Allotropy

“ It is worthy of remark, that we are led precisely to this law of alteration of dimensions when we assume first that, without taking account of molecular motions, in a solid body left to itself the forces of attraction and repulsion acting on each molecule maintain themselves in equilibrium, and secondly — for which there is admittedly no evidence — that the same law applies to these molecular forces, as regards their alteration by convection, that has been demonstrated for the electrostatic attractions of moving charges. ”

“ Polymorphism is thus completely and simply explained as a direct result of the establishment of the geometrical theory of crystal structure as laid down in Chapter IX. The equilibrium of the homogeneous structure is a function of the temperature, and the stereometric arrangement of the atoms 137in the chemical molecule of a substance may be such as permits of two or more homogeneous arrangements of the molecules in assemblages of varying degrees of stability, but each of which has a maximum stability at a particular temperature. ”

“ But when a simple body, such as oxygen, is concerned, this kind of hypothesis is no longer open to us; we have only one kind of particle to deal with; and the theory of altered position is no longer applicable. In short, it does not seem possible to imagine a rational hypothesis to explain the condition of allotropism as regards simple bodies. We can only accept it as a fact, not to be doubted, and add the discovery to that long list of truths which start up in the field of every science, in opposition to our most cherished theories and long received convictions. ”

“ Mass, electric charge, temperature, have the same symmetry, of a type called scalar, that of the sphere. A current of water and a rectilineal electric current have the symmetry of an arrow, of the type polar vector. The symmetry of an upright circular cylinder is of the type tensor. All of the physics of crystals can be expressed in a form in which the particular phenomena in question are not specified, but in which are examined only the geometrical and analytical relations between the types of quantities where certain ones are considered as causes and the other as effects. ”

“ As it is here equally likely that a particular molecule will be found in any one of these unit volumes, it is evident that the increase of volume will add increased variety to the location or configuration of the molecules and by indulging in the swapping of places inherent in the production of complexions, we see that said increment of volume will make the number of complexions in the final state greater than in the initial state, which in turn means that the entropy in the final state is also greater. ”

“ All that is required, therefore, to explain gravity, is a diminution of pressure, or increase of tension, caused by the formation of a matter unit—that is to say of an electron or corpuscle. And although we do not yet know what an electron is—whether it be a strain centre, or what kind of singularity in the ether it may be—there is no difficulty in supposing that a slight, almost infinitesimal, strain or attempted rarefaction should be produced in the ether whenever an electron comes into being—to be relaxed again only on its resolution and destruction. ”

“ There you will see the atoms in the solution flowing towards the negative pole and depositing themselves in minute crystals upon the metallic surface, if the negative pole is a metal, and adhering with great pertinacity; the metallic crystals penetrate the crystalline surface of the substance to be plated in minute arms filling up the infinitesimal space, and the atoms in this process transfer their atomic motion in line with the electric force, and become arrested as in the primitive state when they formed the crystalline foundations of primitive earth. ”

“ If we regard each chemical molecule as represented by a point, disregarding the separate atoms of which it is composed, then these fourteen space-lattices represent the possible arrangements of the molecules in the crystal in all the simpler cases; three of these lattices have cubic symmetry; the tetragonal, hexagonal, trigonal, rhombic and monoclinic systems claim two space-lattices each; while one space-lattice conforms to the lack of symmetry of the triclinic system. ”

“ This is on the hypothesis which is here generally adopted, that the dimensions of the nucleus of an electron are so small, compared with the mutual distances of electrons, that these dimensions are not sensibly involved in the forces between them. If this condition is left out the constancy of volume of the nucleus will have to be taken into consideration in the dimensional transformation, so that k must be unity; and this indefiniteness of linear scale in a material body cannot exist. ”

“ A purely geometrical question, of no importance in respect to the molecular tactics of a crystal but of considerable interest in pure mathematics, is forced on our attention by our having seen (§ 27) that a homogeneous assemblage of solids of given shape, each touched by twelve neighbours, has three freedoms which may be conveniently taken as the three angles specifying the orientation of each molecule relatively to rows of the assemblage as explained in § 30. ”

“ Now in an elastic medium permeated by centres of permanent intrinsic strain, whether it be the rotational æther with its contained electrons, or an ordinary elastic solid permeated by polar strain-nuclei as described above, the specification of the strain-energy of the medium involves a mathematical function, not only of the displacement at each material point of the medium, but also of the positions of these intrinsic strain-centres which can move independently through it. ”

“ It may easily be shown that the quantity denoted by is the work which would be required to form (by a reversible process) the heterogeneous globule in the interior of a very large mass having initially the uniform phase of the exterior mass. For this work is equal to the increment of energy of the system when the globule is formed without change of the entropy or volume of the whole system or of the quantities of the several components. ”

“ What new elements may be found no one may predict. Mendelejeff’s periodic law, however, suggests great possibilities in this field. Allotropism, in which the same element will present entirely different physical aspects, is also a significant and suggestive phenomenon, for in it we see carbon appearing at one time as a crude, black and ungainly mass of coal, and at another it appears as the limpid and flashing diamond. ”

“ Hence, according to the configuration and the molecular forces of the particular crystalline molecule in natural crystallization, there may be necessarily, or almost necessarily, the twin, when growth proceeds simultaneously on the two sides: or the twin growth may be impossible, because the first occupation of the wider hollows on one side, may compel the continuity of the crystalline quality throughout, by leaving only the narrower hollows p q r free for occupation by molecules attaching themselves on the other side. ”

