Thin films II

Kinematic theory - works OK for mosaic crystals & other imperfect matls

Doesn't work for many, more complicated films

Thin films II(see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect

Crystals. Rev. Mod. Phys. 36, p 681 (1964))

The Borrmann effect

Thin films II(see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect

Crystals. Rev. Mod. Phys. 36, p 681 (1964))

The Borrmann effect

!!!

Thin films II(see Batterman & Cole, Dynamical Diffraction of X-rays by Perfect

Crystals. Rev. Mod. Phys. 36, p 681 (1964))

The Borrmann effect

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

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")

Thin films II

For Borrmann effect, dynamical theorypredicts standing wave in diffractingmedium

Two solutions – one for no absorption,one for enhanced absorption

Thin films II

Dynamical theory changes Ewald construction

In dynamical theory, more than one sphere

Thin films II

Dynamical theory changes Ewald construction

In dynamical theory, more than one sphere

Determine loci of permitted Ewald spheres – the "dispersionsurface". Drawing vectors from points on this surface to reciprocal lattice points gives allowed waves

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

Thin films II

Correct for index of refraction in medium

Thin films II

Correct for index of refraction in medium

Nature of dispersion surfaces

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 planewave advancing thru latticenormal to lattice planes

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 planewave advancing thru latticenormal to lattice planes

Dipoles in lattice planeoscillate in phase

Two waves result, one going up, other down

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

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

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

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

Thin films II (see Bowen and Tanner)

K slightly smaller than k

Interaction of incident and diffracted beams takes place atand/or near

O

H

Thin films II (see Bowen and Tanner)

Deviations in dynamical theory are extremely small

Highly magnified view req'd

Thin films II (see Bowen and Tanner)

Deviations in dynamical theory are extremely small

Highly magnified view req'd

Interaction takes place on hyperbolic surfacesnear L

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

Thin films II

MnxHg1-xTe on CdTe on GaAs substrate

Thin films II

Graded layers

Simulated rocking curves for InxGa1-xAs on InP &AlxGa1-xAs on GaAs