Diffuse scattering: Stacking faults Diffuse scattering
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Short range order
Stacking faults

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Stacking faults

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This example shows the effect of stacking faults. They are illustrated by use of a 2-D example. The crystal is build up of rows corresponding to a primitive unit structure. Three different layers are stacked: a at x=0.0, b at x=1/3 and c at x=2/3. The stacking sequence is controlled by two probabilities ALPHA and BETA. The meaning of these probabilities is defined as:

 ``` ALPHA probability of "ab" followed by "a" 1 - ALPHA probability of "ab" followed by "c" BETA probability of "ba" followed by "b" 1 - BETA probability of "ba" followed by "c"``` ``` ALPHA BETA resulting structure 0.0 0.0 pure cubic sequence 0.05 0.05 Sequence of cubic twins 0.5 0.5 random stacking 1.0 1.0 pure hexagonal sequence```

The layers "a","b" and "c" can be rotated cylindrically. The following images show a sequence going from ALPHA=BETA=0, i.e. pure cubic sequence to ALPHA=BETA=1, a pure hexagonal sequence. The left image shows part of the crystal, the right image the corresponding Fourier transform. To get a larger image just click on the corresponding picture.

A note on the indices: The crystal was created on a square coordinate system with the x=1 being the distance between atoms in each row and y=1 being the separation of rows. Accordingly the reciprocal space direction labeled corresponds to the  direction of the three dimensional hexagonal or cubic crystal. The reciprocal k direction corresponds to the hexagonal  and the cubic  direction. The spacing is doubled for a hexagonal crystal and tripled for a cubic crystal.

ALPHA=BETA=0.00  The first structure is a perfect "abc" sequence. Only sharp Bragg reflections of the cubic structure are observed.

ALPHA=BETA=0.05  The probability of a hexagonal sequence is very low. Accordingly if such a sequence occurs, the crystal is continued immediately as the cubic structure in its twined orientation. The Bragg reflections at h=0 and h=3 remain invariant, in addition to the Bragg reflections of the first image, those of the twined orientation appear. Weak diffuse streaks connect the Bragg reflections at h=1 and h=2.

ALPHA=BETA=0.20  The probability for hexagonal stacking increases. The Bragg reflections at h=1 and h=2 are replaced by diffuse streaks that are absent at integer k.

ALPHA=BETA=0.50  The random stacking of cubic and hexagonal sequences causes continuous diffuse streaks at h=1 and h=3. As for all the other situations, the Bragg reflections at h=3n remain invariant.

ALPHA=BETA=0.80  The crystal becomes predominantly hexagonal. The diffuse streaks start to develop sharp maxima at k=n and k=n+0.5.

ALPHA=BETA=1.00  A perfect hexagonal crystal has been created. The diffuse streaks have disappeared, the Bragg reflections of the hexagonal structure result.
© Th. Proffen and R.B. Neder, 2003