Chapter I: Crystalline Form (1)

The fundamental laws governing the form of crystals are:--

1. Law of the Constancy of Angle.

2. Law of Symmetry.

3. Law of Rational Intercepts or Indices.

According to the first law, the angles between corresponding faces of all crystals of the same chemical substance are always the same and are characteristic of the substance.

(a) _Symmetry of Crystals._

Crystals may, or may not, be symmetrical with respect to a point, a line or axis, and a plane; these "elements of symmetry" are spoken of as a centre of symmetry, an axis of symmetry, and a plane of symmetry respectively.

_Centre of Symmetry._--Crystals which are centro-symmetrical have their faces arranged in parallel pairs; and the two parallel faces, situated on opposite sides of the centre (O in fig. 3) are alike in surface characters, such as lustre, striations, and figures of corrosion. An octahedron (fig. 3) is bounded by four pairs of parallel faces. Crystals belonging to many of the hemihedral and tetartohedral classes of the six systems of crystallization are devoid of a centre of symmetry.

_Axes of Symmetry._--Consider the vertical axis joining the opposite corners a3 and a'3 of an octahedron (fig. 3) and passing through its centre O: by rotating the crystal about this axis through a right angle (90 deg.) it reaches a position such that the orientation of its faces is the same as before the rotation; the face a'1a'2a'3, for example, coming into the position of a1a'2a3. During a complete rotation of 360 deg. (= 90 deg. X 4), the crystal occupies four such interchangeable positions. Such an axis of symmetry is known as a tetrad axis of symmetry. Other tetrad axes of the octahedron are a2a'2 and a1a1.

An axis of symmetry of another kind is that which passing through the centre O is normal to a face of the octahedron. By rotating the crystal about such an axis Op (fig. 3) through an angle of 120 deg. those faces which are not perpendicular to the axis occupy interchangeable positions; for example, the face a1a3a2 comes into the position of a'2a1a'3, and a'2a1a'3 to a3a'2a'1. During a complete rotation of 360 deg. (= 120 deg. X 3) the crystal occupies similar positions three times. This is a triad axis of symmetry; and there being four pairs of parallel faces on an octahedron, there are four triad axes (only one of which is drawn in the figure).

Axes and Planes of Symmetry of an Octahedron.]

An axis passing through the centre O and the middle points d of two opposite edges of the octahedron (fig. 4), i.e. parallel to the edges of the octahedron, is a dyad axis of symmetry. About this axis there may be rotation of 180 deg., and only twice in a complete revolution of 360 deg. (= 180 deg. X 2) is the crystal brought into interchangeable positions. There being six pairs of parallel edges on an octahedron, there are consequently six dyad axes of symmetry.

A regular octahedron thus possesses thirteen axes of symmetry (of three kinds), and there are the same number in the cube. Fig. 5 shows the three tetrad (or tetragonal) axes (aa), four triad (or trigonal) axes (pp), and six dyad (diad or diagonal) axes (dd).

Although not represented in the cubic system, there is still another kind of axis of symmetry possible in crystals. This is the hexad axis or hexagonal axis, for which the angle of rotation is 60 deg., or one-sixth of 360 deg. There can be only one hexad axis of symmetry in any crystal (see figs. 77-80).

_Planes of Symmetry._--A regular octahedron can be divided into two equal and similar halves by a plane passing through the corners a1a3a'1a'3 and the centre O (fig. 3). One-half is the mirror reflection of the other in this plane, which is called a plane of symmetry. Corresponding planes on either side of a plane of symmetry are inclined to it at equal angles. The octahedron can also be divided by similar planes of symmetry passing through the corners a1a2a'1a'2 and a2a3a'2a'3. These three similar planes of symmetry are called the cubic planes of symmetry, since they are parallel to the faces of the cube (compare figs. 6-8, showing combinations of the octahedron and the cube).

A regular octahedron can also be divided symmetrically into two equal and similar portions by a plane passing through the corners a3 and a'3, the middle points d of the edges a1a'2 and a'1a2, and the centre O (fig. 4). This is called a dodecahedral plane of symmetry, being parallel to the face of the rhombic dodecahedron which truncates the edge a1a2 (compare fig. 14, showing a combination of the octahedron and rhombic dodecahedron). Another similar plane of symmetry is that passing through the corners a3a'3 and the middle points of the edges a1a2 and a'1a'2, and altogether there are six dodecahedral planes of symmetry, two through each of the corners a1, a2, a3 of the octahedron.

A regular octahedron and a cube are thus each symmetrical with respect to the following elements of symmetry: a centre of symmetry, thirteen axes of symmetry (of three kinds), and nine planes of symmetry (of two kinds). This degree of symmetry, which is the type corresponding to one of the classes of the cubic system, is the highest possible in crystals. As will be pointed out below, it is possible, however, for both the octahedron and the cube to be associated with fewer elements of symmetry than those just enumerated.

(b) _Simple Forms and Combinations of Forms._

A single face a1a2a3 (figs. 3 and 4) may be repeated by certain of the elements of symmetry to give the whole eight faces of the octahedron. Thus, by rotation about the vertical tetrad axis a3a'3 the four upper faces are obtained; and by rotation of these about one or other of the horizontal tetrad axes the eight faces are derived. Or again, the same repetition of the faces may be arrived at by reflection across the three cubic planes of symmetry. (By reflection across the six dodecahedral planes of symmetry a tetrahedron only would result, but if this is associated with a centre of symmetry we obtain the octahedron.) Such a set of similar faces, obtained by symmetrical repetition, constitutes a "simple form." An octahedron thus consists of eight similar faces, and a cube is bounded by six faces all of which have the same surface characters, and parallel to each of which all the properties of the crystal are identical.

Examples of simple forms amongst crystallized substances are octahedra of alum and spinel and cubes of salt and fluorspar. More usually, however, two or more forms are present on a crystal, and we then have a combination of forms, or simply a "combination." Figs. 6, 7 and 8 represent combinations of the octahedron and the cube; in the first the faces of the cube predominate, and in the third those of the octahedron; fig. 7 with the two forms equally developed is called a cubo-octahedron. Each of these combined forms has all the elements of symmetry proper to the simple forms.

