Parabolic microstrain-like line broadening induced by random twin faulting

Abstract

The broadening of diffraction lines of layer faulted and, in particular, of microtwinned microstructures is analysed in detail. Thereby, stacked layer models turn out to be valuable to simulate the diffracted intensity if the metrics of the layers pertaining to the two types of domains are described by an average metrics plus domain-specific deviations from that metrics. Characteristic changes occur for the line-broadening effects if the fault probability and the extent of deviations from the average layer metrics vary. Emphasis is put, in particular, on a regime of line broadening characterised by merging of otherwise split peaks to yield single ones for sufficiently high twin probability. These merged peaks show unusual quadratic/parabolic increase of the integral breadths with the reflection order (parabolic microstrain broadening). This type of broadening has been predicted previously for special types of microstrain broadening, and indeed the (metrical) distortions of the domains with respect to the average structure allow perceiving even the twin-faulted microstructures as, in some sense, microstrained ones. Moreover, the special situation of superstructures is analysed, where the above mentioned merging is only observed for the fundamental but not for the superstructure reflections. Inconsistent (with respect to lattice metrics) peak positions can be observed for fundamental and superstructure reflections, which can affect lattice parameters determined from diffraction patterns.

Keywords:

• diffraction
• domain structure
• twinning
• microstructural characterisation

Acknowledgements

The author wishes to thank Prof. Dr. E.J. Mittemeijer (MPI for Intelligent Systems, formerly Metals Research, Stuttgart, Germany) for his continuous support of and interest in this work.

Notes

Notes

1. WH plots are widely used as a method to separate size and microstrain contributions to line broadening. For that, in the simplest form, the strong assumptions of Lorentzian shapes of both line-broadening contributions as well as of validity of the Stokes-Wilson approximation are made. These severe assumptions led to strong criticism of an application of the original WH procedure for quantitative size-strain separation 6. Nevertheless, if the problem of separation of several broadening contributions is not the principal task, but the analysis of the character of one (main) broadening contribution, the WH method is very powerful.

2. This restriction is not necessary but allows the discussion of the Δ⊥c* stacks to be kept simple because one can use the notions monoclinic for the distorted and orthorhombic for the average structure.

3. In general the transition vector between an i-type to a j-type layer may be given by R _ij , resulting in a matrix of four transition vectors for two types of layers 1 and 2. However, for the scenarios of interest for the present work there is little distortion expected at the fault planes due to a lack of distortion at the fault planes and due to very similar character of the 1,2-type layers this simplification seems justified.

4. Equation (4) results from the average real lattice spanned by a, b, and c and its reciprocal lattice with basis vectors etc. (see Equation (3)), as well as from the two real lattices spanned by a ₁ = a ₂ = a, b ₁ = b ₂ = b and c ₁ = c + Δ, and c ₂ = c − Δ, and the corresponding reciprocal ones. Then one can show by straightforward calculus that , , and .

5. The description of the two lattices as mirrored ones would primarily imply use of basis vectors like a ₂′ = −a ₁, b ₂ = b ₁ and a ₂′ = −c + p a. The presently used choice of the two sets of basis vectors, however, allows having the same Laue indices for the pair of secondary nodes occurring around a primary node. Note, moreover, that in specific cases, in order to arrive at basis vectors with a and b within the twinning plane and c* perpendicular to it, a unit cell transformation involving enlargement of the unit cell may be necessary.

6. The formula (15) contains the term in the case of the F ₂(u) = 0 and F ₁(u) = F ₁(g ₁) option.

7. For L = 0, the line broadening along c* is perpendicular to the diffraction vector and for small broadening the projection of that broadening on the diffraction vector amounts nil in agreement with Equation (18b). For extended broadening along c*, however, so-called Warren peaks arise 24.

8. Taking into account the exponential thickness distribution resulting for small α.

9. Such stacks of laterally structureless layers are only designed to model the diffraction behaviour pertaining to a series of higher-order reflections measured perpendicular to the layers. They are not capable to describe the overall diffraction behaviour of three-dimensional microstructures 6.

10. It should be kept in mind that likely origins to the two types of stacks is symmetry reduction in the course of phase transition for the Δ⊥c stacks and decomposition within a miscibility gap with slight composition-dependent lattice parameters for the Δ||c* stacks. This implies that the considered unstrained (average) states correspond to local maxima in energy, which contrasts the situation where the strain is an elastic one with respect to a strain-free minimum-energy state. This, however, should not interfere with the use of strain relative to these states in the context of analyzing the diffraction phenomena.