CRYSTAL STRUCTURE AND CHEMICAL COMPOSITION.1

V. M. Goldschmidt, Oslo.

I. The historical path of crystal chemistry.

The task of crystal chemistry is to reveal the connection between the material composition and the physical properties of crystalline substances and, first of all, to find correlations between their chemical composition and crystalline form.

The existence of regular correlations between chemical composition and crystalline structure is apparent from many facts long known to us, chemists and crystallographers. The first step along this path was the discovery by Haüy that to every variety of a substance, to every chemically individual body, there belongs an entirely definite complex of crystal faces, depending, obviously, on the internal architecture, the atomic-molecular arrangement of the given substance. The next achievement was Mitscherlich’s discovery of the profound crystallographic similarity often observed in chemically similar substances; this phenomenon was called isomorphism. To Mitscherlich we also owe the subsequent fundamental discovery that the dependence of crystalline structure on material composition is not

unambiguous—the phenomenon of polymorphism. A hundred years later Pasteur discovered the geometrical enantiomorphism of left- and right-rotating tartaric acids.

In the second half of the last century, Hiortdahl (Gh. Hiordahl) and especially P. Groth (P. Groth) engaged in the search for further regularities in this field. Both of them strove, through the planned substitution in series of organic compounds, to connect “morphotropy”—the consecutive variability of crystalline forms—with changes in chemical composition. The works of this period, although in a number of cases they did provide substantial indications of the presence of definite regularities, nevertheless, despite all the labor expended, did not reveal any new general laws comparable with the discoveries of Mitscherlich or Pasteur. Closest to modern ideas concerning the connection between crystalline form and composition are Brogger’s works on isomorphism and morphotropy in the world of minerals. The large factual material collected in this epoch, the compilation and ordering of which constitutes Groth’s enduring merit, is of the greatest value for crystallochemical investigations, after the study of simpler substances has presently laid the foundations of the crystallochemistry of more complex compounds.

To establish the connection between crystalline structure and chemical composition, we must naturally begin with the study of the simplest substances in composition, namely, alongside the free elements—with chemical compounds of the simplest types \(AX\), \(AX_2\), \(AX_3\).1 Only after establishing the sought regularities in these simple substances will it be expedient to take up substances of more complex composition, in order to study on them the limits of applicability of the regularities already found. For a number of years I have been working according to this principle, developing by the inductive method the laws of crystallochemistry. In

In what follows a brief survey of these works is given; in this survey, too, the empirical-inductive approach is maintained, and the crystal-chemical regularities are derived from a direct comparison of empirically, directly measured quantities.

A glance at crystal-chemical handbooks and tables representing the state of our knowledge 5–7 years ago shows that at that time, precisely with regard to the crystal structure of the simplest compounds, the information was in every respect insufficient. It was therefore necessary first to take care of accumulating the experimental material needed for my purposes. Thanks to X-ray analysis and to the micro-optical methods of petrography, it is possible to establish crystal structures with confidence, often even with small quantities of microcrystalline substances. Therefore the work could be extended to very rare substances, and also to substances whose ready decomposability had hitherto hindered their investigation by macrocrystallographic methods. Together with a number of outstanding collaborators I was able, beginning in 1923, to accumulate the factual material necessary for the crystal chemistry of simple substances.

II. Structural types of crystals and their rational classification.

We shall first consider heteropolar compounds with formulas AX and AX₂. Experience shows that compounds with one and the same stoichiometric formula may crystallize in very different ways. For example, for compounds of formula AX we find the following structural types, named after their typical representatives: CsCl, NaCl, NiAs, ZnS (zinc blende), ZnO, BN. For substances of formula AX₂, we know, along with others, the structures: CaF₂, TiO₂ (rutile), TiO₂ (anatase), SiO₂ (structural varieties of quartz, tridymite, and cristobalite), Cu₂O, CO₂, FeS₂ (pyrite), FeAs₂, CdI₂, MoS₂.

In Figs. 1 and 2 the most important structural types AX and AX₂ are compared (pp. 814 and 815).

As the classificatory principle for these different structural varieties of crystals, we choose not crystallographic symmetry, distributing them, for example, as cubic or tetragonal types, as has hitherto been accepted in crystallography; rather, we classify crystals according to the mode of coordination—according to the mode in which the atoms are linked with one another. This point of view, which is directly adjacent to Werner’s structural chemistry, has already been applied by many followers, especially by Ewald and Pfeiffer, to the doctrine of crystalline structures; as the classificatory principle, therefore, we choose the number and the geometrical arrangement of the neighbors around each ion of the crystal lattice.

Fig. 1.

In the case of compounds with formulas \(AX\) and \(AX_2\), the coordination numbers of the most important structures studied up to the present are as follows:

Compounds \(AX\). The coordination number of the atoms \(X\) around \(A\) must be the same as the coordination number of \(A\) around \(X\), although the geometrical arrangement of \(X\) around \(A\) may be different from the arrangement of \(A\) around \(X\).

Coordination numbers	Types of lattices
1	single molecules and molecular lattices of separate molecules;
2	double molecules, molecular chains, and also lattices made up of complexes of a similar kind;
3	boron nitride—graphite structures;
4	structures of wurtzite—zinc blende, diamond, tetragonal layered lattices with the same coordination number;
6	rock-salt type structure, nickel sulfide type structure;
8	cesium chloride type structure.

$CO_2$  $SiO_2$, cristobalite  $TiO_2$, rutile  $CaF_2$, fluorite

$Cu_2O$  $FeS_2$, pyrite  $CdI_2$  $MoS_2$

Fig. 2.

Compounds $AX_2$ and $A_2X$. For any mode of coordination in the family $AX_2$, there holds, of course, the law that each $A$-atom must be surrounded by a number of $X$ atoms twice as large as the number of $A$ atoms around each $X$-atom.

We obtain, therefore, for each mode of coordination two numbers, representing the number of neighboring structural elements in the first sphere of one and the other kind of atoms, and these numbers stand in the ratio $2:1$.

Coordination number.

Types of lattice

2 and 1: isolated molecules and molecular lattices of isolated molecules;

4 and 2: structural types of α- and β-quartz, β-tridymite, β-cristobalite and cuprite;

6 and 3: the anatase structure, the rutile structure, and also layered lattices of the cadmium iodide and molybdenite type;

8 and 4: the fluorite structure.

First of all, we are interested in the question: by what factors is the crystalline composition of a substance determined; why, for example, does magnesium fluoride possess a rutile-type structure, while strontium fluoride has the fluorite structure?

In order to establish what causes the occurrence of one structure or another, we must make clear to ourselves by what operations we can alter or transform a crystalline structure. The best means of causing changes in crystalline structure is chemical substitution. We must therefore seek the laws by which the influence of chemical substitution on crystalline structure is established—the laws of “morphotropy.”

Let us consider the series of fluorides of divalent metals: barium, strontium, calcium, and magnesium, in their natural sequence. Their structures and principal lattice dimensions are as follows:

	Structure	Lattice constants in Å	Lattice constants in Å	Distance of atoms A—X in Å	Coordination number
		a	c
BaF₂	fluorite	6.19	—	2.68	8 and 4
SrF₂	”	5.78	—	2.50	8 and 4
CaF₂	”	5.45	—	2.36	8 and 4
MgF₂	rutile	4.62	3.06	1.99	6 and 3

Here one is struck by the sudden change in the mode of coupling of the atoms between the fluorides of calcium and magnesium, whereas the change in the dimensions of the structure, or, more precisely, the distance of neighboring atoms from one another, changes continuously from substance to substance.

This change in the distance between neighboring atoms can be formally reduced to a difference in the size of the substituting and substituted atoms. In connection with this, with the aid of measured interatomic distances we can establish the following sequence of atomic sizes:

\[ \mathrm{Mg}<\mathrm{Ca}<\mathrm{Sr}<\mathrm{Ba}. \]

With the aid of other series we could establish, in the same way, the following sequences of atomic sizes:

\[ \mathrm{Li}<\mathrm{Na}<\mathrm{K}<\mathrm{Rb}<\mathrm{Cs} \quad \text{and} \quad \mathrm{O}<\mathrm{S}<\mathrm{Se}<\mathrm{Te}. \]

Investigating the distance between the same atoms in several different crystalline structures, we find, in most cases, a very close quantitative agreement. Thus we can estimate the Mg—F distance in the compound \(\mathrm{KMgF}_3\) at \(2.00\ \text{Å}\) according to Arkel, whereas this distance in \(\mathrm{MgF}_2\) is \(1.99\ \text{Å}\). Such constancy of distances between atoms leads to the idea that each atom, each structural element of the lattice, possesses a practically impenetrable sphere of action, and that the distances of atoms in crystals are precisely the sums of the radii of their “spheres of action.”

III. TABLE OF ATOMIC RADII.

Determination of the arrangement of atoms in crystals by means of X-ray methods gives us numerical values of interatomic distances, and each such distance may be regarded as the sum of the radii of the corresponding atoms. Can we, however, calculate from this the radius of each atom separately? The first attempt to derive the radii of individual atoms from the distances of atoms in crystalline structures, under the assumption of compact “packings” of spherical atoms, is due to W. L. Bragg. In this investigation, structures were used in which there was contact of identical—

...atoms, and the distance between the latter was simply divided in half. In this way the radius of the atoms of the kind under consideration was also determined, namely the radius of the carbon atom from the atomic distances in diamond and the radius of the sulfur atom from the distance of the two sulfur atoms in pyrite. With the aid of the radii of several individual atoms thus established, the radii of atoms of other elements were determined from the atomic distances in the corresponding crystals, for example, the radius of the zinc atom from the Zn—S distance in zinc blende, etc. Such a method of calculation is based on the assumption that in all crystals used for the calculation the atomic distances are composed additively of atomic radii, that all of them, as I call it, must be commensurable. This condition, however, was not satisfied in Bragg’s calculation, as was shown at the time by Grimm. A table of radii constructed by consistently carrying out the principle of commensurability was proposed by me in 1926. The very extensive compilations of atomic distances required for this were obtained, in very large part, by measurements in my institute.

