Through studies of external forms and angular relationships between the crystal faces, some fundamental laws have been established, which govern the whole crystallography. They are as follows:
1. Law of constancy of interfacial angle.
2. Law of rational indices.
3. Law of axial ratio.
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4. Law of crystallographic axes.
5. Law of constancy of symmetry.
1. Law of constancy of ‘interfacial-angle’:
Interfacial angle may more generally be defined as the angle between any two adjacent faces of a crystal. In crystallography, however, the interfacial angle to a crystal is the angle subtended between the normal drawn on the two faces concerned.
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It has been observed that the interfacial angles of crystals of a particular mineral remain always constant. Since the atomic structure of the crystals of a particular mineral is fixed, the position of faces of such crystals will also be equal. So the corresponding interfacial angles are constant for all the crystals of a given mineral, provided they have identical chemical composition and are measured at the same temperature.
Law of constancy of interfacial-angle states that ‘measured at the same temperature, similar angles on crystals of the same substance remain constant, regardless the size and the shape of the crystals.
[Contact-goniometers and reflecting-goniometer are used in measuring the interfacial angle of crystals.]
2. Law of rational-indices:
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Two crystals of the same substance may differ considerably in appearance that in number, size and shape of the individual faces. In order to describe the external form of crystals, a mathematical method of relating plane; to certain imaginary lines in space is used.
The position of any plane can be uniquely fixed by the intercepts it makes on the axes of reference. The ratio of the distances from the origin, at which the crystal face cuts the crystallographic axes, is known as the ‘parameter of a crystal face.
In the given figure, let OX, OY, OZ represents 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 length for the purpose of representing the position of any other face such as DEF. In this case OD=OA, 0E=20B, OF= ½OC. Therefore the parameters are of DEF with reference to the standard face ABC.
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The reciprocals of the parameters are known as indices.
According to the crystallographic notation by Miller a law- has been established, which states that “the intercepts that any face makes on the crystallographic axes are either infinite or small rational multiples of the intercepts made by the unit form”.
Hence the ratio between the intercepts on the axes of different faces on a crystal can always be expressed by rational numbers as 1: 2, 1: 3, 1: 4; but not as 1: 42, 1: V3 etc.
3. Law of axial-ratio:
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This law states that ‘the ratio between the lengths of the axes of the crystals of a given substance is- constant. This ratio is termed as ‘axial-ratio’. Axial-ratio which is the ratio of the lengths of the crystallographic expressed in terms of one of the horizontal axes, usually, ‘6’-axis, as unity.
In cubic system, where the three axes are identical, the ratio is 1: 1: 1 or a: a: a.
In tetragonal system, in case of zircon, a: c= 1 : 0’6403, in case of retile, a : c= 1 : 0 64415.
In Hexagonal system, in case of Beryl, a : c= 1 : 04989 and so
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4. Law of crystallographic axes:
The positions of the crystallographic axes are more or less fixed by symmetry of the crystals, for in most crystals they are symmetry axes or normal to symmetry plane. It has been observed that “crystals of a given mineral can be referred to the same set of crystallographic axes”.
For example, all the crystals of galena may be referred to three crystallographic axes, which are of equal length, mutually perpendicular and are inter-convertible and are designated as affront-back axis, a,=right-left axis and a2=top-bottom axis.
5. Law of constancy of symmetry:
From X-ray studies of crystals, a general law regarding symmetry has been propounded; which states that “the symmetry in all crystals of a particular species is constant, though they may not be similar in form”.
The law of symmetry finds expression upon a crystal in the -distribution of- similar angles and faces. It is well known that the geometric locus about which a group of repeating operations acts is the symmetry element, which may be with respect to a plane and is known as a plane of symmetry, or, with respect to a line, where it is said to be an axis of symmetry, or, with respect to a point, in which case it is known to be a centre of symmetry.
For example, the crystals of the mineral-‘galena’, whether it is octahedral, dodecahedral or cubic in shape, it shows the same symmetry elements, like, ‘9’-planes of symmetry, ‘l3’-axes of symmetry and the centre of symmetry is also present. Similarly all the crystals of the mineral-Barite, shows ‘3’-plan«s of symmetry, 3-axes of two fold symmetry and the presence of centre of symmetry.
These afore said laws totally govern all the aspects of crystallography.