Chapter 1Crystal StructurePhys 175ADr. Ray KwokSJSUIdeal Crystal Infinite periodic structure No edge No impurity Translational invariance for all observablesCrystal = Lattice + BasislatticebasiscrystalTranslational invariance {a} With a set of primitive vectors {a} such that r’ = r + u1a1+ u2a2+ u3a3where the u’s are integers and a’s are Primitive Translational Vectors  If the origin is chosen at a lattice point, all other lattice points is given by this Translational Vectors (Translational Operator) T = u1a1+ u2a2+ u3a3Primitive Translational Vectors are not unique Many choices. eg 2D latticeCrystal axes Crystal axes are not necessarily the same as translational vectors. Sometimes it’s more convenience to use axes with more symmetrical properties such as rotation & reflection (called Point Group Symmetry)translationalcrystalCrystallography Characterization and enumeration of crystal structure.  Require knowledge of Group Theory –symmetry operators (eg. The rotational group in the Rubic Cube’s solution…)Space GroupPoint GroupUnit Cell Somewhat arbitrary...  Defined by T only (or multiples). Can have more than 1 lattice point per cellPrimitive Lattice Cells defined by the parallelepiped of {a}  one Lattice point per primitive cell.  volume = | a1· a2x a3| or its cyclic permutationWigner-Seitz Cell bisectors planes of the nearest neighbors.  unique, does not depend on choice of {a}Common Crystals simple cubic: all a’s are equal, α β γ are 90 deg. Tetragonal: 2 a’s are equal, α β γ are 90 deg. Orthorhombicall different a, α β γ are 90 deg. Trigonalall a’s are equal, α β γ are equal but not 90 deg. Hexagonal2 a’s are equal, α β are 90 deg. γ is 120 deg.( not 60??) (convention)Cubic conventional cells: sc, bcc, fcc Not primitive cells. # of atoms per cell is 1, 2 & 4.bcc & fccfcc()( )( )+=+=+=xˆzˆazˆyˆayˆxˆa213212211aaarrr()( )( )+−=++−=−+=zˆyˆxˆazˆyˆxˆazˆyˆxˆa213212211aaarrrbccPrimitive CellConventional Primitive VectorsBasis Crystal Structure = lattice point + basis Basis = rj= xja1+ yja2+ zja3for the jth-atom in the cell, where |x, y, z| ≤ 1  Examples are NaCl (Fig 17 & 18).  Space lattice is fcc, basis Cl-at 000, and Na+½ ½ ½ In primitive cell: Cl at 000, ½ ½ 0, ½ 0 ½ , 0 ½ ½ (fcc) Na at ½ ½ ½ , 0 0 ½ , 0 ½ 0, ½ 0 0Point Group Rotation & Reflection Most lattice has 2, 3, 4 or 6 fold rotational symmetry. Basis can be a lot more complicated.(flip two edges).(twist two corners)Crystal Plane Index System for Crystal Planes. See Fig 15. Take intercepts on (a’s ) eg (3,2,2) Inverse of them (1/3, ½, ½ ), multiply with LCD 6 to get (233).  It describes the set of planes. e.g. (100) set, (0ī0) set, (012)…etc(100), (200), (300), (ī00)…etcMiller Indices In cubic primitive cell, this is the direction of the normal [233] of the (223) planes. For a (hkl) plane, [hkl] is the direction vector perpendicular to plane (normal). Use top-bar for negative. Family of planes with crystal symmetry is denoted by {hkl}. e.g. {100} means (100), (010), (001), (ī00), (0ī0), (00ī) in sc structures.Example – simple cubic (233) planeExample – fcc (233) plane fcc (233) plane, intercepts are (3,2,2)  With unit vectors defined in Fig 13 (for example),  the normal [233] is calculated to be[ ] [ ] [ ][ ]zyxaNzxazyayxaaaaNˆ3ˆ5ˆ52ˆˆ23ˆˆ23ˆˆ332321++=+++++=++=rrrrr()( )( )+=+=+=xˆzˆazˆyˆayˆxˆa213212211aaarrrClosed-Packed Structures hcp is basically hexagonal with off-set layers. For identical spheres: fcc & hcp are the most dense packing 0.74 of volume occupied.  These are the closest packed structures. (should try to convince yourself that this is true)HCP & FCC Fig 21 – offset hexagonal every other layer ABABAB = hcp Offset hexagonal twice ABCABCABC = fccABCACoordination number # of nearest neighbors (touching spheres) As in Table 2, coordination # = 6 for sc, 8 for bcc, 12 for fcc.Other structures Diamond structure. Space lattice is fcc. 2-pt basis 000, ½ ½ ½ (Fig 25) No primitive cell of single atom is possible. Zinc Blend…etcExample – simple cubic Sketch (201) planes  Calculate the shortest distance between planesIntercepts (1, infinity, 2)daCrystal planesxzdaa/2aa/2θθθ(√5)ad = a sinθ = a [a/(√5)a]d = a/√5Direction of normal ?[201]Example – fcc planes Sketch the (111) planes in the standard fcc primitive vectors representation. What’s the normal of the planes in Cartesian coordinates? What’s the shortest distance between planes?Solution – from symmetryNormal = (1,1,1) in CartesianThe adjacent planes are shown in the diagram (left)The planes are re-drawn to show the symmetry of the cube.The body diagonal is separated by the 2 planes so the shortest distance isd = √3a/3 = a/√3Solution – from calculationa1=a/2(1,1,0); a2=a/2(0,1,1); a3=a/2(1,0,1) in Cartesian CoordinatesLine 12: a/2(-1, 0, 1) Line 13: a/2(0, -1, 1)The normal is given by the cross-product of these 2 vectors:zˆyˆxˆ110101zˆyˆxˆ++=−−Un-normalized normal vector = (1,1,1) in CartesianSolution – from calculation (cont)a1=a/2(1,1,0); a2=a/2(0,1,1); a3=a/2(1,0,1)Equation of planes : Ax + By + Cz = k (constant)Point 1: (a/2, a/2, 0); A + B = 2k/aPoint 2: (0, a/2, a/2); B + C = 2k/a, so A = CPoint 3: ((a/2, 0 a/2); A + C = 2k/a, so A = B = C = k/aTherefore, equation of the first plane (blue plane in previous diagram)x + y + z = a [and normal = (A,B,C) = (1,1,1) un-normalized]The next plane has 3 corners at (a, a, 0), (0, a, a) and (a, 0, a) and the corresponding coefficients A = B = C = k/2a. The equation of the (red) plane is:x + y + z = 2a or x + y + z = na for all the planesThe points at which the normal go thru these planes from origin are:a/3(1,1,1), and 2a/3(1,1,1)The distance between these 2 points is just d = (a/3)√3 = a/√3Homework Ch. 1, Problem # 1, 2, 3  + Proof of Table