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In this article we will discuss about:- 1. Meaning of a Crystal 2. Elements of a Crystal 3. Crystallographic Axes 4. Parameters, Indices, Symbols 5. Forms.
Meaning of a Crystal:
A crystal may be defined as a solid polyhedral form of a substance bounded by smooth geometrical surfaces called faces.
Crystals may be natural or synthetic. Natural crystals occurring in thousands of varieties are formed in the Earth from natural fluids by their cooling under different conditions of temperature and pressure. Most minerals occur in the form of crystals or crystalline aggregates. Synthetic crystals are obtained by cooling saturated solutions under controlled conditions of temperature and pressure.
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In either case, the process of formation of crystals is called crystallisation. It results in a very orderly, perfect internal arrangement of the atoms of the elements or compounds during the cooling phase. The external shape of a crystal is, therefore, only an outward manifestation of the regular internal atomic structure.
Crystallography is that branch of science which deals with all the aspects of crystals, that is, their formation from the melts, their internal structure and their external shape or morphology. An elementary knowledge of crystallography is essential for all those who have to deal with minerals in one way or another, such as mining engineers, civil engineers, metallurgists and engineering geologists. For mineralogists and geologists, an in-depth study of crystallography is indispensable.
Elements of a Crystal:
In the study of a crystal, few definite characters are sought for. These include crystal faces, the angle between the crystal faces, the relationship of these faces with reference to certain assumed lines passing through the crystal and also the arrangement of these faces. All these characters form elements of a crystal.
These are discussed below briefly:
1. Crystal Faces:
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Any crystal will have one or more types of external surfaces which may be regular or modified geometrical figures such as a square, a rectangle, a triangle, a trapezium or a rhombus. Such an external regular surface on a crystal is called a face.
A crystal may have only two faces of the same geometrical shape or it may have up to forty-eight faces of a combination of geometrical shapes. (Fig. 10.1). A given number of similar faces on a crystal when studied together make a form. In Fig. 10.1, obviously there are three crystal forms—a, b and c on the same crystal.
2. Interfacial Angle:
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There are always a number of faces on a crystal. The angle at which any two adjacent faces are placed on the crystal with respect to each other is called an interfacial angle (Fig. 10.2). A Danish Scientist, N. Steno observed in 1869 that the interfacial angles for all the faces of a particular mineral species are always constant.
This observation was later on enunciated as a law of Constancy of Interfacial Angles and stated as below:
“The angles between the corresponding faces of a crystal of a given substance, measured at the same temperature, have a constant value, irrespective of size, shape and number of these faces.”
Since 1869, the validity of this law, which is considered the first fundamental law of crystallography, has been established for all the substances that are crystallised.
A simple instrument used for the measurement of interfacial angles is known as goniometer. It is simply a protractor having a rotating bar pivoted at its centre. For taking measurements, the edge between the two faces of the given crystal is made to lie parallel to the axis of rotation of the bar.
When the protractor has double graduations, internal and external, measurements with goniometer will give two values, total being 180°. Of these two values, one is called the internal solid angle and the other is external angle between the face and the extension of the other. This latter is termed the polar angle and is generally mentioned as interfacial angle.
Measurements with contact goniometer are accurate to the extent of 1/2°. For greater accuracy, optical goniometers such as one circle reflecting goniometers or theodolite goniometers are available.
Symmetry of a Crystal:
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By crystal symmetry is understood a sort of regularity in the arrangement of faces on the body of a crystal. For instance, if a crystal could be cut into two halves, one half being the mirror image of the other half, it has a symmetry with reference to a plane along which it has been cut or imagined to be cut.
Symmetry is a property of fundamental importance for a crystal. It can be studied with reference to three different characters, commonly called elements of symmetry.
These are:
1. A plane of symmetry;
2. An axis of symmetry, and,
3. Centre of symmetry.
1. A Plane of Symmetry:
Any imaginary plane passing through the centre of a crystal in such a way that it divides the crystal in two exactly similar halves is called a plane of symmetry. In other words, a plane of symmetry is said to exist in a crystal when for each face, edge or solid angle there is another similar face, edge or solid angle occupying identical position on the opposite side of this plane. (Fig. 10.4)
A crystal may possess one, two or more planes of symmetry, the highest number being 9 (nine) occurring in the normal class of isometric system. Further, a plane of symmetry may be described as axial, horizontal, vertical or diagonal depending upon its position with reference to the outline of the crystal.
2. An Axis of Symmetry:
It is defined as an imaginary line in a crystal passing through its centre in such a way that when a crystal is given a complete rotation along this line a certain crystal face (taken as a reference face) comes to occupy the same position at least twice. In fact, it may occupy the same position more than twice also.
The angle of rotation needed to bring a reference face to the same position, that is, to repeat itself, defines the nature of the axis of symmetry into one of four types:
(i) Axis of Binary or Two Fold Symmetry:
This requires that a crystal must be rotated by an angle of 180° to bring the reference face occupy the same position. In other words, when the crystal is rotated about such an axis, a given face makes its appearance only twice in one complete rotation.