The starting point of this “table of radii,” since it includes ionic radii, was the determinations of the atomic radii of singly negative fluorine (1.33 Å) and doubly negative oxygen (1.32 Å), carried out by Wasastjerna as early as 1923 on the basis of optical data.

IV. Variability of Atomic Sizes

The importance that ionic radii have in structures makes it desirable to determine as accurately as possible the radii of the individual structural units of a crystal. This task confronts us with the question: with what accuracy, in general, is the constancy of ionic radii maintained? If one confines oneself only to such groups of crystals in which the ions are in a comparable state, to “commensurable” crystals, as I have called them, then it turns out that, although

STRUCTURE OF CRYSTALS AND CHEMICAL COMPOSITION

As a first approximation the radii of ions can be considered constant; nevertheless, there are deviations from their additivity, and the deviations are strictly regular. This variability of ionic distances, established by me experimentally on a whole series of crystalline series, is partly conditioned by the environment of the given ion, i.e., by the number and arrangement of neighboring ions, and partly depends on the individuality of the latter. According to my investigations, the influence of the coordination number and coordination arrangement for compounds \(AX\) and \(AX_2\) is as follows:

Transition	Transition	Change in coordination number	Decrease in distance
from type	to type
CsCl	NaCl	\(8 \longrightarrow 6\)	3%
NaCl	ZnS	\(6 \longrightarrow 4\)	5—7%
CaF\(_2\)	Rutile	\(8\) and \(4 \longrightarrow 6\) and \(3\)	3%

These figures show, first, that the fluctuations in the magnitudes of ionic radii are small if they are compared with the radii themselves; secondly, that the distances of ions decrease regularly with a lowering of the coordination number. The more neighbors a given ion has, the farther removed it is from these latter.

This empirically discovered variability of the distances between ions has, as Pauling has shown, special significance in the calculation of the electrostatic energies of lattices. It is clear that the forces of electrostatic attraction increase rapidly with an increase in the coordination number, so that in nature structures with large coordination numbers would always have to be preferred if the mutual separations of the ions did not also depend on the structural geometry. The decrease in the distance between ions with a lowering of the coordination number creates a precondition for the stability of structures with smaller coordination numbers; this is also aided by the fact that in structures with anionic contacts, but without contacts between anions and cations, the distances between atoms are, of course, greater than in the structure with the nearest smaller coordination number.

V. M. GOLDSCHMIDT

In the critical assessment of the empirical material that served for the compilation of my tables, special attention was paid to ensuring that the radii of ions corresponded, as far as possible, to comparable states; as such a state, the state of ions in a lattice of the rock-salt type was chosen, while avoiding such combinations of ions in which polarization phenomena were especially strongly manifested. L. Pauling (L. Pauling)1 has recently published a table of ionic radii, which he calculated theoretically on the basis of Schrödinger wave mechanics, likewise taking as his basis an ionic lattice of the rock-salt type and avoiding cases of stronger polarization effects than in the alkali halides. It is noteworthy that almost all the theoretically calculated ionic radii of Pauling agree satisfactorily with the radii that I derived a year earlier, based directly on my collection of interatomic distances.

In the table (p. 821) I compare my empirical ionic radii and the corresponding theoretical radii of Pauling, in order to show the degree of agreement between the two. Significant discrepancies occur only in the case of singly negative hydrogen, and also for divalent and tetravalent anions, which, as Pauling also notes, cannot be directly compared with empirical quantities. The radii of a large number of cations, calculated by Pauling directly from empirical data, naturally agree fully with the empirical radii of my tables (see the table on p. 821).

Both my radii and those calculated by Pauling are radii of the “spheres of action” of ions in crystals. Quite different are the ionic radii that Grimm calculated for a large number of ions, having investigated in a series of works both their ratios and their connections with many chemical and physical properties of ions. Grimm’s radii were obtained

1 L. Pauling. The Sizes of Ions and the Structure of Ionic Krystal Journ. Amer. chem. Soc. 49, 765 (March, 1927).

Comparison of empirical ionic radii according to Goldschmidt (1916) and theoretical ionic radii according to Pauling (1927).

	1− H	0 He	1+ Li	2+ Be	3+ B	4+ C maximum	5+ N maximum
Goldschmidt empir.	1,27		0,78	0,31		0,2	0,1—0,2
Pauling theor.	2,08	1,22	0,60	0,31	0,20	0,15	0,11

	2− O	1− F	0 Ne	1+ Na	2+ Mg	3+ Al	4+ Si	5+ P	6+ S
G. emp.	1,32	1,33		0,98	0,78	0,57	0,39	0,3—0,4	0,34
P. theor.	1,40	1,36	1,52	0,95	0,65	0,50	0,41	0,34	0,29

	2− S	1− Cl	0 Ar	1+ K	2+ Ca	3+ Sc	4+ Ti	5+ V	6+ Cr	1+ Cu	2+ Zn	3+ Ga	4+ Ge	6+ Se
G. emp.	1,74	1,81		1,33	1,06	0,83	0,64	0,4	0,3—0,4		0,83	0,62	0,44	0,3—0,4
P. theor.	1,84	1,81	1,92	1,33	0,99	0,81	0,68	0,59	0,52	0,96	0,74	0,62	0,53	0,42

	2− Se	1− Br	0 Kr	1+ Rb	2+ Sr	3+ J	4+ Zr	5+ Nb		1+ Ag	2+ Cd	3+ In	4+ Sn
G. emp.	1,91	1,96		1,49	1,27	1,06	0,87	0,69		1,13	1,03	0,92	0,74
P. theor.	1,98	1,95	2,1	1,48	1,13	0,09	0,80	0,70		1,26	0,97	0,81	0,71

	2− Te	1− J	0 X	1+ Cs	2+ Ba	3+ La	4+ Ce			1+ Au	2+ Hg	3+ Tl	4+ Pb
G. emp.	2,11	2,20		1,65	1,43	1,22	1,02				1,12	1,05	0,84
P. theor.	2,21	2,16	2,3	1,69	1,35	1,15	1,01			1,37	1,10	0,95	0,84

	1+ NH₄	1+ Tl	2+ Mn	2+ Fe	2+ Co	2+ Ni	2+ Pb	3+ Cr	3+ Fe	3+ Rh
G. emp.	1,43	1,49	0,91	0,83	0,82	0,78	1,32	0,65	0,67	0,69
P. theor.		1,44	0,80	0,75	0,72	0,69	1,21

	3+ La	3+ Ce	3+ Pr	3+ Nd	3+ Sm	3+ Eu	3+ Gd	3+ Tb	3+ Dy	3+ Ho	3+ Er	3+ Tu	3+ Yb	3+ Cp
G. emp.	1,22	1,18	1,16	1,15	1,13	1,13	1,11	1,09	1,07	1,05	1,04	1,04	1,00	0,99

	4+ V	4+ Mn	4+ Nb	4+ Mo	4+ W	4+ U	4+ Rn	4+ Os	4+ Jr	4+ Te	4+ Pr	4+ Tb	4+ Th
G. emp.	0,61	0,52	0,69	0,68	0,68	1,05	0,65	0,67	0,66	0,89	1,00	0,89	1,10
P. theor.	0,59	0,50	0,67	0,66	0,66	0,97	0,63	0,65	0,64	0,81	0,92		1,02

by means of the method of calculation proposed by Fajans and Herzfeld, in which a static arrangement of the eight outer electrons in ions constructed analogously to the atom of a noble gas is assumed. This calculation is no longer justified by modern conceptions of the structure of atoms. The radii of cations calculated by this method, nevertheless, in almost all cases have almost the same trend as the values we use. Grimm’s investigations of the relations between the properties and dimensions of ions, as well as his highly valuable investigations of isomorphism, therefore retain all their inherent significance; but for the study of morphotropy it is expedient to use only the values of atomic dimensions given by me.

V. Morphotropic series connected with the geometrical properties of ions.

We shall use the atomic dimensions established by me first of all for considering morphotropic transformations in the group of compounds of the formula \(AX_2\). Let us arrange the difluorides and dioxides studied up to now according to the magnitude of the ratio of the ion radii \(R_A : R_x\), as I did originally in the VI communication on the geochemical laws of distribution at the beginning of 1926 (see p. 817).

It turns out that a sudden change of structural type is connected with a definite limiting value of the ratio of ionic radii: both in the difluorides and in the dioxides the change of structural type occurs when the indicated ratio reaches the numerical value \(\sim 0.7\).

In order to find out whence this quantity arises, let us turn to the geometrical laws of spatial architectures built from spherical bodies. For this purpose we take, as a model of a heteropolar crystal, a structure made of spheres of different radius, by which space must be filled so that each sphere of one kind is in contact with the greatest possible number of spheres of the other kind. Which spatial arrangement will most satisfy this condition depends only on

numerical ratio of the spheres of both kinds and on the ratio of their magnitudes. Considering the environment of any one sphere, we may disregard the numerical ratio of the different kinds of spheres in an infinitely extended crystal and consider the manner of combination of the spheres only as a function of the ratio of their sizes. This will be clarified by the following example from planimetry. Let two kinds of circles of different radius be arranged in such a way that one circle of radius \(B\) is in contact with the greatest possible number of circles of radius \(X\), but so that the circles do not overlap one another. As is easily seen from Fig. 3, the arrangement of three \(X\) around \(B\) is possible only under the condition that

\[ R_B : R_X > 0.15; \]

the arrangement of four \(X\) around \(B\) requires

\[ R_B : R_X > 0.22. \]

Fig. 3.

