结晶学 chap4Atomic Scattering.ppt

Atomic Scattering (factor) (r) ：electron density around nucleus dq = (r) dV q: charge V : volume d r Crystallography If the atom is a point charge (compared w. r. t the wavelength), it scatters as Z (atomic number) Scattering amplitude amplitude scattered by atom Eatom amplitude scattered by free e Ee Crystal Diffraction Braggs Equation 2d sin = n Difference in phase 2 X-ray Beam Atom Atomic Scattering Intl Table of X-ray CrystallographyVol C mostly from HF p.477 sin/ fZ For example Analytical form : Vol C.p500. f0 for atoms from Z = 1 to 90 coefficients of analytical form in atomic scattering factors Gd f” f 170517101715 30 20 10 0 10 20 30 () 30 20 10 0 10 20 30 184185 () Sm f” f Anomalous Scattering 3020100 0 10 20 30 Gd f” f 302010 10 20 Sm f 30 f” 0 0 (a)(b) (c) (d) Fig. : Anomalous scattering terms f and f” for： (a) gadolinium near the L3 edge； (b) samarium near the L3 edge Fig. : Plot in the complex plane of f i f”： (c) gadolinium near the L3 edge； (d) samarium near the L3 edge Atomic f and f” pt. charge atomic sphere (fixed atom at ri) T=0K with thermal vibration T as function of B : Thermal parameter Z B = 0 B 0 Atomic amplitude with anomalous scattering Nuclear thermal parameter: B; thermal vibration T = B (sin/)2 Considering nuclear thermal vibration thermal vibration electron density smearing if isotropic spherically symmetric As a point scatterer, fj 2 Tanisotropic thermal vibration as an ellipsoid 33 matrix 6 elements uij symmetric: u12= u21; u23 = u32; u13 = u31 based on a, b, c,-axis diagonalize eigen value eigen function u3 u1 u2 (three principle axes of the ellipsoid) Thermal ellipsoid u1 u2 u3 In case of mirror plane symmetry (ab-plane): U11 , U22 , U33 ， U12 U13=U23=0 x x y y z -z U11 ; U22 ; U33 ; U12 ; U13 ; U23 Constraint from U12=U12 U13 -U13；U23 -U23 U13=U23=0 Constraint in Thermal vibration Minimum Symmetry Elements for LS Constraint Sym. Element Fixed Parameters Special Relations x,y,z,- 1 1 2x,y,u13, u23- 1 1 mz,u13, u23- 1 2 1x,z,u12, u23- 1 m 1y,u12, u23- 2 1 1y,z,u12, u13- m 1 1x,u12, u13- 2 y,z,u22, u231, 4, 7 m y,u22, u232, 4, 7 2 y,z,u22, u232, 4, 6 m y,u22, u231, 4, 6 1 1 3x,y,u22, u12, u13, u234, 10 1 1 4x,y,u22, u12, u13, u234 x,y,z,u22, u12, u13, u234 3 y,z,u22, u33, u13, u231, 3, 4, 5, 8, 9 10 possible special relations 1y = x 2y = x 3z = x 4u22 = u11 5u33 = u11 6u23 = u13 7u23 = u13 8u13 = u1