Part 39

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:--

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.)

Scalenohedron (fig. 68), bounded by twelve scalene triangles, and with the general indices {hkl}. The zig-zag lateral edges coincide with the similar edges of a rhombohedron, as shown in fig. 69; if the indices of the inscribed rhombohedron be {100}, the indices of the scalenohedron represented in the figure are {201'}. The scalenohedron {201'} is a characteristic form of calcite, which for this reason is sometimes called "dog-tooth-spar." The angles over the three edges of a face of a scalenohedron are all different; the angles over three alternate polar edges are more obtuse than over the other three polar edges. Like the two sets of rhombohedra, there are also direct and inverse scalenohedra, which may be similar in form and angles, but different in orientation and indices.

Hexagonal bipyramid (fig. 70), bounded by twelve isosceles triangles each of which are equally inclined to two planes of symmetry. The indices are {210}, {412'}, &c., or in general (_hkl_), where h - 2k + l = 0.

Hexagonal prism of the first order (21'1'), consisting of six faces parallel to the principal axis and perpendicular to the planes of symmetry; the angles between (the normals to) the faces are 60 deg.

Hexagonal prism of the second order (101'), consisting of six faces parallel to the principal axis and parallel to the planes of symmetry. The faces of this prism are inclined to 30 deg. to those of the last prism.

Dihexagonal prism, consisting of twelve faces parallel to the principal axis and inclined to the planes of symmetry. There are two sets of angles between the faces. The indices are {32'1'}, {53'2'} ... {hk'l}, where h + k + l = 0.

Basal pinacoid {111}, consisting of a pair of parallel faces perpendicular to the principal axis.

Fig. 71 shows a combination of a hexagonal prism (m) with the basal pinacoid (c). For figures of other combinations see CALCITE and CORUNDUM. The relation between rhombohedral forms and their indices are best studied with the aid of a stereographic projection (fig. 72); in this figure the thicker lines are the projections of the three planes of symmetry, and on these lie the poles of the rhombohedra (six of which are indicated).

Numerous substances, both natural and artificial, crystallize in this class; for example, calcite, chalybite, calamine, corundum (ruby and sapphire), haematite, chabazite; the elements arsenic, antimony, bismuth, selenium, tellurium and perhaps graphite; also ice, sodium nitrate, thymol, &c.

DITRIGONAL PYRAMIDAL CLASS

(Hemimorphic-hemihedral).

Here there are three similar planes of symmetry intersecting in the triad axis; there are no dyad axes and no centre of symmetry. The triad axis is uniterminal and polar, and the crystals are differently developed at the two ends; crystals of this class are therefore pyro-electric. The forms are all open forms:--

Trigonal pyramid {hkk}, consisting of the three faces which correspond to the three upper or the three lower faces of a rhombohedron of the holosymmetric class.

Ditrigonal pyramid {hkl}, of six faces, corresponding to the six upper or lower faces of the scalenohedron.

Hexagonal pyramid (hkl) where (h - 2k + l = 0), of six faces, corresponding to the six upper or lower faces of the hexagonal bipyramid.

Trigonal prism {21'1'} or {2'11}, two forms each consisting of three faces parallel to principal axis and perpendicular to the planes of symmetry.

Hexagonal prism {101'}, which is geometrically the same as in the last class.

Ditrigonal prism {hk'l'} (where h + k + l = 0), of six faces parallel to the principal axis, and with two sets of angles between them.

Basal pedion (111) or (1'1'1'), each consisting of a single plane perpendicular to the principal axis.

Fig. 73 represents a crystal of tourmaline with the trigonal prism (21'1'), hexagonal prism (101'), and a trigonal pyramid at each end. Other substances crystallizing in this class are pyrargyrite, proustite, iodyrite (AgI), greenockite, zincite, spangolite, sodium lithium sulphate, tolylphenylketone.

TRAPEZOHEDRAL CLASS

(Trapezohedral-hemihedral).

Here there are three similar dyad axes inclined to one another at 60 deg. and perpendicular to the triad axis. There are no planes or centre of symmetry. The dyad axes are uniterminal, and are pyro-electric axes. Crystals of most substances of this class rotate the plane of polarization of a beam of light.

FIG. 74.--Trigonal Trapezohedron.

FIG. 75.--Trigonal Bipyramid.

In this class the rhombohedra {hkk}, the hexagonal prism {21'1'}, and the basal pinacoid {111} are geometrically the same as in the holosymmetric class; the trigonal prism {101'} and the ditrigonal prisms are as in the ditrigonal pyramidal class. The remaining simple forms are:--

Trigonal trapezohedron (fig. 74), bounded by six trapezoidal faces. There are two complementary and enantiomorphous trapezohedra, {hkl} and {hlk}, derivable from the scalenohedron.

Trigonal bipyramid (fig. 75), bounded by six isosceles triangles; the indices are {hkl}, where h - 2k + l = 0, as in the hexagonal bipyramid.

The only minerals crystallizing in this class are quartz (q.v.) and cinnabar, both of which rotate the plane of a beam of polarized light transmitted along the triad axis. Other examples are dithionates of lead (PbS2O6.4H2O), calcium and strontium, and of potassium (K2S2O6), benzil, matico-stearoptene.

