1 / 7

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

hdixon
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related