D2. Mechanical Properties of solids

Extension of concept of symmetry

So far, we have only considered visible symmetry. However, the concept of symmetry extends further. Starting from visible regularity of the surface of a body, we find again corresponding regularity in the physical and chemical properties of the interior of bodies. Crystals display in directions, which appear externally to have equal growth velocities, equal strength and equal behaviour in the face of chemical and physical actions. Also, the internal structure of bodies displays often externally visible symmetry and one might suspect that the regularity of the internal structure causes the regularity of the outer form as well as the same behaviour in symmetric directions. An investigation of biological objects yields indeed externally visible symmetry also in the rough lay-out of the soft parts of bodies. In contrast, the examination of crystals has shown that the same symmetry extends the external form right into the finest details of the atomic structure and the interatomic field.

In order to give this concept of symmetry a strict formulation, we imagine that in addition to the external shape also all physical properties of a substance and finally also its atomic configuration and inter-atomic force field have been measured. Properties like density are given at each point by a single number (scalar). In this case, you can plot from the fixed point in all directions the same segment as measure of the density and thus obtain a spherical surface from the end points of the segments. However, properties like conductivity of heat cannot, in general, be expressed by a single number, since they differ in different directions. You can at each point plot the conductivity as a vector and thus obtain about is as end points of the vectors a more or less complicated surface instead of a sphere (Fig. 158). If you now imagine the physical properties represented in this manner by a spatial image, you can examine the symmetry of these surfaces and hence that of the physical properties in the same manner as the external shapes of bodies. If you want to examine the symmetry of the arrangement of atoms, you imagine the atoms represented by material points and otherwise proceed as above.

We will apply simply the concept of symmetry in this extended sense, that is, demand of every symmetry operation of a body that it does not only take it with respect to its external shape into an indiscernibly other position, but also with respect to its atomic configuration and its interatomic force field as well as with respect to all its physical and chemical properties. - Now we can discuss the different kinds of symmetry in detail and make them physically intuitive.

Homogeneity. Isotropy. Orthomorphy

Corresponding to the three kinds of symmetry operations, you distinguish three kinds of symmetry. Depending on whether the symmetry group of a body contains translations, rotations or reflections, we speak of homogeneity, isotropy and orthomorphy. In empty space, you can displace arbitrarily the rigid scale (metric), rotate and reflect it whithout bending or distorting it, that is, without changing the metric. This is why one says that the metric of the empty Euclidean space, or the empty space is completely homogeneous, isotropic and orthomorphous; its symmetry group contains all the symmetry operations, mentioned earlier under 1., 2. and 3., as well as their combinations. In the case of material bodies, we relate symmetry, and with it also homogeneity, isotropy and orthomorphy in the strict sense always to their fine structure (that is, atomic configurations and inter-atomic force fields).

You call a body homogeneous or inhomogeneous depending on whether its symmetry group contains or does not contain translations, that is, whether it comes through translations into a position which is physically indistinguishable from its initial position or not. Since you can go from one point of a body to any other always by a parallel displacement of the coordinate system, all points in a completely homogenous body are indistinguishable.

Totally homogeneous , that is, continuously homogeneous is only the empty (that is, without matter and fields) Euclidean space or, better still, its metric. In contrast, a material body, better, its fine structure cannot be continuously homogeneous, since it contains, as a consequence of its discontinuous structure, discernible points, for example, those with and those without matter (spaces between atoms). However, these inhomogeneities exist necessarily only in atomic measures, whence they contradict only continuous homogeneity and not homogeneity altogether.

In fact, the fine structure can be, for example, so regular, that it repeats itself in all three dimensions periodically and translatorily, that is, that its symmetry group contains finite translations; such a regularly built body is called spatially periodically homogeneous or, briefly homogeneous. It behaves with respect to the small parallelepipeds, which are formed out of the translation periods as edges, exactly homogeneously as the empty space with respect to single points; the corners of the parallelepipeds form a space lattice, and one understands then that the body attains a physically undiscernible position when it is displaced parallel to itself by arbitrary multiples of the edges of the parallelepiped.

Crystals are such periodically homogeneous discontinua (see below). Most methods of physical examination measure the properties of a substance only as a mean value taken over a (related to the atomic distances) large range, not at single points. If inside such a range the fine structure changes statistically without order in space, only a mean value is measured at each point and the substance appears to be, depending on the method of examination, homogeneous (more exactly, statistically homogeneous) or heterogeneous, depending on whether the mean value remains the same or changes from point to point. A geometric line is a strictly homogeneous continuum; if it is covered by equal, statistically disordered atoms; it becomes a discontinuum, but a statistically homogeneous one (because in a finite definable interval lie statistically the same number of atoms).

