Symmetry. The primitive lattice vectors are:

Transcription

1 Syetry Previously, we noted all crystal structures could be specified by a set of Bravais lattice vectors, when describing a lattice you ust either use the priitive vectors or add a set of basis vectors (e.g., ato positions) to the conventional vectors, The cubic F lattice For cubic F lattice, the conventional lattice vectors are: a ax b ay c az The priitive lattice vectors are: a a a ( y z) b ( z x) c 2 2 a ( x 2 y) In practice, the nuber of basis atos can be quite large and siply listing the is cubersoe., e.g. Al 2 O 3 has 12 Al & 18 O=30 basis vectors with no translational centering By acknowledging the syetry of the atoic configuration, which ust eet iniu syetry criteria, it is possible to describe the basis down to a sall nuber of paraeters. Thus, crystal structure data are always presented with reference to 32 point groups (acroscopic syetry operations) or crystal classes and 230 space groups (icroscopic syetry operations) or crystal structures. or the underlying syetry of the structure. We need to define syetry operators and groups of operators that are used to describe long-range configuration of atos in a crystal, such as you will see in diffraction data. Knowledge of syetry is extreely iportant for understanding the theral, optical, echanical, agnetic, electrical, properties of solids, e.g. the presence of an inversion center 1 eliinates piezoelectricity - electric polarization is induced by applying external forces/stresses.

2 Syetry Operators A syetry operator describes an action that can be used to develop a pattern by changing the position and/or orientation of an object in space. The 7 syetry operators are : translation, rotation, reflection (irror), inversion (center of syetry), roto-inversion (inversion axis), glide (translation + reflection), and screw (translation + rotation). Syetry eleents are iaginary objects that perfor the syetry operation. It s used to specify the reference point about which an operation is perfored. When perfored on any object, the syetry operation will bring equivalent points into coincidence. The above, first 5 acroscopic (for point groups) syetry eleents to consider are: 1. Translation vectors. 2. Rotation axes: syetry about an axis of rotation (line). 3. Mirror planes: syetry across a reflecting plane. 4. Centers of syetry (inversion points): syetry about a point. 5. Inversion axes (roto-inversion): cobination of rotation and inversion. Reeber the operators in a point syetry group leave at least one point in the pattern unchanged. Excellent site for 3-D visualization of the 32 point groups: 2

3 Syetry Operators (continued) 1. Translation is the replication of an object at a new spatial coordinate. If the operator is the vector, R, and if there is an object (e.g. ato) located at a position specified by the vector r xa yb zc then we know that an identical object will be located at This operator is sae as previously discussed Bravais lattice vector: R ua vb wc Translation is used to build a crystal structure by replicating an object (basis) at each of the Bravais lattice points. r R Lattice points at (0,0,0) and (1/3 x, 2/3 y, 1/2 z): 3

4 Syetry Operators (continued) 2. Rotation is otion through an angle about an axis, syetry about a line. Rotation about a certain axis brings the lattice into a position indistinguishable fro itself. Since repeated operations ust eventually place the object in its original position, the possible angles are constrained by the condition that na = 2p, where n is integer nuber (n=1, 2, 3, 4, and 6) of rotations, or ties to bring it back to original position, and a is the angle of each rotation, in radians. (a=2p, p, 2p/3, p/2, p/3) The five rotation operators that are consistent with translational syetry. The solid object in center shows the position of the rotation axes and sall circle is the object which is repeated to for the pattern These are 5 of the 32 point groups and easiest to visualize. Syetry about a line: (Note: we saw these previously for 10 2-D point groups) 4

5 Syetry Operators (continued) 3. Reflection describes operation of a irror, or syetry about a plane. The syetry eleent is the irror plane denoted by : L.H. R.H. The reflection operator. The axis of a irror refers to its noral. The right hand (R.H.) object is specified by an open circle and left hand (L.H.) replica is specified by a coa. Reflection converts a right-handed object into a left-handed replica: Equivalent points are brought into coincidence by reflection across the plane (irror). The positions of irror planes on 32 point group stereogras are specified by bold lines. When a irror plane is noral to an axis of rotation, is placed in the denoinator and a 1,2,3,4 or 6 is in the nuerator,e.g. 4/ vs. when a irror plane is parallel to rotation axis: 4 4. Inversion this operation occurs through an eleent called a center of syetry, syetry about a point. A center of syetry at the origin transfors an object at (x,y,z) to the position (-x,-y,-z): Projection (along z) of the pattern fored by an inversion center (a). The (+) and (-) sign indicate sall vertical displaceents above (+) and below (-) the plane of the paper/slide. Written sybol is i or ī. Like the irror, inversion also creates a lefthand replica. 5

6 Syetry Operators (continued) 5. Roto-inversion operator rotates an object in a pattern about its axis and then inverts the object through a center of syetry on the axis: Equivalent points are brought into self-coincidence by a cobined rotation and inversion. A A A : A is now on top of A The roto-inversion axes produce only one pattern that could not be produced by other operators used alone or in cobination. The roto-inversion operators written sybols is the sae as n-fold rotation axes but with bar on top, e.g. ī: These are 5 ore of the 32 point groups to total 10 so far: coordinates: x = y z 6

7 Suarize the first 10 Operators fall into 2 classes, those producing a right hand replica (proper) and those with left hand or irror iage replica (iproper). 5: 5: 7

8 Syetry Operators (continued) Transforation Matrix: coordinates: x = y z 8

9 Rest of the Point Groups In principle, there are a treendous nuber of point syetry groups that can be constructed fro the 5 operators we just defined. However, we are only interested in the ones that can be cobined with Bravais lattice vectors to build a crystal. Since the Bravais lattice translations theselves already have soe syetry, this liits the possibility. For exaple, tetragonal crystals have a rotation tetrad parallel to [001]. Thus, this operator ust be part of any point syetry group used in conjunction with tetragonal Bravais lattices vectors. There are 32 (distinct) point groups that are copatible with one of the 14 Bravais lattice translations. Let s look at the other 22: A A then reflect (irror) & A A then reflect (irror) RH + LH - Bold line on edge of stereogra ( is to rotation axis) 6 / ==2 =6 (up or down) These are 3 ore of the 32 point groups to total 13 so far. 9

10 Rest of the Point Groups (continued) A A then reflect (irror) & A A then reflect (irror) Bold line inside stereogra ( is to rotation axis) These are 4 ore of the 32 point groups to total 17 so far. (Note: we saw these previously for 10 2-D point groups) 10