Mineralogy

There are certain conventions with regard to the lettering and order of the crystallographic axes. In the most general case, that in which the unit form cuts all three axes at unequal lengths and in which none of the axes is at right angles to any other, the crystallographic axis which is taken as the vertical axis is called c, that running from right to left is b, and that running from front to back is a. One end of each axis is positive, and the other end is negative, and the rule with regard to this is illustrated in Fig. 17. The angle between + a and + b is called y, that between + b and + c is called a, and that between + c and + a is B.

In this most general case, the unit form cuts the three axes at unequal lengths from the origin, and this fact is often indicated loosely by stating that the crystallographic axes of this type of crystal are of unequal lengths.

In some crystals the unit form cuts two axes at an equal distance and the third at a different distance. In this case, the axes cut at equal distances are both called a and the third, placed vertical, is called c. It is customary to say here that the two axes are equal and the third different.

Again, in other crystals, the unit form cuts all three axes at the same distance, so that all the axes are interchangeable; in this case the axes are all called a, and are loosely said to be equal.

The position in space of the faces of a great number of crystals can be referred to three crystallographic axes, but in one group four axes are used.

The planes in which two of the crystallographic axes lie are called the axial planes.

Crystallographic Notation

Crystallographic notation is a concise method of writing down the relation of any crystal face to the crystallographic axes. The most widely used systems depend upon either parameters or indices Of these systems of notation, the chief are two,-the Parameter System of Weiss, and the Index System of Miller (modified by Bravais).

Parameter System of Weiss

In this system of crystallographic notation, the axes are taken in the order explained above,-that is, a, b , c, for unequal axes, a, a, c, for two axes equal, and a, a, a, for three axes equal. The intercept that the crystal face under discussion makes on the a-axis is then written before a, the intercept on the b-axis before b, and the intercept on the c-axis before c. These intercepts are of course measured in terms of the intercepts made by the unit form on the corresponding crystallographic axes.

The most general expression for a crystal face in the Weiss notation is

na, mb, pc,

where n, m, p are the lengths cut off by. the face on the a, b , c axes as compared with the corresponding lengths cut off by the unit form. It is usual to reduce either n or m to unity.

If a crystal face is parallel to an axis, it can be imagined as cutting that axis at an infinite distance, and accordingly the sign of infinity, 00, is placed as its parameter before the corresponding axial letter . Thus a face cutting the a-axis at a distance 1 unit,-that is at the same distance as the unit form cuts this same axis,-and cutting the b-axis at a distance 2 units or twice the distance cut off by the unit form along the b-axis, and running parallel to the c-axis has the Weiss symbol

a, 2b, c.

A face cutting the a-axis and parallel to the b-axis and c-axis obviously has the symbol

a, b, c.

Crystallographic symmetry must not be confused with geometrical symmetry. Crystallographic symmetry depends upon the internal atomic structure of the crystal, and as the arrangement of the atoms is the same for parallel planes, it follows that the angular position is the only factor concerned, and that the sizes of like faces and their distances from a plane or centre of symmetry are of no importance in this connection. This is illustrated in Fig. 13, which shows a regular octahedron with faces equally developed and a distorted octahedron with some faces larger than others. If such a distorted octahedron was examined with a goniometer it would be found that the interfacial angles were exactly the same as in the regular octahedron.

Crystals in which like faces are equally developed and are equal distances from the centre of the crystal are rare; but for convenience of study and of representation by diagrams, it is necessary to deal with crystals in their simplest and due to some restraint on directions or to a greater supply of material being available in one direction as compared with another, most intelligible form, and that is when they have perfect geometrical symmetry.

Most crystals occur in distorted forms, having like faces not of the same size and not in the same geometrically symmetrical position. In many cases of distorted crystals the crystallographic symmetry has been made out from the fact that like faces have like properties. Etch-marks produced by solvents acting on the crystal faces, the behaviour towards heat and electricity, the hardness, luster, etc., of the faces, have revealed the true symmetry of the distorted crystals. This is illustrated in the quartz crystal shown in Fig. 14, where the etch-marks are similar on like faces.

