Equivalent Positions in 3-D Figure (a) shows equivalent points generated by a 4-fold rotation axis. Start at x,y,z; a counterclockwise rotation of 90o generates a point with coordinates y,x,z; then x,y,z; and finally to y,x,z. Because the points are related via a symmetry operation, they are equivalent. What if the operator is a 42 screw axis (¼ rotation & translation by ½c)? Figure (b). The equivalent positions will be: x,y,z; y,x,z+½; x,y,z; y,x,z+½ _ _ _ _ _ _ _ _

Equivalent Positions in 3-D For the Space Group P42/m, in addition to the 42 screw axis there is a mirror plane perpendicular to that axis. So, for each of the four equivalent positions, there is an additional reflected point at –z. So, if we know that there is an atom at x = 0.1, y = 0.2, & z = 0.3, there are seven additional (equivalent) atoms at: 0.1 0.2 0.3 0.1 0.2 0.7 0.8 0.1 0.8 0.8 0.1 0.2 0.9 0.8 0.3 0.9 0.8 0.7 0.2 0.9 0.8 0.2 0.9 0.2 note: 0.3 reflected yields -0.3, which is the same as 0.7.

Special Positions Look at a particular point in P42/m, x,y,0, which lies on the mirror plane. The eight equivalent points (using the general formula) would be: obviously the four points on the right are identical to the four on the left. so, there are only four distinct points. Whenever the number of general equivalent points is reduced, usually because the point lies on a symmetry element, it is called a Special Position. x y 0 x y 0 y x ½ y x ½ _ _ _ _ _ _ _ _

Table of Equivalent Positions multiplicity location of equivalent positions e.g. if atoms are occupying position 2f, this means that there are two equivalent atoms at ½ ½ ¼ and ½ ½ ¾, and the symmetry at that position is 4. lowest letter = General Position a-j = Special Positions a = position with highest symmetry symmetry at that position 1 = none _

Using the Table of Equivalent Positions PdS has a tetragonal crystal structure with a = b = 6.429Ǻ & c = 6.608Ǻ; its Space Group is P42/m, and there are 8Pd and 8S per unit cell. k 2e 2c 4j the sulfurs occupy the General Position there are 3 crystallographically different sets of Pd 2e = edges above corners 2c = edges along base; ½ up a face 4j = four interior positions P42/m PdS a = b = 6.429Ǻ & c = 6.608Ǻ 8S in 8k 0.19 0.32 0.23 2 Pd(1) in 2e 2Pd(2) in 2c 4Pd(3) in 4j with x = 0.48, y = 0.25 Note that all of the above information can be completely described by the following: Note: Nothing at lattice points!

International Tables for Crystallography primitive inversion of 230 o = symbol for inversion center. location of symmetry elements in ab plane; may be other projections---here ac & bc planes. location of equivalent points in the general position. guide lines y x + ≡ general point within the unit cell. - means below this unit cell.* , means the handedness has changed. *if + = 0.1, 0.1, 0.1, then - = -0.1, -0.1, -0.1, or 0.9, 0.9, 0.9 within this unit cell. location of the motif

International Tables for Crystallography a c d f b a e d General Position Special Positions b g a c Not shown on right. Why not?

International Tables for Crystallography x: C2 y: C2 z: screw axis a ≠ b ≠ c C2 axis located ¼ up z-axis║ to y-axis. 2-fold axis in page 2-fold axis ┴ to page _ + y z x x 21 screw axis ┴ to page 21 screw axis in page + + _ z generated by () generated by ↑ () y generated by ↑

International Tables for Crystallography C2/m unit cell centered on C face mirror plane in this plane axial glide plane in direction of arrow located ¼ above plane inversion center & C2 rotation ┴ to plane lattice pts on corners lattice pt located ½ above this plane inversion center ¼ above plane & 21 screw axis ┴ to plane • • • • • • two atoms lie on top of each other in this projection. _ , + _ ,

International Tables for Crystallography C2/m unit cell centered on C face add these values to all coordinates; due to centered, rather than primitive unit cell. note multiplicity of 8, but only four coordinates given.

Pearson’s Crystallographic Data Index Listing of many known crystals, sorted by Space Group. TN690.P361 1991 a = triclinic P = primitive 24 = 24 atoms/unit cell 4 atoms in formula so, 6 units in cell unit cell dimensions most atoms in general position, 2(i): x,y,z; x, y, z. _ _ _ although two in special positions 24