**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 ).

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 *anisotropic*^{1}. 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.

^{1}A
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).

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 e*very 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.*

According to Schönflies 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 lattice*^{1}.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.*

^{1}Material 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**.
*

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.

T**he 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. T**he 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*.

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. René Juste
Haüy 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
growth^{1}- 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 (Romé 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 (Haüy, 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 preferred^{1}; 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.

^{1}*The 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).