Elementary MineralogyGS 122

Crystal Symmetry

Symmetry

The seven symmetry operators are 1)Translation2) Rotation3) Reflection4) Inversion (center of symmetry)5) Roto-inversion (inversion axis)6) Glide (translation + reflection)7) Screw (rotation + translation).

-spatial transformations or the spatial relationships between objects in a pattern

Symmetry Elements

- The primary function is to specify the reference point about which an action occurs.

The first five symmetry elements that we consider are 1) Translation vectors2) Rotation axes3) Mirror planes4) Centers of symmetry (inversion points)5) Inversion axes

Symmetry operations:

1. Translation

- the replication of an object at a new spatial coordinate - Shift in a specified direction by a specified length

Based on a, b and c vectors of unit cell, a translation vector t can be expressed as t = ua + vb + wc, where u, v and w are positive or negative integers.

Symmetry operations:

1. Translation

- used to build a crystal structure by replicating an object (the basis) at each of the Bravais lattice points

Figure. The lattice translation vector moves the object to an equivalent position in a different unit cell.

2. Rotation - motion through an angle about an axis - symmetry element is an N-fold rotation axis. -The multiplicity N is an integer.

-After having performed the rotation N times the object has returned to its original position.

2. Rotation

Figure. The five rotation operators that are consistent with translational symmetry.

- The large circles are lines of construction to guide the eye. - The solid object in the center shows the position of the

rotation axis and the small circle is the object which is repeated to form the pattern.

The 2 axis is referred to as a diad, the 3 axis as a triad, the 4 axis as a tetrad, and the 6 axis as a hexad.

2. Rotation

3. Reflection. -describes the operation of a mirror -The symmetry element is a reflection plane. Hermann–

Mauguin symbol: m.

Figures with the axes of symmetrydrawn in. The figure with no axes is asymmetric.

3. Reflection. -In 2D there is a line of symmetry, in 3D a plane 0f symmetry.

- An object or figure which is indistinguishable from its transformed image is called mirror symmetric.

Reflection converts a right-handed object into a left-handed or enantiomorphous (in opposite shape) replica.

4. Inversion‘- Reflection’ through a point. - This point is the symmetry element and is called inversion

center or center of symmetry.

5. Rotoinversion. -The symmetry element is a rotoinversion axis or, for short, an inversion axis. - This refers to a coupled symmetry operation which involves two motions: take a rotation through an angle of 360/N degrees immediately followed by an inversion at a point located on the axis

rotoreflection is a coupled symmetry operation of a rotation (2π/n) and a reflection at aplane perpendicular to the axis.

The operation of a two-fold rotoreflectionimproper rotation axis 2bar: (a) the initial atom 　 position; (b) rotation by 180 counter clockwise;

(c) reflection across a mirror normal to the axis;

(d) the operation of a centre of symmetry

Correspondence of rotoreflection and rotoinversion axes

6. Screw rotation. -The symmetry element is a screw axis. -It can only occur if there is translational symmetry in the direction of the axis.

-The screw rotation results when a rotation of 360/N degrees is coupled with a displacement parallel to the axis.

The Hermann– Mauguin symbol is NM (‘N subM’); N expresses the rotational component and the fraction M/ N is the displacement component as a fraction of the translation vector.

Screw axes and their graphical symbols. The axes 31, 41, 61, and 62 are right-handed; 32, 43, 65, and 64 are the corresponding left-handed screw axes.

The operation of a 42 screw axis parallel to the z direction; (a) atom A at z =0 is repeated at z=T and then rotated counter clockwise by 90; (b) the atom is translated parallel to z by a distance of t =2T/4, i.e. T/2 to create

atom B;(c) atom B is rotated counter clockwise by 90 and translated parallel to z by a distance of t = 2T/4, i.e. T/2, 　 to give atom C;

Construction of 42 screw operation:

(d) atom C is at z=T, the lattice repeat, and so is repeated at z = 0; (e) repeat of the symmetry operation produces atom D at z = T/2; (f) standard crystallographic depiction of a 42 screw axis viewed alongthe axis

Construction of 42 screw operation:

In this figure, the motif is represented by a circle, the + means that the motif is situated above the plane of the paper and ½+ indicates the position of a motif generated by screw operation.

