187, 1929.

10. v. Göler F. and Wassermann C., Naturwiss., 17, 309, 1929; Metallwirtsch., 8, 871, 1929.

11. Schmid, E. and Wassermann C., Metallwirtsch., 9, 421, 1930.

12. Portervin, A. and Chevenard, P., Rev. d. Metallurg., 27, 412, 1930.

13. Hengstenberg, J. and Wassermann, G., Z. f. Metallkunde, 23, 114, 1930.

14. Kokubo, S. and Honda, K., Sc. Reports Tohoku Univ., 19, 365, 1930.

15. Tammann, G., Z. f. Metallkunde, 22, 365, 1930.

*

To Section II, B

1. Graf, L., Z. f. Physik, 67, 888, 1931. The literature cited here is not cited further.
2. Hanawalt, I. D. and Ingersoll, L. K., Physik. Rev., 34, 972, 1929.
3. Dembinska, S., Z. f. Physik, 51, 46, 1929.
4. Büssem, W., Gross, F. and Herrmann, K., Z. f. Physik, 64, 537, 1930.
5. Schmid, E., Metallwirtsch., 8, 631, 1929; Z. Metallkunde, 21, 72, 1928.

To Section II, C

1. Graf, L., Ann. d. Physik, 1931.

To Section II, D and II, E

1. Bericht von E. Schmid and C. Wassermann, Handbuch d. physik u. techn. Mechan., 4, 2, 319, 1931. The literature cited here is not cited further.
2. Sachs, G. and Woerst, J., Z. f. Physik, 64, 344, 1930.
3. Landolt-Börnstein, 2, Ergbd., 41.
4. Burgers, W. G., Z. f. Physik, 67, 606, 1931.
5. Dehlinger, U., Z. Metallkunde, 23, 147, 1931; Z. Krist., 65, 615, 1927.
6. Boas, W. and Schmid, E., Z. f. techn. Physik, 12, 71, 1931.
7. Wever, F. and Schmid, W. E., Mitt. K. W. Inst. f. Eisenf., 11, 109, 1929; Z. f. Metallkunde, 22, 133, 1930.
8. Schmid, W. E., Z. f. techn. Physik, 1931.
9. Masing, G. and Polanyi, M., Erg. d. exakt. Naturwiss., 2, 177, 1923.
10. Dehlinger, U. Ann. d. Physik, 2, 49, 1929; Z. f. Metallkunde, 22, 221, 1930.
11. Tammann, G., Z. f. anorg. Chem., 185, 1, 1930; Z. f. Metallkunde, 22, 224, 1930.
12. van Liempt, J. A. M.; Z. f. anorg. Chem., 195, 306, 1931.
13. Kossel, W., Naturwiss., 18, 901, 1930.
14. Dehmek, J., Z. f. Physik, 65, 138, 1930.
15. Dehlinger, U., Naturwiss., 17, 545, 1929; Metallwirtsch. 26, 1931.
16. Burgers, W. G., Z. f. Physik, 68, 11, 1929.
17. Agte, C. and Becker, K., Physik. Z., 32, 65, 1931.
18. Althertum, H., Physik. Z., 32, 315, 1930.
19. Kurdjumow, G. and Sachs, G., Z. f. Physik, 62, 592, 1930.
20. Hergstenberg, J. and Mark, C., ibid., 61, 435, 1930.
21. Brill, N., ibid., 61, 445, 1930.

1. Ergebn. der exakt. Naturwiss., 10, 325, 1931; translated by N. A. Shishakov. ↩