Gases and fluids are statistically homogeneous: Examined under a microscope, they do not display a structure, that is, light passes through them at every point in the same manner, whence they are said to be optically homogeneous; if you examine their density, their mechanical properties, etc., you discover them to be homogeneous during all sufficiently rough physical examinations . However, during X-ray illumination, you observe in them inhomogeneities of the arrangement of the atoms. If a body under a microscope exhibits structure, it means that its has at different points different transmissibility of light, it is optically inhomogeneous, heterogeneous. For most purposes, many materials can be considered to be homogeneous (for example, water, glass), but none is perfectly homogeneous as is shown by the colour dispersion of light.

We consider water to be perfectly homogeneous. However, if you were to distribute the amount of water, which can fill a football in a sphere of the size of Earth - or in other words, if you were to expand a sphere of water of the size of a football to the size of Earth, - then you would discover inhomogeneities in that the individual water molecules would be separated by spaces, which change between the diameters of the finest grist and footballs ( Lord Kelvin ).

Isotropy, anisotropy

You call a body isotropic or anisotropic at a point depending on whether each rotation about this point brings it into (in the method of examination in use) into an undistinguishable position or not. In an isotropic body, all directions at a point of examination are undiscernible. The empty space is completely istropic, that is continuously isotropic. In contrast, the structure of real bodies can just as little be continuously isotropic as it can be completely continuously homogeneous; only finite rotations corresponding to finite translations are compatible with the discontinuous fine structure of matter. If you cannot display by available means a difference between directions, a body is called statistically isotropic (in spite of its fine structure anisotropy).

In statistically homogeneous bodies (gases, fluids), the difference between directions will change statistically disordered from point to point, so that there arises statistical isotropy in the mean. In air, in water, in glass, there exist no preferred directions. Imagine yourself placed at any point in a room filled with air and start to move from this point; irrespectively of your direction, you will always find the same conditions - qualitatively and quantitatively: The same cohesion of the particles, the same elasticity, the same heat conductivity, etc. - more briefly, equality in every respect. Such a material is said to be isotropic.

It is quite different in a crystal. If you move from some point inside a crystal, you will find, in general, in each direction a different fine structure and, if you have sufficient experimental means of making a judgement, also other cohesion of the particles, other electric conductivity, etc. Such bodies are said to be anisotropic1. In other words: You call a body anisotropic or isotropic depending on the directional differences in the sense just described or depending on whether it has vectorial or scalar properties. [However, the anisotropy, the characteristic of crystals, is not restricted to the solid state; there exist also liquid crystals (Otto Lehmann 1855-1922 1889).We restrict ourselves here to this remark.] A special form of anisotropy is polarity in which direction and counter-direction differ.

1A proposal of Tammann makes a distinction between aggregate states (not gaseous, fluid, solid): Kinds of glass, fluids, gases, on the one hand (isotropic) and crystals, on the other hand (anisotropic). (Glass is considered to be a sub-cooled fluid with specially large internal friction, towards which all fluids trend as the temperature decreases.) His special justification is: Isotropic states can always be somehow continuously converted into each other (like isotropic fluid by cooling into amorphous glass), but not the others (however, just the last statement is not considered to be certain).

Ortho-morphy, enantio-morphy

You call a body (Fig. 152) depending on whether a reflection or asymmetry operation, that is, a symmetry operation involving a reflection, can bring it into an indistinguishable situation under the method of examination used or not - that is, depending on whether its symmetry group contains a reflection axis or not. Also the external form of a body can be orthomorphic or enantiomorphic.

For example, orthomorphic are all plane figures, as is shown in Fig. 153 (right) by a triangle in the XY-plane of the XYZ-system and in Fig.153 (left) by a triangle which lies in a plane inclined to all three axes. The execution of the construction involves reflection of the XYZ-system in the plane of the plane figure (triangle) , whence the system X'Y'Z' is obtained. Spheres, cylinders and cones are orthomorphic, that is, they are identical with their mirror images; in contrast, a tetrahedron with different sides is enantio-morphic. Therefore a non-symmetrical body is always enantio-morphic, but also a symmetric body can be enantio-morphic; in fact, if its symmetry group contains only rotation- and screw-axes, but no rotation-reflection-axis. For example, a straight pyramid, the base of which is a general, that is, a non-equilateral and non-orthogonal, parallelogram, is enantio-morphous and has the symmetry of a two-fold axis of rotation, Bodies with enantiomorphic symmetry can, analogously to the coordinate system, occur in two with respect to each other enantio-morpic-like forms (Fig. 152).