The simple forms, though referable to the same type of symmetry and axes of reference, are quite independent, and cannot be derived one from the other by symmetrical repetition, but, after the manner of Rome de l'Isle, they may be derived by replacing edges or corners by a face equally inclined to the faces forming the edges or corners; this is known as "truncation" (Lat. _truncare_, to cut off). Thus in fig. 6 the corners of the cube are symmetrically replaced or truncated by the faces of the octahedron, and in fig. 8 those of the octahedron are truncated by the cube.

(c) _Law of Rational Intercepts._

For axes of reference, OX, OY, OZ (fig. 9), take any three edges formed by the intersection of three faces of a crystal. These axes are called the crystallographic axes, and the planes in which they lie the axial planes. A fourth face on the crystal intersecting these three axes in the points A, B, C is taken as the parametral plane, and the lengths OA : OB : OC are the parameters of the crystal. Any other face on the crystal may be referred to these axes and parameters by the ratio of the intercepts

OA OB OC
-- : -- : --.
h k l

Thus for a face parallel to the plane A Be the intercepts are in the ratio OA : OB : Oe, or

OA OB OC
-- : -- : --
1 1 2

and for a plane fgC' they are Of : Og : OC' or

OA OB OC'
-- : -- : ---.
2 3 1

Now the important relation existing between the faces of a crystal is that the denominators h, k and l are always rational whole numbers, rarely exceeding 6, and usually 0, 1, 2 or 3. Written in the form (hkl), h referring to the axis OX, k to OY, and l to OZ, they are spoken of as the indices (Millerian indices) of the face. Thus of a face parallel to the plane ABC the indices are (111), of A Be they are (112), and of fgC' (231'). The indices are thus inversely proportional to the intercepts, and the law of rational intercepts is often spoken of as the "law of rational indices."

The angular position of a face is thus completely fixed by its indices; and knowing the angles between the axial planes and the parametral plane all the angles of a crystal can be calculated when the indices of the faces are known.

Although any set of edges formed by the intersection of three planes may be chosen for the crystallographic axes, it is in practice usual to select certain edges related to the symmetry of the crystal, and usually coincident with axes of symmetry; for then the indices will be simpler and all faces of the same simple form will have a similar set of indices. The angles between the axes and the ratio of the lengths of the parameters OA: OB: OC (usually given as a: b: c) are spoken of as the "elements" of a crystal, and are constant for and characteristic of all crystals of the same substance.

The six systems of crystal forms, to be enumerated below, are defined by the relative inclinations of the crystallographic axes and the lengths of the parameters. In the cubic system, for example, the three crystallographic axes are taken parallel to the three tetrad axes of symmetry, i.e. parallel to the edges of the cube (fig. 5) or joining the opposite corners of the octahedron (fig. 3), and they are therefore all at right angles; the parametral plane (111) is a face of the octahedron, and the parameters are all of equal length. The indices of the eight faces of the octahedron will then be (111), (1'11), (11'1), (1'1'1), (111'), (1'11'), (11'1'), (1'1'1'). The symbol {111} indicates all the faces belonging to this simple form. The indices of the six faces of the cube are (100), (010), (001), (1'00), (01'0), (001'); here each face is parallel to two axes, i.e. intercepts them at infinity, so that the corresponding indices are zero.

(d) _Zones._

An important consequence of the law of rational intercepts is the arrangement of the faces of a crystal in zones. All faces, whether they belong to one or more simple forms, which intersect in parallel edges are said to lie in the same zone. A line drawn through the centre O of the crystal parallel to these edges is called a zone-axis, and a plane perpendicular to this axis is called a zone-plane. On a cube, for example, there are three zones each containing four faces, the zone-axes being coincident with the three tetrad axes of symmetry. In the crystal of zircon (fig. 88) the eight prism-faces a, m, &c. constitute a zone, denoted by [a, m, a', &c.], with the vertical tetrad axis of symmetry as zone-axis. Again the faces [a, x, p, e', p', x"', a"] lie in another zone, as may be seen by the parallel edges of intersection of the faces in figs. 87 and 88; three other similar zones may be traced on the same crystal.

The direction of the line of intersection (i.e. zone-axis) of any two planes (hkl) and (h1k1l1) is given by the zone-indices [uvw], where u = kl1 - lk1, v = lh1 - hl1, and w = hk1 - kh1, these being obtained from the face-indices by cross multiplication as follows:--

h k l h k l
X X X
h1 k1 l1 h1 k1 l1.

Any other face (h2k2l2) lying in this zone must satisfy the equation

h2u + k2v + l2w = 0.

This important relation connecting the indices of a face lying in a zone with the zone-indices is known as Weiss's zone-law, having been first enunciated by C. S. Weiss. It may be pointed out that the indices of a face may be arrived at by adding together the indices of faces on either side of it and in the same zone; thus, (311) in fig. 12 lies at the intersections of the three zones [210, 101], [201, 110] and [211, 100], and is obtained by adding together each set of indices.

(e) _Projection and Drawing of Crystals._

The shapes and relative sizes of the faces of a crystal being as a rule accidental, depending only on the distance of the faces from the centre of the crystal and not on their angular relations, it is often more convenient to consider only the directions of the normals to the faces. For this purpose projections are drawn, with the aid of which the zonal relations of a crystal are more readily studied and calculations are simplified.

The kind of projection most extensively used is the "stereographic projection." The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zone-circles. The calculation of the angles between the normals of faces and between zone-circles is then performed by the ordinary methods of spherical trigonometry. The stereographic projection, however, represents the poles and zone-circles on a plane surface and not on a spherical surface. This is achieved by drawing lines joining all the poles of the faces with the north or south pole of the sphere and finding their points of intersection with the plane of the equatorial great circle, or primitive circle, of the sphere, the projection being represented on this plane. In fig. 10 is shown the stereographic projection, or stereogram, of a cubic crystal; a^1, a^2, &c. are the poles of the faces of the cube. o^1, o^2, &c. those of the octahedron, and d^1, d^2, &c. those of the rhombic dodecahedron. The straight lines and circular arcs are the projections on the equatorial plane of the great circles in which the nine planes of symmetry intersect the sphere. A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig. 11, in which the faces are lettered the same as the corresponding poles in the projection. From the zone-circles in the projection and the parallel edges in the drawing the zonal relations of the faces are readily seen: thus [a^1o^1d^5], [a^1d^1a^5], [a^5o^1d^2], &c. are zones. A stereographic projection of a rhombohedral crystal is given in fig. 72.

