We are now in a position to approach the conclusion of the long controversy as to the constancy or otherwise of crystal angles in the cases of greatest similarity, those of isomorphous substances, and to appreciate how the conflicting views of Haüy and Mitscherlich and their schools of thought have at length been reconciled. As the result of a comprehensive study, on the part of the author, of the sulphates and selenates of the rhombic series R₂S
SeO₄, and of the double sulphates and selenates of the monoclinic series R₂M(S
SeO₄)₂.6H₂O, in which R represents the alkali metals, potassium, rubidium and cæsium, and in which M may be magnesium, zinc, iron, nickel, cobalt, manganese, copper or cadmium, four facts of prime significance have been definitely established.

(1) The crystals of the different members of an isomorphous series exhibit slight but real differences in their interfacial angles, the magnitude of the angle changing regularly with the alteration of the atomic weight of the interchangeable metals or negative elements of the same family group which give rise to the series, as one metal or acid-forming element is replaced by another. The amount of the difference increases as the symmetry of the system diminishes. Thus the maximum difference for the more symmetrical rhombic series of sulphates and selenates is 56′, which occurs in the case of one angle between potassium and cæsium selenates, and it is usually much less than this; in the case of the less symmetrical monoclinic series of double salts the maximum angular difference observed was 2° 21′, between potassium and cæsium magnesium sulphates.

(2) The physical properties of the crystals, such as their optical and thermal constants, are also functions of the atomic weights of the elements of the same family group which by their interchange produce the series.

(3) The dimensions of the elementary parallelepipedon of the space-lattice, or in other words, the separation of the molecular centres of gravity, the points or nodes of the space-lattice, along the three directions of the crystal axes, also vary with the atomic weight of the interchangeable elements.

(4) Specific chemical replacements are accompanied by clearly defined changes in the crystal structure along equally specific directions. Thus, when the metal, say potassium, in an alkali sulphate or selenate is replaced by another of the same alkali-family group, rubidium or cæsium, there is a marked alteration in the crystal angles and in the dimensions of the space-lattice, corresponding to elongation of the vertical axis; and when the acid-forming element sulphur is replaced by selenium, its family analogue, a similar very definite change occurs, but the expansion in this case takes place in the horizontal plane of the crystals.

Confirmatory results have also been obtained as regards the morphological constants, the investigations not extending to the optical or thermal properties, by Muthmann for the permanganates, and by Barker for the perchlorates, of the alkali metals. Hence, there can be no doubt whatever that, as regards the various series investigated, which are such as would be expected to afford the most definite results owing to the electro-positive nature of metals being at its maximum strength in the alkali group, the above rules are definite laws of nature.

Thus it is clear that in the cases of isomorphous substances, which were the only possible exceptions to the generalisation that to every chemically distinct solid substance of other than perfect cubic symmetry there appertains a specific crystalline form, endowed with its own particular angles and morphological crystal elements, which are absolutely constant for the same temperature, the law does really hold, and isomorphous substances are no exceptions. The law of progression of the crystal properties according to the atomic weight of the interchangeable elements affords indeed at the same time both an amplification of the generalisation and a precise explanation of its mode of operation in these cases.

The discovery of the local effect produced by the two kinds, positive and negative, of chemical replacement, has a profound bearing on crystal structure. For it is thereby rendered certain that the atoms are fixed in the crystal edifice, and therefore in the molecule in the solid state. It becomes obvious that the atoms—in their stereometric positions in the molecule, being thus fixed in the solid crystal when the molecules set themselves rigidly in the regular organisation of the space-lattice—form the points of the regular point-system of the crystal structure, which determines to which of the thirty-two classes of symmetry the crystal shall belong. Any movement of the atoms in the crystal, other than that which accompanies change of temperature, and possibly change of pressure, is thus improbable; and this experimental proof of their fixity, afforded by the fact that definitely orientated changes accompany the replacement of particular atoms, also doubtless indicates that the latter are located in the particular directions along which the changes of exterior angle and of internal structural dimensions are observed to occur. Stereo-chemistry, which has made such enormous advances during the last few years, thus becomes of even greater importance than Wislicenus and its other originators ever dreamt of.

Within the atoms in the crystal the constituent electronic corpuscles may be and probably are in rapid movement, and such physical effects as have hitherto been ascribed to movement of the atoms within the crystal are doubtless due to movement of the electronic corpuscles within them, the sphere of influence of the atom itself being fixed in space in the solid crystal, and being doubtless defined by the area within which the corpuscular movements occur.