In every crystal system, there exist classes of crystals with and without reflection. Among those with reflection are all those crystals, which during totally undisturbed crystallization assume forms which are undistinguishable from their reflection images olike cubes, octohedra, regular tetrahedra, etc. Among the crystal classes without reflection - enantio-morphic crystal classes - are all crystals of shapes which differ from their mirror images like unequal-sided tetrahedra, etc. A single enantio-morphic crystal on its own has no plane of symmetry (as, for example, your right hand by itself has none; only together with the left hand, the pair of hands has mirror symmetry.). However, there exists always another crystal of the same material which behaves with respect to it like the left hand to the right hand. Every substance which generally enantiomorphicly crystalizes, thus always yields two forms of crystal which are enantio-morphic with respect to each other, that is, are interrelated like an image and its mirror image.

Symmetry of crystals

According to SchŲnflies 1853-1928 and Mikhail Mikhailovich v. Fedorow 1867-1929, one understands by a crystal a solid body, the fine structure (atomic configuration and ineratomic field of force) of which is so regular that it contains a symmetry group with a translation lattice1.The presence of the symmetry group is here emphasized, because the Morphology, Physics and Chemistry of crystals appear at first to be three different , but also cohesive subjects. However, the theory of symmetry joins them into one whole; those (partly empirical) fundamental laws of the three subjects, especially the laws of rationality (the morphological one of X-ray interference of Laue, the chemical measurement theory of Dalton and, finally, Weissenberg's analogue of Avogadro's law turn out to be different formulations of the same laws of symmetry, which rules all crystals. Prior to dealing with the morphology, Physics and Chemistry of crystals in detail, we must discuss the translation lattice as the most important part of crystal symmetry.

1Material bodies which are strictly homogeneous discontinua are called ideal crystals.

Lattice Theory of Crystals (Bravais 1848, Sohncke 1879, Schoenflies 1891, v. Fedorow 1894, v.Laue 1912, Weissenberg 1925)

Translation lattices can be viewed to have arisen in three steps: Let a material point execute a translation a and generate by repetition of the translation a chain of points. If a translation b follows, you obtain a lattice of points. If a translation c follows, you obtain a space point lattice. All of the translations of a chain of points - a lattice of points - form a chain of translations - the translation lattice, already referred to (Fig. 154). If you choose a coordinate system S in such a manner that its origin is a lattice point and its three axes, in the directions of the translations a, b, c, have as measurement scales the translation chains A, B, C, formed out of a, b, c, then the spatial image of the lattice (that is, of its base, elevation and side view) is given by the three lattices formed out of a, b, out of b, c, and out of c, a, whence the space lattice is fundamentally reduced to plane lattices. You employ as basic figures of the lattices only parallelograms and regular hexagons, because you can cover a plane periodically, homogeneously and completely only by means of these periodically repeated translations . Since other figures, especially 5-, 7-, and higher n-fold polygons cannot do so, 5-, 7-, and higher n-fold axes of symmetry are not compatible with a lattice structure.

These considerations, applied to lattices, yield that
1. the 5-, 7- and more-fold axes of symmetry are also incompatible with lattice structure and
2. only parallelograms and the regular 6-faced prism can serve as fundamental lattice figure and hence of crystals.

The fundamental figures can - as cannot be justified here in detail - have 7 and only 7 different lattices, Correspondingly, lattices and crystals are subdivided into seven symmetry groups (crystal systems). These are the 7 macroscopically distinguishable symmetries of translation lattices, but the symmetry possibilities of lattices are not thereby exhausted. The totality of the possible lattices is as follows:

The sets of parallel translation chains (g) and (h) (Fig. 155) decompose the plane into congruent parallelograms, all corners of which form a point lattice N; the same lattice can also be generated by the sets (g) and (d). Also the sets (d) and (e) generate a lattice; it contains all points of N (for at each point of N one d- and one e-line intersect), but contains yet more points - namely the centres of the faces of the parallelograms of N.

Hence there arises through (e) and (d) a new lattice N': You generate it out of N by addition of the centres M of the parallelograms, formed by (g) and (h), that is, the original parallelogram centred as the fundamental figure of N. Hence, applied to the lattices, you can derive from the fundamental figures 7 translation lattices by viewing the 3 edges which meet at one corner as lattice generating translations (Fig. 156, 4-th row).

