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					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 theory
predicts standing wave in diffracting
medium

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 "dispersion
surface". 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 plane
wave advancing thru lattice
normal 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 plane
wave advancing thru lattice
normal to lattice planes

Dipoles in lattice plane
oscillate 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 at
and/or near




                                             H




                                             O
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 surfaces
near 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

				
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posted:9/29/2012
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