DiffractionDiffraction fromfrom CrystalsCrystals ““StructureStructure FactorFactor”” LectureLecture 55 CoherentCoherent elasticelastic scatteringscattering Detector r’s - position vectors -ki k’s - wave vectors K r kd r - r’ kd θ r’ ki r r r r Incident plane wave:Ψ = exp⎡ 2πi(k ⋅r′ −ωt)⎤ = exp⎡ 2πi(k ⋅r)′ ⎤ incident ⎣ i ⎦ ⎣ i ⎦ Scattered wave: r r r r r r exp⎡ 2πik ⋅ r − r′ ⎤ exp⎡ 2πik ⋅ r − r′ ⎤ ⎣ d ( )⎦ ⎣ d ( )⎦ Ψscattered = f ()ki,kd r r = f ()θ r r r − r′ r − r′ ScatteringScattering fromfrom aa latticelattice r [R j ]- atom locations r R n r r′ −R n r r ′ r V r ′ = ∑V r ′ −R atom ( ) at ( j ) r r Ψscattered ()K, r = r exp 2 i k r −m []π ()d ⋅ r v 3 V r ′ −R exp −2πi K ⋅ r ′ d r′ 2 r ∫ ∑ at ()j []() 2πh r ScatteringScattering fromfrom aa latticelattice r [R j ]- atom locations r R n r r′ −R n r r ′ r r r v fel (R j,K) ≈ fel (R j ) r ≡ r ′ −R (i.e. r ′ = r +R′ ) j j Ignore the 1/r2 term to give: r −m v r r 3r Ψ K = ∑V r r exp −2πi K ⋅ r +R d r′ scatt () 2 ∫ at,R j () []()()j 2πh ⎛ ⎞ −m v r 3r v r = ∑ V r r exp −2πi K ⋅ r d r′ exp −2πi K ⋅R ⎜ 2 ∫ at,R j () []() ⎟ [ ()j ] Rv ⎝ 2π ⎠ j h ScatteringScattering fromfrom aa latticelattice r [R j ]- atom locations r R n r r′ −R n r r ′ So scattered wave from an array of N atoms: r N r v r Ψscatt ()K = ∑fel ()R j exp[−2πi()K ⋅R j ] j Notes: – Now have Ψ(K) only, Rj are fixed. Means scattered wave depends on incident wave vector. – This is a Fourier Transform (FT) Diffracted intensity is proportionproportionalal to the Fourier Transform of the scattering factor distribution in the material ScatteringScattering fromfrom aa latticelattice InIn simplestsimplest form,form, aa crystalcrystal is:is: Crystal = lattice + basis + defect displacements r R = r + r + r = r + r + r j lattice basis defects l b d ForFor now,now, considerconsider aa defectdefect freefree crystal:crystal: Crystal = lattice + basis r R = r + r = r + r j lattice basis l b ScatteredScattered wave:wave: r r r v r r Ψscatt ()K = ∑∑fel (rl + rb )exp[−2πi(K ⋅(rl + rb ))] rl rb BasisBasis mustmust bebe samesame forfor allall unitunit cells:cells: f r + r = f r ( l b ) ( b ) r v r r v r Ψscatt ()K = ∑exp[−2πi(K ⋅ rl )]∑fel ()rb exp[−2πi(K ⋅ rb )] rl rb ScatteringScattering fromfrom aa latticelattice Repeating:Repeating: r v r v r Ψscatt ()K = ∑exp[−2πi(K ⋅ rl )]∑fel ()rb exp[−2πi(K ⋅ rb )] r r rl rb r r r Ψ K = S K F K scatt ( ) ( ) ( ) where:where: v lattice v r S()K = ∑ exp[]−2πi()K ⋅ rl “Shape Factor” r rl v basis v r F()K = ∑ fel ()rb exp[−2πi()K ⋅ rb ] “Structure Factor” r rb FF(K)(K) samesame forfor allall latticelattice points:points: v lattice v v r Ψ()K = ∑ F()K exp[−2πi()K ⋅ rl ] r rl StructureStructure FactorFactor StructureStructure Factor:Factor: – Scattering from all atoms in the unit cell v basis r v r F()K = ∑ fel ()rb exp[−2πi()K ⋅ rb ] r rb InIn essence,essence, StructureStructure FactorFactor determinesdetermines whetherwhether oror notnot constructiveconstructive interferenceinterference occursoccurs atat aa givengiven reciprocalreciprocal latticelattice pointpoint ThisThis meansmeans ourour priorprior conditioncondition -- KK == gg -- isis aa necessarynecessary condition,condition, butbut isis notnot sufficientsufficient – There may not be intensity at a given