CRYSTAL. - The chemical elements and compounds, inorganic and organic, when they originate in the solid state, can assume the form of polyhedra: these natural polyhedra are called crystals. Their nature and structure are of considerable interest both for theoretical speculations about their presumed life, and because in living beings the crystalline structure underlies many morphological arrangements which, while not representing life, constitute its supposed material (v. COLLOIDS; PROTOPLASM). The polyhedral form of crystals is not random, but a consequence of the rate of growth, which varies with direction in a discontinuous manner, and this through the effect of a three-periodic ordering by which the atoms or ions that enter into the chemical constitution of the crystal are grouped together. The number and type of possible forms are governed by rules of symmetry and by the law of the rationality of indices.
A substance is isotropic with respect to a vectorial physical property (growth, solubility, cohesion, refraction of light, magnetic susceptibility, etc.) if the value of that property remains constant in all directions, and anisotropic if instead the value changes with direction in a continuous or discontinuous manner.
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 vectorial physical properties, and this character is clearly brought out by various phenomena: formation of crystals, cleavage, etc. Amorphous substances (glasses, certain colloids) are isotropic with respect to all vectorial physical properties, and therefore exhibit none of the above-mentioned phenomena.
Tracing back to the search for the cause that determines such a different behavior, it has been necessary to admit that it lies in a different structure, which can be expressed as follows: amorphous substances are made up of leptons (a generic term for atoms, ions, molecules) arranged in space without order. Crystalline substances, on the other hand, are made up of leptons ordered with regular periodicity in the three directions of space. Consider, in fact, a disordered assemblage of molecules or atoms; a line passing through it will encounter, after a path very great compared with the dimensions of its constituents, a number of atomic particles independent of direction; a disordered assemblage is therefore isotropic because all directions turn out to be equivalent. If the same atoms or the same molecules are aligned in space with regular periodicity, some directions turn out to be denser than others, and to the different densities there will correspond a different behavior with respect to one or more physical properties; the ordered assemblage must therefore turn out to be anisotropic.
The first hypothesis, organically developed, on the structure of crystals is due to the abbé R. J. Haüy (1743-1822). He had established that certain crystals, on breaking, resolved into fragments which, however small, were not formless but small polyhedra of constant form for each species of mineral (cubic in rock salt, rhombohedral in calcite, octahedral in fluorite). And from this he deduced that, since the subdivision could be pushed as far as particles of molecular size, even these would have to present the form of the cleavage polyhedron. Haüy had also established that subdivision by cleavage took place along planes always parallel to one another: he therefore formulated the hypothesis that crystals were constituted by the juxtaposition, in perfect contact, of parallelepipedal molecules, called integral molecules, having the form of the cleavage solid.
The three forms that a crystal of rock salt can assume may be supposed derived one from another by truncating edges and vertices with the suppression of a decreasing number of rows in the successive layers of molecules; the rhombic dodecahedron, for example, is obtained from the cube by eliminating, for each edge, a few rows of integral 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, contrasted too openly with the principle of the compressibility of matter. The phenomenon of anisotropy of crystals therefore had to be related not to the form of the molecules but to their regular and periodic distribution in space. Once Haüy's hypothesis was reduced to this simpler expression, one was free to conceive for the molecules forms such as to allow the existence of intermolecular voids. The hypothesis of a regular and continuous structure was therefore replaced by that of a regular and discontinuous structure upheld by Seeber, Delafosse, and L. Frankenheim.
If we imagine concentrating the masses of the molecules at their centers of equilibrium, the crystalline edifice is reduced to an assemblage of points or nodes that repeat in space with the same periodicity as the molecules themselves; this ordered assemblage is given the name of crystal 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 edifice by clothing the nodes with material particles (atoms, in the case of the chemical elements; molecules or complexes of atoms in compounds). To establish in how many ways the material particles repeat in a crystalline edifice 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 atomic masses around each of them turns out perfectly equal, so that the one can be considered derived from the other by translation (parallel displacement) of a certain segment T. The nodes of a space lattice are all identical, and the distance T between two nodes is constant for every bundle of parallel rows. A crystal lattice is therefore decomposable into as many equal, contiguous and parallel parallelepipeds, and its geometric, physical and chemical properties are identifiable with those of one of these elementary parallelepipeds.
The elementary parallelepiped, or fundamental cell, therefore 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.
From what precedes it follows that the problem of defining the different 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, who was the first to tackle systematically the study of space lattices, established that for crystals 14 types of elementary cells are possible, having their sides parallel to the crystallographic axes.
Diffraction teaches that an ordered system of small material particles (a diffraction grating), struck by a thin beam of light, produces light rays that propagate not only in the direction of incidence but also in other directions; thus one obtains a diffraction spectrum consisting of a central luminous image, due to the rays that propagate in the direction of incidence, surrounded by other images that correspond to the deflected rays. For the phenomenon to appear, it is not enough that the diffracting bodies have an ordered constitution; 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 on the order of a few Angstrom units (1 Å = 10⁻⁸ cm.); to obtain diffraction spectra with crystals and thus give experimental confirmation to the lattice theory, it was necessary to resort to wave radiations consisting of wavelengths much smaller (about a thousand times) than those of light.
Max von Laue, a physicist at the University of Munich, guided by this principle, in 1912 conceived the idea of irradiating a rock-salt crystal with X-rays, for which a wavelength on the order of magnitude of the atomic particles was in fact predicted.
The experiment, carried out 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 starting point of new conquests in the field of morphological, physical and chemical crystallography.
The English physicists W. L. Bragg and L. Bragg, father and son, in 1913, having modified Laue's experimental apparatus, demonstrated that the diffraction effects of X-rays could also be considered as images reflected by layers of atoms, parallel to the possible faces of crystals, for determined angles of incidence (law of selective reflection).
Such an interpretation so simplified the analytical expression given by Laue that the two English physicists, starting from Bravais's theoretical premises, were able to point the way to establishing the first criteria for the determination of crystal structure.