Another kind of projection in common use is the "gnomonic projection" (fig. 12). Here the plane of projection is tangent to the sphere, and normals to all the faces are drawn from the centre of the sphere to intersect the plane of projection. In this case all zones are represented by straight lines. Fig. 12 is the gnomonic projection of a cubic crystal, the plane of projection being tangent to the sphere at the pole of an octahedral face (111), which is therefore in the centre of the projection. The indices of the several poles are given in the figure.

In drawing crystals the simple plans and elevations of descriptive geometry (e.g. the plans in the lower part of figs. 87 and 88) have sometimes the advantage of showing the symmetry of a crystal, but they give no idea of solidity. For instance, a cube would be represented merely by a square, and an octahedron by a square with lines joining the opposite corners. True perspective drawings are never used in the representation of crystals, since for showing the zonal relations it is important to preserve the parallelism of the edges. If, however, the eye, or point of vision, is regarded as being at an infinite distance from the object all the rays will be parallel, and edges which are parallel on the crystal will be represented by parallel lines in the drawing. The plane of the drawing, in which the parallel rays joining the corners of the crystals and the eye intersect, may be either perpendicular or oblique to the rays; in the former case we have an "orthographic" ([Greek: orthos], straight; [Greek: graphein], to draw) drawing, and in the latter a "clinographic" ([Greek: klinein], to incline) drawing. Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. 11) a cube face a^5 is first placed parallel to the plane on which the crystal is to be projected and with one set of edges vertical; the crystal is then turned through a small angle about a vertical axis until a second cube face a^2 comes into view, and the eye is then raised so that a third cube face a^1 may be seen.

(f) _Crystal Systems and Classes._

According to the mutual inclinations of the crystallographic axes of reference and the lengths intercepted on them by the parametral plane, all crystals fall into one or other of six groups or systems, in each of which there are several classes depending on the degree of symmetry. In the brief description which follows of these six systems and thirty-two classes of crystals we shall proceed from those in which the symmetry is most complex to those in which it is simplest.

1. CUBIC SYSTEM

(Isometric; Regular; Octahedral; Tesseral).

In this system the three crystallographic axes of reference are all at
right angles to each other and are equal in length. They are parallel
to the edges of the cube, and in the different classes coincide either
with tetrad or dyad axes of symmetry. Five classes are included in
this system, in all of which there are, besides other elements of
symmetry, four triad axes.

In crystals of this system the angle between any two faces P and Q
with the indices (hkl) and (pqr) is given by the equation

hp + kq + lr
COS PQ = ----------------------------------------
[root] [(h^2 +k^2 +l^2) (p^2 +q^2 +r^2)].

The angles between faces with the same indices are thus the same in
all substances which crystallize in the cubic system: in other systems
the angles vary with the substance and are characteristic of it.

HOLOSYMMETRIC CLASS

(Holohedral ([Greek: holos], whole); Hexakis-octahedral).

Crystals of this class possess the full number of elements of symmetry
already mentioned above for the octahedron and the cube, viz. three
cubic planes of symmetry, six dodecahedral planes, three tetrad axes
of symmetry, four triad axes, six dyad axes, and a centre of symmetry.

There are seven kinds of simple forms, viz.:--

Cube (fig. 5). This is bounded by six square faces parallel to the
cubic planes of symmetry; it is known also as the hexahedron. The
angles between the faces are 90 deg., and the indices of the form are
{100}. Salt, fluorspar and galena crystallize in simple cubes.

Octahedron (fig. 3). Bounded by eight equilateral triangular faces
perpendicular to the triad axes of symmetry. The angles between the
faces are 70 deg. 32' and 109 deg. 28', and the indices are {111}.
Spinel, magnetite and gold crystallize in simple octahedra.
Combinations of the cube and octahedron are shown in figs. 6-8.

Rhombic dodecahedron (fig. 13). Bounded by twelve rhomb-shaped faces
parallel to the six dodecahedral planes of symmetry. The angles
between the normals to adjacent faces are 60 deg., and between other
pairs of faces 90 deg.; the indices are {110}. Garnet frequently
crystallizes in this form. Fig. 14 shows the rhombic dodecahedron in
combination with the octahedron.

In these three simple forms of the cubic system (which are shown in
combination in fig. 11) the angles between the faces and the indices
are fixed and are the same in all crystals; in the four remaining
simple forms they are variable.

Triakis-octahedron (three-faced octahedron) (fig. 15). This solid is
bounded by twenty-four isosceles triangles, and may be considered as
an octahedron with a low triangular pyramid on each of its faces. As
the inclinations of the faces may vary there is a series of these
forms with the indices {221}, {331}, {332}, &c. or in general {hhk}.

Icositetrahedron (fig. 17). Bounded by twenty-four trapezoidal faces,
and hence sometimes called a "trapezohedron." The indices are {211},
{311}, {322}, &c., or in general {hkk}. Analcite, leucite and garnet
often crystallize in the simple form {211}. Combinations are shown in
figs. 18-20. The plane A Be in fig. 9 is one face (112) of an
icositetrahedron; the indices of the remaining faces in this octant
being (211) and (121).

Tetrakis-hexahedron (four-faced cube) (figs. 21 and 22). Like the
triakis-octahedron this solid is also bounded by twenty-four isosceles
triangles, but here grouped in fours over the cubic faces. The two
figures show how, with different inclinations of the faces, the form
may vary, approximating in fig. 21 to the cube and in fig. 22 to the
rhombic dodecahedron. The angles over the edges lettered A are
different from the angles over the edges lettered C. Each face is
parallel to one of the crystallographic axes and intercepts the two
others in different lengths; the indices are therefore {210}, {310},
{320}, &c., in general {hko}. Fluorspar sometimes crystallizes in the
simple form {310}; more usually, however, in combination with the cube
(fig. 23).