g! StructureStructure factorfactor calculationcalculation Consider i atoms in the unit cell v v r F(K) = ∑fi exp[−2πi()K ⋅ ri ] i Location of ith atom (with respect to real lattice): r r = x ar + y b + z cr i i i i Using defn of a*, Diffraction condition (K = g): b* & c* r r r r K = ha * +kb * +lc * General from: v F()K = ∑fi exp[−2πih( xi +kyi +lzi )] i StructureStructure FactorFactor simplesimple cubiccubic F = ∑ fi exp[−2πihx( i +kyi + lzi )] i One atom basis: r = ()0,0,0 F = f{ exp[]−2πi0()⋅h+ 0 ⋅k + 0 ⋅l } = f{} exp[]−2πi0() = f{} exp[] 0 = f Simple result: intensity at every reciprocal lattice point StructureStructure FactorFactor bodybody--centeredcentered cubiccubic ⎛ 1 1 1⎞ Two atom basis : r = ()0,0,0 & r= ⎜ , , ⎟ ⎝ 2 2 2⎠ ⎧ ⎡ ⎛ 1 1 1 ⎞⎤ ⎫ F = f⎨ exp[]−2πi0()⋅h+ 0 ⋅k + 0 ⋅l + exp⎢− 2πi⎜ ⋅h+ ⋅k + ⋅l⎟⎥ ⎬ ⎩ ⎣ ⎝ 2 2 2 ⎠⎦ ⎭ = f1{}+ exp[]−πih()+k + l h, k & l are integers, so h+k + l = N (where N is an integer) The exponential can then take one of two values: exp[]−πih()+k + l =+1 if N = even exp[]−πih()+k + l =−1 if N = odd So : F = 2f if N = even F = 0 if N = odd StructureStructure FactorFactor bodybody--centeredcentered cubiccubic Allowed low order 002 022 reflections are: – 110, 200, 220, 310, 222, 321, 112 400, 330, 411, 420 … 011 202 Forbidden reflections are: 121 – 100, 111, 210 000 101 020 Origin of forbidden 211 reflections? 110 – Identical plane of atoms halfway between causes destructive interference 200 220 Real bcc lattice has an fcc reciprocal lattice StructureStructure FactorFactor faceface centeredcentered cubiccubic ⎛ 1 1 ⎞ ⎛ 1 1⎞ ⎛ 1 1⎞ Four atom basis: r = ()0,0,0 , r = , ,0 , r = ,0, & r = 0, , ⎝⎜ 2 2 ⎠⎟ ⎝⎜ 2 2⎠⎟ ⎝⎜ 2 2⎠⎟ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ F = f1{}+ exp ⎣−πih()+ k ⎦ + exp ⎣−πik()+ l ⎦ + exp ⎣−πih()+ l ⎦ So: F=4f if h,k,l all even or odd F=0 if h,k,l are mixed even or odd StructureStructure FactorFactor faceface centeredcentered cubiccubic Allowed low order 002 022 reflections are: – 111, 200, 220, 311, 222, 400, 331, 310 202 Forbidden reflections: – 100, 110, 210, 211 000 111 020 Real fcc lattice has a bcc reciprocal lattice 200 220 StructureStructure FactorFactor NaClNaCl (rock(rock salt)salt) structurestructure Na on each fcc site, but with a two atom basis : ⎛ 1 1 1⎞ rNa = ()0,0,0 & rCl = ⎜ , , ⎟ ⎝ 2 2 2⎠ F = {}fNa + fCl exp[]−πih()+k + l × {1+ exp[]−πih()+k + exp[]−πik()+ l + exp[]−πih()+ l } F = 4f()Na + fCl if h,k,l all even F = 4f()Na − fCl if h,k,l all odd F = 0 if h,k,l mixed If h,k,l all odd have ‘chemically sensitive’ reflections StructureStructure FactorFactor NiNi3AlAl (L1(L12)) structurestructure Simple cubic lattice, with at four atom basis ⎛ 1 1 ⎞ ⎛ 1 1⎞ ⎛ 1 1⎞ r = 0,0,0 , r = , ,0 , r = ,0, & r = 0, , Al ()Ni Ni Ni ⎝⎜ 2 2 ⎠⎟ ⎝⎜ 2 2⎠⎟ ⎝⎜ 2 2⎠⎟ F = f + f exp ⎡−πih+ k ⎤ + exp ⎡−πik+ l ⎤ + exp ⎡−πih+ l ⎤ al Ni ⎣ ()⎦ ⎣ ()⎦ ⎣ ()⎦ So : F=fAl +3fNi if h,k,l all even or odd F=f -f if h,k,l are mixed even or odd Al Ni Again, since simple cubic, intensity at all points. But each point is ‘chemically sensitive’. ChemicallyChemically sensitivesensitive reflectionsreflections Dark field images formed from chemically sensitive reflections produce a real space map of (highly) local chemistry LongLong periodperiod superlatticessuperlattices.