Symmetry, Groups and Crystal Structures

Symmetry, Groups and Crystal Structures

Symmetry, Groups and Crystal Structures The Seven Crystal Systems Minerals structures are described in terms of the unit cell The Unit Cell The unit cell of a mineral is the smallest divisible unit of mineral that possesses the

symmetry and properties of the mineral. It is a small group of atoms arranged in a box with parallel sides that is repeated in three dimensions to fill space. It has three principal axes (a, b and c) and Three interaxial angles (, , and ) The Unit Cell is angle between b and c is angle between a and c

is angle between a and b Seven Crystal Systems The presence of symmetry operators places constraints on the geometry of the unit cell. The different constraints generate the seven crystal systems.

Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Hexagonal Cubic (Isometric)

Seven Crystal Systems Triclinic a b c; 90 120 Monoclinic a b c; = 90 120 Orthorhombic a b c; = Tetragonala = b c; = Trigonal a = b c; = ; 120 Hexagonal a = b c; = ; 120 Cubic a = b = c; =

Symmetry Operations A symmetry operation is a transposition of an object that leaves the object invariant. Rotations 360 , 180 , 120 , 90 , 60 Inversions (Roto-Inversions) 360 , 180 , 120 , 90 , 60

Translations: Unit cell axes and fraction thereof. Combinations of the above. Rotations

1-fold 2-fold 3-fold 4-fold 6-fold 360 180

120 90 60 I 2 3 4 6

Identity Roto-Inversions (Improper Rotations)

1-fold 2-fold 3-fold 4-fold 6-fold 360 180 120 90

60 Translations Unit Cell Vectors Fractions of unit cell vectors (1/2, 1/3, 1/4, 1/6) Vector Combinations Groups

A set of elements form a group if the following properties hold: Closure: Combining any two elements gives a third element Association: For any three elements (ab)c = a(bc). Identity: There is an element, I such that Ia = aI = a Inverses: For each element, a, there is another element b such that ab = I = ba Groups

The elements of our groups are symmetry operators. The rules limit the number of groups that are valid combinations of symmetry operators. The order of the group is the number of elements. Point Groups (Crystal Classes) We can do symmetry operations in two

dimensions or three dimensions. We can include or exclude the translation operations. Combining proper and improper rotation gives the point groups (Crystal Classes) 32 possible combinations in 3 dimensions 32 Crystal Classes (Point Groups) Each belongs to one of the (seven) Crystal Systems Space Groups

Including the translation operations gives the space groups. 17 two-dimensional space groups 230 three dimensional space groups Each space group belongs to one of the 32 Crystal Classes (remove translations) Crystal Morphology A face is designated by Miller indices in

parentheses, e.g. (10 0 ) (111) etc. A form is a face plus its symmetric equivalents (in curly brackets) e.g {10 0 }, {111}. A direction in crystal space is given in square brackets e.g. [10 0 ], [111]. Halite Cube Miller Indices Plane cuts axes at

intercepts (,3,2). To get Miller indices, invert and clear fractions. (1/, 1/3, 1/2) (x6)= (0 , 2, 3) General face is (h,k,l) Miller Indices The cube face is (10 0 )

The cube form {10 0 } is comprises faces (10 0 ),(0 10 ), (0 0 1), (-10 0 ),(0 10 ),(0 0 -1) Halite Cube (100) Stereographic Projections Used to display crystal

morphology. X for upper hemisphere. O for lower. Stereographic Projections We will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalents (general form hkl).

Illustrated above are the stereographic projections for Triclinic point groups 1 and -1.

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