Diffraction

1. Diffraction Analysis of crystal structure x-rays, neutrons and electrons MENA3100

2. 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 MENA3100

3. 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 r*hkl (hkl) k-k0 The Laue equations Two lattice points separated by a vector r r = a, b or c and IkI=Ik0I=λ gives the Laue equations: Δ=hλΔ=kλΔ=lλ If (k-k0) = r*, then φ= 2πn r*= ha*+kb*+lc* Δ=r .(k-k0) MENA3100

4. y θ d x θ r*hkl (hkl) ghkl = r*hkl k-k0 Bragg’s law • 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 MENA3100

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

6. 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’ x Allowed and forbidden reflections θ 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. MENA3100

7. z wjc rj c a b vjb uja The intensity of a reflection is proportional to: y x Structure factors X-ray: 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) . 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. MENA3100

8. eiφ = cosφ + isinφ enπi = (-1)n eix + e-ix = 2cosx Example: Cu, fcc 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) MENA3100