CRISTALLO. — Chemical elements and compounds, inorganic and organic, when they originate in the solid state, may assume the form of polyhedra: these natural polyhedra are called crystals. Their nature and structure are of considerable interest both for theoretical speculations concerning their presumed life and because, in living organisms, crystalline structure underlies many morphological arrangements which, although they do not represent life, constitute its supposed material (v. COLLOIDI; PROTOPLASMA). The polyhedral form of crystals is not accidental, but is a consequence of the rate of growth, which varies discontinuously according to direction; this results from a three-dimensional periodic arrangement by which the atoms or ions entering into the chemical constitution of the crystal are joined. The number and type of possible forms are governed by rules of symmetry and by the law of rationality of indices.
A substance is isotropic with respect to a vector physical property (growth, solubility, cohesion, refraction of light, magnetic susceptibility, etc...) if the value of that property remains constant in all directions; it is anisotropic if, instead, the value changes with direction, either continuously or discontinuously.
The solid state comprises two phases or modifications of matter: the crystalline phase and the amorphous phase. Crystalline substances are discontinuously anisotropic with respect to at least one of the vector physical properties, and this characteristic is clearly revealed by various phenomena: crystal formation, cleavage, etc. Amorphous substances (glasses, some colloids) are isotropic with respect to all vector physical properties and therefore exhibit none of the phenomena mentioned above.
In investigating the cause determining such different behaviour, it became necessary to admit that it lies in a different structure, which may be expressed as follows: amorphous substances consist of leptons (a generic term denoting atoms, ions, and molecules) arranged in space without order. Crystalline substances, on the other hand, consist of leptons ordered with regular periodicity in the three directions of space. Let us consider, in fact, a disordered assemblage of molecules or atoms; a line passing through it will encounter, after a path very long in relation to the dimensions of its constituents, a number of atomic particles independent of direction; a disordered assemblage is therefore isotropic because all directions are equivalent. If the same atoms or the same molecules are aligned in space with regular periodicity, some directions will be denser than others, and the different densities will correspond to different behaviour with respect to one or more physical properties; the ordered assemblage must therefore be anisotropic.
The first organically developed hypothesis concerning the structure of crystals is due to the abbot R. J. Haüy (1743–1822). He had observed that certain crystals, when broken, resolved into fragments which, however small, were not formless but small polyhedra of constant shape for each mineral species (cubic in rock salt, rhombohedral in calcite, octahedral in fluorite). He therefore inferred that, if the division could be carried as far as particles of molecular size, these too would have to display the form of the cleavage polyhedron. Haüy had also observed that division by cleavage took place along planes always parallel to one another; he consequently formulated the hypothesis that crystals were constituted by the juxtaposition, in perfect contact, of parallelepipedal molecules, called integrating molecules, having the form of the cleavage solid.
The three forms that a rock-salt crystal may assume can be regarded as derived from one another by truncating edges and vertices through the removal of a decreasing number of rows in successive layers of molecules; the rhombic dodecahedron, for example, is obtained from the cube by eliminating, along each edge, several rows of integrating molecules, so that each edge is replaced by a new face.
The conception of a solid state characterized by polyhedral molecules in perfect contact with one another, without intermolecular voids, conflicted too openly with the principle of the compressibility of matter. The phenomenon of anisotropy in crystals therefore had to be related not to the shape of the molecules but to their regular and periodic distribution in space. Once Haüy’s hypothesis had been reduced to this simpler expression, it became possible to conceive of molecules having shapes that allowed intermolecular voids to exist. The hypothesis of a regular and continuous structure was consequently replaced by that of a regular and discontinuous structure, advocated by Seeber, Delafosse, and L. Frankenheim.
If we imagine the masses of the molecules concentrated at their centres of equilibrium, the crystalline structure is reduced to an assemblage of points or nodes that repeat themselves in space with the same periodicity as the molecules themselves; this ordered assemblage is called a crystal lattice or space lattice because the points are distributed in space like the nodes of a three-dimensional net. From the abstract space lattice one passes to the concrete crystalline structure by coating the nodes with material particles (atoms, in the case of chemical elements; molecules or complexes of atoms, in compounds). To establish in how many ways the material particles repeat themselves in a crystalline structure is therefore equivalent to establishing which types of lattices are compatible with the morphological properties of crystals. Two nodes are said to be identical if the distribution of the atomic masses around each of them is
The elementary parallelepiped, or fundamental cell, thus represents the smallest particle of crystalline substance that can be imagined, in the same way that the molecule represents the smallest quantity of a liquid or amorphous substance.
It follows from the foregoing that the problem of defining the various possible types of lattices is reduced to that of determining the possible types of elementary cells compatible with the morphological properties of crystals. A. Bravais, a pupil of Haüy and the first to undertake a systematic study of spatial lattices, established that 14 types of elementary cells are possible for crystals, with their sides parallel to the crystallographic axes.
Diffraction teaches that an ordered system of small material particles (diffraction lattice), when struck by a narrow beam of light, gives rise to luminous rays that propagate not only in the direction of incidence but also in other directions; a diffraction spectrum is thus obtained, consisting of a central luminous image, produced by the rays propagating in the direction of incidence, surrounded by other images corresponding to the deviated rays. For the phenomenon to occur, it is not enough for the diffracting bodies to have an ordered structure; it is also necessary that the dimensions of the individual material elements and of the intervals separating them be of the same order of magnitude as the wavelengths of the incident beam. The atoms and the empty spaces that they delimit, when they are in contact with one another, have dimensions of the order of a few ångströms (1 Å = 10⁻⁸ cm.); in order to obtain diffraction spectra with crystals and thus provide experimental confirmation of the lattice theory, it was necessary to resort to wave radiations having wavelengths much smaller—approximately one thousand times smaller—than those of light.
Max von Laue, a physicist at the University of Munich, guided by this principle, conceived in 1912 the idea of irradiating a rock-salt crystal with X-rays, for which
a wavelength of the same order of magnitude as atomic particles was indeed predicted.
The experiment, brought to completion with a very simple experimental apparatus, fully confirmed the validity of the theory concerning the regular and discontinuous structure of crystals. Laue’s experiment marks a very important date in the history of science, because it represents the point of departure for new advances in the field of morphological, physical, and chemical crystallography.
The English physicists W. L. Bragg and L. Bragg, father and son, in 1913, modifying Laue’s experimental apparatus, demonstrated that the diffraction effects of X-rays could also be regarded as images reflected from layers of atoms parallel to the possible faces of crystals, at certain angles of incidence (law of selective reflection).
This interpretation so greatly simplified the analytical expression given by Laue that the two English physicists, starting from Bravais’s theoretical premises, were able to indicate the way toward establishing the first criteria for determining crystal structure.