Hexakis-octahedron (fig. 24). Here each face of the octahedron is
replaced by six scalene triangles, so that altogether there are
forty-eight faces. This is the greatest number of faces possible for
any simple form in crystals. The faces are all oblique to the planes
and axes of symmetry, and they intercept the three crystallographic
axes in different lengths, hence the indices are all unequal, being in
general {hkl}, or in particular cases {321}, {421}, {432}, &c. Such a
form is known as the "general form" of the class. The interfacial
angles over the three edges of each triangle are all different. These
forms usually exist only in combination with other cubic forms (for
example, fig. 25), but {421} has been observed as a simple form on
fluorspar.

Several examples of substances which crystallize in this class have
been mentioned above under the different forms; many others might be
cited--for instance, the metals iron, copper, silver, gold, platinum,
lead, mercury, and the non-metallic elements silicon and phosphorus.

TETRAHEDRAL CLASS

(Tetrahedral-hemihedral; Hexakis-tetrahedral).

In this class there is no centre of symmetry nor cubic planes of
symmetry; the three tetrad axes become dyad axes of symmetry, and the
four triad axes are polar, i.e. they are associated with different
faces at their two ends. The other elements of symmetry (six
dodecahedral planes and six dyad axes) are the same as in the last
class.

Of the seven simple forms, the cube, rhombic dodecahedron and
tetrakis-hexahedron are geometrically the same as before, though on
actual crystals the faces will have different surface characters. For
instance, the cube faces will be striated parallel to only one of the
diagonals (fig. 90), and etched figures on this face will be
symmetrical with respect to two lines, instead of four as in the last
class. The remaining simple forms have, however, only half the number
of faces as the corresponding form in the last class, and are spoken
of as "hemihedral with inclined faces."

Tetrahedron (fig. 26). This is bounded by four equilateral triangles
and is identical with the regular tetrahedron of geometry. The angles
between the normals to the faces are 109 deg. 28'. It may be derived
from the octahedron by suppressing the alternate faces.

Deltoid[1] dodecahedron (fig. 27). This is the hemihedral form of the
triakis-octahedron; it has the indices {hhk} and is bounded by twelve
trapezoidal faces.

Triakis-tetrahedron (fig. 28). The hemihedral form {hkk} of the
icositetrahedron; it is bounded by twelve isosceles triangles arranged
in threes over the tetrahedron faces.

Hexakis-tetrahedron (fig. 29). The hemihedral form {hkl} of the
hexakis-octahedron; it is bounded by twenty-four scalene triangles and
is the general form of the class.

Corresponding to each of these hemihedral forms there is another
geometrically similar form, differing, however, not only in
orientation, but also in actual crystals in the characters of the
faces. Thus from the octahedron there may be derived two tetrahedra
with the indices {111} and {1'11}, which may be distinguished as
positive and negative respectively. Fig. 30 shows a combination of
these two tetrahedra, and represents a crystal of blende, in which the
four larger faces are dull and striated, whilst the four smaller are
bright and smooth. Figs. 31-33 illustrate other tetrahedral
combinations.

Tetrahedrite, blende, diamond, boracite and pharmacosiderite are
substances which crystallize in this class.

PYRITOHEDRAL[2] CLASS

(Parallel-faced hemihedral; Dyakis-dodecahedral).

Crystals of this class possess three cubic planes of symmetry but no
dodecahedral planes. There are only three dyad axes of symmetry, which
coincide with the crystallographic axes; in addition there are three
triad axes and a centre of symmetry.

Here the cube, octahedron, rhombic dodecahedron, triakis-octahedron
and icositetrahedron are geometrically the same as in the first class.
The characters of the faces will, however, be different; thus the cube
faces will be striated parallel to one edge only (fig. 89), and
triangular markings on the octahedron faces will be placed obliquely
to the edges. The remaining simple forms are "hemihedral with parallel
faces," and from the corresponding holohedral forms two hemihedral
forms, a positive and a negative, may be derived.

Pentagonal dodecahedron (fig. 34). This is bounded by twelve
pentagonal faces, but these are not regular pentagons, and the angles
over the three sets of different edges are different. The regular
dodecahedron of geometry, contained by twelve regular pentagons, is
not a possible form in crystals. The indices are {hko}: as a simple
form {210} is of very common occurrence in pyrites.

Dyakis-dodecahedron (fig. 35). This is the hemihedral form of the
hexakis-octahedron and has the indices {hkl}; it is bounded by
twenty-four faces. As a simple form {321} is met with in pyrites.

Combinations (figs. 36-39) of these forms with the cube and the
octahedron are common in pyrites. Fig. 37 resembles in general
appearance the regular icosahedron of geometry, but only eight of the
faces are equilateral triangles. Cobaltite, smaltite and other
sulphides and sulpharsenides of the pyrites group of minerals
crystallize in these forms. The alums also belong to this class; from
an aqueous solution they crystallize as simple octahedra, sometimes
with subordinate faces of the cube and rhombic dodecahedron, but from
an acid solution as octahedra combined with the pentagonal
dodecahedron {210}.

PLAGIHEDRAL[3] CLASS

(Plagihedral-hemihedral; Pentagonal icositetrahedral; Gyroidal[4]).

In this class there are the full number of axes of symmetry (three
tetrad, four triad and six dyad), but no planes of symmetry and no
centre of symmetry.

Pentagonal icositetrahedron (fig. 40). This is the only simple form in
this class which differs geometrically from those of the holosymmetric
class. By suppressing either one or other set of alternate faces of
the hexakis-octahedron two pentagonal icositetrahedra {hkl} and {khl}
are derived. These are each bounded by twenty-four irregular
pentagons, and although similar to each other they are respectively
right- and left-handed, one being the mirror image of the other; such
similar but nonsuperposable forms are said to be enantiomorphous
([Greek: enantios], opposite, and [Greek: morphe], form), and crystals
showing such forms sometimes rotate the plane of polarization of
plane-polarized light. Faces of a pentagonal icositetrahedron with
high indices have been very rarely observed on crystals of cuprite,
potassium chloride and ammonium chloride, but none of these are
circular polarizing.

TETARTOHEDRAL CLASS

(Tetrahedral pentagonal dodecahedral).