Three illustrations of the law of change of the crystal properties with variation of the atomic weight of the determinative elements of an isomorphous series may be given, and will serve to render the practical meaning of the generalisation clearer. The first is a diagrammatic representation, in Fig. 60 (in a very exaggerated manner as the real change would be inappreciable on the scale drawn), of the change of angle on replacing the potassium in potassium sulphate, K₂SO₄, or selenate, K₂SeO₄, by rubidium or cæsium. The inner crystal outline, a vertical section, is that of the potassium salt. The vertical lines represent the intersections of the two faces of the brachypinakoid b = {010} with a vertical plane parallel to the macropinakoid a = {100}; the horizontal lines represent the intersection of the two faces of the basal plane c = {001} with the same vertical plane; and the oblique lines represent the intersection of the vertical plane with the four faces of the dome form q = {011}, which are inclined to both b and c planes. The diagram is thus designed to show the variation of the inclination of these latter dome faces to the two rectangular axial plane faces b and c. The outer crystal outline represents a similar section of a crystal of the corresponding cæsium salt, and the middle outline that of a crystal of the rubidium salt.

The progressive alteration of the angle of the q-face will be obvious, the direction of the change being correct, but the amount of change, as already stated, being much exaggerated; in reality it never reaches a degree between the two extreme (potassium and cæsium) salts. It will be remembered that the respective atomic weights of potassium, rubidium, and cæsium are 38·85, 84·9 and 131·9, when hydrogen equals 1, that of rubidium being almost exactly the mean.

The second illustration is taken from the optical properties. Fig. 61 represents graphically the regular diminution of double refraction (the difference between the two extreme indices of refraction α and γ) which accompanies increase of the atomic weight of the metal present. The diagram exhibits the closing up of the two spectra afforded by three analogously orientated 60°-prisms, one of each of the three salts, such as was used in determining two of the refractive indices of the salt. Each prism produces two refracted rays from the single ray furnished by the collimator of the spectrometer, and consequently two images of the signal-slit of the collimator when monochromatic light is used, or two spectra if white light be employed. The Websky signal-slit is narrow at the centre to enable an accurate allocation to the vertical cross-wire of the telescope to be made, but wide at its top and bottom ends, in order to transmit ample light, and Fig. 61 shows four images of this signal produced by each prism, namely, one R in red C-hydrogen light and another B in greenish-blue F-hydrogen light belonging to each of the two spectra, in order to locate the two ends of each of the latter, coloured monochromatic light of each of the two colours in turn and of the exact C and F wave-lengths having been fed to the spectrometer from the spectroscopic illuminator. It will be observed in the case of the top row that the two spectra, each indicated by the adjacent red and greenish-blue images, are well apart, the relative distance being about that actually observed in the case of potassium sulphate. They are nearer together, however, in the second row, which indicates what is observed in the case of the analogous rubidium salt, and in the lowest row representing the relative distances of the two spectra apart in the case of the cæsium salt, they are so close together as to overlap; for in this latter case the greenish-blue image of the left-hand spectrum, corresponding to the a index of refraction, occupies the same position as the image for yellow sodium light of the right-hand spectrum corresponding to γ would occupy in the case of cæsium sulphate, the a refractive index for F-light being 1·5660 and the γ index for Na-light being 1·5662. The progression of the alteration of the amount of the double refraction is thus very striking, as the atomic weight of the metal is varied.

The third illustration of the law of progression with atomic weight is also an optical one, and is taken from the monoclinic series of double sulphates and selenates. It indicates the rotation, with increase of the atomic weight of the metal, of the ellipsoid which graphically represents the optical properties, about the unique axis of symmetry, which is likewise an axis of optical symmetry, of the crystal. In the potassium salt the ellipsoid occupies the position indicated by the ellipse drawn in continuous line in Fig. 62, the section of the ellipsoid by the symmetry plane; the outline of a tabular crystal parallel to the symmetry plane is also given, as well as the axes of the crystal and of the ellipsoid lying in that plane.

In the rubidium salt the ellipsoid has rotated over to the left, as indicated by the dotted ellipse, for a few degrees, the number of which varies slightly for the different groups of double salts; while in the cæsium salt it has swung over much more still, to the place marked by the ellipse drawn in broken line. In both this and the last illustration it will be remarked that the optical change is greater between the rubidium and cæsium salts than it is between the potassium and rubidium salts, the reason being that the optical properties are usually functions (of the atomic weight of the interchangeable elements) which are of an order higher than the first corresponding to simple proportionality.

These three ocular illustrations may serve to render this interesting law of progression, according to the atomic weight of the interchangeable elements which give rise to the isomorphous series, clearer to the mind, by placing before it concrete instances of the operation of the law.

