Solid State Chemistry Principles

The Solid State Chemistry The universe consists only of atoms and the void: all else is opinion and illusion. Edward Robert Harrison

Conventional & Primitive Unit Cells Un t Cell Types Conventional (Non-primitive) Primitive More than one lattice point per cell Volume (area) = integer multiple of that for primitive cell A single lattice point per cell The smallest area in 2 dimensions, or The smallest volume in 3 dimensions Body Centered Cubic (BCC) Conventional Cell Primitive cell Simple Cubic (SC) Conventional Cell = Primitive cell

2d sin = n (Bragg Equation) Extra distance = BC + CD =

Summing the waves The overall scattering intensity depends on Atom types (as above) - electron density Their position relative to one another. hkl j = + + F exp 2 ( ) f i hx ky lz j j j j Centrosymmetric means that there is a centre of symmetry , and for every atom at (x,y,z) there is an identical atom at (-x, -y, -z) Or for simple (centrosymmetric) structures: hkl = + + F f cos 2 ( hx ky lz ) j j j j j This is the sum of the (cos) waves, where: - fj is the atomic scattering factor for atom j - hkl are the Miller indices - xj, yj, zj are the atomic (fractional) coordinates

= + + F f cos 2 ( hx ky lz ) hkl j j j j j The expression 2 (hx+ky+lz) = phase difference or phase factor Where F(hkl) is the structure factor for the hkl reflection of the unit cell, f is the atomic scattering factor for each of j atomic planes and is the repeat distance between atomic planes measured from a common origin (called the phase factor) Scattering by an unit cell = f(position of the atoms, atomic scattering factors) Amplitude of wave scattered atoms all by in uc = = F Structure Factor Amplitude of wave scattered electron an by n n j j j [2 ( + + )] i i h x k y l z = = hkl n F f e f e j For n atoms in the UC j j = = 1 1 j j Structure factor is independent of the shape and size of the unit cell! F Fhkl

Structure factor calculations Some useful relations = ( ni n ) 1 = = e Now we perform calculation of structure factor calculations for some simple crystal structure unit cells. We basically perform the summation of the function fexp(i ) for the different atoms ions in the unit cell and work out the restrictions on the allowed values of hkl (and the corresponding intensities). i + ( ) odd n 1 1 e e i ( ) even n = ni ni e e Atom at (0,0,0) and equivalent positions A i i + e e = ( ) Cos Simple Cubic 2 n i = n ( 1) e Note: UC in reciprocal space n i n i = e e a n is an integer 1/a We note that in simple cubic crystals there is no restrictions on the allowed values of hkl (i.e. for all values of hkl reflections are present). j j j [2 ( + + )] i i h x k y l z = = hkl F f e f e j j j [2 ( 0 + + = = = 0 0)] 0 i h k l F F = f e f e f All reflections are present 2 2 f F is independent of the scattering plane (h k l)

Atoms at (0,0,0) & (, , 0) and equivalent positions B C- centred Orthorhombic For this case there is one additional lattice point with an associated atom. Hence, there will be two terms in the summation for the structure factor. 1/c i = hkl F f e j UC in reciprocal space j 1/b j j j [2 ( + + )] i h x k y l z = f e 1/a j 1 2 1 2 [2 ( + + 0)] i h k l Important note: The 100, 101, 210, etc. points in the reciprocal lattice exist (as the corresponding real lattice planes exist), however the intensity decorating these points is zero. [2 ( 0 + + = + 0 0)] hkl i h k l F f e f e Real + h k [ 2 ( )] i h k + = + = + 0 ( ) i [1 ] f e f e f e 2 = F = 2 2 2 F f 4 f e.g. (001), (110), (112); (021), (022), (023) + = + ( ) ie h k 1 [ f ] F = 2= 0 F 0 F F is independent of the l index e.g. (100), (101), (102); (031), (032), (033)