If the screw axis runs parallel to the c-axis, the heights could be written as +z and +z + ½c, where c is the lattice parameter.

the total displacement is represented by the vector nt, running parallel to the rotation axis

where p is an integer, and T is the lattice repeat in a direction parallel to the rotation axis.

the repeat translations for a threefold screw axis are:

7. Glide reflection. -- The symmetry element is a glide plane. It can only occur if translational symmetry is present parallel to the plane. At the plane, reflections are performed, but every reflection is coupled with an immediate displacement parallel to the plane.

Hermann–Mauguin symbol is a, b, c, n, d or e, the letter designating the direction of the glide referred to the unit cell. a, b and c refer to displacements parallel to the basis vectors a, b and c, the displacements amounting to 1/2a, 1/2 b and 1/2c, respectively.

The glide planesn and d involve displacements in a diagonal direction by amounts of ½ and ¼ of the translation vector in this direction, respectively. e designates two glide planes in one anotherwith two mutually perpendicular glide directions.

Top left:Perspective illustration of a glide plane. Other images: printed and graphical symbols for glide planes perpendicular to a and c with different glide directions.z = height of the point in the unit cell

The combination of a twofold rotation and a reflection at a plane perpendicular to therotation axis results in an inversion

Point Groups-- Possible combinations of symmetry operations excluding translations

The seventeen plane groups:

derived by combining the translations inherent in the five plane lattices with the symmetry elements present in the ten plane point groups, together with the glide operator, represent, in a compact way, all possible planar repeating patterns.

CRYSTALLOGRAPHIC SPACE GROUPS:The 230 crystallographic space groups summarise the total number of three-dimensional patterns that result from combining the 32 point groupswith the 14 Bravais lattices and including the screw axes.

Each space group is given a unique symbol and number,

Rotation, inversion and screw axes allowedin crystals

Full Hermann–Mauguin symbols for some point groups

** Mirror planes always normal to the axis it refers to.

Rules for Point group nomenclature

-- to easily interpret the meaning of the symbols

(1) Each component in the name refers to a different direction. For example, the symbol for the orthorhombic group, 222, refers to the symmetry around the x, y, and z axes, respectively.

(2) The position of the symbol m indicates the direction perpendicular to themirror plane.

(3) Fractional symbols mean that the axes of the operators in the numerator and denominator are parallel. For example, 2/m means that there is a mirror plane perpendicular to a rotation diad.

(4) For the orthorhombic system, the three symbols refer to the three mutuallyperpendicular x, y, and z axes, in that order.

(5) All tetragonal groups have a 4 or 4 bar rotation axis in the z-direction and this is listed first. The second component refers to the symmetry around the mutually perpendicular x and y axes and the third component refers to the directions inthe x–y plane that bisect the x and y axes.

(6) In the trigonal systems (which always have a 3 or 3bar axis first) and hexagonal systems (which always have a 6 or 6bar axis first), the second symbol describes the symmetry around the equivalent directions (either 120° or 60° apart) in the plane perpendicular to the 3, 3bar, 6, or 6bar axis.

(7) A third component in the hexagonal system refers to directions that bisect the angles between the axes specified by the second symbol.

(8) If there is a 3 in the second position, it is a cubic point group. The 3 refers to rotation triads along the four body diagonals of the cube. The first symbol refers to the cube axis and the third to the face diagonals.

Merohedry--according to the presence of symmetry elements other than the major(or unique) axis.

Another classification of point groups is based on their action. Thus, centrosymmetric point groups, or groups containing a center of inversion are shown in Table 2.8 in bold, while the groups containing only rotational operation(s) and, therefore, not changing the enantiomorphism (all hands remain either left of right

Projected representation of patterns formed by the tetragonal point groups

A)

B)

The cubic point groups

Example: Point group 23 has the minimum number of symmetry operations-- three diad axes as joining the midpoints of opposite edges of a tetrahedron, which serve the reference axes-- four triads are directed from the four vertices of the tetrahedron to the centers of the opposite faces

Examples of three point groups. The letters under the Hermann– Mauguin symbols indicate to which directions the symmetry elements refer.

