Analysis of crystal structure x-rays, neutrons and electrons Diffraction Analysis of crystal structure x-rays, neutrons and electrons 29/1-08 MENA3100

Radiation: x-rays, neutrons and electrons Elastic scattering of radiation No energy is lost The wave length of the scattered wave remains unchanged Regular arrays of atoms interact elastically with radiation of sufficient short wavelength CuKα x-ray radiation: λ=0.154 nm Scattered by electrons ~from sub mm regions Electron radiation (200kV): λ=0.00251 nm Scattered by atomic nuclei and electrons Thickness less than ~200 nm Neutron radiation λ~0.1nm Scattered by atomic nuclei Several cm thick samples 29/1-08 MENA3100

gives the Laue equations: Two lattice points separated by a vector r Waves scattered from two lattice points separated by a vector r will have a path difference in a given direction. The scattered waves will be in phase and constructive interference will occur if the phase difference is 2π. The path difference is the difference between the projection of r on k and the projection of r on k0, φ= 2πr.(k-k0) k0 r k k-k0 r*hkl (hkl) r = a, b or c and IkI=Ik0I=λ gives the Laue equations: Δ=hλ Δ=kλ Δ=lλ If (k-k0) = r*, then φ= 2πn r*= ha*+kb*+lc* Δ=r . (k-k0) 29/1-08 MENA3100

The path difference: x-y Bragg’s law d θ y x nλ = 2dsinθ Planes of atoms responsible for a diffraction peak behave as a mirror 1/d2=(h/a)2+(k/b)2+(l/c)2 Orthorhombic lattice The path difference: x-y Y= x cos2θ and x sinθ=d cos2θ= 1-2 sin2θ k-k0 r*hkl (hkl) ghkl = r*hkl 29/1-08 MENA3100

The limiting-sphere construction Vector representation of Bragg law IkI=Ik0I=λ λx-rays>> λe k = ghkl (hkl) k0 k-k0 2θ Diffracted beam Incident beam Reflecting sphere Limiting sphere 29/1-08 MENA3100

Allowed and forbidden reflections Bravais lattices with centering (F, I, A, B, C) have planes of lattice points that give rise to destructive interference for some orders of reflections. Forbidden reflections y’ y x θ x’ d θ In most crystals the lattice point corresponds to a set of atoms. Different atomic species scatter more or less strongly (different atomic scattering factors, fzθ). From the structure factor of the unit cell one can determine if the hkl reflection it is allowed or forbidden. 29/1-08 MENA3100

Structure factors X-ray: The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc. h, k and l are the miller indices of the Bragg reflection g. N is the number of atoms within the crystal unit cell. fj(n) is the x-ray scattering factor, or x-ray scattering amplitude, for atom j. The structure factors for x-ray, neutron and electron diffraction are similar. For neutrons and electrons we need only to replace by fj(n) or fj(e) . rj uja a b x z c y vjb wjc The intensity of a reflection is proportional to: 29/1-08 MENA3100

Example: Cu, fcc eiφ = cosφ + isinφ enπi = (-1)n eix + e-ix = 2cosx Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ] Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l)) What is the general condition for reflections for fcc? If h, k, l are all odd then: Fhkl= f(1+1+1+1)=4f What is the general condition for reflections for bcc? If h, k, l are mixed integers (exs 112) then Fhkl=f(1+1-1-1)=0 (forbidden) 29/1-08 MENA3100