Distortion in crystals may be growth of the crystal in certain directions or to a greater supply of material being available in one direction as compared with another.

The term habit is used to denote the characteristic shapes of crystals arising from variations in the number, size and shape of the faces; the distorted octahedron shown in Fig. 13 has a tabular habit; in Fig. 15 are shown two habits of apophyllite crystals.

Crystallographic Axes

In solid geometry the position of a plane in space is given by the intercepts (or the lengths cutoff) that the plane makes on three given lines called axes. This method of treatment is employed in crystallography, and the axes are termed the crystallographic axes. Whenever there is present a suitable number of axes of symmetry they are chosen as the crystallographic axes. The crystallographic axes intersect at the origin.

Parameters

The parameters of a crystal face are the ratios of the distances from the origin at which the face cuts the crystallographic axes, - that is, the parameters are the ratios of the intercepts. In Fig. 16, OX, OY, OZ, represent the crystallographic axes, and ABC is a crystal face making intercepts of OA on OX, OB on OY, and OC on OZ. The parameters of the face ABC are given by the ratio of OA, OB, and OC. It is convenient to take the relative intercepts of this face as standard lengths for the purpose of representing the position of any other face, such as DEF. In the case of the face DEF, OD is equal to OA, OE is twice OB, and OF is half OC, and therefore 1/1, 2/1, 1/2 are the parameters of DEF with reference to the standard face ABC.

The form whose face is taken as intersecting the axes at the unit lengths which are to be used for measuring the intercepts made by other forms on the same axes is called the fundamental, parametral or unit form. The selection of a suitable unit form depends on the properties and nature of the crystals. A form well developed, or commonly occurring, or parallel to which there is a good cleavage, is usually selected for this purpose.

The parameters of the unit form can be obtained by measurement, and can be expressed as multiples of one of their number. Take, for example, gypsum. It is found that the most commonly occurring form in gypsum crystals which makes intercepts with all three crystallographic axes does so in the ratio of 0′6899:1 :0,4124. This expression is called the axial ratio, and simply means that the standard or unit form cuts one axis at a distance represented by 0′6899, the second axis at a distance represented by 1, and the third axis at a distance represented by 0′4124. When we use this unit form to measure the intercepts, or to obtain the parameters, of any other form that cuts all three axes we shall do so by taking 0•6899 as our unit of measurement along the first axis, 1 along the second axis, and 0′4124 along the third axis.

Indices

The reciprocals of the parameters are called the indices and are of use for purposes of crystallographic notation.

It has already been mentioned that crystals are built up of an orderly arrangement of the atoms or atomic groups of the mineral. Examination of crystals by X-rays has led to the determination of the relative positions of the different atoms in the structure. If the atoms are represented by points, their arrangement in the crystal can be shown by a geometrical pattern or framework which is called the space-lattice or point-system. In this, the atoms are arranged in innumerable parallel rows which intersect in a regular pattern. The rows lie in planes to form what may be called a net-plane. Crystal faces are parallel to these net-planes, and crystal edges to the rows of atoms occurring at the intersections of net-planes.

We have seen that the atomic structure for the crystals of anyone mineral is fixed, so that it follows that the positions of the faces of such crystals are fixed also. This leads to the enunciation of the important law of the Constancy of Interfacial Angles. The corresponding interfacial angles are constant for all crystals of a given mineral, provided, of course, that the crystals have identical chemical compositions and that the measurements are made at the same temperatures.

Zones

Inspection of many crystals shows that their faces are so arranged that the edges formed by the intersections of certain of the faces are parallel with one another. Such a set of faces constitutes a zone, and the line with which the edges are parallel is called the zone-axis. For instance, the common crystals of quartz or rock crystal such as are illustrated in Fig. 121, show six faces meeting in parallel edges, and terminated by a set of six usually triangular faces which do not meet in parallel edges; the first set of six faces forms a zone.

