If the same chemical compound occurs in several crystalline forms (as in the case of SiO 2 which forms α- and β-Quartz, Cristobalite, Tridymite) these are distinct thermodynamical phases and the substance is called polymorphic. It is true that the same angles occur in all crystals belonging to the so-called cubic system (see below), but apart from these crystals, angular measurements may be used to identify each kind of crystal. (ii) Whereas the size of the crystal, the relative sizes of its faces, and thence the overall shape or 'habit' of a particular kind of crystal may vary widely according to the circumstances of its formation, the angles between the faces are characteristic for the chemical composition. (i) Crystals grow naturally with plane faces. The goniometer led to the discovery of the three fundamental laws of descriptive or morphological crystallography: In order to measure the other angles the crystal has either to be re-set with a different zone axis coinciding with the axis of rotation, or a two-circle goniometer has to be used where this adjustment can be made without resetting. With one setting of the crystal, only the normals of one 'zone,' which all lie in the plane at right angles to the zone axis, can be obtained. If the crystal is rotated until a second face reflects into the telescope, then the angle of rotation, which can be read accurately on a graduated circle, is the angle formed by the normals of the two planes. the edge direction which is common to two or more crystal faces, coincides with the axis of rotation. The crystal is mounted in soft wax on an axis at right angles to the plane of the light path so that a 'zone axis,' i.e. In this instrument light made parallel after passing through a slit in the focal plane of a collimating telescope is reflected by a crystal face, and focussed by a second telescope, so that the observer sees an image of the slit. Wollaston's construction of an optical goniometer (1809). The accuracy of this angular measurement was greatly increased, and extended to smaller crystals, by W. Quantitative crystallography began with Carangeot's invention of the contact goniometer (1780), an instrument with which the angles between the faces of a crystal could be crudely measured.
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