RHOMBOHEDRAL CLASS

(Parallel-faced hemihedral).

The only elements of symmetry are the triad axis and a centre of symmetry. The general form {hkl} is a rhombohedron, and is a hemihedral form, with parallel faces, of the scalenohedron. The form {hkl}, where h - 2k + l = 0, is also a rhombohedron, being the hemihedral form of the hexagonal bipyramid. The dihexagonal prism {hk'l'} of the holosymmetric class becomes here a hexagonal prism. The rhombohedra (hkk), hexagonal prisms {21'1'} and {101'}, and the basal pinacoid {111} are geometrically the same in this class as in the holosymmetric class.

Fig. 76 represents a crystal of dioptase with the fundamental rhombohedron r {100} and the hexagonal prism of the second order m {101'} combined with the rhombohedron s {031'}.

Examples of minerals which crystallize in this class are phenacite, dioptase, willemite, dolomite, ilmenite and pyrophanite: amongst artificial substances is ammonium periodate ((NH4)4I2O9.3H2O).

TRIGONAL PYRAMIDAL CLASS

(Hemimorphic-tetartohedral).

Here there is only the triad axis of symmetry, which is uniterminal. The general form {hkl} is a trigonal pyramid consisting of three faces at one end of the crystal. All other forms, in which the faces are neither parallel nor perpendicular to the triad axis, are trigonal pyramids. All the prisms are trigonal prisms; and perpendicular to these are two pedions.

The only substance known to crystallize in this class is sodium periodate (NaIO4.3H2O), the crystals of which are circularly polarizing.

TRIGONAL BIPYRAMIDAL CLASS

Here there is a plane of symmetry perpendicular to the triad axis. The trigonal pyramids of the last class are here trigonal bipyramids (fig. 75); the prisms are all trigonal prisms, and parallel to the plane of symmetry is the basal pinacoid. No example is known for this class.

DITRIGONAL BIPYRAMIDAL CLASS

Here there are three similar planes of symmetry intersecting in the triad axis, and perpendicular to them is a fourth plane of symmetry; at the intersection of the three vertical planes with the horizontal plane are three similar dyad axes; there is no centre of symmetry.

The general form is bounded by twelve scalene triangles and is a ditrigonal bipyramid. Like the general form of the last class, this has two sets of indices {hkl, p'q'r'}, (hkl) for faces above the equatorial plane of symmetry and (p'q'r') for faces below: with hexagonal axes there would be only one set of indices. The hexagonal bipyramids, the hexagonal prism {101'} and the basal pinacoid {111} are geometrically the same in this class as in the holosymmetric class. The trigonal prism {21'1'} and ditrigonal prisms {hkl} are the same as in the ditrigonal pyramidal class.

The only representative of this type of symmetry is the mineral benitoite (q.v.).

_Hexagonal Division._

In crystals of this division of the hexagonal system the principal axis is a hexad axis of symmetry. Hexagonal axes of reference are used: if rhombohedral axes be used many of the simple forms will have two sets of indices.

HOLOSYMMETRIC CLASS

(Holohedral; Dihexagonal bipyramidal).

Intersecting in the hexad axis are six planes of symmetry of two kinds, and perpendicular to them is an equatorial plane of symmetry. Perpendicular to the hexad axis are six dyad axes of two kinds and each perpendicular to a vertical plane of symmetry. The seven simple forms are:--

Dihexagonal bipyramid, bounded by twenty-four scalene triangles (fig. 77; v in fig. 80). The indices are {213'1}, &c., or in general {hikl}. This form may be considered as a combination of two scalenohedra, a direct and an inverse.

Hexagonal bipyramid of the first order, bounded by twelve isosceles triangles (fig. 70; p and u in fig. 80); indices {101'1}, {202'1} ... (hoh'l). The hexagonal bipyramid so common in quartz is geometrically similar to this form, but it really is a combination of two rhombohedra, a direct and an inverse, the faces of which differ in surface characters and often also in size.

Hexagonal bipyramid of the second order, bounded by twelve faces (s in figs. 79 and 80); indices {112'1}, {112'2} ... {h.h.2'h'.l}.

Dihexagonal prism, consisting of twelve faces parallel to the hexad axis and inclined to the vertical planes of symmetry; indices {hiko}.

Hexagonal prism of the first order {1010}, consisting of six faces parallel to the hexad axis and perpendicular to one set of three vertical planes of symmetry (m in figs. 71, 78-80).

Hexagonal prism of the second order {112'0}, consisting of six faces also parallel to the hexad axis, but perpendicular to the other set of three vertical planes of symmetry (a in fig. 78).

Basal pinacoid {0001}, consisting of a pair of parallel planes perpendicular to the hexad axis (c in figs. 71, 78-80).

Beryl (emerald), connellite, zinc, magnesium and beryllium crystallize in this class.

BIPYRAMIDAL CLASS

(Parallel-faced hemihedral).