The generalisation itself may be very concisely expressed in the statement that:

The whole of the properties, morphological and physical, of the crystals of an isomorphous series of salts are functions of the atomic weights of the interchangeable chemical elements of the same family group which give rise to the series.

The fact that this law extends to the structural dimensions, equally with all other morphological properties, as stated under (3) at the beginning of this chapter, is of especial interest. For it has actually been found possible to determine the relations of the dimensions of the unit parallelepipeda of the space-lattices of the various salts, that is, the separation of the molecular points of the space-lattice in the directions of the three crystal axes, for the various salts of the isomorphous series. This is achieved by combining in suitable formulæ the volume of the unit cell of the space-lattice with the relative lengths of the three crystal axes, a, b, c.

The axial ratios a : b : c are calculated from the measurements of the crystal angles, as explained in Chapter VI., page 68, and the volume is the physical constant long known as “molecular volume,” but now for the first time understood as regards its meaning in the case of solid substances. It is the quotient of the chemical constant molecular weight (the sum of the atomic weights, taking into account the number of atoms of each element present) by the specific gravity of the substance, here the solid crystal. Very great care has been taken to obtain absolutely accurate determinations of the specific gravities of the salts, as much depends on this now very valuable physical constant, and all the values obtained were reduced to the constant reference temperature of 20°, as the density notoriously alters rapidly with change of temperature.

We have thus arrived at morphological constants of very considerable importance, which are best termed “Molecular Distance Ratios,” as they express the relative distances apart in the three directions of space of the centres of gravity or other representative points of contiguous chemical molecules. They are dependent on three experimental determinations, atomic weight, specific gravity, and crystal angles, all of which have now been brought to the highest pitch of refinement and accuracy; hence the molecular distance ratios are particularly trustworthy constants. If it were only known how much is matter and how much is space in the molecular parallelepipedal cell, we should actually have in these constants a relative measure of the sizes of the molecules. They do give us, however, the relative directional dimensions of the molecular unit parallelepipedal cells of the space-lattices of the various members of the isomorphous series, just as the molecular volumes give us the relative volumes of these cells. For in an isomorphous series we are absolutely sure that the plan on which the space-lattice is constructed, its style of architecture, is identical for all the members of the isomorphous series. Hence, the molecular distance ratios are in these cases absolutely valid and strictly comparable. The ratios are generally expressed by the Greek letters χ : ψ : ω.

On comparing the molecular distance ratios for a potassium, a rubidium, and a cæsium salt of any of the series of sulphates, selenates, permanganates, perchlorates, double sulphates or double selenates investigated, we invariably find that the values of χ, ψ, and ω for the rubidium salt (rubidium having the intermediate atomic weight) lie between the analogous sets of three values for the potassium and cæsium salts respectively, in complete accordance with the law.

For the generalisation to apply absolutely it is essential that the interchangeable elements shall belong strictly to the same family group of the periodic classification of Mendeleéff. Potassium, rubidium, and cæsium fulfil this condition absolutely, and so the law of progression of the crystal properties with the atomic weight of the interchangeable elements applies rigidly to their salts. Now there are two bases, the metal thallium and the complex radicle group ammonium NH₄, which are not thus related to the group of three alkali metals just mentioned, but which are yet capable of replacing those metals isomorphously in their crystals without more change of angle or of structural constants than is provoked by the replacement of potassium by cæsium; and often indeed the amount of change has been singularly like the lesser amount observed when rubidium has been interchanged for potassium. But although this is so, the directions of the changes are irregular, being sometimes the same as when rubidium or cæsium is introduced, and sometimes contrariwise, and in the case of thallium there are also striking optical differences, the thallium salts being exceptionally highly refractive. Still, morphologically the ammonium and thallium salts may legitimately be included in the same isomorphous series with the salts of potassium, rubidium, and cæsium, and a somewhat wider interpretation has to be given to the term “isomorphism” in order to admit these cases. To distinguish the inner group formed by family analogues, that is, the more exclusive group obeying the law of progression according to the atomic weight, the term “eutropic” is employed.

Thus the “isomorphous series” of rhombic sulphates, selenates, permanganates, and perchlorates, and the monoclinic series of double sulphates and double selenates, comprise the potassium, rubidium, cæsium, thallium and ammonium salts and double salts of sulphuric, selenic, permanganic, and perchloric acids, while the inner more exclusive “eutropic series,” following the law absolutely, comprises in each case only the salts containing the family analogues, potassium, rubidium, and cæsium.

In this beautiful manner has the controversy between the schools of Haüy and Mitscherlich now been settled, the interesting law described in this chapter having definitely laid down the true nature and limitations of isomorphism, while at the same time absolutely proving as a law of nature the constancy and specific character of the crystal angles of every definitely chemically constituted substance.