This method of theoretical analysis was already used quite long ago by A. Magnus1 to explain the formation of one or another complex ion or molecule depending on the ratio of the ionic radii, Magnus having derived all possible configurations of coordinating ions around a single central ion. For these configurations the following critical values of the ratio of ionic radii are established:

Number of coordinating ions \(X\)	Configuration of ions \(X\)	Critical \(R_A/R_X\)
2	opposite one another	0.000
3	at the vertices of an equilateral triangle	0.15
4	at the vertices of a tetrahedron	0.22
4	at the vertices of a square	0.41
6	at the vertices of an octahedron	0.41
8	at the vertices of a cube	0.73

Let us assume, as a preliminary hypothesis, that a necessary condition for the stability of a crystalline structure—

tures of heteropolar compounds is the contact of anions and cations with one another. Then the table also indicates to us the configurations of ions that may arise in the crystal depending on one or another ratio of the geometrical dimensions of its structural units; for example, the critical value of this ratio between the rutile structure and the fluorspar structure is 0.73. As I have already indicated, such an approach can

Influence of the ratio of the radii \(R_A : R_X\) on the occurrence of various structures.

	Rutile structure	Rutile structure	Rutile structure	Rutile structure	Rutile structure	Fluorite structure	Fluorite structure	Fluorite structure	Fluorite structure	Fluorite structure
	MgF\(_2\)	NiF\(_2\)	FeF\(_2\)	ZnF\(_2\)	MnF\(_2\)	CdF\(_2\)	CaF\(_2\)	HgF\(_2\)	PbF\(_2\)	BaF\(_2\)
\(R_A : R_X\)	0.59	0.59	0.62	0.62	0.68	0.77	0.80	0.84	0.99	1.08
	MnO\(_2\)	RuO\(_2\)	MoO\(_2\)	PbO\(_2\)	TeO\(_2\)	ZrO\(_2\)	PrO\(_2\)	CeO\(_2\)	UO\(_2\)	ThO\(_2\)
\(R_A : R_X\)	0.39	0.49	0.52	0.64	0.67	0.68	0.76	0.77	0.80	0.84

give, at best, only a very approximate representation of the conditions on which one or another structure of a crystal depends, and it is necessary first of all to consider in the case of which substances this representation corresponds most closely to reality. These are, obviously, those substances whose structural elements may, with sufficient approximation, be regarded as incompressible spheres. According to modern views of the structure of the atom, the conception of spherical formations is met to the greatest extent by ions built like the atom of a noble gas, perhaps with the sole exception of helium-type ions,\(^{1}\) and at the same time cations of this kind, especially those with large charges, satisfy the condition of “incompressibility” rather well. Among the anions of this same family, those that especially correspond to this extremely simplified picture are the smallest anions with small charges.

It is therefore clear that it is precisely on fluorides and oxides of metals, whose ions are constructed according to the type of the atom,

\(^{1}\) According to wave mechanics, helium-like ions may also be regarded as spherical.

of the noble gases, a regular quantitative dependence of the crystalline structure on the ratio of the radii was first discovered in my sixth communication on the geochemical laws of distribution, 1926 (see table on p. 824).

Consideration of the crystalline structures of heteropolar compounds as packings of incompressible spheres satisfying the condition of mutual contact clearly reveals the meaning of the concept of “critical ratios.” It is extremely interesting that some of the empirically found “critical ratios” can also be derived theoretically, proceeding from wave mechanics. The “critical ratio” between the rutile structure and the fluorite structure may serve as an example of the influence of the magnitude of the ions on crystallization in the family \(AX_2\). Let us consider one more example from the series of \(AX\) structures: the transition from structures of the zinc-blende—wurtzite type to the rock-salt structure, in the binary compounds of divalent cations and anions of groups II and VI. The crystalline structure and lattice dimensions of these compounds have been established by our investigations without any gaps.

The geometrical prerequisite of the rock-salt type structure is a radius ratio lying between the limits 0.41 and 2.41. Let us form the quantities \(R_A : R_X\).

	Mg	Ca	Sr	Ba
O	0.59	0.80	0.96	1.06
S	0.49	0.61	0.73	0.82
Se	0.41	0.56	0.66	0.75
Te	0.37	0.50	0.60	0.68

Magnesium telluride is here the only compound for which the ratio of the radii of the cation and anion lies outside the allowed limits, 0.41–2.41, and precisely this compound has the structure not of rock salt, but of wurtzite.

It is remarkable that in many crystals, in particular in lithium iodide,1 morphotropy still does not set in, despite

1 Here it should be mentioned that only from the X-ray data is the rock-salt structure for lithium iodide not yet definitely ...

to reach the critical value 0.41, between cubic and octahedral configurations. Whether this depends on the deviation of the ions from sphericity, or whether the ratio of ionic radii must pass somewhat through the critical value, still seems unclear to me. Pauling, with the aid of wave mechanics and electrostatics, states that the rock-salt structure in the case of lithium iodide still remains stable, although here the geometrical limiting ratio has been considerably exceeded. It seems to me extremely important that the structural lability of lithium iodide, if one may so express oneself, is also manifested, so to speak, in the unsaturated chemical character of this compound—in its exceptionally great tendency to form hydrates and ammoniates.

In lithium chloride and lithium bromide, in magnesium sulfide and selenide, the geometrical “limiting ratio” of the rock-salt structure is also exceeded, if the radii accepted by Pauling are adopted. In these compounds, consequently, contact of the anions must also occur. Cases of this kind are of special importance for establishing ionic radii, since from such structures there arises the possibility of a direct determination of the radius of the anions, as Pauling has shown for a whole series of crystals.

Below I compare these and certain other cases of ionic lattices, which demonstrate the great closeness between my data for ionic sizes and Pauling’s theoretical figures (see the table on p. 827).

Wasastjerna and the two Braggs have shown, for a number of other oxygen compounds, that the value adopted by Wasastjerna for oxygen is confirmed if contact of the anions is allowed.

I have shown empirically that the regular influence of the ratio of radii on the type of crystal struc-

it follows that only the arrangement of iodine in a face-centered lattice has been proved. The influence of the lithium ions on the interference phenomenon is so negligible that their position in the lattice has not yet been established. It is possible that here there is a molecular lattice.

ture is valid not only for monatomic ions, but also for ionic formations of the complex type, such as NH₄, alkyl-substituted ammoniums, hydrates, and ammoniates of metallic ions.

Radii of anions,
calculated under the (hypothetical) assumption of anionic contacts.

Anion	Determined from	R	according to Goldschmidt	according to Pauling
F	MgF₂—MnF₂	1.28—1.33	1.33	1.36
Cl	LiCl	1.81	1.81	1.81
Cl	SrCl₂	1.74
Br	LiBr	1.94	1.96	1.95
Br	TlBr	1.99
J	LiJ	2.14	2.20	2.16
J	AgJ	2.16
J	TlJ	2.09
O	TiO₂	1.28	1.32	1.40
O	SiO₂	1.26—1.28
O	Al₂O₃	1.35
O	MgAl₂O₄	1.45
S	MgS	1.83	1.74	1.84
S	MnS	1.83
Se	MgSe	1.93	1.91	1.98
Se	MnSe	1.93
Te	MgTe	2.26	2.11	2.21
Te	CaTe	2.24
Te	SnTe	2.22
Te	PbTe	2.28

Whereas even the barium ion is not voluminous enough to satisfy, with the iodine ion, the condition of the fluorite structure, we can nevertheless construct a diiodide with the fluorite structure if we are able, so to speak, artificially to build a sufficiently large divalent cation; such is, for example, the divalent hexaammine nickel ion with a radius of 2.57 Å. With regard to structures of the fluorite type from such complex ions, Pauling also later confirmed my empirical conclusions by theoretical calculations.

The electrostatic energies of lattices of the fluorite type and of the rutile type, reduced to the same ion separation, are in the ratio 5.04 to 4.82, i.e. as 1.05. This means: in order to motivate, from the energy point of view, the morphotropy from the fluorite type to the rutile type, a decrease in the ion separation by \(\sim 5\%\) is required. The decrease in the ion separation associated with the change in coordination amounts in this transition, according to experiment, to 3%; the remaining 2% must be obtained as a result of passing through the limiting value of the geometrical ratio of the given radii. Accordingly, morphotropy should occur not exactly at the geometrically limiting ratio of the radii 0.73, but approximately at 0.70, which is in good agreement with my empirical limiting value.

An analysis of this kind, recently carried out by Pauling on many examples, explains why there is no exact coincidence between my empirical values of the limiting ratio and the geometrical limiting values for constructions from spheres; but always, as a general rule, the former are somewhat larger than the latter, if the ratio of the smaller radius to the larger is taken.

For coordination lattices we may state the following rule, which is, however, self-evident in itself but can nevertheless be used in calculations: in order for morphotropy to occur (a change of crystal structure), the accompanying change in ion separation must be at least sufficient to compensate the difference in the electrostatic energies of the lattices calculated for a constant distance between the ions.

Since the lattice energy in first approximation is inversely proportional to the ion separation (the distance between oppositely charged ions), a very simple method is obtained for estimating, from experimental data on ion separations, the possibility of morphotropy.