Here, in addition to four polar triad axes, the only other elements of
symmetry are three dyad axes, which coincide with the crystallographic
axes. Six of the simple forms, the cube, tetrahedron, rhombic
dodecahedron, deltoid dodecahedron, triakis-tetrahedron and pentagonal
dodecahedron, are geometrically the same in this class as in either
the tetrahedral or pyritohedral classes. The general form is the
Tetrahedral pentagonal dodecahedron (fig. 41). This is bounded by
twelve irregular pentagons, and is a tetartohedral or quarter-faced
form of the hexakis-octahedron. Four such forms may be derived, the
indices of which are {hkl}, {khl}, {h'kl} and {k'hl}; the first pair
are enantiomorphous with respect to one another, and so are the last
pair. Barium nitrate, lead nitrate, sodium chlorate and sodium bromate
crystallize in this class, as also do the minerals ullmannite (NiSbS)
and langbeinite (K2Mg2(SO4)3).

2. TETRAGONAL SYSTEM

(Pyramidal; Quadratic; Dimetric).

In this system the three crystallographic axes are all at right
angles, but while two are equal in length and interchangeable the
third is of a different length. The unequal axis is spoken of as the
principal axis or morphological axis of the crystal, and it is always
placed in a vertical position; in five of the seven classes of this
system it coincides with the single tetrad axis of symmetry.

Tetragonal Bipyramids.]

The parameters are a : a : c, where a refers to the two equal
horizontal axes, and c to the vertical axis; c may be either shorter
(as in fig. 42) or longer (fig. 43) than a. The ratio a : c is spoken
of as the axial ratio of a crystal, and it is dependent on the angles
between the faces. In all crystals of the same substance this ratio is
constant, and is characteristic of the substance; for other substances
crystallizing in the tetragonal system it will be different. For
example, in cassiterite it is given as a : c = 1 : 0.67232 or simply
as c = 0.67232, a being unity; and in anatase as c = 1.7771.

HOLOSYMMETRIC CLASS

(Holohedral; Ditetragonal bipyramidal).

Crystals of this class are symmetrical with respect to five planes,
which are of three kinds; one is perpendicular to the principal axis,
and the other four intersect in it; of the latter, two are
perpendicular to the equal crystallographic axes, while the two others
bisect the angles between them. There are five axes of symmetry, one
tetrad and two pairs of dyad, each perpendicular to a plane of
symmetry. Finally, there is a centre of symmetry.

There are seven kinds of simple forms, viz.:--

Tetragonal bipyramid of the first order (figs. 42 and 43). This is
bounded by eight equal isosceles triangles. Equal lengths are
intercepted on the two horizontal axes, and the indices are {111},
{221}, {112}, &c., or in general {hhl}. The parametral plane with the
intercepts a : a : c is a face of the bipyramid {111}.

Tetragonal Bipyramids of the first and second orders.]

Tetragonal bipyramid of the second order. This is also bounded by
eight equal isosceles triangles, but differs from the last form in its
position, four of the faces being parallel to each of the horizontal
axes; the indices are therefore {101}, {201}, {102}, &c., or {hol}.

Fig. 44 shows the relation between the tetragonal bipyramids of the
first and second orders when the indices are {111} and {101}
respectively: ABB is the face (111), and ACC is (101). A combination
of these two forms is shown in fig. 45.

Ditetragonal bipyramid (fig. 46). This is the general form; it is
bounded by sixteen scalene triangles, and all the indices are unequal,
being {321}, &c., or {hkl}.

Tetragonal prism of the first order. The four faces intersect the
horizontal axes in equal lengths and are parallel to the principal
axis; the indices are therefore {110}. This form does not enclose
space, and is therefore called an "open form" to distinguish it from a
"closed form" like the tetragonal bipyramids and all the forms of the
cubic system. An open form can exist only in combination with other
forms; thus fig. 47 is a combination of the tetragonal prism {110}
with the basal pinacoid {001}. If the faces (110) and (001) are of
equal size such a figure will be geometrically a cube, since all the
angles are right angles; the variety of apophyllite known as tesselite
crystallizes in this form.

Tetragonal prism of the second order. This has the same number of
faces as the last prism, but differs in position; each face being
parallel to the vertical axis and one of the horizontal axes; the
indices are {100}.

Ditetragonal prism. This consists of eight faces all parallel to the
principal axis and intercepting the horizontal axes in different
lengths; the indices are {210}, {320}, &c., or {hko}.

Basal pinacoid (from [Greek: pinax], a tablet). This consists of a
single pair of parallel faces perpendicular to the principal axis. It
is therefore an open form and can exist only in combination (fig. 47).

Combinations of Tetragonal Prisms and Pyramids.]

Combinations of holohedral tetragonal forms are shown in figs. 47-49;
fig. 48 is a combination of a bipyramid of the first order with one of
the second order and the prism of the first order; fig. 49 a
combination of a bipyramid of the first order with a ditetragonal
bipyramid and the prism of the second order. Compare also figs. 87 and
88.

Examples of substances which crystallize in this class are
cassiterite, rutile, anatase, zircon, thorite, vesuvianite,
apophyllite, phosgenite, also boron, tin, mercuric iodide.

SCALENOHEDRAL CLASS

(Bisphenoidal-hemihedral).

Here there are only three dyad axes and two planes of symmetry, the
former coinciding with the crystallographic axes and the latter
bisecting the angles between the horizontal pair. The dyad axis of
symmetry, which in this class coincides with the principal axis of the
crystal, has certain of the characters of a tetrad axis, and is
sometimes called a tetrad axis of "alternating symmetry"; a face on
the upper half of the crystal if rotated through 90 deg. about this
axis and reflected across the equatorial plane falls into the position
of a face on the lower half of the crystal. This kind of symmetry,
with simultaneous rotation about an axis and reflection across a
plane, is also called "composite symmetry."

In this class all except two of the simple forms are geometrically the
same as in the holosymmetric class.

Bisphenoid ([Greek: sphen], a wedge) (fig. 50). This is a double
wedge-shaped solid bounded by four equal isosceles triangles; it has
the indices {111}, {211}, {112}, &c., or in general {hhl}. By
suppressing either one or other set of alternate faces of the
tetragonal bipyramid of the first order (fig. 42) two bisphenoids are
derived, in the same way that two tetrahedra are derived from the
regular octahedron.

Tetragonal scalenohedron or ditetragonal bisphenoid (fig. 51). This is
bounded by eight scalene triangles and has the indices {hkl}. It may
be considered as the hemihedral form of the ditetragonal bipyramid.