In this process, the fundamental figures of the lattice (Elementary bodies) remain uncentred, except for the hexagonal prism, the base- and cover-face becomes thus centred (during the translation of the hexagon by one side length, one of its corners moves to the centre of the initial position). You may derive from these 7 lattices by further centralisation of the fundamental figures at the space- and face-centres further lattices, and, (according to Bravais) yet 7 more, altogether 14, and, in fact, only 14. [You arrive thus at three kinds of centred lattices:
1, If one centres
only one pair of side faces of the fundamental figure, the simple face centred lattice is obtained.
2. If you centre two side faces which meet at a corner of the fundamental figure, the
centralisation of the third face at that corner takes place automatically. You obtain the three-fold face-centred or simpler the face-centred lattice.
3. If you centre the fundamental figure,
its body centre, you obtain the body-centred or simpler the centred lattice (Fig. 156, the last three lattices of the rhombish system).

Fig. 156 shows all of them, in the top row, the simple and below the centred lattices. Those lattices, arranged below each other, agree in all of the directions of possible lattice-lines and -planes and only differ in the number of lattice points.

The rationality law of the translation lattice of Schoenflies.

The translation lattice as a part of the symmetry of crystals is the expression of the periodic homogeneity of the fine structure of crystals. Every lattice element (lattice-point, lattice-line, lattice-plane) must therefore also periodically homogeneously recur in space, repeat itself parallel to itself periodically in space and can therefore always be denoted by integral multiples of the given fundamental periods a, b, c , one says: It is rationally indexed. The corresponding mathematical formulation yields the rationality law of crystal lattices as a purely geometrical concept. It is the most general formulation of all empirical rationality laws of Crystal Physics and Crystal Chemistry, which can be derived from the morphology of crystals. We will present the law for points, lines and planes separately under the headings 1 - 3:

Projection of the lattice elements on to the coordinate planes and axes yields:

1. The three coordinates x y z of each lattice point are integral multiples of a b c.
2. The three projections
a b g of each translation of a lattice line on to three axes are integral multiples of a b c, that is, equal to ha and kb and lc. Their ratio a :b :g = ha : kb : lc determines their direction in the lattice; h, k, l are called the indices of the lines.
3. The three projections of each translation parallelogram of a lattice
plane of the three coordinate planes are integral multiples of the parallelograms formed in the planes of the axes from ab and bc and ca, that is, equal to wab and ubc and vca. The direction of the lattice-plane is dretermined by their ratios, The u v w are integers - the indices of the plane.

A plane's direction in space can also still be determined by the direction of its perpendicular, that is, by the ratio l : m : n of its 3 projections on to the axes or through the ratio A : B : C of the segments (measured from the origin) which they cut off from the axes. You have the relation

bc : vca : wab = l : m : n = 1/A : 1/B : 1/C = u/a : v/b : w/c.

Crystal systems, crystal classes and space groups.

Corresponding to the 7 symmetries of the fundamental figures, crystals are aligned in 7 crystal systems. In order to achieve a yet finer classification of the crystal lattices, one looks for all symmetry groups which can be distinguished macroscopically and are compatible with the lattice structure. Macroscopically viewed, the translations of a crystal must be considered to be vanishingly small. Therefore the difference between screw- and rotation-axes as well as between reflection- and sliding-reflection-planes vanishes. We find the sought for symmetry groups by production of all spatial combinations of the 1-, 2-, 3-, 4- and 6- fold rotation-screw- and rotation-reflection-axes. You find thus 32 symmetry groups (including, of course, the 7 symmetry groups of the crystal systems) - 32 crystal classes.

If you want to obtain all symmetry groups, which are compatible with one lattice (that is all combinations of 1-, 2-, 3-, 4- and 6- fold rotation-screw- and rotation-reflection-axes as well as slide-reflections), you obtain 230 different symmetry groups, the space groups, the finest possible subdivision (Schoenflies and Fedorow). If you neglect their translations, you obtain from them the above mentioned 32 symmetry classes.

Morphology of crystals

Every crystal can only have plane and straight line boundaries, which correspond to its translations; the regular polyhedra, which thus arise, have different forms depending on whether the crystal originated by evaporation, fusion or solution. If you make sure that in the experiment no direction in space is preferred (relative to the crystal) and you allow the crystal to float as freely as possible in steam, solution or melt, all these forms have the same symmetry group. Thus, the symmetry group of a crystal does not depend on its creation and is not only valid for an accidental form, but for all possible forms of boundaries. As a single symmetry group rules all forms of one crystal, the crystal symmetry groups rule all possible forms of boundaries of all crystals. Renť Juste HaŁy 1743-1822 and Neumann found first empirically as basic law of crystal morphology the Law of rational indices of edges and faces, J.F.C. Hessel 1830 discovered that crystal symmetries with 2-, 3-, 4- and 6- fold axes of rotation occur frequently, while 5- and 7- and all higher folds axes of rotation were never observed; this absence was the more remarkable, since in biological structures of animals and plants, for example, 5-fold symmetres are frequent.

