Analysis of crystal structure x-rays, neutrons and electrons Diffraction Analysis of crystal structure x-rays, neutrons and electrons 2/2-10 MENA3100

The reciprocal lattice g is a vector normal to a set of planes, with length equal to the inverse spacing between them Reciprocal lattice vectors a*,b* and c* These vectors define the reciprocal lattice All crystals have a real space lattice and a reciprocal lattice Diffraction techniques map the reciprocal lattice 2/2-10 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 Neutron radiation λ~0.1nm Scattered by atomic nuclei Several cm thick samples Electron radiation (200kV): λ=0.00251 nm Scattered by atomic nuclei and electrons Thickness less than ~200 nm 2/2-10 MENA3100

Interference of waves Sound, light, ripples in water etc etc Constructive and destructive interference =2n =(2n+1) 2/2-10 MENA3100

Nature of light Newton: particles (corpuscles) Huygens: waves Thomas Young double slit experiment (1801) Path difference  phase difference Light consists of waves ! Wave-particle duality 2/2-10 MENA3100

Discovery of X-rays Wilhelm Röntgen 1895/96 Nobel Prize in 1901 Particles or waves? Not affected by magnetic fields No refraction, reflection or intereference observed If waves, λ10-9 m Røntgen jobbet med en Crookes tube med sort papp rundt. Oppdaget at en fluoriserende skjerm i rommet lyste opp. Fant ut at strålingen gikk gjennom papp og bøker. Så når han holdt et objekt at han kunne se skjelettet sitt. Fotograferte sin kones hånd. 2/2-10 MENA3100

Max von Laue The periodicity and interatomic spacing of crystals had been deduced earlier (e.g. Auguste Bravais). von Laue realized that if X-rays were waves with short wavelength, interference phenomena should be observed like in Young’s double slit experiment. Experiment in 1912, Nobel Prize in 1914 2/2-10 MENA3100

Laue conditions Scattering from a periodic distribution of scatters along the a axis a ko k The scattered wave will be in phase and constructive interference will occur if the phase difference is 2π. Φ=2πa.(k-ko)=2πa.g= 2πh, similar for b and c 2/2-10 MENA3100

The Laue equations The Laue equations give three conditions for incident waves to be diffracted by a crystal lattice 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) r k k0 k-k0 r*hkl (hkl) Δ=a.(k-ko)=h Δ=b.(k-ko)=k Δ=c.(k-ko)=l If (k-k0) = r*, then φ= 2πn r*= ha*+kb*+lc* Δ=r . (k-k0) 2/2-10 MENA3100

Bragg’s law William Henry and William Lawrence Bragg (father and son) found a simple interpretation of von Laue’s experiment Consider a crystal as a periodic arrangement of atoms, this gives crystal planes Assume that each crystal plane reflects radiation as a mirror Analyze this situation for cases of constructive and destructive interference Nobel prize in 1915 2/2-10 MENA3100

Derivation of Bragg’s law θ x dhkl Path difference Δ= 2x => phase shift Constructive interference if Δ=nλ This gives the criterion for constructive interference: Bragg’s law tells you at which angle θB to expect maximum diffracted intensity for a particular family of crystal planes. For large crystals, all other angles give zero intensity. 2/2-10 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 The path difference: x-y Y= x cos2θ and x sinθ=d cos2θ= 1-2 sin2θ 2/2-10 MENA3100

von Laue – Bragg equation θ Vector normal to a plane θ 2/2-10 MENA3100

The limiting-sphere construction Vector representation of Bragg law IkI=Ik0I=1/λ λx-rays>> λe k = ghkl (hkl) k0 k-k0 2θ Diffracted beam Incident beam Reflecting sphere Limiting sphere 2/2-10 MENA3100

The Ewald Sphere (’limiting sphere construction’) Elastic scattering: k k’ The observed diffraction pattern is the part of the reciprocal lattice that is intersected by the Ewald sphere g 2/2-10 MENA3100

The Ewald Sphere is flat (almost) Cu Kalpha X-ray:  = 150 pm => small k Electrons at 200 kV:  = 2.5 pm => large k 2/2-10 MENA3100

50 nm 2/2-10 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. 2/2-10 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: 2/2-10 MENA3100

Example: 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 ] What is the general condition for reflections for fcc? Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l)) What is the general condition for reflections for bcc? If h, k, l are all odd then: Fhkl= f(1+1+1+1)=4f If h, k, l are mixed integers (exs 112) then Fhkl=f(1+1-1-1)=0 (forbidden) 2/2-10 MENA3100

The structure factor for fcc The reciprocal lattice of a FCC lattice is BCC What is the general condition for reflections for bcc? 2/2-10 MENA3100

The reciprocal lattice of bcc Body centered cubic lattice One atom per lattice point, [000] relative to the lattice point What is the reciprocal lattice? 2/2-10 MENA3100