The crystal of chalcopyrite (CuFeS2) represented in fig. 52 is a
combination of two bisphenoids (P and P'), two bipyramids of the
second order (b and c), and the basal pinacoid (a). Stannite
(Cu2FeSnS4), acid potassium phosphate (H2KPO4), mercuric cyanide, and
urea (CO(NH2)2) also crystallize in this class.

BIPYRAMIDAL CLASS

(Parallel-faced hemihedral).

The elements of symmetry are a tetrad axis with a plane perpendicular
to it, and a centre of symmetry. The simple forms are the same here as
in the holosymmetric class, except the prism {hko}, which has only
four faces, and the bipyramid {hkl}, which has eight faces and is
distinguished as a "tetragonal pyramid of the third order."

Fig. 53 shows a combination of a tetragonal prism of the first order
with a tetragonal bipyramid of the third order and the basal pinacoid,
and represents a crystal of fergusonite. Scheelite (q.v.), scapolite
(q.v.), and erythrite (C4H10O4) also crystallize in this class.

PYRAMIDAL CLASS

(Hemimorphic-tetartohedral).

Here the only element of symmetry is the tetrad axis. The pyramids of
the first {hhl}, second {hol} and third {hkl} orders have each only
four faces at one or other end of the crystal, and are hemimorphic.
All the simple forms are thus open forms.

Examples are wulfenite (PbMoO4) and barium antimonyl dextro-tartrate
(Ba(SbO)2(C4H4O6).H2O).

DITETRAGONAL PYRAMIDAL CLASS

(Hemimorphic-hemihedral).

Here there are two pairs of vertical planes of symmetry intersecting
in the tetrad axis. The pyramids {hhl} and {hol} and the bipyramid
{hkl} are all hemimorphic.

Examples are iodosuccimide (C4H4O2NI), silver fluoride (AgF.H2O), and
penta-erythrite (C5H12O4). No examples are known amongst minerals.

TRAPEZOHEDRAL CLASS

(Trapezohedral-hemihedral).

Here there are the full number of axes of symmetry, but no planes or
centre of symmetry. The general form {hkl} is bounded by eight
trapezoidal faces and is the tetragonal trapezohedron.

Examples are nickel sulphate (NiSO4.6H2O), guanidine carbonate
((CH5N3)2H2CO3), strychnine sulphate ((C21H22N2O2)2.H2SO4.6H2O).

BISPHENOIDAL CLASS

(Bisphenoidal-tetartohedral).

Here there is only a single dyad axis of symmetry, which coincides
with the principal axis. All the forms, except the prisms and basal
pinacoid, are sphenoids. Crystals possessing this type of symmetry
have not yet been observed.

3. ORTHORHOMBIC SYSTEM

(Rhombic; Prismatic; Trimetric).

In this system the three crystallographic axes are all at right
angles, but they are of different lengths and not interchangeable. The
parameters, or axial ratios, are a: b: c, these referring to the axes
OX, OY and OZ respectively. The choice of a vertical axis, OZ = c, is
arbitrary, and it is customary to place the longer of the two
horizontal axes from left to right (OY = b) and take it as unity: this
is called the "macro-axis" or "macro-diagonal" (from [Greek: makros],
long), whilst the shorter horizontal axis (OX = a) is called the
"brachy-axis" or "brachy-diagonal" (from [Greek: brachus], short). The
axial ratios are constant for crystals of any one substance and are
characteristic of it; for example, in barytes (BaSO4), a: b: c =
0.8152 : 1 : 1.3136; in anglesite (PbSO4), a: b: c = 0.7852: 1 :
1.2894; in cerussite (PbCO3), a : b : c = 0.6100 : 1 : 0.7230.

There are three symmetry-classes in this system:--

HOLOHEDRAL CLASS

(Holohedral; Bipyramidal).

Here there are three dissimilar dyad axes of symmetry, each coinciding
with a crystallographic axis; perpendicular to them are three
dissimilar planes of symmetry; there is also a centre of symmetry.
There are seven kinds of simple forms:--

Orthorhombic Bipyramids.]

Bipyramid (figs. 54 and 55). This is the general form and is bounded
by eight scalene triangles; the indices are {111}, {211}, {221},
{112}, {321}, {123}, &c., or in general {hkl}. The crystallographic
axes join opposite corners of these pyramids and in the fundamental
bipyramid {111} the parametral plane has the intercepts a: b: c. This
is the only closed form in this class; the others are open forms and
can exist only in combination. Sulphur often crystallizes in simple
bipyramids.

Prism. This consists of four faces parallel to the vertical axis and
intercepting the horizontal axes in the lengths a and b or in any
multiples of these; the indices are therefore {110}, {210}, {120} or
{hko}.

Macro-prism. This consists of four faces parallel to the macro-axis,
and has the indices {101}, {201} ... or {hol}.

Brachy-prism. This consists of four faces parallel to the brachy-axis,
and has the indices {011}, {021} ... {okl}. The macro- and
brachy-prisms are often called "domes."

Basal pinacoid, consisting of a pair of parallel faces perpendicular
to the vertical axis; the indices are {001}. The macro-pinacoid {100}
and the brachy-pinacoid {010} each consist of a pair of parallel faces
respectively parallel to the macro- and the brachy-axis.

Figs. 56-58 show combinations of these six open forms, and fig. 59 a
combination of the macro-pinacoid (a), brachy-pinacoid (b), a prism
(m), a macro-prism (d), a brachy-prism (k), and a bipyramid (u).

Holohedral Orthorhombic Combinations.]

Examples of substances crystallizing in this class are extremely
numerous; amongst minerals are sulphur, stibnite, cerussite,
chrysoberyl, topaz, olivine, nitre, barytes, columbite and many
others; and amongst artificial products iodine, potassium
permanganate, potassium sulphate, benzene, barium formate, &c.

PYRAMIDAL CLASS

(Hemimorphic).

Here there is only one dyad axis in which two planes of symmetry
intersect. The crystals are usually so placed that the dyad axis
coincides with the vertical crystallographic axis, and the planes of
symmetry are also vertical.

