Crystallographic Axes and its Symmetry Operations
Symmetry operations are used to describe the crystal’s outward symmetry. Symmetry operations help to define the manner in which a crystal can repeat the facets or faces on their crystal’s surface.
The plane that is used to reflect a face from one side of the crystal to the other is termed as mirror plane. It is important to note that while being reflected using the concept of mirror plane, the face of reflection that is maintained is identical but reversed in orientation. That is, for example if the original face has any right handed characteristics, then the reflected face should represent all the same attributes as the original face with a left handed attribute.
Center Symmetry Operation
Another symmetry operation that is worth knowing is called ‘center’. The center symmetry operation refers to an operation which would invert the original face of a crystal through the center of the crystal. In fact, the resulting effect stands similar to the operation named as roto inversion axis which is explained in detail below. That is, by this operation, every point of the crystal is inverted to the other side of the crystal. The center operation is mostly applied in triclinic system which follows a single fold rotational axis by which with just a single rotation the crystal face returns to the original face of rotation.
Rotational axis is an imaginary line which acts an axis and is drawn on a crystal. By rotating the crystal along the axis, it is possible to repeat a crystal face. Thus it is possible to generate new crystal faces at consistent intervals of rotation done as explained above. It is also vital to note that the resulting face should be identical to the original face only if the orientation is reversed. The next point of consideration is the determination of interval for rotation of crystal face. The determination of interval for rotation of crystal face is done by dividing the full turn into equal segment intervals. That is, for instance 360 degrees is divided into a segment of four 90 degrees that results in four fold rotational axis. Thus the numbers of folds the rotational axes can have are one, two, three, four or six. This means that a singe fold axis of rotational axis of rotation would rotate the crystal in 360 degree intervals. The two fold interval of rotational axis of rotation would rotate the crystal in 180 degrees, three fold in 120 degrees each, four fold in 90 degrees as explained before and six fold of rotational axis of rotation would rotate the crystal in 60 degrees.
Rotoinversion does the functionality of both rotational axis and inversion along with this. That is, the rotoinversion axis after performing the functionality of rotation once would invert the face of crystal along the center of crystal to the opposite side. Thus, the out coming face would be totally flipped. For instance, if the original face is up, the resulting face would be down and if the original face is right the resulting face would be left. This operation of rotoinverion axis is done until the operation returns to the original face. Apart from this, the rules, determination of interval for rotation namely folds explained above for rotational axis all holds good even for rotoinversion axis.
The crystallographic axes are used mainly by crystallographers. These axes are similar to the geometric axes and are used for plotting the orientations of faces and symmetry elements in crystals. It is important to know that it is not vital that the crystallographic axes should be part of symmetry of the crystals. It can also be present or not within the symmetry of the crystals. Generally, it would be present in the symmetry of crystal because crystallographers would try the orientation mostly along the planes and axes of symmetry to study the operations and orientations of crystals in depth.
The seven system of crystallography along with the folds required for the axis of rotation are given below:
- ISOMETRIC: The isometric crystal system requires four three fold axes of rotation. Instances of minerals and crystals in this system are spinel, lazurite, analcime, galena, gold, fluorite, almandine, halite, cobaltite, diamond, tetrahedrite, bixbyite and so on.
- TETRAGONAL: The tetragonal crystal system requires a single four fold axis of rotation. Instances of minerals and crystals in this system are zircon, carletonite, rutile, scapolite, anatase, vesuvianite, narsarsukite, autunite, xenotime, thorite, zeunerite and so on.
- HEXAGONAL: This crystal system need a single six fold axis of rotation. Instances of minerals and crystals in hexagonal system are aquamarine, gmelinite, pyrrhotite, apatite, ettringite, molybdenite, hanksite, thaumasite, vanadinite and so on.
- TRIGONAL: This crystal system need a single three fold axis of rotation. Instances of minerals and crystals in trigonal system are sapphire, ankerite, ruby, sturmanite, magnesite, pyrargyrite, hematite,rhodochrosite, cinnabar,elbaite and so on.
- ORTHORHOMBIC: The orthorhombic crystal system should possess three two fold axes of rotation or one two fold axis of rotation with two mirror planes along with this. Instances of minerals and crystals in this system are topaz, staurolite, barite, anhydrite, chrysoberyl, olivine, celestite and so on.
- MONOCLINIC: The monoclinic crystal system requires either a single two fold axis of rotation or a single mirror plane. Instances of minerals and crystals in this system are brazilianite, aegirine, azurite, borax, catapleiite, muscovite, huebnerite, crocoite and so on.
- TRICLINIC: This crystal system should possess either a center operation of symmetry or translational symmetry. Instances of minerals and crystals in triclinic system are babingtonite, inesite, bytownite, kyanite, turquoise, albite, rhodonite, oligoclase and so on.
Generally the substances that are non-crystalline are amorphous. They do not have any symmetry and so could not be classified under any crystallographic system.
The symmetry of the lattice is thus used for determining the angular relationships between crystal faces. The measurements of the angles between crystal face is used for calculating the relative lengths of sthe crystallographic axes or unit cell edges. The crystallographic axes help to define a coordinate system within the crystal.