If we systematically study the influence of ionic sizes on crystal structure, sta-

comparatively limiting ourselves here to such ions as correspond as closely as possible to the ideal image of incompressible electrically charged spheres, we can, as I have shown, embrace all the relations in whole series of structures from one single point of view and then compare these series with model constructions made of electrically charged spheres, in order to satisfy our legitimate need for not merely a formal, but a deeper understanding of the established facts. But this method of analysis is limited only to those crystals whose structural elements correspond sufficiently closely to the model representation of incompressible spheres. In considering crystals for which this is not the case, the ratio of radii loses its exceptional significance for determining the structural type of the crystal lattice.

VI. Morphotropic series connected with the polarization properties of ions.

The deviations of ions in their properties from the representation of incompressible spheres can be explained by means of the concept of the polarization of ions, i.e. by the notion of a displacement of the positive ionic nucleus and the negative electron shell relative to one another under the influence of an external electrostatic field. Like an ion, a neutral atom too can be polarized.

Whereas in the case of weakly polarizing and weakly polarizable structural elements of a crystal their mutual arrangement tends to become as symmetrical as possible, each ion being surrounded by as large a number as possible of ions of the other kind at as nearly equal distances as possible, the phenomena of polarization lead to a lowering of the coordination number, to unequal distances between neighboring ions, and thereby—in many cases—to a lower symmetry of the crystalline formation as a whole. The limiting case is the formation of an isolated molecule, for example, from two ions A and X, in which the coordination number reaches its lowest value—unity—and the distance between the ions

decreases to a much smaller value than in the case of the above-described crystalline formations.

There have been a number of attempts to predict the intensity of polarization effects on the basis of atomic properties. In particular, Fajans, in many important works, proved the existence of a connection between polarizability or deformability and refraction; Born and Heisenberg calculated polarization coefficients from spectral data. For us, the clearest and most direct path is the establishment of the polarization characteristics of ions on the basis of the properties of crystals, just as earlier we established, from the dimensions of the lattice, the magnitudes of its structural elements.

By the polarization properties of ions I mean the totality of the deformations which they undergo under the action of electrical forces. The simplest case of a polarization effect is the formation of a dipole under the influence of an external field. Typically, under these conditions a negatively monovalent iodine ion is polarized. Owing to this, in the case of cadmium iodide we find not the structure of the rutile type, which would have been expected according to the laws of sphere packing, and not the structure of one of the varieties of silica, but a structural type very strongly different from the lattices considered up to now. This is the special type of cadmium iodide. Its characteristic surroundings may be described as follows: although each cadmium ion is surrounded by six iodine ions in a highly symmetrical manner (a rhombohedral configuration), each iodine ion is in contact with three cadmium ions only on one side, i.e. is subjected to a one-sided polarizing action. The geometry of the crystalline formation as a whole is characterized in particular by the fact that a layered structure of the following kind predominates in it: the crystal is composed of mutually parallel packings of ions, each consisting of three layers; these triple layers enclose in the middle a layer of weakly polarizing ions, to which, on both sides, layers of strongly polarizable ions adjoin—in the present case, iodine ions.

iodine. Structures of this kind were first noted in a very important work by F. Hund,¹ as a particularly remarkable crystalline structure; he gave them the name “layer lattices” (Schichtengitter).

Along with the structure of cadmium iodide, a whole series of other varieties of “layer lattices” is now known. Each combination of three layers in them becomes, as it were, an independent physical individual. These three-layer formations are connected with one another by forces of the second order, in contrast to the all-round connection of all structural units in a typical coordination lattice. A consequence of this peculiarity of the layer lattice is excellent cleavage along the ionic layers; each three-layer formation, being electrically neutral as a whole, may to a certain extent be regarded as one gigantic molecule, which may have an arbitrarily large extent in the direction of two of its dimensions.

If the polarization effects exceed the magnitude characteristic of layer lattices, separate molecules arise in the crystal, corresponding either to the chemical gross formula or to some multiple of it: we arrive at molecular lattices. If the bond between the structural elements of the latter, for example by means of sublimation, is broken, we arrive at the lowest stage of coordination—at the most isolated molecule.

Thus, morphotropic series arising as a result of increasing polarization effects, beginning with typical coordination structures, lead through layer lattices to molecular lattices.

The polarization in the case of cadmium iodide consists in the induction of dipoles under the influence of one-sided electric actions and in the reverse action of these resulting dipoles on the force field. Such polarization may arise not only in ions, as in the polarization

¹ F. Hund. Z. Physik 34, 833, 1925.

when in cadmium iodide, but also in neutral atoms and molecules devoid of their own dipole moment, such as the molecule \(J_2\); but in particular such molecules, radicals, or ions which possess a dipole moment in themselves, without the action of an external field, such as \(H_2O\), \(CN\), \(OH\), may also function as dipoles. In cases of the latter kind, the action of a unilateral field consists above all in orienting the dipole, and then in increasing its dipole moment.

Let us consider a number of cases of lowering the symmetry of a crystal structure—a typical consequence of the occurrence of polarization effects.

Let us take compounds of the formula \(AX_2\) and again use the method of chemical substitution to alter the properties of the structural elements of the crystal in the desired direction. Suppose that initially we are given cadmium fluoride—a body with the fluorite structure. We replace fluorine by iodine and obtain, as has already been mentioned, instead of the rutile structure the layered lattice of cadmium iodide. We now ask ourselves: should the reason for so profound a change of structure be sought in the unequal magnitude of the iodine and fluorine ions, or in some other distinction between them? To decide this question, let us try to replace the halide by the radical \(OH\)—that is, let us study the structure of cadmium hydroxide. In size the hydroxide ion is considerably closer to the fluorine ion than to the iodine ion, as is evident from the following comparison: radius \(F'\)—\(1.33\,\text{Å}\); radius \(I'\)—\(2.20\,\text{Å}\); radius \(OH'\)—\(1.4\text{—}1.5\,\text{Å}\). But in contrast to fluorine, hydroxyl is a natural dipole. It thus turns out that the replacement of fluorine by hydroxyl likewise causes the transition of cadmium fluoride into a layered lattice of the cadmium-iodide type. Therefore the special kind of morphotropy \(CdF_2 \longrightarrow CdI_2\) is connected precisely with the strong polarizability of the iodine ion.

In our example, by means of an intensification of polarization effects, we obtain the transition from fluorite to the cadmium-iodide type. Starting from the rutile structure of cassiterite, or titanium dioxide, it is likewise possible

by means of morphotropy to pass to the structure of cadmium iodide, replacing the weakly polarizing oxygen by the strongly polarizing sulfur. With further replacement of sulfur by still more strongly polarizing anions, the cadmium iodide structure retains its stability over a considerable interval, as is shown by the series: TiS₂, TiSe₂, TiTe₂, all members of which possess the CdI₂ structure, as do the compounds ZrS₂ and ZrSe₂ investigated by Van Arkel.

Thus, alongside such morphotropic transitions, which depend on the ratio of the geometrical dimensions of the structural elements of the crystal, as, for example, the transition from the fluorite structure to the rutile structure, we also have transitions for which the decisive role is played not by the ratio of the dimensions of the structural units, but by the polarization properties of the atoms and of their combinations.

In the following schemes, for some of the most important types of substances, the actions of both factors determining the crystalline structure of a substance are compared. The structural types are “mounted” on the coordinate system: radius ratio—polarizability.

                         R_A : R_X

                         4.45   }  Fluorite
                                   structure  →  { Cadmium iodide and       →  Molecular
                                                   molybdenum sulfide           lattices
                                                   structure
                         0.73   ↓

                              }  Rutile structure →

                         0.41   ↓                 ↓
                              }  Structure of     Molecular
                                 varieties of     lattices
                                 SiO₂

                         0.22   ↓
                              }  CO₂ structure,
                                 molecular lattices

R_A : R_X increases upward ↑
Radius ratio increases upward ↑

Polarization increases to the right →

VII. THE FUNDAMENTAL LAW OF CRYSTALLOCHEMISTRY

Up to now, in the examples considered, we have studied separately the significance of the ratio of the radii of ions and their

polarization properties. We have managed, on the one hand, to find cases in which the question could concern almost exclusively the influence of the relative size of the atoms; on the other hand, examples in which polarization properties almost exclusively played the role. In the majority of morphotropic series encountered in practice, both factors participate simultaneously, and one must carefully weigh the action of both in order correctly to predict the structure of an unknown substance.

In principle, for every formula \(AX\), \(AX_2\), \(AX_3\), \(A_2X_3\), \(AXY\), \(ABX_3\), we can uncover the connection between crystalline structure and chemical composition by first studying, if possible, on suitable examples the significance of the sizes of atoms and of polarization effects separately, and then drawing conclusions about the result of the combined action of these factors. The possibility of carrying out such an investigation for every kind of stoichiometric formula by the same methods is based on the fact that the geometrically possible structures made up of heterogeneous atoms are determined in a very general way from the relative amounts of atoms of different kinds. Here we welcome the theory of space groups, for which we are indebted to Schoenflies and Fedorov, as an exceptionally valuable, indeed directly irreplaceable, tool in the analysis of structures. The new investigations of Weissenberg likewise lead to extremely important and general ideas concerning the geometry of point systems.