(ii) Axis of Trigonal or Three Fold Symmetry:
It is that axis (imaginary line) on which a crystal must be rotated by an angle of 120° for a reference face to occupy the same position again in space. In other words, a reference face makes its appearance thrice in one complete rotation about such an axis.
(iii) Axis of Tetragonal or Four fold Symmetry:
It is that axis on which the crystal must be rotated by an angle of 90° to bring a reference face in the same position in space. Obviously, in such a case, reference face will occupy the same position at least four times in one complete rotation.
(iv) Axis of Hexagonal or Six fold Symmetry:
In which a rotation of 60° is required to fulfill the condition of repetition of a reference face. In such a case, a reference face will appear six times in one complete rotation about the axis.
In a given crystal, there may be possible one or more axes of symmetry, some of which may be of binary nature, other of trigonal and tetragonal nature and so on. The highest number of axes of symmetry is thirteen (13), observed in normal class of isometric system.
Centre of Symmetry:
A crystal is said to possess a centre of symmetry if on passing an imaginary line from some definite face, edge or corner on one side of the crystal through its centre, another exactly similar face or edge or corner is found on the other side at an equal distance from the centre. Many crystals have no planes or axes of symmetry but do possess a centre of symmetry. In other cases, the centre of symmetry may not be there whereas the crystal may be symmetrical to a plane of symmetry.
Crystallographic Axes:
These are also termed as axes of reference, and are simply certain imaginary lines arbitrarily selected in such a way that all of them pass through center of an ideal crystal. The concept of axes of reference is based on the fact that exact mathematical relations exist between all the faces on a given crystal with reference to its centre. It has been, therefore, possible to define their position with reference to these assumed straight lines passing through the centre of a given crystal.
In crystallography, following general assumptions have been universally agreed upon regarding these crystallographic lines:
(a) Three Straight Lines, essentially passing through a common centre, and varying in mutual relationships with respect to their lengths and angular inclinations from- all equal, interchangeable and at right to all unequal and inclined with each other. (Fig. 10.6)
(b) Four Straight Lines, essentially passing through a common centre; one vertical, being unequal to the other three but at right angles to them. The three horizontal axes are separated from each other at 120°. (Fig. 10.6)
The concept of crystallographic axes has been the basis of classifying all the crystalline substances into six crystal systems.
While fixing a crystal in its study position with reference to the crystallographic axes, it is important to align it in such a way that one of the axes runs front to back from the observer, another runs right to left and one is passing from top to bottom of the crystal.
Obviously, in such a case, each of the three (or four) axes will have two ends, which must be distinguished by some convention. It is the usual practice to designate the axis running from front to back as ‘a1‘ axis and its end nearer to the observer as (+) whereas the farther end is marked (-).
The second axis running from right to left is designated as ‘a2‘ axis if its length is same as that of first axis; otherwise it is designated as ‘b’ axis. It will be given (+) sign on its right terminus and (-) sign on the left terminus. Similarly, the vertical axis may be designated simply as ‘a3‘ if it has length equal to the other two, otherwise it is ‘c’ axis, with (+) sign placed at the top and (-) sign at the bottom.
Axial Ratio:
It is the numerical expression for the relation (ratio) existing between the length of different crystallographic axes in a given crystal and is always constant for that crystal.
When all the three crystallographic axes have the same length (as in the crystals of isomeric system), axial ratio need not be determined. When, however, the lengths of one or more axes are unequal, then this ratio becomes of great significance and has to be determined. In such cases, the length of one of the three (or four) axes is taken as unity and those of remaining axes is expressed in relation to that axis.
There are well established conventions for selecting the axis for fixing the unit length in different crystal systems. Example- In a crystal that can be referred to three crystallographic axes of which two horizontal axes (a1, a2) are equal and interchangeable, and the vertical axis is either longer or shorter, the axial ratio is- (Fig. 10.8)
a1: a2: c = a: c (since a1 = a2)
Parameters, Indices, Symbols used for Representing Crystals:
The exact mathematical relations exist between various faces of a crystal. In actual practice these mutual relations of the faces are determined and expressed with reference to three (in one case, four) crystallographic axes. The numerical expression for such a relationship of a given crystal face with the crystallographic axes is variously termed as parameters, index (pl. indices) or a symbol, each term having its own significance.
Parameters:
The relative intercepts made by a crystal face on the three (or four) crystallographic axes are known as its parameters. For instance in Fig. 10.9, the three crystallographic axes are represented by XOX’, YOY’ and ZOZ’, with their relative lengths as Ox = a, OY = b and OZ = c.
Let us consider a face represented by a plane ijk occurring on this crystal. The intercepts of this plane (face) with the three axes are – oi, oj and ok, which expressed in terms of the lengths of the axes, are- 1a, 1/3/b, 1/2c.
Therefore, 1a, 1/3b and 1/2c are parameters of the face. These are generally expressed as parametrical ratios, as 1a: 1/3b: 1/2c.
One can draw a number of other planes with reference to the above three axes and determine their relationship in the same manner.
