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Flashcards in Symmetry Deck (31)
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Space group

Space groups represent the various ways motifs (such as atoms in xtals) can be arranged in space in a homogeneous array.

Each motif is equivilant to every other motif in the pattern

230 possible combinations of symmetry operations and translation operations in space

  1. Based on 1 of the 14 Bravais lattices
  2. Isogonal w/ 1 of the 32 point groups




Point group

The 32 possible combinations of rotational symmetry operations - both proper and improper

Point indicates that the symmetry operations are fixed around an "unmoved" center axis

  • 21 without a center of symmetry
  • 11 with a center of symmetry


Identical to the 32 crystal classes that are assigned based on morphology

The translation-free "residue" of a family of possible isogonal space groups



The rotation and screw axes have the same rotational angle

ex: 60° in a six-fold rotation (6) of six-fold screw axis (61)



The study of the external form and internal atomic arrangement of crystalline solids and the principles that govern their growth, external shape, and internal structure.


Unit Cell

The smallest unit of structure (or pattern) that can be infinitely repeated to generate the entire structure

Maintains the diagnostic properties of the mineral



An imaginary pattern of points (or nodes) in which every point/node has an environment that is identical to that of every other point/node in the pattern

No specific origin as it can be shifted parallel to itself


Law of Bravais

Faces are most likely to develop parallel to lattice planes that have a high density of lattice points

The larger the number of points/nodes, the more common the face


Steno's Law of the constancy of interfacial angles

The angles between equivilant faces of crystals of the same substance, measured at the same T, are constant


Symmetry Operations

The processes of rotation about an axis, of refection by a mirror, and of inversion about a central point



Rotation is established by rotating a motif about an imaginary axis that intersects the crystal center

When rotational symmetry is present, a specific face on the crystal is repeated n times in a complete rotation of the crystal



Reflection (mirror)

A mirror plane (m) is an imaginary plane that divides a crystal into two halves, each of which, in a perfectly developed crystal, is the mirror image of the other

A reflection across a mirror plane produces a mirror image



A center of symmetry (i) is present in a crystal if an imaginary line can be passed from any point on its surface thru its center and the same point is found on the line at an equal distance beyond the center of the opposite side

Tested by confirming that for each face of the crystal there is some parallel face

Produces an enantiomorphic pair



  1. First rotate the motif thru α (the rotation angle)
  2. Then invert the motif thru the center

Best seen and evaluated in 3D objects


Hermann-Maugin (HM) notation

The symbols for the overall symmetry content of a crystal


i, 3A4, 4A3, 6A2, 9m 

i = center of symmetry

3A4 = three four-fold rotations

4A3 = four three-fold rotations

6A2 = six two-fold rotations

9m = nine mirror planes



Miller Indices

The Miller indices of a face consist of a series of whole numbers that have been derived from the intercepts by inverting and, if necessary, the subsequent clearing of fractions.



Morphology is commonly used to indicate the general outward appearance or shape of crystals


Crystal Habit

In crystallography, external shape is denoted by the term, habit whereas form is used in a special and restricted sense


Crystal Form

Crystal habit = external shape

Crystal Form is used in a special and restricted sense

A form consists of a group of like crystal faces, all of which have the same relation to the elements of symmetry



An ordered pattern can be generated by taking a motif and repeating it or translating it a specific distance t1

The external form of a crystal, although an expression of its internal structure, lacks any expresion of translation


Space Lattices

When the five planar nets are stacked in various ways along a third direction (z), periodic arrays in three-dimensions results



A primitive space lattice is a parallelpiped with lattice points only at the corners



Nodes may be centered on a pair of opposite faces of the unit cell

When nodes are centered on a pair of opposite faces of the unit cell

A-centered = along the a (x) axial direction

B-centered = along the b (y) direction

C-centered = along the c (z) direction



When nodes are centered on all faces of the unit cell



When the node is present in the center of the unit cell, it is refered to as body-centered


Bravais Lattices

There are 14 lattices that are compatible with the 32 point groups

They represent the only possible wys in which nodes can be arranged in three dimensions

The varios lattices can be arranged among the seven crystal systems


Screw Operation

The combination of a rotarional operation with a translation (t) parallel to the axis of rotation

Any additional motifs are generated from the original motif in a 3-D, helical path - as in a screw motion

possible screws: 2, 21, 3, 31, 32, 4, 41, ...

When the ratio of the subscript to the number of rotation is less than 1/2 the screw is right-handed

When the ratio of the subscript to the number of rotation is  greater than 1/2 the screw is left-handed


Glide Operation

The combination of a mirror reflection with a translation component (t/2 or t/4) parallel to the mirror

also known as glide reflection

It relates motifs on either side of the line with a periodicity of hald the lattice translation


Glide Planes

3-D glide movements expressed in terms of the crystallographic axes: a, b, and c

Example: If it is parallel to the a-axis, then it is an a-glide. Similarly, if it is parallel to the b-axis, a b-glide, and a along the c-axis, a c-glide

diagonal (n) glides and diamond (d) glides exist



When the ratio of the subscript to the number of a rotation axis is less than 1/2

Defined as one that advances away from the observer when rotated clockwise



When the ratio of the subscript to the number of the rotation axis is greater than 1/2

Defined as one that advances toward the observer when rotated ?