At present, in connection with my empirical investigations, a practically important problem arises for the theory of space groups, which is apparently in principle easy to solve: namely, to derive from the point systems the possible structures (Kugelpackungen) and to tabulate the latter, calculating the geometrically permissible radius ratios for each structure separately. In the cases examined up to now I found the limiting values of the radius ratios empirically and only afterward confirmed them by calculation; now it would be expedient, once and for all, to find these limiting values for all structural types, which would make possible the prediction of new, hitherto ...

unknown structural types. As an example, one may indicate that for compounds \(AX_2\) a structure is possible in which each atom \(A\) is surrounded by 12 atoms \(X\), and each atom \(X\) by six atoms \(A\). The atoms \(A\) occupy the nodes of a simple hexagonal lattice, while the atoms \(X\) occupy all the midpoints between each six atoms \(A\). This structure can arise, for a certain ratio of the sizes of the \(A\)- and \(X\)-atoms, instead of the fluorite structure, and it would be the structure with the greatest coordination number for compounds \(AX_2\).

In connection with these inductive conclusions about the factors by which the crystalline structure of a given substance is predetermined, we can formulate a general proposition combining the conclusions to which the study of heteropolar structures leads us: the structure of a crystal is predetermined by the ratio of the number of structural units, by the ratio of their size to their polarization properties. By structural units are meant here atoms (or ions) and atomic groups.

I have called this law the fundamental law of crystallochemistry; in its most general form it formulates our conception of those factors through which chemical composition predetermines crystalline structure.

I would like to note here that the formulation of this law does not include the atomic weight of the participating atoms. In older crystallography, a correlation was often sought between crystalline structure and the atomic weight of the kinds of atoms participating in them. But this was an error, since the weights of the structural units have no relation whatever to the crystalline structure; a crystal does not weigh its component parts, it merely arranges them according to their requirements in the cubature. One may refer, for example, to the close crystallographic similarity between monovalent thallium and rubidium, divalent lead and strontium, trivalent bismuth and cerium, or to the extraordinarily deep similarity, in a crystallochemical respect, between yttrium and holmium, zirconium and hafnium [cf. by

concerning these questions, the experimental investigations of Hevesy (G. v. Hevesy), as well as my own and those of my collaborators.

It was just as erroneous to seek a direct connection between the valence numbers of the structural units and the crystal structure; the sum of the valences determines the strength of the crystalline formation, but not its structure.

We can verify the fundamental law of crystal chemistry, established by us for the simplest crystalline formations, also in more complex structures of heteropolar compounds. A very instructive series of chemical compounds is, for example, the series of sesquioxides, $A_2O_4$, studied by me and my collaborators many years ago. The influence of the ratio of radii is manifested most vividly in the transition from the structural type of corundum to the structural type C of the sesquioxides—the lanthanides.

Very demonstrative examples of the relations I have revealed between crystal structure and chemical composition are found in compounds of the formula $ABX_3$, as, for example, in the transition from the aragonite type to the calcite type, which is prepared by a decrease of the structural element $A$, as is shown by the following series of compounds:

$\mathrm{LiNO_3}$		Calcite type
$\mathrm{NaNO_3}$	$\mathrm{MgCO_3}$	Calcite type
$\mathrm{KNO_3}$	$\mathrm{CaCO_3}$	Calcite type
$\mathrm{KNO_3}$	$\mathrm{CaCO_3}$	Aragonite type
	$\mathrm{SrCO_3}$	Aragonite type
	$\mathrm{BaCO_3}$	Aragonite type

The last example can likewise justify the existence of polymorphism from the point of view of the fundamental law. We can replace in the lattice of a given structural type one of its structural elements, by means of an isomorphous series, only up to a certain limit. Beyond this limit further isomorphous replacement in the lattice under consideration is no longer possible; a morphotropic transformation occurs, i.e. the last step of replacement leads to a rearrangement of the entire crystalline formation. Each series of isomorphous replacements thus has its own completion, its own

boundary beyond which morphotropy begins. If, however, we are at the boundary of an isomorphous series, in many cases a single change in the thermodynamic conditions—if only the temperature—is sufficient to bring about morphotropy, as is revealed with extraordinary clarity in the aragonite–Iceland spar family.

Regularities similar to those I have presented for inorganic compounds are also valid for organic crystal chemistry; but it would take us too far if we were to touch here upon this field, which deserves extremely detailed development.

The examples considered by us up to now of the connection between chemical structure and chemical composition have referred to the so-called ionic lattices, to the crystalline structures of heteropolar compounds, whose structure and properties closely correspond to the conception of ions as atoms, charged with one or several units of electric charge, occupying individual nodes of the crystal lattice.

VII. Morphotropic Series of Binary Compounds of Heavy Metals.

We must now consider crystalline formations of another kind, using the same inductive approach that proved so suitable in the case of ionic lattices. We proceed from the monoxides of metals from calcium to nickel and study the action of chemical substitution, replacing oxygen first by sulfur, then by selenium and tellurium, and finally by antimony. The study of the structures of these compounds reveals a morphotropic transformation between the structure of rock salt and the structure known under the name of the nickel-arsenide type. In the nickel-arsenide type the coordination number is 6—the same as in the case of rock salt—but the geometrical arrangement of the nickel atoms around each arsenic atom is different from the arrangement of the arsenic atoms around the nickel atoms.

In the following table the structures of these compounds are compared, since complete series of them are now known:

	Ca	Mn	Fe	Co	Ni
O	NaCl	NaCl	NaCl	NaCl	NaCl
S	NaCl	NaCl	NiAs	NiAs	NiAs
Se	NaCl	NaCl	NiAs	NiAs	NiAs
Te	NaCl	NiAs	NiAs	NiAs	NiAs
Sb	?	NiAs	NiAs	NiAs	NiAs

Structures of the rock-salt type in these series pass morphotropically into nickel-arsenide structures if their structural elements are replaced in the following order: \(O \to S \to Se \to Te\) or \(Ca \to Mn \to Fe\).

The interatomic distances in nickel-arsenide structures are considerably smaller than would follow from the sums of the normal ionic radii in rock-salt structures, as the following table shows:

	Ca	Mn	Fe	Co	Ni
O \(\Sigma R\)	2.38	2.23	2.15	2.14	2.10
O observed	2.40	2.22	2.14	2.13	2.09
S \(\Sigma R\)	2.80	2.65	2.57	2.56	2.52
S observed	2.84	2.59	2.45	2.33	2.38
Se \(\Sigma R\)	2.97	2.82	2.74	2.74	2.69
Se observed	2.98	2.73	2.55	2.47	2.38
Te \(\Sigma R\)	3.17	3.02	—	—	2.89
Te observed	3.17	2.91	—	—	2.65

Together with the change in structure and in ionic distances under morphotropy, there appears a change in the physical appearance of the crystals. Instead of transparent or translucent substances, metallic-looking substances are formed. The nickel-arsenide structures of this series clearly belong to an entirely different class of bodies than the structures of the rock-salt type. Whereas in structures of the rock-salt type an isomorphous replacement, for example of calcium by oxygen,

completely excluded as contradicting the very essence of the structure—we encounter extremely extensive isomorphous series of mixtures between a pure compound and its components. The classical example of this is magnetic pyrites (Magnetkies), an isomorphous mixture \( \mathrm{FeS}_{1+n} \), in which, according to the excellent investigations of Alsen, excess sulfur can replace part of the iron in the lattice atom for atom. According to Alsen, FeSe behaves in the same way; we find the same in CoSe, MnSb, FeSb, so that it would be better to write the formulas as \( \mathrm{Fe}_x\mathrm{S} \), etc.

These facts definitely speak in favor of the view that here the lattices are not of an ionic character, since in an ionic lattice it is in no way possible isomorphously to replace ions of one kind by ions with the opposite charge; we must therefore assume that here we are dealing with structural units of some other kind.

Let us examine the mechanism of morphotropic transformations in our new series of crystals, taking at least the manganese compounds. MnO is still a fairly normal ionic lattice. In MnS, contact of the anions is probably already attained; in MnSe this is noticeably clearer, and MnTe would have had to acquire a structure with a smaller coordination number than that of the rock-salt lattice, if the new crystalline formation were still built of ions, i.e. a lattice of the wurtzite or zinc-blende types. The necessary prerequisite for the occurrence of these types (see below), however, is not fulfilled in the case of manganese, and there arises a morphotropy of a fundamentally different kind, in which the ionic nature of the structural units is already being lost. Let us see in which particular bodies a structure of the nickel-arsenide type is observed. Alsen observed crystals of this type in binary compounds of iron, nickel, and cobalt; I have also obtained manganese and chromium compounds belonging to this type. What all these substances have in common is that their metallic constituent belongs to the elementary series scandium–nickel, the atomic (or ionic) arrangement of all the members of which is characterized by unfilled gaps (in other words, by a reduced charge density) in the \(M\)-level,

which, among other things, manifests itself in the magnetic properties of these bodies. Crystalline varieties of the nickel-arsenide type occur only in binary compounds with such anions as are relatively large and strongly polarized. It seems probable to me that there is a causal connection between the existence of this kind of crystalline structure and the presence of an \(M\)-defect. I would like to think that the polarization of the anion goes so far that the negative charges of the anion partially pass over to the cation and directly or indirectly diminish the \(M\)-defect.

If one is naturally to seek the condition for a nickel-arsenide structure in polarization phenomena of this kind, the same structural type might also have been expected for analogous compounds of the palladium group and of the platinum group. Up to the present, however, this has not been found. The idea arose to synthesize such a structure by combining platinum with an element whose atom is sufficiently large and polarizable for the desired effect to be expected. Tin seemed suitable to me for this purpose; tin combines with platinum, with a large evolution of heat, in PtSn, and this body, metallic in appearance, possesses—as Oftedal and I found on my preparation—indeed a nickel-arsenide structure. Analogous observations have been made by Thomassen on PdSb, PdTe and by Preston and Owen on AuSb. Thus, the condition for the nickel-arsenide type from the point of view developed by me may be the presence of an \(M\)-defect, or of an analogous defect in the \(N\)- or \(O\)-level, and, in addition, strong polarizability and a sufficient size of the electronegative partner. The individuality of the individual atom in such an arrangement would have disappeared to a certain degree, and it seems to me very probable that precisely this circumstance accounts for the appearance of metallic properties.

