1. Thin films II Kinematic theory - works OK for mosaic crystals & other imperfect matls Doesn't work for many, more complicated films Kinematic theory - works OK for mosaic crystals & other imperfect matls Doesn't work for many, more complicated films

2. Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) The Borrmann effect

3. Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) The Borrmann effect !!!

4. Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) The Borrmann effect

5. Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) Thin films II (see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect Crystals. Rev. Mod. Phys. 36, p 681 (1964)) Past discussions of diffraction – 2 beams, in & out ("kinematic theory") But these beams coherently coupled – energy swapped back & forth betwn them Past discussions of diffraction – 2 beams, in & out ("kinematic theory") But these beams coherently coupled – energy swapped back & forth betwn them

6. Thin films II Past discussions of diffraction – 2 beams, in & out ("kinematic theory") But these beams coherently coupled – energy swapped back & forth betwn them Must consider all of field as a unit ("dynamical theory") Past discussions of diffraction – 2 beams, in & out ("kinematic theory") But these beams coherently coupled – energy swapped back & forth betwn them Must consider all of field as a unit ("dynamical theory")

7. Thin films II For Borrmann effect, dynamical theory predicts standing wave in diffracting medium Two solutions – one for no absorption, one for enhanced absorption For Borrmann effect, dynamical theory predicts standing wave in diffracting medium Two solutions – one for no absorption, one for enhanced absorption

8. Thin films II Dynamical theory changes Ewald construction In dynamical theory, more than one sphere Dynamical theory changes Ewald construction In dynamical theory, more than one sphere

9. Thin films II Dynamical theory changes Ewald construction In dynamical theory, more than one sphere Determine loci of permitted Ewald spheres – the "dispersion surface". Drawing vectors from points on this surface to reciprocal lattice points gives allowed waves Dynamical theory changes Ewald construction In dynamical theory, more than one sphere Determine loci of permitted Ewald spheres – the "dispersion surface". Drawing vectors from points on this surface to reciprocal lattice points gives allowed waves

10. Thin films II Main problem – solve Maxwell's eqns. for medium with periodic, anisotropic, complex dielectric constant assume solutions consistent with Braggs' law obtain solns of waves w/ permitted wave vectors tips of these vectors form dispersion surface dispersion surface used to generate all diffraction effects Main problem – solve Maxwell's eqns. for medium with periodic, anisotropic, complex dielectric constant assume solutions consistent with Braggs' law obtain solns of waves w/ permitted wave vectors tips of these vectors form dispersion surface dispersion surface used to generate all diffraction effects

11. Thin films II Correct for index of refraction in medium

12. Thin films II Correct for index of refraction in medium Nature of dispersion surfaces

13. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Each lattice point occupied by a dipole set into oscillation by radiation field of electromagnetic wave passing thru crystal Oscillation of dipoles produces radiation and create radiation field Oscillation is itself a plane wave advancing thru lattice normal to lattice planes Each lattice point occupied by a dipole set into oscillation by radiation field of electromagnetic wave passing thru crystal Oscillation of dipoles produces radiation and create radiation field Oscillation is itself a plane wave advancing thru lattice normal to lattice planes

14. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Each lattice point occupied by a dipole set into oscillation by radiation field of electromagnetic wave passing thru crystal Oscillation of dipoles produces radiation and create radiation field Oscillation is itself a plane wave advancing thru lattice normal to lattice planes Dipoles in lattice plane oscillate in phase Two waves result, one going up, other down Each lattice point occupied by a dipole set into oscillation by radiation field of electromagnetic wave passing thru crystal Oscillation of dipoles produces radiation and create radiation field Oscillation is itself a plane wave advancing thru lattice normal to lattice planes Dipoles in lattice plane oscillate in phase Two waves result, one going up, other down

15. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Think now of two waves: scattered wave shown in diagram, wave vector k, velocity = c dipole wave, wave vector K, velocity = nearly c Think now of two waves: scattered wave shown in diagram, wave vector k, velocity = c dipole wave, wave vector K, velocity = nearly c

16. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Think now of two waves: scattered wave shown in diagram, wave vector k, velocity = c dipole wave, wave vector K, velocity = nearly c Can be shown that: K = k(1+  ),  small Think now of two waves: scattered wave shown in diagram, wave vector k, velocity = c dipole wave, wave vector K, velocity = nearly c Can be shown that: K = k(1+  ),  small

17. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Actually, K is an infinite set of vectors In reciprocal space Actually, K is an infinite set of vectors In reciprocal space

18. Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Thin films II (see James, Optical Principles of the Diffraction of X-rays,(1962)) Actually, K is an infinite set of vectors In reciprocal space In real space Actually, K is an infinite set of vectors In reciprocal space In real space

19. Thin films II (see Bowen and Tanner) Thin films II (see Bowen and Tanner) K slightly smaller than k Interaction of incident and diffracted beams takes place at and/or near K slightly smaller than k Interaction of incident and diffracted beams takes place at and/or near O H

20. Thin films II (see Bowen and Tanner) Thin films II (see Bowen and Tanner) Deviations in dynamical theory are extremely small Highly magnified view req'd Deviations in dynamical theory are extremely small Highly magnified view req'd

21. Thin films II (see Bowen and Tanner) Thin films II (see Bowen and Tanner) Deviations in dynamical theory are extremely small Highly magnified view req'd Interaction takes place on hyperbolic surfaces near L Deviations in dynamical theory are extremely small Highly magnified view req'd Interaction takes place on hyperbolic surfaces near L

22. Thin films II (see Bowen and Tanner) Thin films II (see Bowen and Tanner) Unfortunately, cannot use dynamical theory to extract structure directly from rocking curves But, can use it to simulate rocking curves These then compared to experimental curves and refined Unfortunately, cannot use dynamical theory to extract structure directly from rocking curves But, can use it to simulate rocking curves These then compared to experimental curves and refined

23. Thin films II Mn x Hg 1-x Te on CdTe on GaAs substrate

24. Thin films II Graded layers Simulated rocking curves for In x Ga 1-x As on InP & Al x Ga 1-x As on GaAs Graded layers Simulated rocking curves for In x Ga 1-x As on InP & Al x Ga 1-x As on GaAs