Here there is a plane of symmetry perpendicular to the hexad axis; there is also a centre of symmetry. All the closed forms are hexagonal bipyramids; the open forms are hexagonal prisms or the basal pinacoid. The general form {hikl} is hemihedral with parallel faces with respect to the general form of the holosymmetric class.

Apatite (q.v.), pyromorphite, mimetite and vanadinite possess this degree of symmetry.

DIHEXAGONAL PYRAMIDAL CLASS

(Hemimorphic-hemihedral).

Six planes of symmetry of two kinds intersect in the hexad axis. The hexad axis is uniterminal and all the forms are open forms. The general form {hikl} consists of twelve faces at one end of the crystal, and is a dihexagonal pyramid. The hexagonal pyramids {hoh'l} and (h.h.2'h'.l) each consist of six faces at one end of the crystal. The prisms are geometrically the same as in the holosymmetric class. Perpendicular to the hexad axis are the pedions (0001) and (0001').

Iodyrite (AgI), greenockite (CdS), wurtzite (ZnS) and zincite (ZnO) are often placed in this class, but they more probably belong to the hemimorphic-hemihedral class of the rhombohedral division of this system.

TRAPEZOHEDRAL CLASS

(Trapezohedral-hemihedral).

Six dyad axes of two kinds are perpendicular to the hexad axis. The general form {hikl} is the hexagonal trapezohedron bounded by twelve trapezoidal faces. The other simple forms are geometrically the same as in the holosymmetric class. Barium-anti-monyldextro-tartrate + potassium nitrate (Ba(SbO)2(C4H4O6)2.KNO3) and the corresponding lead salt crystallize in this class.

HEXAGONAL PYRAMIDAL CLASS

(Hemimorphic-tetartohedral).

No other element is here associated with the hexad axis, which is uniterminal. The pyramids all consist of six faces at one end of the crystal, and prisms are all hexagonal prisms; perpendicular to the hexad axis are the pedions.

Lithium potassium sulphate, strontium-antimonyl dextro-tartrate, and lead-antimonyl dextro-tartrate are examples of this type of symmetry. The mineral nepheline is placed in this class because of the absence of symmetry in the etched figures on the prism faces (fig. 92).

(g) _Regular Grouping of Crystals._

Crystals of the same kind when occurring together may sometimes be grouped in parallel position and so give rise to special structures, of which the dendritic (from [Greek: dendrou], a tree) or branch-like aggregations of native copper or of magnetite and the fibrous structures of many minerals furnish examples. Sometimes, owing to changes in the surrounding conditions, the crystal may continue its growth with a different external form or colour, e.g. sceptre-quartz.

Regular intergrowths of crystals of totally different substances such as staurolite with cyanite, rutile with haematite, blende with chalcopyrite, calcite with sodium nitrate, are not uncommon. In these cases certain planes and edges of the two crystals are parallel. (See O. Mugge, "Die regelmassigen Verwachsungen von Mineralien verschiedener Art," _Neues Jahrbuch fur Mineralogie_, 1903, vol. xvi. pp. 335-475).

But by far the most important kind of regular conjunction of crystals is that known as "twinning." Here two crystals or individuals of the same kind have grown together in a certain symmetrical manner, such that one portion of the twin may be brought into the position of the other by reflection across a plane or by rotation about an axis. The plane of reflection is called the twin-plane, and is parallel to one of the faces, or to a possible face, of the crystal: the axis of rotation, called the twin-axis, is parallel to one of the edges or perpendicular to a face of the crystal.

In the twinned crystal of gypsum represented in fig. 81 the two portions are symmetrical with respect to a plane parallel to the ortho-pinacoid (100), i.e. a vertical plane perpendicular to the face b. Or we may consider the simple crystal (fig. 82) to be cut in half by this plane and one portion to be rotated through 180 deg. about the normal to the same plane. Such a crystal (fig. 81) is therefore described as being twinned on the plane (100).

An octahedron (fig. 83) twinned on an octahedral face (111) has the two portions symmetrical with respect to a plane parallel to this face (the large triangular face in the figure); and either portion may be brought into the position of the other by a rotation through 180 deg. about the triad axis of symmetry which is perpendicular to this face. This kind of twinning is especially frequent in crystals of spinel, and is consequently often referred to as the "spinel twin-law."

In these two examples the surface of the union, or composition-plane, of the two portions is a regular surface coinciding with the twin-plane; such twins are called "juxtaposition-twins." In other juxtaposed twins the plane of composition is, however, not necessarily the twin-plane. Another type of twin is the "interpenetration twin," an example of which is shown in fig. 84. Here one cube may be brought into the position of the other by a rotation of 180 deg. about a triad axis, or by reflection across the octahedral plane which is perpendicular to this axis; the twin-plane is therefore (111).

Since in many cases twinned crystals may be explained by the rotation of one portion through two right angles, R. J. Hauy introduced the term "hemitrope" (from the Gr. [Greek: hemi]-, half, and [Greek: tropos], a turn); the word "macle" had been earlier used by Rome d'Isle. There are, however, some rare types of twins which cannot be explained by rotation about an axis, but only by reflection across a plane; these are known as "symmetric twins," a good example of which is furnished by one of the twin-laws of chalcopyrite.