Topic 4: Crystal Systems and Forms

Symmetry

Used to characterize crystals, identify repeating parts of molecules and simplify data collection and calculation; described with respect to a transformation if the object appears to be in a state that is identical to the initial state after the transformation

Symmetry Element

A geometric feature that expresses the symmetry of an ordered arrangement

Symmetry Operation

The process (or apparent movement) of the object that results in this symmetry element

Mirror Plane (m)

These reflect an object, or a specific crystal face or a unit of a structure into its mirror image (operation is a reflection through a plane/axis) and can go through edges of crystals but also faces (reflection through x/y-axis)

Rotation Axes (R)

Imaginary lines about which an object, or a crystal face, or an atomic arrangement (inside the crystal structure) is rotated and repeats itself once or several times during a complete (360 degrees) rotation

Centre of Symmetry (i)

Also known as inversion, it is where every face on a crystal with a centre of symmetry has an equivalent, parallel face on the opposite side (basically its reflected and flipped upside down) (reflection through x=y axis)

Rotoinversion Axes (r̄)

Imaginary lines about which a crystal face or an atom (or atom cluster) in a crystal structure is rotated as well as inverted

Higher Symmetry

The combination of two or more symmetry elements/operations; more than one symmetry can be applied to a crystal

Crystal Systems

The grouping of the 32 symmetry combinations based on: how many axes it has, the lengths of the axes (a=shortest, c=longest, b=intermediate), and the angles at which the axes meet

Isometric (Cubic) System

Includes the presence of four 3-fold axes (through corners, 4R3, 4r̄3) (all same lengths); eg. cubic, octahedron, dodecahedron

Hexagonal Crystal System

Includes one 6-fold axis or one 3-fold axes (R6, r̄6, R3, r̄3); eg. hexagonal prism, hexagonal dipyramid

Tetragonal Crystal System

Includes one 4-fold axis (R4, r̄4), where a1=a2 but NOT a3; eg. tetragonal prism, tetragonal dipyramid

Orthorhombic Crystal System

Includes three 2-fold axes (R2) that are mutually perpendicular with binary (m) symmetry (all different lengths); eg. prisms, pyramids

Monoclinic Crystal System

Includes one two-fold axis (1R2) or one mirror plane (1m), all different lengths with two at 90 degrees, and one not; eg. clinopinacoid

Triclinic Crystal System

Either has no symmetry, or one centre of symmetry (i), three different axes at different lengths with none at 90 degrees

Twinning

A symmetrical intergrowth of two or more crystals of the same substance

Growth Twins

Result of an emplacement of atoms, or ions on the outside of a growing crystal in such a way that the regular arrangement of the original structure (and hence the crystal lattice) is interrupted

Primary Twinning

These are considered “accidents” or nucleation errors that occur during the free growth of a crystal

Twin Element

A new symmetry element not present in the untwinned form; eg. a mirror plane or twinned plane, a rotation axis or twin axis, an inversion about a point or a twin-centre

Composition Plane/Surface

The surface on which two individuals are united

Contact Twins

A regular composition plane joins the two individuals and such individuals are separated by a twin plane (common in orthoclase and staurolite)

Penetration Twins

Appear to be intergrown with the individuals interpenetrating, and an irregular composition surface; usually produced by rotation and defined by an axis of rotation or a twin axis (eg. Carlsbad twinning in orthoclase or iron cross in pyrite)

Multiple/Repeated Twins

Three or more individual crystals that are twinned according to the same twin law

Polysynthetic Twins

If all the composition surfaces are parallel, common in plagioclase feldspar (striations reflecting twins)

Cyclic Twins

Occurs when all the compositional surfaces are not parallel (eg. rutile)

Secondary Twinning

Twinning which occurs once the crystal growth has been completed

Deformation Twins

Form in response to mechanical stress and the crystal relieves stress on its structure by twinning

Inversion Twins

Results from the structural change that happens when an earlier-formed crystal transforms into a lower symmetry as a result of decreasing temperature

Polymorphism

The ability of a specific mineral (or chemical compound) to occur in more than one type of structure, as a function of changes in temperature, pressure, or both

Crystal Classes

32 unique combinations of symmetry elements; also known as point group

Crystallographic Axes

A set of axes that is compatible with each of the six crystal systems, or point groups

Translation

A shift in position without rotation

Miller Index

A set of three digits (or four, as in the hexagonal system) that describe the orientation of a crystal plane with respect to a set of crystallographic axes

Crystal Form

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

Unit Face

A crystal face (or plane) that intersects all crystallographic axes at unit distances

General Form

A form of a crystal in which the faces intersect all of the crystallographic axes at different lengths

Open Form

A crystal form that does not enclose space, such as a four-faced prism

Closed Form

A crystal form that encloses space, such as a cube in the isometric system

Zone

A collection of crystal faces with parallel edges

Twin Law

A statement about how two or more identical crystals are related to each other by a new symmetry element

Reconstructive Polymorphism

This involves extensive structural rearrangement in going from one atomic structure to another

Displacive Polymorphism

In this type of polymorphism, the overall structure is left completely intact and no bonds between atoms are broken; there is only a slight displacement of the atoms (or ions) and some readjustment of bond angles
between the atoms

Transformation Twins

A twin pattern that results when a crystal (or mineral grain) that was formed at high temperature is cooled and rearranges its crystal structure from a high- to a low-temperature architecture

Order-Disorder Polymorphism

This type of polymorphism ranges from a state of perfect order (at 0 K), through partial disorder, to total disorder and relates to the site occupancy by a specific atom (or ion) in a specific atomic site

Perfect Order

This describes the 100% probability of finding a specific atom (or ion) in a specific atomic site in a crystal structure and only occurs at absolute zero

Partial Order

This describes the distribution of a specific atom (or ion) equally over two equivalent atomic sites in a crystal structure

Total Disorder

This describes the equal probability of finding one of two different atoms (or ions) in a specific atomic site in a crystal structure

Polytypism

A special kind of polymorphism in which different structural arrangements result from the different ways in which identical sheets (or layers) of structure are stacked in three dimensions

Polytype

Also known as stacking polymorph, it is type of polymorph in which different structural arrangements result from the different stackings of nearly identical atomic structural units, such as layers