Examples of space-group type symbols and 　 their meanings.

The 230 crystallographic space groups arranged according to seven crystal systems and 32 crystallographic point groups as they are listed in the International Tables for Crystallography,vol. A.

The centrosymmetric groups are in bold, while the noncentrosymmetric groups that do not invert an object are in italic. The remaining are noncentrosymmetric groups that invert an object(contain inversion axis or mirror plane).

Uses of the table above:

--Monoclinic system: the rotation axis is parallel to the primary direction, which is the [010] direction.

--Trigonal system, the primary direction is the [111], so that the three-fold axis in this system is parallel to [111]. The two-fold axis lies along the tertiary [1-10] directions, since these directions are at right angles to [111].

--Hexagonal system: The six-fold axis lies along the primary [00.1] directions

--Tetragonal system: the four-fold rotation axis lies along [001] the two-fold axes lie along both secondary and tertiary directions whereas the orthorhombic system has a two-fold axis along each of the 3 symmetry directions.

Space group symbol, the particular sequence of the symmetry elements listed describes their orientation in space relative to the three crystallographic axes; (rule for listing symmetry elements:1. Triclinic system, trivial point

2. Monoclinic system, there is a choice between calling the unique axis c or b. In giving the complete space-group symbol, if the symmetry elements are listed in the sequence abc, the two symbols for P2 are, respectively, P112 (first setting) and P121 (second setting)

3. Orthorhombic system, list the symmetry elements in the order abc. Example, space groups belonging to the crystal class 2mm are represented by c as the unique axis, namely, Pmm2.

nontrivial nature: two different space groups

Pmna P2/m2/n21/a and PnmaP21/n21/m21/a

4. Tetragonal system, the c axis is the 4-fold axis.

The sequence for listing the symmetry elements is c, a, [110] since the two crystallographic axes orthogonal to c are equivalent.

Ex: P-4m2 states that the unique c axis in a primitive tetragonal lattice had the symmetry -4, the two a axes each are parallel to m, and the [110] direction has the symmetry 2.

5. Hexagonal system, c is the unique axis and a1=a2.

P denotes the primitive hexagonal lattice, whereas R denotes the centered hexagonal lattice in which the primitive rhombohedral cell has been chosen as the unit cell.

4. Tetragonal system, the c axis is the 4-fold axis.

The sequence for listing the symmetry elements is c, a, [110] since the two crystallographic axes orthogonal to c are equivalent.

Ex: P-4m2 states that the unique c axis in a primitive tetragonal lattice had the symmetry -4, the two a axes each are parallel to m, and the [110] direction has the symmetry 2.

5. Hexagonal system, c is the unique axis and a1=a2.

P denotes the primitive hexagonal lattice, whereas R denotes the centered hexagonal lattice in which the primitive rhombohedral cell has been chosen as the unit cell.

6. Cubic system, all three crystallographic axes are equivalent, and the order of listing the symmetry elements is a, [111], [110]. Since [111] must have the symmetry 3 or -3, the appearance of a 3 in the second position serves to distinguish the cubic system from the hexagonal system.

Unit cell of the body-centered cubic 　 packing of spheres and the coordination 　 around one sphere

Unit cells of the rutile and the trirutile structures. The positions of the twofold rotation axes have been included.

Unit cells for 　 hexagonal (left) 　 and cubic 　 closest-packing of 　 spheres. Top row: 　 projections in the 　 stacking direction. 　

Spheres are drawn 　 smaller than their 　 actual size. Spheres 　 with the samecoloring form 　 hexagonal layers as 　 in Fig. 14.1.

Fig. 14.1 Arrangement of 　 spheres in a 　 hexagonal layer 　 and the relative 　 position of the 　 layer positions A, B 　 and C