From the point of view of lattice theory: The natural growth1- and solution-forms of crystals are to start with determined by the translation lattices; indeed, the edges and faces of crystals arise - as a consequence of the sharply pronounced anisotropy of their growth and solution-rates - parallel to the with matter most densely covered translation chains and lattices. The size of the edges and faces of crystals depend on the accidental conditions of growth and solution, respectively; however, the magnitude of the angles, which they enclose between them, depends on them, since the lattice can only increase or shrink during solution of the crystal without change of direction. Every crystal is therefore, to start with, determined by its characteristic angle between its faces and edges (Romť de l'Isle 1783).

1 You can shape crystals by grinding, pressing or other mechanical means arbitrarily; however, if you let them then grow or dissolve without application of external forces, they will resume a natural, regular external shape which corresponds to their internal fine structure.

The rationality law (HaŁy, Franz Neumann), derived from the purely geometric lattice for the directions of the translation-chains and -lattices, applies to the limiting-planes and -edges. It says for the macroscopically observed crystal forms: Choose the directions of 3 edges of a crystal as coordinate axes. You can then always find a proportion of the measure units of the three axes (axes ratio) a:b:c in such a manner that you can derive the ratio of the three projections a::b:: g of each crystal edge as well as the ratio of the 3 reciprocal sections of the axes A:B:C of each crystal plane , and hence also the ratio l : m : n of the three projections of the perpendicular of the crystal planes, as ratios, that is, through three integers out of the ratio of the axes. In other words: You find then at the crystal as earlier at the translation lattice :

ha : kb : lc = a::b:: g ,
(u/a) : (v/b) : (w/c) = (1/A) : (1/B): 1/C =
l : m : n .

For a practical evaluation, you simplify the ratio of the axes by dividing a:b:c by b, that is, you write instead of a:b:c:

(a/b) : 1 :( c/b) = a' : 1 : c',

whence

ha' : 1 : l/c' = a::b:: g ,
(u/a') : v : (w/c') = (1/A) : (1/B) : (1/C) =
l : m : n .

If you measure on a crystal the projections and axes cut-offs, respectively, of all crystal-edges and -faces, respectively, you can compute from the equations above to start with a' and c' (that is, the ratio of the axes) and then the integers hkl and uvw, respectively, that is, the indices of all edges and faces. In this manner, all the crystal-edges and -faces are indexed as integers. You choose the coordinate system, that is, the three edges of the crystal so that all edges and faces of the crystal are indexed by the smallest possible integers. The sytematics of the crystal shapes from the point of view of the 7 crystal systems and the rationality law are given in Fig. 156 above.

The crystal systems allow only differentiation with respect to the crystal angles and their rationality law. A finer differentiation is obtained by allowing the crystals to grow under ideal conditions, so that during their growth no direction in space is preferred1; then not only the angles, but also the face- and edge-quantities of the symmetry of the fine structure of a crystal form correspondingly in a regular manner. In the case of such crystal forms, you can fundamentally distinguish all 32 crystal classes, since then the directions and counter-directions as well as the left- and right-forms become distinguishable; such a differentiation is not possible neither in the case of the fundamental figures nor in the translation lattices nor by means of the rationality law. If each crystal face is developed equally large in all positions, as is to be expected according to the symmetry of the crystal system, you speak of holohedrism. Correspondingly, you call forms of formation which have only fully formed one half or one quarter , respectively, of the faces expected according to the crystal system , hemihedrism and tetrahedrism.

1The preference for a definite direction in which a crystal grows enables you to influence a metal crystal, which grows in the melt (wolfram, bismuth, zinc, tin, lead, cadmium, aluminium) in such a manner that it only grows in one direction: In the cooling melt crystallisation nuclei form. The nuclei grow in that molecules deposit themselves to them and form grains. If you extract a crystal grain from the melt at the same velocity, at which it grew by addition of molecules, the crystal only grows in the direction of this motion. You call such a one-dimensionally grown crystal a single crystal wire. (Method of production of wolfram wires for incandescent lamps).

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