Symmetry

Examination of a crystal either with the eye or a gonimeter shows that there is a certain regularity of position of like faces, edges, etc. This regularity constitutes the symmetry of the crystal. The degree of symmetry varies in different minerals and is employed, as seen later, in the classification of crystals. It is defined with reference to three criteria of symmetry:

• Plane of Symmetry
• Axis of Symmetry
• Centre of Symmetry

Plane of Symmetry

A plane of symmetry divides a crystal into two similar and similarly placed halves. In other- words, such a plane divides the crystal so that one half is the mirror-image of the other. Planes of symmetry can be illustrated by considering a cube. A cube has nine planes which divide it into two halves so that one half is the reflection of the other. The traces of these nine planes are indicated on the faces of the cube in Fig. 9 and the dissected planes are shown in Fig. 10.

The geometrical symmetry of a matchbox or a brick is obviously lower than that of a cube for, as inspection shows, there are only three planes that divide the object into similar and similarly placed halves.

Axis of Symmetry

If a crystal, on being rotated, comes to occupy the same position in space more than once in a complete turn, the axis about which rotation has taken place is called an axis of symmetry. Depending upon the degree of symmetry, a crystal may come to occupy the same position two, three, four or six times in a complete rotation. The terms applied to these different classes of axes are as follow:-

Two times: two-fold, diad, half-turn or digonal axis.

Three times: three-fold, triad, one-third-turn or trigonal axis.

Four times: four-fold, tetrad, quarter-turn or tetragonal axis.

Six times: six-fold, hexad, one-sixth-turn or hexagonal axis.

We can again use the cube and our brick to illustrate axes of symmetry. In the cube, as shown in Fig. ii, there are axes of four-fold, three-fold and two-fold symmetry. Rotation of the cube about the axis of four-fold symmetry shown in the figure causes the cube to take up the same position in space four times during a complete rotation, about the three-fold axis three times, and about the two-fold axis twice. It ‘is clear, moreover, that there are three axes of four-fold symmetry, four of three-fold symmetry and ,six of two-fold symmetry in the cube. This is expressed in the following way:

Axes of Symmetry of the cube - 3^iv, 4ⁱⁱⁱ, 6ⁱⁱ.

In our brick there are only three axes of symmetry and these are of two-fold type; they connect the middle points of the pairs of opposite faces of the brick.

Centre of Symmetry

A crystal has a centre of symmetry when like faces, edges, etc., are arranged in pairs in corresponding positions and on opposite sides of a central point. The cube and our brick obviously have centers of symmetry.

The Symmetry of Gypsum as an Illustration

A crystal of gypsum may be taken to illustrate these definitions of symmetry. The usual form of such a crystal is shown in Fig. 12.

There is one plane which divides the crystal into two similar and similarly placed halves. This plane is the only plane of symmetry for this crystal. At right angles to this plane is an axis of symmetry. Rotation about this axis causes the crystal to take up the same position twice in a complete rotation, and this axis is therefore an axis of two-fold symmetry. Lastly, for every face, edge or corner that occurs in one half of the crystal there is a similar face, edge or corner in a corresponding position in the other half. Therefore the crystal has a centre of symmetry.

Thus the symmetry of this gypsum crystal may be expressed in the following way:

• Planes of symmetry
• Axes of symmetry
• Centre of symmetry

In was noticed by the ancient Greeks that a certain mineral, quartz, usually occurred in forms having a characteristic shape, being bounded by flat faces. From the transparency of this mineral and the occurrence in it of included material, it was thought that quartz resulted from the freezing of water under intense cold, and hence the name krustallos-meaning clear ice-was given to the substance. There were, however, numerous other minerals known to the ancients which occurred in forms bounded by flat faces, and so, by a natural extension of the term, krustallos came to signify any mineral showing such forms.

By the researches of Steno, De l’Isle and Hauy, the study of crystals gradually evolved from mere speculation. It is to Hauy that we are indebted for an illuminating theory of the structure of crystals. Hauy argued that crystals were built up of minute bricks of the mineral, different modes of arrangement of the bricks producing different crystal forms. By more recent investigations Hauy’s notion of the arrangement of material bricks has been replaced by that of the arrangement of atomic groups. It is therefore apparent that chemical constitution has an important influence on crystalline form, and, indeed, Von Federov issued a list of some ten thousand substances, the chemical composition of which he was able to tell with certainty from an examination of their crystals.

