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Understanding Displacive Disorder in 1-D Lattice Modeling

Explore the impact of random longitudinal displacements in 1-D lattice models, focusing on the diffusion of scattering and lattice spacing calculations. Analyze substitutional and continuously variable displacement models. Investigate joint distribution and correlation coefficients of neighboring displacement values.

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Understanding Displacive Disorder in 1-D Lattice Modeling

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  1. Diffuse scattering - displacive disorder

  2. Diffuse scattering - displacive disorder 1-D model lattice spacing ao random longitudinal displacements xi Then spacing di = xi - xi-1 + ao

  3. Diffuse scattering - displacive disorder 1-D model lattice spacing ao random longitudinal displacements xi Then spacing di = xi - xi-1 + ao Note: substitutional model xi = 0 or 1 displacement model xi continuously variable

  4. Diffuse scattering - displacive disorder 1-D model lattice spacing ao random longitudinal displacements xi Then spacing di = xi - xi-1 + ao All xi normally distributed in same manner P(xi) = K exp (-xi2/2L2)

  5. Diffuse scattering - displacive disorder 1-D model lattice spacing ao random longitudinal displacements xi Then spacing di = xi - xi-1 + ao All xi normally distributed in same manner P(xi) = K exp (-xi2/2L2) Joint distribution of neighboring xi also normal P(xi|xi-1) = K exp (-(xi-12 + xi2 - 2rxi-1xi)/(2L2(1 -r2)))

  6. Diffuse scattering - displacive disorder 1-D model All xi normally distributed in same manner P(xi) = K exp (-xi2/2L2) joint distribution of neighboring xi also normal P(xi,xi-1) = K exp (-(xi-12 + xi2 - 2rxi-1xi)/(2L2(1 -r2))) r is correlation coeff. r = <xi-1xi>/L2

  7. Diffuse scattering - displacive disorder 1-D model All xi normally distributed in same manner P(xi) = K exp (-xi2/2L2) joint distribution of neighboring xi also normal P(xi,xi-1) = K exp (-(xi-12 + xi2 - 2rxi-1xi)/(2L2(1 -r2)) P(xi|xi-1) = P(xi,xi-1)/P(xi) = K exp (-(xi- 2rxi-1)2/(2L2(1 -r2)) ith site defined by i-1th site

  8. Diffuse scattering - displacive disorder 1-D model

  9. Diffuse scattering - displacive disorder 1-D model

  10. Diffuse scattering - displacive disorder 1-D model

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