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The Debye-Waller factor Start w/ structure factor F hkl =  f n exp (2 π i r *  r n )

The Debye-Waller factor Start w/ structure factor F hkl =  f n exp (2 π i r *  r n ). hkl. unit cell. S / . (S -S o )/ . 2 . S o / . | r *| = | s - s o |/  = (2 sin  )/ . r * = ( s - s o )/  = h a * + k b * + l c * r = x a + y b + z c. For small  r n

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The Debye-Waller factor Start w/ structure factor F hkl =  f n exp (2 π i r *  r n )

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  1. The Debye-Waller factor Start w/ structure factor Fhkl =  fn exp (2πi r* rn) hkl unit cell S/ (S -So)/ 2 So/ |r*| = |s - so|/ = (2 sin )/ r* = (s - so)/ = ha* + kb* + lc* r = xa + yb + zc

  2. For small rn exp (2πi r* rn) = 1+ 2πi r* rn + 1/2 (2πi r* rn)2 + … hkl hkl hkl Averaging over time exp (2πi r* rn) ≈ 1- 2π2 |r* |2 <ux> = 1 - 8π2(sin /)2 <ux> 2 2 hkl hkl 2 exp (- 8π2(sin /)2 <ux>) = exp (-B(sin /)2) Debye-Waller factor 2 B = 8π2 <ux> The Debye-Waller factor

  3. The Debye-Waller factor For anisotropic motion exp {-(11h2 + 22k2 + 33l2 + 212hk+ 213hl+ 223kl)} exp {-1/4(B11h2a*2 + B22k2b*2 + B33l2c*2 + 2B12hka*b* + 2B13hl a*c*+ 2B23kl b*c*)} exp {-2π2(U11h2a*2 + U22k2b*2 + U33l2c*2 + 2U12hka*b* + 2U13hl a*c*+ 2U23kl b*c*  = 2π2 Ua* B = 8π2 U

  4. The Debye-Waller factor For anisotropic motion exp {-(11h2 + 22k2 + 33l2 + 212hk+ 213hl+ 223kl)} exp {-1/4(B11h2a*2 + B22k2b*2 + B33l2c*2 + 2B12hka*b* + 2B13hl a*c*+ 2B23kl b*c*)} exp {-2π2(U11h2a*2 + U22k2b*2 + U33l2c*2 + 2U12hka*b* + 2U13hl a*c*+ 2U23kl b*c* Bii are lengths of thermal ellipsoid semi-major and semi-minor axes All Bs describe orientation of ellipsoids wrt lattice vectors

  5. The Debye-Waller factor Bii are lengths of thermal ellipsoid semi-major and semi-minor axes All Bs describe orientation of ellipsoids wrt lattice vectors Need: Bii > 0 Bii Bjj > Bij2 B11 B22 B33 + B122 B132 B232 > B11 B232 + B22B132 + B33 B122

  6. Then, scatt ampl due to displacement only is • 1/<b> iQunbn exp (iQrno) n un(rno) = 1/N1/2 uq exp (iqrno) q i/<b>N1/2 bnQuq exp (i(Q-q)rno) q,n Thermal diffuse scattering (see extensive discussion in James: Optical Principles of the Diffraction of X-rays, chapter V) Following Egami & Billinge, p. 33: Define |Q| = 4π (sin )/ rn = rno + un

  7. i/<b>N1/2 bnQuq exp (i(Q-q)rno) q,n Thermal diffuse scattering Summing over all unit cells, and separating out Bragg peak scattering Idiffuse(Q) = |1/N  bj exp (2πir*rjo)|2 |QuQ-2πr*|2 j

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