The study of crystals is called crystallography. Crystals are bodies bounded’ by surfaces, usually flat, arranged on a definite plan which is an expression of the internal arrangements of the atoms. They are formed by the solidification of minerals from the gaseous or liquid states or from solutions,-a process known as crystallization.

From the definition of a crystal just given we see that the internal atomic structure is their fundamental property. Though we could construct a model of a crystal in glass or some other amorphous material, such a model would not be a crystal since it would lack the essential atomic structure. In this book, however, we are chiefly concerned with the determination of minerals, so that for us the external form of crystals demands most attention. In this chapter our crystallography will be almost entirely morphological. The atomic structure of crystals is considered in the next chapter.

Characteristics of Crystals

Faces

Crystals are bounded by a number of surfaces which are usually perfectly flat, but may be curved as in some specimens of siderite and diamond. These surfaces are called faces. Faces are of two kinds, like and unlike. Some crystals are limited by faces that are all alike. For instance, fluor-spar commonly crystallizes in cubes, and any one face of the fluor-spar cube is like all the other faces in its properties. Such faces that have the same properties are called like faces, whilst faces having different properties are unlike faces.

Forms

A crystal made up entirely of like faces is termed a simple form. For example, the cube and the octahedron are each of them simple forms, since all the faces of each have the same properties. The front face shown in the drawing of a cube in Fig. 5 can be replaced by any other of the cube faces without altering the drawing. A crystal which consists of two or more simple forms is called a combination. In Fig. 5, the cube and the octahedron are shown as simple forms and also as a combination such as occurs in crystals of galena.

Some simple forms occur by themselves in crystals as they can enclose space, but others can only occur in combinations, since they have too few faces to enclose space by themselves. Such latter forms are called open.

Edge

An edge is formed by the intersection of any two adjacent faces. The position in space of an edge depends, of course, upon the positions of the faces whose intersection gives rise to it.

Solid Angle

A solid angle is formed by the intersection of three or more faces.

Interfacial Angle

The angle between any two faces of a crystal ‘is termed the interfacial angle. In crystallography, the interfacial angle is the angle between the normals, or perpendiculars, to the two faces. The interfacial angle between the two faces shown in section is A. Interfacial angles are of great importance in crystallography and are recorded in works of reference in the following way, if the angle between the normals to two faces which we will call m and mIII is 630° 48′ it is recorded as mIII = 630° 48′.

Measurement of Interfacial Angle

The interfacial angles of crystals are measured by the goniometer (or angle measurer). Two types of this instrument are used, one termed the contact-goniometer, the other the reflecting goniometer.

The contact-goniometer consists of two straight-edged arms movable on a pivot or screw, and connected by a 3 graduated are, as shown in Fig. 7. These two arms are brought accurately into contact with adjacent faces of the crystal, and the angle between them read off on the graduated arc. In the illustration, the angle actually measured is the internal angle between the two faces, and this must be subtracted from 1800 to give the interfacial angle of the crystallographer.

Reflecting goniometers are rather elaborate instruments used with crystals possessing perfectly smooth or flawless faces. In general, the smaller the crystal, the more suitable for use with the reflecting goniometer will it be.

A common form of .reflecting goniometer consists of a vertical circle, graduated and capable of rotation, and a horizontal arm fixed at right angles to the plane of the circle. A mirror is fixed on the horizontal arm. The crystal is placed at the centre of the graduated circle with an edge parallel to the horizontal arm. The image of a distant signal is observed by reflection from the mirror, and also by reflection from the crystal face. By rotating the graduated circle and with it the crystal, the two images are made to lie in the same straight line. The circle is then rotated until an image is obtained by reflection from the adjacent face. The amount of rotation gives the angle between the normals to the two crystal faces, that is, the interfacial angle, as shown in Fig. 8. Here light reflected from the face AB of the crystal in the ABCD position is seen by the eye. If the crystal is rotated about the edge between AB and AD so that the face AD takes up the new position dA where dA and AB are in the same straight line, then the signal is again seen. The crystal has been rotated through the angle dAD, which is the supplement of the internal angle between the faces B AB and AD, and is therefore the interfacial angle.