3XRD MLL701 2023 L13-14 XRD L5-6.pdf

02 09 2023 13.. ~Recit procal Lattice -> Fixh+ 1. De-inition ChRK -/Tl= - dak W 4 Weiss Zone Law 5. Vector...

02 09 2023 13.. ~Recit procal Lattice -> Fixh+ 1. De-inition ChRK -/Tl= - dak W 4 Weiss Zone Law 5. Vector Bragg's law - Next Claus. Ewald's 6. Sphere Lattice periodic arrangement of points i is ⑧ it ⑧ ⑧ ⑧ ⑧ · ⑧ 18 14 ⑧ ⑧ 90 ⑧ ⑧ ⑧ ⑳ ⑧ ⑧ ⑳ 54↑ A I 12 13 T = 1a + 25 ⑱ · ⑧ ⑧ ⑱ Tit aftI ⑧ ⑧ ⑧ ⑧ j = 76 8 5 ↳ 1 a - a · 0 O ⑳ ⑧ ⑧ 0 O a ⑧ 9 2 3 ⑧ · 0 ⑧ ⑧ · 0 ⑧ Dimitive basis 99 , 5} Nonprimitive bain 59 53 , "Real"Lattere Crystal Motif + =  Crystal I I Real Lattice -> S Reciprocal Lattice (Crystal structure) (Diffraction pattern) Lattice ↳real If a 5 3 , , is a Primitive basis of a then the primitive bain lattice Sa*, b A, * 3 the lattere is of corresponding reciprocal ax. I = 1 &A B. = 0 at. = 0 JA. = 0 5*. 5 = 1 5A. = 0 IA. = 0 A 5. = 0 A I. = 1 of Ex -action IA. = 0 A 5. = 0 A. I = 1 I ↓ c *+ T T I* 1 -- - - - b - S TC I --> & (1 to plane a 5) , x5  I* 11 ax5 - = k F > & Scalar It = k(ax5) where k is a scalon k ? have Using 1 we A. = k(ax5). I = 1 m =k = I I. I V I volume - of the cell unit nee 24 = k(ax5) = (ax5) a* There are primitive +(5xi) - basis vectors of = - lattice the reciprocal +(exa) + 5 = + RB*+ I ix = (ax5) hk1 where h, k 1 E , *A L bat 5 - Ch · a I A = Fak unit vector normal Is to (4K1) Op = Projection of Ox along dri = A. ate -. -'. Ta. (ha * +k5x+(c) -I - IF*= Ink L14 04. 09. 2023 slast Class before Mid Term) law Vector Form of Bragg's E wald's construction  formulation of Bragg's -me Law Elastic scattering B No loss · N i To of - energy "10 -- ok I h5 E= ko C -A k ko - - E · - D - x E = I 11 18 - -· Ol (4K1) A O 156 I IE1 = ~Pri = - Inelastic scattering loss of energy - k E - wave vector , 0 vector in - the direction of the beam length , with a IR1 = 1 X Uk = R - Ro rector = scattering ↓R 1 (hk1) IR1 = 2x OA SinQ = 2x x sinc  = Sino- Jak Cusing Bragg's law) -E 1 ChKK > UK # = ITE) = iss Vector form of Bragg's law W e of Bragg's law : The R R Fo scattering rector = - is reciprocal vector Fi lattice a R = FIA hk1 -Construction (Reciprocal Space) Meas Ewald's Sphere Reflection Sphere - x = 2 d Sints , 02 IO1 ~ ↑ ⑰ O id ⑧ x 2 d SinOz · O , L ⑧ de 00 ↑ S Problem : Given an incident beam (ko) nee and a crystal find all possible diffracted beams (E). -solutio : Find Draw ChkD , dukl and Onky and check X = 20nk SinOnk1 Infinitely many ChKI) to check. -solution in Reciprocal space ↳P ⑧ E. ↑ ~ E Ro+OR= E & --- = UR = R - Ko Fo ⑧ ⑧ ⑰ · OR Op = OE has ⑧ ⑧ to be a I sorigin reciprocal lattice oflattice reciprocal vector Maks ( A - steps of Ewald Sphere Constructions. 1 Draw = Fo (incident wave rector). 2 Draw a sphere of radius co=Ikol= This Ewald's is sphere Since vector E also has the 3. diffracted same length Celastic scattering ( IR1 151 = = I also will lie somewher on the Ewald's sphere. 4. Draw reciprocal lattice with O as the origin. 5. By Bragg's law in reciprocal space R if = ot is a diffracted rector then R = P = F I P is a reciprocal Caller point -. Reconsider - 1. Draw Ewald's Sphere with Ro Tail of to as centre Length of 50 (5) as radius. lattice 2. Draw reciprocal with to head of as origin. 3. All reciprocal lattice fronts lying are the sphere will define diffracted beams /Drawn from centre c of Ewald's sphen to the Reciprocal point ) lattice.