Presentation on theme: "Equivalent Positions in 3-D"— Presentation transcript:
1Equivalent 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__ ____ __
2Equivalent 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:note: 0.3 reflected yields -0.3, which is the same as 0.7.
3Special PositionsLook 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 0y x ½ y x ½_ __ _ _ __ _
4Table of Equivalent Positions multiplicitylocation of equivalent positionse.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 Positiona-j = Special Positionsa = position with highest symmetrysymmetry at that position1 = none_
5Using 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.k2e2c4jthe sulfurs occupy the General Positionthere are 3 crystallographically different sets of Pd2e = edges above corners2c = edges along base; ½ up a face4j = four interior positionsP42/m PdSa = b = 6.429Ǻ & c = 6.608Ǻ8S in 8k2 Pd(1) in 2e2Pd(2) in 2c4Pd(3) in 4j with x = 0.48, y = 0.25Note that all of the above information can be completely described by the following:Note: Nothing at lattice points!
6International Tables for Crystallography primitiveinversionof 230o = 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 linesyx+ ≡ 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, 0.9within this unit cell.location of the motif
7International Tables for Crystallography a cd fb ae dGeneral PositionSpecial PositionsbgacNot shown on right.Why not?
8International Tables for Crystallography x: C2y: C2z: screw axisa ≠ b ≠ cC2 axis located ¼ up z-axis║ to y-axis.2-fold axis in page2-fold axis ┴ to page_+yzxx21 screw axis ┴ to page21 screw axis in page_zgenerated by()generated by ↑()ygenerated by ↑
9International Tables for Crystallography C2/munit cell centeredon C facemirror planein this planeaxial glide planein direction of arrowlocated ¼ above planeinversion center &C2 rotation ┴ to planelattice pts on cornerslattice pt located ½ above this planeinversion center¼ above plane &21 screw axis ┴ to plane• • •• • •two atoms lie on top of each other in this projection._ ,+_ ,
10International Tables for Crystallography C2/munit cell centeredon C faceadd these values to all coordinates; due to centered, rather than primitive unit cell.note multiplicity of 8,but only four coordinates given.
11Pearson’s Crystallographic Data Index Listing of many known crystals, sorted by Space Group.TN690.Pa = triclinicP = primitive24 = 24 atoms/unit cell4 atoms in formulaso, 6 units in cellunit cell dimensionsmost atoms in generalposition, 2(i): x,y,z; x, y, z._ _ _although two in special positions24