“ Ether being incompressible, the density of fluid inside and outside an electron must be the same. So, dealing with it in this simplest fashion, the resultant inertia is half as great again as that of the volume of fluid corresponding to the electron: that is to say the effective mass is 2πρα3, where ρ is the uniform density. If an electron is of some other shape than a sphere, then the numerical part is modified, but remains of the same order of magnitude, so long as there are no sharp edges. ”

“ These diatomic compounds, like their elements, are to a considerable degree characterized by the prevalence among them of allotropism; or, as it is more usually called when displayed by compound bodies—isomerism. Professor Graham finds reason for thinking that a change in atomic arrangements of this nature, takes place in water, at or near the melting point of ice. In the various series of hydro-carbons, differing from each other only in the ratios in which the elements are united, we find not simply isomerism but polymerism occurring to an almost infinite extent. ”

“ Further, since rotation of the body without deformation cannot alter the potential energy of mutual configuration of the molecules, it follows that for a rectangular element of ordinary solid matter the tangential components of these tractions must be self-conjugate, as they are taken to be in the ordinary theory of elasticity. On the other hand, for a medium not molecularly constituted we can hardly treat at all of mutual configuration of parts, and the self-conjugate stress-relation will not be a necessary one. ”

“ We shall see that metallic alloys undergo continual physical and chemical changes. They are always seeking a more or less elusive equilibrium. Physicists in modern times have given their attention [Pg 264] to this internal activity of material bodies, to the pursuit of stability. Wiedemann, Warburg, Tomlinson, MM. Duguet, Brillouin, Duhem, and Bouasse have revived the old experimental researches of Coulomb and Wertheim on the elasticity of bodies, the effects of pressures and thrusts, the hammering, tempering, and annealing of metals. ”

“ The view I advocate is that the ether is a perfect continuum, an absolute plenum, and that therefore no rarefaction is possible. The ether inside matter is just as dense as the ether outside, and no denser. A material unit—say an electron—is only a peculiarity or singularity of some kind in the ether itself, which is of perfectly uniform density everywhere. What we "sense" as matter is an aggregate or grouping of an enormous number of such units. ”

“ If R converges to a point (x, y, z, t) , then the quotient of this mass, and the volume of R approaches a limit μ (x, y, z, t) , which is known as the mass-density at the space-time point (x, y, z, t) .
The principle of conservation of mass says—that for an infinitely thin space-time filament, the product μdJ, where μ = mass-density at the point (x, y, z, t) of the filament (i.e., the principal line of the filament) , dJ = contents of the cross-section normal to the t axis, and passing through (x, y, z, t) , is constant along the whole filament. ”

“ But in all probability, the dynamical polarity, or asymmetry of the cell is a very complicated phenomenon: for the obvious reason that, in any system, one asymmetry will tend to beget another. A chemical asymmetry will induce an inequality of surface-tension, which will lead directly to a modification of form; the chemical asymmetry may in turn be due to a process of electrolysis in a polarised electrical field; and again the chemical heterogeneity may be intensified into a chemical “polarity,” by the tendency of certain substances to seek a locus of greater or less surface-energy. ”

“ One knows how the cells of bees are formed by pressure, and how by pressure elastic spheres may be converted into angular, such as hexagonal-shaped, bodies.
As remarked, the gravitation theory (and many independent facts) demand that the molecules of bodies shall possess an open structure; which also satisfies the conditions of lightness and economy of material. As crystals exist, it is sometimes supposed that the molecules of bodies (whose open structure is often illustrated by cubes and other figures formed of wire) themselves represent the shapes of the crystals. ”

“ I must now call your attention to a purely geometrical question3 of vital interest with respect to homogeneous assemblages in general, and particularly the homogeneous assemblage of molecules constituting a crystal:—what can we take as ‘the’ boundary or ‘a’ boundary enclosing each molecule with whatever portion of space around it we are at liberty to choose for 15it, and separating it from neighbours and their portions of space given to them in homogeneous fairness? ”

“ Bryan: “The case of a polyatomic molecule, whose atoms are capable of vibrating relative to one another, affords an interesting field for investigation and speculation. Is the Boltzmann distribution still unique, or do other permanent distributions exist in which the kinetic energy is unequally divided?” ”

“ The micro-observer needs only to assimilate in his theory the physical hypothesis that all those special cases in which special exceptional conditions exist among the neighboring configurations of interacting atoms do not occur in nature, or, in other words, that the micro-states are in elementary disorder. Then the uniqueness of the macroscopic process is assured and with it, also, the fulfillment of the principle of increase of entropy in all directions. ”

“ We start building on a seed crystal of alum and continue until we reach a precise thickness. Here is a solution of chrome alum. You'll note the insulated tanks. Room temperature is maintained within half a degree. The solutions are held to within one-tenth of a degree. Crystal dimensions must be held to tolerances of little more than the thickness of a molecule—"
The gimmick to fool him and cheat him. ”

“ As regards any of the vectors , ..., we also have to consider the derivatives of its components with respect to the coordinates.
If we now eventually let fall our simplifying presupposition, and consider any molecule as a formation of, maybe, very complicated structure that contains several movable ions, then it is near at hand to assume, that still a relation like the one represented in (58) does exist. Our next task shall be, to simplify as much as possible the relation by means of certain, general considerations.
Simplification for transparent bodies. ”