Indices:
In common practice, the relationship of a crystal face with the crystallographic axes is expressed in simple whole numbers, which are called indices. These indices are, however, always based on and derived from the parameters. There are a number of methods for deriving indices from parameters such as the Neumann system, the Goldschmidt system, the Weiss system and the Miller system. Of these, the Miller System has been extensively used in the descriptive crystallography.
In the Miller System, the indices for a given crystal face are derived from its parameters in two simple steps:
(i) By taking the reciprocals of the parameters actually obtained for the given face;
(ii) By clearing fractions, if any, by simplification and omitting the letters for the axes in the final expression.
Example:
Let us take case of four faces with the following parameters:
(i) 1a: 1b: 2c (ii) 1/2a: ∞b: 1c (iii) 1/2a: 1/3b: 1c (iv) 1a: ∞b: ∞c.
Reciprocals for these parameters are respectively:
(i) 1/1a: 1/1b: 1/2c (ii) 2/1a: 1/∞b: 1/1c (iii) 2/1a: 3/1b: 1/1c (iv) 1/1a: 1/∞b: 1/∞c
By simplifying and omitting the letters a, b, c, we get,
(i) 2:2:1 (ii) 2:0:1 (iii) 2:3:1 (iv) 1:0:0
In their simplest form, we usually omit the sign for ratio also and write the indices as 221, 201, 231 and 100.
Symbol:
It is the simplest and most representative of the indices for a set of similar faces that constitute a crystallographic form. For instance, in Fig. 10.10, there are six exactly identical crystal faces, which have same mathematical relationship with all the three crystallographic axes.
Their indices (Miller’s System) are:
100, 010, 1̅00, 01̅0, 001, 001̅.
Obviously, the six faces together make a form in which each face has an identical mathematical relationship with the three axes. This may be read as “each axis being parallel to two axes and intercepting the third at a unit length.”
This statement can also be mathematically written as 100, which is a generalization for all the six faces of this particular form.
Hence 100 (one zero zero) is the symbol for the form in question and is written as (100), i.e. in brackets. This also happens to be the indices of one of the faces of the form. Hence a symbol may also be defined as the simplest of all the indices of a form.
Forms of Crystals:
Every crystal system shows typical group or groups of faces developed on individual crystals, which besides occurring in an orderly manner show identical mathematical relations with the crystallographic axes. Any group of similar faces showing identical mathematical relations with crystallographic axes makes a form.
Obviously, minimum number of faces required to make a form are two. In practice, forms having upto 24 and 48 faces are also known to occur. A crystal may show development of only one form (e.g. cube) on it, or there may be a number of forms occurring together making the crystal a really beautiful natural creation.
Forms are further distinguished into following types:
1. Holohedral form,
2. Hemihedral form,
3. Hemimorphic form,
4. Enantiomorphic form,
5. Fundamental form,
6. Open and closed form.
1. Holohedral Form:
It is that form in a crystal system, which shows development of all the possible faces in its domain. For instance, octahedron is a holohedral form because it shows all the eight faces developed on the crystal. Generally, holohedral forms develop in the crystals of highest symmetry in a crystal system. Such class of highest symmetry in a system is called its normal class (Fig. 10.11).
2. Hemihedral Form:
It shows, as the name indicates, only half the number of possible faces of a corresponding holohedral form of the normal class of the same system. As such, all hemihedral forms may be assumed to have been derived from holohedral forms.
Obviously, two complimentary hemihedral forms (termed positive and negative or right and left) will necessarily embrace all the faces and characters of the parent holohedral form.
A hemihedral form develops due to decrease in the symmetry of a crystal. (Fig. 10.12)
Example:
Octahedron is the holohedral form and Tetrahedron (only four faces) is a hemihedral form developed from it. It has only four faces and occurs in crystals of Tetrahedrite class of isometric system, which has a lower symmetry than normal class.
3. Hemimorphic Form:
It is also derived from a holohedral form and has only half the number of faces as in hemihedral form. In this case, however, all the faces of the form are developed only on one extremity of the crystal, being absent from the other extremity. In other words, such a crystal will not be symmetrical with reference to a center of symmetry. (Fig. 10.13)
4. Enantiomorphous Form:
An enantiomorphous form is composed of faces placed on two crystals of the same mineral in such a way that faces on one crystal become the mirror image of the form of faces on the other crystal. Despite that, each form is independent, that is, though having an identical mathematical relationship, and one form cannot be interchanged with its counterpart on the other crystal. As right hand and left hand having similar relation to the body axis are not interchangeable, so is the case with enantiomorphous forms.
The forms and the corresponding crystals showing these forms are referred as left and right handed.
Crystals of quartz show best-developed enantiomorphous forms.
5. Fundamental Form:
It is also called a unit form. It designates that type of any given form in which the parameters essentially correspond to the (assumed) unit lengths of the crystallographic axes. A form having a symbol (111) is essentially a fundamental form whereas another form also intercepting the three axes at different lengths (e.g. 123, 213, 321) is not a fundamental form.
6. Open and Closed Forms:
A form is defined as closed when on full development it makes a fully enclosed solid. In an open form, space cannot be fully enclosed by it and entry from one or more sides is possible.