The pyramid {hkl} has only four faces at one end or other of the
crystal. The macro-prism and the brachy-prism of the last class are
here represented by the macro-dome and brachy-dome respectively, so
called because of the resemblance of the pair of equally sloped faces
to the roof of a house. The form {001} is a single plane at the top of
the crystal, and is called a "pedion"; the parallel pedion {001'}, if
present at the lower end of the crystal, constitutes a different form.
The prisms {hko} and the macro- and brachy-pinacoids are geometrically
the same in this class as in the last. Crystals of this class are
therefore differently developed at the two ends and are said to be
"hemimorphic."

Fig. 60 shows a crystal of the mineral hemimorphite (H2Zn2SiO5) which
is a combination of the brachy-pinacoid {010} and a prism, with the
pedion (001), two brachy-domes and two macro-domes at the upper end,
and a pyramid at the lower end. Examples of other substances belonging
to this class are struvite (NH4MgPO4.6H2O), bertrandite (H2Be4Si2O9),
resorcin, and picric acid.

BISPHENOIDAL CLASS

(Hemihedral).

Here there are three dyad axes, but no planes of symmetry and no
centre of symmetry. The general form {hkl} is a bisphenoid (fig. 61)
bounded by four scalene triangles. The other simple forms are
geometrically the same as in the holosymmetric class.

Examples: epsomite (Epsom salts, MgSO4.7H2O), goslarite (ZnSO4.7H2O),
silver nitrate, sodium potassium dextro-tartrate (seignette salt,
NaKC4H4O6.4H2O), potassium antimonyl dextro-tartrate (tartar-emetic,
K(SbO)C4H4O6), and asparagine (C4H8N2O8.H2O).

4. MONOCLINIC[5] SYSTEM

(Oblique; Monosymmetric).

In this system two of the angles between the crystallographic axes are
right angles, but the third angle is oblique, and the axes are of
unequal lengths. The axis which is perpendicular to the other two is
taken as OY = b (fig. 62) and is called the ortho-axis or
ortho-diagonal. The choice of the other two axes is arbitrary; the
vertical axis (OZ = c) is usually taken parallel to the edges of a
prominently developed prismatic zone, and the clino-axis or
clino-diagonal (OX = a) parallel to the zone-axis of some other
prominent zone on the crystal. The acute angle between the axes OX and
OZ is usually denoted as [beta], and it is necessary to know its
magnitude, in addition to the axial ratios a : b : c, before the
crystal is completely determined. As in other systems, except the
cubic, these elements, a : b : c and [beta], are characteristic of the
substance. Thus for gypsum a : b : c = 0.6899 : 1 : 0.4124; [beta] =
80 deg. 42'; for orthoclase a : b : c = 0.6585 : 1 : 0.5554; [beta] =
63 deg. 57'; and for cane-sugar a : b : c = 1.2595 : 1 : 0.8782;
[beta] = 76 deg. 30'.

HOLOSYMMETRIC CLASS

(Holohedral; Prismatic).

Here there is a single plane of symmetry perpendicular to which is a
dyad axis; there is also a centre of symmetry. The dyad axis coincides
with the ortho-axis OY, and the vertical axis OZ and the clino-axis OX
lie in the plane of symmetry.

All the forms are open, being either pinacoids or prisms; the former
consisting of a pair of parallel faces, and the latter of four faces
intersecting in parallel edges and with a rhombic cross-section. The
pair of faces parallel to the plane of symmetry is distinguished as
the "clino-pinacoid" and has the indices {010}. The other pinacoids
are all perpendicular to the plane of symmetry (and parallel to the
ortho-axis); the one parallel to the vertical axis is called the
"ortho-pinacoid" {100}, whilst that parallel to the clino-axis is the
"basal pinacoid" {001}; pinacoids not parallel to the arbitrarily
chosen clino- and vertical axes may have the indices {101}, {201},
{102} ... {hol} or {1'01}, {2'01}, {1'02} ... {h'ol}, according to
whether they lie in the obtuse or the acute axial angle. Of the
prisms, those with edges (zone-axis) parallel to the clino-axis, and
having indices {011}, {021}, {012} ... {okl}, are called
"clino-prisms"; those with edges parallel to the vertical axis, and
with the indices {110}, {210}, {120} ... {hko}, are called simply
"prisms." Prisms with edges parallel to neither of the axes OX and OY
have the indices {111}, {221}, {211}, {321} ... {hkl} or {1'11} ...
{h'kl}, and are usually called "hemi-pyramids" (fig. 62); they are
distinguished as negative or positive according to whether they lie in
the obtuse or the acute axial angle [beta].

Fig. 63 represents a crystal of augite bounded by the clino-pinacoid
(l), the ortho-pinacoid (r), a prism (M), and a hemi-pyramid (s).

The substances which crystallize in this class are extremely numerous:
amongst minerals are gypsum, orthoclase, the amphiboles, pyroxenes and
micas, epidote, monazite, realgar, borax, mirabilite (Na2SO4.10 H2O),
melanterite (FeSO4.7H2O) and many others; amongst artificial products
are monoclinic sulphur, barium chloride (BaCl2.2H2O), potassium
chlorate, potassium ferrocyanide (K4Fe(CN)6.3H2O), oxalic acid
(C2O4H2.2H2O), sodium acetate (NaC2H3O2.3H2O) and naphthalene.

HEMIMORPHIC CLASS

(Sphenoidal).

In this class the only element of symmetry is a single dyad axis,
which is polar in character, being dissimilar at the two ends.

The form {010} perpendicular to the axis of symmetry consists of a
single plane or pedion; the parallel face is dissimilar in character
and belongs to the pedion {01'0}. The pinacoids {100}, {001}, {hol}
and {h'ol} parallel to the axis of symmetry are geometrically the
same in this class as in the holosymmetric class. The remaining forms
consist each of only two planes on the same side of the axial plane
XOZ and equally inclined to the dyad axis (e.g. in fig. 62 the two
planes XYZ and X'YZ'); such a wedge-shaped form is sometimes called a
sphenoid.

Fig. 64 shows two crystals of tartaric acid, a a right-handed crystal
of dextro-tartaric acid, and b a left-handed crystal of laevo-tartaric
acid. The two crystals are enantiomorphous, i.e. although they have
the same interfacial angles they are not superposable, one being the
mirror image of the other. Other examples are potassium
dextro-tartrate, cane-sugar, milk-sugar, quercite, lithium sulphate
(Li2SO4.H2O); amongst minerals the only example is the hydrocarbon
fichtelite (C5H8).