Extremely interesting are the ferromagnetic properties of crystals of this type; it should be noted that \(Mn_xSb_y\) is strongly ferromagnetic, as is the \(Cr_xTe_y\) obtained by me. A highly important task of atomic physics is the study and magnetic measurements on these substances.

The case of nickel-arsenide structures shows an example

that, for the occurrence of one or another structural type, polarizational properties of a certain special kind are necessary; exactly the same we shall now find for certain other structures. These are structures of the type fluorite—zinc blende—diamond.

In order that, instead of the rock-salt structure, in the case of the compound \(AX\) there should result the fluorite structure or the zinc-blende structure, a number of conditions must be fulfilled. The ratio of the radii must lie between 0.22 and 4.45; the distance between the atoms must, in the fluorite—zinc-blende structures, be at least 6% smaller than in a normal ionic lattice with the rock-salt structure; and, finally, as experience shows, the fulfillment of yet a third condition is required, one which concerns the positions of the components \(A\) and \(X\) in the periodic system. This condition, originally formulated by M. Huggins and A. Sommerfeld, requires that the element \(A\) should stand in the periodic system just as many (up to three) places before one of the elements: C, Si, Ge, Sn, Pb, as the element \(X\) stands after any of these elements; in other words, the number of outer electrons of the two partners \(A\) and \(X\) together must be the same as that in a pair of atoms of any of the named tetravalent elements C, Si, etc. In the case of the morphotropic transformation MgSe (rock-salt structure) into MgTe (fluorite structure) we saw that this transition had quite obviously been prepared by the change in electrostatic energies, upon passage through the limiting value of the radius ratio, and the distances of the radii and the distances of the atoms in MgTe are in extraordinary agreement with the conception that in MgTe we have an ionic lattice.

The matter is quite different with a large number of other structures of the fluorite—zinc-blende type, such as ZnO, CdS, CdSe, CdTe, etc. True, here also the first requirement is fulfilled—the possibility of contact between the \(A\)- and \(X\)-atoms—as well as the third condition, concerning the number of electrons. But as for the second requirement, the decrease in the distance between the atoms, although experience shows that it is fulfilled, we cannot, while remaining within the framework of ordinary ionic

lattices, to give a model justification for the shortening of distances, since the ratios of the radii of the corresponding ionic lattices must certainly lie within the limits of stability of the rock-salt structure. We must therefore accept that here too the morphotropic transformation is conditioned by polarization properties of a special kind, as was recently stated by Pauling.

There are very strong arguments for the view that in these cases not ordinary ionic lattices arise. In this connection let us consider part of my empirical material, set out in the following table:

Atomic numbers	Formula	Lattice constant	Atomic distance
50,50	SnSn	6,46	2,79
49,51	InSb	6,452	2,793
48,52	CdTe	6,463	2,799
47,53	AgI	6,491	2,811

Let the lattice of gray tin be the starting point; this is a diamond-type structure in which each atom is tetrahedrally surrounded by four identical neighbors. The atomic number of tin is 50. We replace half of the tin atoms by atoms of indium, an element with atomic number 49, and the other half by atoms of antimony, whose atomic number is 51; in doing so the sum of the atomic numbers, as well as the total number of electrons, remains the same. Under such a replacement the crystalline structure remains unchanged, and moreover not only in the sense of its construction, but also with respect to its dimensions. Replacing indium again by its neighbor—cadmium, and correspondingly antimony by tellurium, we once again obtain the same crystalline structure with almost the same dimensions; and once more we can repeat the same replacement with the same consequences, forming silver iodide.

It must be admitted that in these series of compounds the mode of bonding differs from that in ionic lattices and is very closely akin to the mode of bonding in diamond-like elements; hence the conclusion should be drawn that here too the individuality of the separate structural units recedes into the background before the whole construction of the crystal as a whole,

and the dimensions of the latter are determined almost exclusively by the total number of negative charges, and not by the manner in which the positive charges are distributed among the individual atomic nuclei.

In crystals of the fluorite–zinc-blende–diamond types, the sum of the radii of both joined atoms is, apparently, a considerably more important constant than the individual atomic radius. It would be more correct, instead of the “radius” of zinc atoms in ZnS, to speak of zinc’s “share” (Beitrag). In this sense the “share” of C, S, and P in these lattices is numerically one and the same and is equal to half the distance between silicon atoms; the “share” of Cu, Zn, Ga, As, Se, Br is likewise exactly equal and, moreover, coincides with half the distance between germanium atoms.

This kind of mode of consideration is also justified for the distances between atoms in intermetallic compounds.

Earlier, there was a strong inclination to oppose the fluorite–zinc-blende–diamond group to typical ionic lattices as an “atomic” lattice. I specifically subjected this question to investigation in the sixth paper on the laws of geochemical distribution and came to the conclusion that the difference in the state of both kinds of structural units of a crystal does not fully correspond to the difference between atoms and ions. Proceeding from the present state of our knowledge on this subject, I think that the difference between the group of ionic lattices and the zinc-blende–fluorite–diamond group consists above all in the fact that in ionic lattices each individual ion is characterized by greater independence than in typical lattices of zinc blende–fluorite–diamond. In this same sense W. H. Bragg (U. H. Bragg) and W. L. Bragg (W. L. Bragg) speak of the diamond lattice as a molecular one.

VIII. Morphotropic Series of Intermetallic Compounds.

We must now consider the last group of crystalline structures—metallic lattices, i.e., crystals of metallic compounds.

Crystallochemistry of metals is a field of the greatest scientific and technical importance. For metals comprise much more than half of all free elements and a large number of intermetallic compounds, or, more generally speaking, intermetallic “phases.”

The most widespread typical metals, in the free state, belong to three very simple structural types, which are compared in Fig. 4.

These structural types are: a) a regular body-centered cubic lattice, in which each atom has 8 neighbors—examples: Na, Cr; b) a regular face-centered cubic lattice, in which each atom has 12 neighbors—examples: Cu, Al; c) the densest hexagonal lattice, in which each atom has 12 equidistant neighbors—examples: Mg, Os. Finally, one should mention a variant of the third type—the zinc type, which may be represented as a lattice of the third kind stretched along its length; in this case the distances between atoms cease to be equal to one another.

Fig. 4.

In investigating the relations between crystalline form and chemical formula in the case of intermetallic compounds, just as we did with heteropolar compounds, we must take into account that, in the case of intermetallic crystals, what occur are, generally speaking, not strictly definite stoichiometric compounds, but phases of variable composition, mixed crystals, solid solutions, and so forth. In addition, here we must use with great caution our conclusions from the analysis of ionic lattices. Thus the metals magnesium, cobalt, nickel, and zinc are very similar to one another in ionic lattices; in metallic crystals, however, a certain similarity exists only between cobalt and nickel.

Let us apply the inductive method in the present case as well. We shall again investigate—by what paths we

we can alter the crystalline structure of metallic substances, so as thereby to reveal the causal connection between composition and structure.

As the starting point of our considerations we choose the type of cubic lattice with centered faces, for example that of silver, and investigate in what way we can pass from this type to another structure. The resolution of this question has been advanced far by Westgren and Phragmén in Stockholm. In studying the copper–zinc system, these investigators found that an admixture of zinc up to 36 atomic percent enters into the composition of copper crystals as a solid solution. With larger amounts of zinc there arises a lattice of centered cubes, persisting in the interval from 45 to 48 atomic percent zinc, in which, as in the given structure of cesium chloride, the copper atoms preferentially occupy the nodes of one simple cubic lattice, and the zinc atoms the nodes of the lattice centering the first. Between 62 and 68 atomic percent zinc there is formed a highly peculiar phase, $\gamma$-brass, with a very complex structure of the cubic type, with 52 atoms in the elementary cube. Further, between 79 and 85 atomic percent zinc there again arises a simply constructed phase, strictly corresponding to the closest-packed hexagonal lattices of metals (a lattice similar to that of magnesium), and, finally, beginning with 98 atomic percent zinc, a solid solution of copper in zinc crystals is obtained, another variant of the same hexagonal type.

An entirely similar picture was uncovered by Westgren and Phragmén in the silver–zinc and gold–zinc systems, but the limits of existence of the separate phases were delimited somewhat differently, in accordance with the somewhat different mutual solubility of the components. The analogy of the crystalline types in the three metallic systems Cu-Zn, Ag-Zn, Au-Zn, according to modern crystallographic concepts, can be explained as simple isomorphism. But Westgren and Phragmén found, in addition—and this is the most important point—that quite analogously constructed, or at least very similar, phases also arise in the systems Cu-Al,

Cu-Sn, Ag-Al, and Ag-Sn, but with clearly regular shifts in atomic-percent composition. Thus, the phase corresponding to the densest hexagonal lattice arises in the Ag-Zn system within the limits of 71–85 atomic percent zinc, in the Ag-Al system at 28–45 atomic percent aluminum, and in the Ag-Sn system at 11–23 atomic percent tin.

Such a circumstance could not be understood if the stoichiometric ratios and the concept of isomorphism were formulated by us in the same way as in the case of heteropolar compounds. Here, as Westgren and Phragmén have shown, the crystallographic analogy is determined not by the ratio of the numbers of like atoms, but by the ratio between the number of atoms and the number of valence electrons. The higher the valence of the metal alloyed with silver, the smaller an amount of its admixture is sufficient to obtain one and the same crystalline variety.

