Part 38

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.

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 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 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:--

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).

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).