TIPS FOR SOLID STATE
The Seven Crystal Systems
Triclinic The triclinic unit cell is defined by three axes, a, b, and c, of unequal lengths.
None of the angles, alpha, beta, and gamma, between these axes are exactly 90°.
The standard convention for assigning axes is c < a < b.
The angle alpha lies between the b and c axes; beta lies between a and c,
and gamma lies between a and b. The cell is usually chosen
so that alpha and beta are obtuse (between 90º and 180º).
Orthorhombic
The orthorhombic unit cell is defined by three axes, a, b, and c, of unequal lengths.
The angles between all axes are exactly 90°.
The axes are chosen to correspond to 2-fold axes of rotational symmetry axis
or to be perpendicular to mirror symmetry planes. The standard convention is that
c < a < b.
Hexagonal
The hexagonal unit cell can be defined by three axes. Two of them, labeled a are equal
in length and at an angle of 120º to one another. The third axis, labeled c, is
perpendicular to the a axes and of a different length.
The c axis corresponds to a 3-fold or 6-fold symmetry axis.
To highlight the presence of 3-fold or 6-fold symmetry,
usual practice is to include a third a axis at 120° to the other two,
and correspondingly to use a redundant 4th integer in the Miller index.
(The extra integer is placed in the 3rd position and equals the negative of the
sum of the first two).
Trigonal
A subgroup of the hexagonal crystal system characterized by one three-fold symmetry axis.
The remaining crystal systems in the hexagonal crystal system have a six-fold symmetry axis
Monoclinic
The monoclinic unit cell is defined by three axes, a, b, and c, of unequal lengths.
The angles between the a and b axes and between the c and b axes are exactly 90°.
The b axis is chosen to correspond to a 2-fold axis of rotational symmetry axis
or to be perpendicular to a mirror symmetry plane.
The standard convention for assigning the other axes is c < a.
The unit cell is also chosen so that the angle beta, lying between the a and c axes
is obtuse (between 90º and 180º).
Tetragonal
The tetragonal unit cell is defined by three axes. Two of them, labeled a, are equal
in length; and the c axis is of a different length. All angles between axes are 90°.
The c axis corresponds to a symmetry axis of either 4-fold rotation or
4-fold rotation inversion. The c axis can be either longer or shorter than the a axes.
Isometric, Cubic
The isometric (=cubic) unit cell is defined by three axes of equal length, all labeled a.
All angles between axes are 90°. Because of the equivalence of all axes and angles.
The isometric system can contain combinations of many different types of symmetry elements:
2-, 3-, and 4- fold rotation axes, 3- and 4- fold rotation-inversion axes,
mirror planes, and centers of symmetry
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