Somewhat earlier Hume-Rothery pointed out that the great similarity of the three phases: CuZn, Cu$_3$Al, and Cu$_5$Sn could be connected with the circumstance that the numerical ratio between valence electrons and atoms in all three cases exactly corresponds to the ratio 3:2.

These observations appear extremely important for understanding morphotropy in metals and, in general, for the theoretical interpretation of metallic crystals.

The transition of one structure into another is achieved here by changing the quantitative ratio between atoms and valence electrons. The extraordinarily precise and careful investigations of Westgren and Phragmén definitely prove that the region of existence of individual phases is not limited by a definite stoichiometric ratio of ponderable atoms, and structures arise not exactly at rational ratios of valences to atoms, but somewhat earlier or later, as though it were a matter of establishing certain states of equilibrium between atoms and especially weakly bound electrons, while the very kind of atoms is of no fundamental significance. Thus a certain, often considerable, freedom of fluctuation is possible

of composition without disturbance of the structure, as in crystals of the nickel-arsenide type.

These observations lead, as noted, to the view that the total concentration of valence electrons required for the occurrence of definite lattices in metals is a most important structure-forming factor. The presence of whole extended regions of homogeneous crystalline phases is a fact of great crystallographic interest. If all this is so, then by introducing foreign atoms one can obtain any of the crystalline phases considered above from gold, copper, or silver, provided only that some of the monovalent atoms of these metals are replaced by atoms with a large reserve of valence electrons. This prompted me to multiply the examples known hitherto by several new ones. Thus I readily obtained the centered lattice of γ-brass with such metallic combinations as silver and cadmium, gold and cadmium, with an atomic ratio of 1:1, as in brass. To the hexagonal structures hitherto known in the systems Cu-Zn, Ag-Zn, Au-Zn, Ag-Cd, Ag-Al, Au-Al, Cu-Sn, and Ag-Sn I was able to add the following: \(\sim\mathrm{AgCd}_3-\mathrm{AgCd}_4\), \(\mathrm{Ag}_3\mathrm{Sb}\), \(\mathrm{Cu}_3\mathrm{Sb}\). In order to test whether a typical trivalent metal of a side series, such as indium, also has a tendency to produce a morphotropic transformation of silver into the hexagonal type of crystals, I obtained \(\mathrm{Ag}_3\mathrm{In}\), and the expectation was justified. From the typical metallic-crystalline variety \(\mathrm{Cu}_3\mathrm{Sb}\), the compounds \(\mathrm{Cu}_3\mathrm{As}\) and \(\mathrm{Cu}_3\mathrm{P}\) lead to the boundary of ordinary heteropolar compounds.

An especially interesting structural type is represented by the so-called γ-brass, a crystalline variety which is interposed between the centered cubic and the closest-packed hexagonal lattice. Its elementary cube contains 52 atoms, the arrangement of which has recently been studied in detail by Bradley and Thewlis. The setting here is extremely peculiar: each atom is surrounded by 11, 12, or 13 neighbors at approximately equal distances, and these neighbors are preferably atoms of the other kind. Such crystals and their modifications

became known thanks to the work of Westgren and Phragmén also in the systems Cu-Zn, Ag-Zn, Au-Zn, as well as Cu-Al and Cu-Sn. Guided by considerations concerning electron concentrations, I succeeded in proving the presence of the γ-type in the series of amalgams Cu-Hg, Ag-Hg, and also in the Cu-Cd system. The peculiar, sometimes odd, coordination numbers in these varieties of crystals recall the structure of many silicides; similar entourages probably also occur in the α-variety of metallic manganese.

The crystalline varieties of brass are the prototype of very many intermetallic morphotropic series; so far, larger or smaller fragments of such series of about 14 elements are known to us. But other morphotropic series of successive substitution also present extremely curious problems, such as, for example, the origin, studied by myself and Barth, of regular centered crystal varieties, or varieties with centered faces, in the thallium-bismuth system.

It should be noted that in this field a number of extremely important problems are connected with the tendency toward a regular arrangement of components in metallic mixed crystals, predicted by Tammann and proved experimentally in particular by Johansson and Linde.

Comparing everything known to us about the connection between the structure and composition of metallic crystals, we may characterize our present conceptions as follows: first of all, the arrangement of atoms in the metallic varieties of crystals is determined by the ratio of the external parts of the electronic surroundings of the atoms, while the significance of the stoichiometric factor recedes into the background. In this connection the structure depends on the polarization properties of the atoms. The ratios of the dimensions of the structural elements of the crystal, on the contrary, have only a subordinate significance. There is manifested the tendency (but not the necessity) for each structural element of the crystal to be surrounded predominantly by structural elements of another kind; this tendency is often satisfied only upon tempering of the crystal.

The structure of metallic phases can be compared with the structure of such chemical radicals (like ammonium), in which numerous positive nuclei are enclosed in a common electronegative shell. How is the possibility of such constructions to be explained? In the metallic state the structural elements must possess considerably less independence than in ordinary ionic lattices, and even less than in crystals of the nickel-arsenide type or in types of wurtzite–zinc blende. The outer electronegative regions of each atom, to which, according to generally accepted notions, the so-called valence electrons belong, in the metallic state apparently do not belong to each atom separately, but form a common framework enclosing the entire crystal, possibly even sometimes extending beyond the boundaries of the crystalline aggregates; it is also retained in molten metals. This framework of negative atomic shells, in my opinion, is what determines the essence of the metallic state, in particular the property of electrical conductivity; in its structure it probably follows the same quantum laws as an isolated atom. I should therefore like to call a metal a “polynuclear pseudo-atom”; into a structure of negative charges, relatively mobile (as indicated by substitution reactions, diffusion processes in metals, and the easy weldability of the latter), positive atomic nuclei or atomic groups are set, in particular also such positive nuclei as hydrogen, positively tetravalent carbon, and positively trivalent boron.

The number of negative charges by which this framework is created is in many cases quite obviously connected with the number of chemical valences characteristic of the given kinds of atoms, but in no case is it identical with it. This follows quite definitely from the behavior of various alloys. The solubility of hydrogen in solid metallic crystals is in agreement with such a conception. The electron emission of incandescent metals, from this point of view, would then correspond to ioni-

tion; a piece of metal, positively charged as a result of the loss of electrons, would become like a microscopic cation.

Swihne’s views on the electronic isomerism and passivity of metals, as well as investigations in the field of allotropy of metallic systems, would agree very well with such a view of metallic states.

What, then, from our point of view, occurs when a metal melts? We must regard even the molten metal as a multinuclear pseudo-atom; only upon evaporation is the pseudo-atom destroyed, and the individual atoms recover their full independence.

Thus we have analyzed the following structural types of crystals: the structures of typically heteropolar (dualistic) substances, in particular ionic lattices; then a group of structures uniting the lattices of the nickel-arsenide type and of wurtzite—zinc blende—diamond; and finally metallic crystals. These lattices differ above all in the following respects.

In ionic lattices, each structural element is characterized by a more or less independent existence; in metallic lattices, the individual structural units combine into one monstrously large pseudo-atom; the structures of wurtzite, zinc blende, and nickel arsenide occupy a certain intermediate position between these extremes; the former are probably still somewhat closer to the ionic lattice, the latter more closely related to metallic lattices.

As a consequence, the fundamental law of crystal chemistry must manifest itself in these three regions of the world of crystals in somewhat different ways.

The law states: “the structure of a crystal is determined by the ratios of the numbers of its structural units, by the ratios of their sizes, and by their polarization properties. The structural units are atoms (and ions) and atomic groups. In heteropolar crystals, especially in ionic lattices, the structure is determined primarily by the ratio of the numbers and by the ratio of the sizes

structural elements, in a number of cases also by polarizational properties. In crystals of the nickel-arsenide type the importance of the stoichiometric moment is lost in comparison with that of the polarizational properties; in crystals of wurtzite–zinc blende, before the polarizational properties, the ratio of the sizes of the structural units recedes into the background. In metallic crystals, finally, the importance of the polarizational properties becomes quite exceptional, in comparison with both other factors, which retain significance only insofar as they influence the manifestation of polarizational properties. This is also understandable, from my point of view, for a metal. The relative sizes and relative numbers of the structural units have significance only in structures of such a kind where the structural elements retain a certain independence. But when the entire crystalline edifice becomes a single unified individual, the importance of the form-determining factor passes to the polarizational properties of the atoms, connected with their electrical interactions.

X. Crystals with Predetermined Technical Qualities

In the preceding chapters we have found a rational interpretation of the most important crystalline types by connecting them with the structural peculiarities of the atoms or ions composing them. We have established under what prerequisites one or another structure arises, and have thus discovered the possibility of creating these structures at will, “to measure,” since at our disposal there was a sufficient choice of building materials—atoms and ions with various “technical” qualities, i.e., since we freely disposed of three factors predetermining the structure of a crystal: the number of atoms, their sizes, and their polarizational properties. A table of atomic sizes was for us approximately what a catalogue of building materials is for an engineer, and we made use of polarizational properties in the same way,

as an engineer takes into account the mechanical properties of materials. But, in constructing a crystal with predetermined properties, we seek to satisfy not only aesthetic needs; we build the crystal not for decorative purposes, but so as to carry out with its aid a definite technical task—just as we build a bridge not to adorn the landscape, but to improve the means of communication of a given locality.