CLINOHEDRAL CLASS

(Hemihedral; Domatic).

Crystals of this class are symmetrical only with respect to a single
plane. The only form which is here geometrically the same as in the
holosymmetric class is the clino-pinacoid {010}. The forms
perpendicular to the plane of symmetry are all pedions, consisting of
single planes with the indices {100}, {1'00}, {001}, {001'}, {hol},
&c. The remaining forms, {hko}, {okl} and {hkl}, are domes or
"gonioids" ([Greek: gonia], an angle, and [Greek: eidos], form),
consisting of two planes equally inclined to the plane of symmetry.

Examples are potassium tetrathionate (K2S4O6), hydrogen trisodium
hypophosphate (HNa3P2O6.9H2O); and amongst minerals, clinohedrite
(H2ZnCaSiO4) and scolectite.

5. ANORTHIC SYSTEM

(Triclinic).

In the anorthic (from [Greek: an], privative, and [Greek: orthos],
right) or triclinic system none of the three crystallographic axes are
at right angles, and they are all of unequal lengths. In addition to
the parameters a : b : c, it is necessary to know the angles, [alpha],
[beta], and [gamma], between the axes. In anorthite, for example,
these elements are a : b : c = 0.6347 : 1 : 0.5501; [alpha] = 93 deg.
13', [beta] = 115 deg. 55', [gamma] = 91 deg. 12'.

HOLOSYMMETRIC CLASS

(Holohedral; Pinacoidal).

Here there is only a centre of symmetry. All the forms are pinacoids,
each consisting of only two parallel faces. The indices of the three
pinacoids parallel to the axial planes are {100}, {010} and {001};
those of pinacoids parallel to only one axis are {hko}, {hol} and
{okl}; and the general form is {hkl}.

Several minerals crystallize in this class; for example, the
plagioclastic felspars, microcline, axinite (fig. 65), cyanite,
amblygonite, chalcanthite (CuSO4.5H2O), sassolite (H3BO3); among
artificial substances are potassium bichromate, racemic acid
(C4H6O6.2H2O), dibrom-para-nitrophenol, &c.

ASYMMETRIC CLASS

(Hemihedral, Pediad).

Crystals of this class are devoid of any elements of symmetry. All the
forms are pedions, each consisting of a single plane; they are thus
hemihedral with respect to crystals of the last class. Although there
is a total absence of symmetry, yet the faces are arranged in zones on
the crystals.

Examples are calcium thiosulphate (CaS2O3.6H2O) and hydrogen strontium
dextro-tartrate ((C4H4O6H)2Sr.5H2O); there is no example amongst
minerals.

6. HEXAGONAL SYSTEM

Crystals of this system are characterized by the presence of a single
axis of either triad or hexad symmetry, which is spoken of as the
"principal" or "morphological" axis. Those with a triad axis are
grouped together in the rhombohedral or trigonal division, and those
with a hexad axis in the hexagonal division. By some authors these two
divisions are treated as separate systems; or again the rhombohedral
forms may be considered as hemihedral developments of the hexagonal.
On the other hand, hexagonal forms may be considered as a combination
of two rhombohedral forms.

Owing to the peculiarities of symmetry associated with a single triad
or hexad axis, the crystallographic axes of reference are different in
this system from those used in the five other systems of crystals. Two
methods of axial representation are in common use; rhombohedral axes
being usually used for crystals of the rhombohedral division, and
hexagonal axes for those of the hexagonal division; though sometimes
either one or the other set is employed in both divisions.

Rhomobohedral axes are taken parallel to the three sets of edges of a
rhombohedron (fig. 66). They are inclined to one another at equal
oblique angles, and they are all equally inclined to the principal
axis; further, they are all of equal length and are interchangeable.
With such a set of axes there can be no statement of an axial ratio,
but the angle between the axes (or some other angle which may be
calculated from this) may be given as a constant of the substance.
Thus in calcite the rhombohedral angle (the angle between two faces of
the fundamental rhombohedron) is 74 deg. 55', or the angle between the
normal to a face of this rhombohedron and the principal axis is 44
deg. 36(1/2)'.

Hexagonal axes are four in number, viz. a vertical axis coinciding
with the principal axis of the crystal, and three horizontal axes
inclined to one another at 60 deg. in a plane perpendicular to the
principal axis. The three horizontal axes, which are taken either
parallel or perpendicular to the faces of a hexagonal prism (fig. 71)
or the edge of a hexagonal bipyramid (fig. 70), are equal in length
(a) but the vertical axis is of a different length (c). The indices of
planes referred to such a set of axes are four in number; they are
written as {hikl}, the first three (h + i + k = 0) referring to the
horizontal axes and the last to the vertical axis. The ratio a : c of
the parameters, or the axial ratio, is characteristic of all the
crystals of the same substance. Thus for beryl (including emerald) a :
c = 1 : 0.4989 (often written c = 0.4989); for zinc c = 1.3564.

_Rhombohedral Division._

In the rhomobohedral or trigonal division of the hexagonal system
there are seven symmetry-classes, all of which possess a single triad
axis of symmetry.

HOLOSYMMETRIC CLASS

(Holohedral; Ditrigonal scalenohedral).

In this class, which presents the commonest type of symmetry of the
hexagonal system, the triad axis is associated with three similar
planes of symmetry inclined to one another at 60 deg. and intersecting
in the triad axis; there are also three similar dyad axes, each
perpendicular to a plane of symmetry, and a centre of symmetry. The
seven simple forms are:--

Direct and Inverse Rhombohedra.]

Rhombohedron (figs. 66 and 67), consisting of six rhomb-shaped faces
with the edges all of equal lengths: the faces are perpendicular to
the planes of symmetry. There are two sets of rhombohedra,
distinguished respectively as direct and inverse; those of one set
(fig. 66) are brought into the orientation of the other set (fig. 67)
by a rotation of 60 deg. or 180 deg. about the principal axis. For the
fundamental rhombohedron, parallel to the edges of which are the
crystallographic axes of reference, the indices are {100}. Other
rhombohedra may have the indices {211}, {41'1'}, {110}, {221'},
{111'}, &c., or in general {hkk}. (Compare fig. 72; for figures of
other rhombohedra see CALCITE.)