Therefore, from the construction of a crystal with given crystallographic properties we must now pass to the problem of designing crystals with predetermined properties—physical and chemical ones. These properties of crystals depend, above all, on the manner in which their structural units are combined and on the intensity of the forces binding the latter to one another. In the case of so-called ionic lattices, whose structural units are electrically charged ions, the generally recognized binding forces are the forces of Coulomb electrostatic attraction between oppositely charged ions; and, accordingly, the strength of the interionic bonds must, according to the laws of electrostatics, increase as the distance between the ions decreases and as the charges of the latter increase. If the strength of the bonds is measured by the hardness of the crystals, both of the noted dependences appear very sharply. This is shown by the following tables, of which the first shows the influence of the interionic distance, and the second the valency of the ions. Both of them cover those substances which it is customary to regard as ionic lattices.

	Mg	Ca	Sr	Ba
O	2.10 6.5	2.40 4.5	2.57 3.6	2.77 3.3
S	2.59 4.5–5	2.84 4.0	3.00 approx. 3.3	3.18 approx. 3
Se	2.74 3.5	2.96 3.2	3.12 approx. 2.9	3.31 approx. 2.7
Te	— —	3.17 2.9	3.32 approx. 2.8	3.49 2.6

I investigated the hardness of a whole series of bodies with the rock-salt structure; all measurements were carried out on pure, previously fused substances, and moreover on the same specimens that were used also for determining the lattice constant. The data are given on the Mohs scale; in the table, in addition to hardness, the interionic distances are given (in ångströms).

Now we shall compare, in pairs, substances differing in the magnitude of the ionic charges.

	LiF	MgO	NaF	CaO	LiCl	SrO	NaCl	BaO	LiCl	MgS
Distance between ions	2.02	2.10	2.31	2.40	2.57	2.57	2.81	2.77	2.57	2.59
Hardness	3.?	6.5	3.2	4.5	3	3.5	2.5	3.3	3	4.5–5

	NaCl	CaS	LiBr	MgSe	NaBr	CaSe	KCl	CaTe	KJ	BaTe
Distance between ions	2.81	2.84	2.75	2.73	2.98	2.96	3.14	3.17	3.53	3.49
Hardness	2.5	4.0	2.5	3.5	2.4	3.2	2.3	2.0	2.2	2.6

Friedrich’s (E. Friedrich) merit is the discovery that bonding laws, evidently similar to electrostatic behavior, are valid also for the lattices of elements. In the most striking form this is expressed in the following comparisons, which I likewise draw from my own works and which contain partly free elements in crystals of the diamond type, and partly the corresponding compounds in crystals of the zinc-blende type.

Finally, there are two more examples, in which equality of the distance between ions is not so well maintained.

The comparisons show a strong increase in hardness with increasing valency of the constituent ions.

It seems to me especially important that this increase extends in a regular way to the crystals of the free elements—carbon, silicon, and germanium. Friedrich also furnished proof that

the hardness of elements follows the same laws as the hardness of simple compounds.

Valence	1 AgJ	2 CdTe	3 JnSb
Distance between ions	2,81	2,80	2,79
Hardness	1,5	2,8	3,8

Valence	1 CuBr	2 ZnSe	3 GaAs	4 GeGe
Distance between ions	2,46	2,45	2,44	2,43
Hardness	2,6	3—4	4—5	6

Whether the crystal-binding forces are of purely electrostatic origin or not, their intensity increases with the number of valence bonds, and on these depend not only hardness, but also a whole series of other properties of primary importance: melting point, solubility, optical properties, chemical stability. The prediction of these properties may proceed in two ways. One may be guided, on the one hand, by purely theoretical considerations and predict the properties of structures, in particular their strength, from the properties of the elements used to create the structures. On the other hand, it is often necessary to apply a purely practical method, well known also in macro-construction, namely the method of preliminary study of a model of the building.

Valence	1 CuCl	2 ZnS	3 GaP	4 AsAl
Distance between ions	2,34	2,35	2,35	2,44
Hardness	2,5	4	5	6

Valence	1 CuJ	2 ZnTe	3 GaSb	4 SeCd
Distance between ions	2,62	2,64	2,64	2,62
Hardness	2,4	3,0	4,5	3,0

Valency	3 AlP	4 SiSi
Distance between ions	2.36	2.35
Hardness	5.5	7

Valency	2 BeO	4 CC	2 CdS	3 SbAl
Distance between ions	1.65	1.54	2.52	2.64
Hardness	9	10	3.2	4.8

If some structure is stable, then any enlarged or reduced model of it will be stable, provided only that it is built strictly similarly and with a change in the properties of the building materials corresponding to the change in the dimensions of the structure.

This principle is applied with great success and in many ways also in crystalline structures. Thus, considering the crystal lattices of ZnS and CdSe, we may recognize in one of these substances a model of the other; although the scales are changed, the strength of the lattice is obviously the same. It is a different matter with the family CuBr, ZnSe, GaAs, and GeGe. The magnitude of the atomic constant here fluctuates within narrow limits, 5.6—5.7 Å, but the valencies of the structural elements are different, and the strength of the structures changes concomitantly with them. We may therefore assert that CuBr is a weakened model of germanium, namely—weakened by a factor of 4. Similarly, we could regard GaAs as a threefold “strengthened” model of CuBr. We use the expressions “weakened,” “strengthened,” in order to emphasize more clearly the increase or weakening of the crystal-binding forces. Correspondingly, we may designate MgO as a doubly strengthened model of LiF, CdI₂—as a twofold weakened model of ZnS₂ (in this case the scales also differ markedly).

Bearing in mind that, as was indicated above, chemical activ-

ness, melting temperature, solubility, refraction, and other fundamental properties of crystals, let us trace this connection in several concrete examples. Let us set ourselves, first of all, the goal of constructing, in the form of a model, such a substance as in the “original” does not display desirable qualities. For example, we can construct models of silicates, titanates, zirconates that would be “weakened” as compared with the originals, and therefore considerably less hard, more readily fusible, more soluble, and would not possess that often very undesirable chemical inertness which is characteristic of silicates. On such models we could study the chemistry of silicates and titanates with considerably greater convenience than on the originals. For modeling silicates one may choose salts of fluoroberyllium acids. I have studied lithium orthofluoroberyllate as a model of zinc orthosilicate, willemite.

The following table shows the extraordinary similarity of both substances—the original Zn₂SiO₄ and its twice-weakened model Li₂BeF₄.

Lattice constants.	Lattice constants.
Zn₂SiO₄ $a = 8.63\ \text{Å}$ $\alpha = 107^\circ 45'$	Li₂BeF₄ $a = 8.15\ \text{Å}$ $\alpha = 107^\circ 40'$
Crystallographic properties.	Crystallographic properties.
Symmetry rhombohedral. Habit prismatic. Cleavage parallel to 1010 and 0001, distinct.	Symmetry rhombohedral. Habit prismatic. Cleavage parallel to 1010 good, parallel to 0001 evident.
Optical properties.	Optical properties.
Positive double refraction rather weak, $\gamma-\alpha$ about 0.02, refractive index about 1.70.	Positive double refraction extremely weak, $\gamma-\alpha$ about 0.006, refractive index about 1.3.
Hardness.	Hardness.
5.5.	3.8.
Melting temperature.	Melting temperature.
1509.5°	About 470°.
Solubility.	Solubility.
Insoluble in water.	Extremely readily soluble; easily recrystallizes from dilute hydrofluoric acid.

Similarly, one may construct a doubly weakened model of the calcium salt of the complex magnesium-silicic acid—diopside, $\mathrm{CaMgSi_2O_6}$—in the form of the compound $\mathrm{NaLiBe_2F_6}$.

Thus, by the method of modeling we can create crystals with a predetermined structure and predetermined properties.

This constructive method, however, is not limited to crystalline phases alone; analogous considerations may be applied, for example, to “amorphous-solid” substances, to “glasses,” and in this way obtain a weakened fluoroberyllate model of ordinary silicate glasses. These are glasses with a very low softening temperature, whose refraction and scattering of light are exceptionally low. In such glasses refractive indices lower than that of water have been observed, and it is not difficult to prepare a fluoroberyllate glass with a refractive index so close to that of water that these glasses are almost invisible in water. Unfortunately, most fluoroberyllate glasses are extremely hygroscopic; it is understandable that they dissolve in water incomparably more readily than the corresponding silicates, which is likewise a consequence of the model “weakening.” It is probable that “strengthened” models of silicate glasses can also be obtained in the form of double nitrides and double carbides (strengthening relative to oxide glasses in the ratios $3:2$ and $4:2$). It is also possible that glassy phases of complex carbides may be found in many specimens of steel; these would be doubly strengthened models of oxygen glasses.

In conclusion let us consider one more example. Suppose that the following are given as material: carbon, nitrogen, oxygen, phosphorus, sulfur, chlorine, sodium, calcium, i.e. the varieties of atoms most abundantly represented in the human organism, and that the task is posed of obtaining from them the hardest possible crystal, one that would be capable of crystallizing out of water at an ordinary temperature, about $37^\circ$.

To achieve the greatest hardness we choose a cation and an anion of the greatest valency—namely, calcium with two

positive charges and the trivalently negative $\mathrm{PO}_4$ and form calcium phosphate. In order thereby to prevent the formation of acidic or water-containing salts, we introduce calcium phosphate as a complex into the halide salt of calcium itself—we build an apatite crystal. Nature does the same when it forms dental cement, as an especially hard substance among all that is contained in the organism. Recently R. Gross in Greifswald has proved that dental cement is nothing other than crystalline apatite, which I can only confirm. But nature goes further. According to Reiss and Friedrich, hardness increases with decreasing distances between ions; preferably, therefore, for apatite one should choose the smallest halide ion, and precisely for this reason the choice must inevitably stop at fluorine, which indeed corresponds to reality.

1. A. Magnus, Die chem. Komplex. Z. f. anorg. Chem. 124, 288 (1922). ↩↩↩