Some considerations concerning Wilkens’ theory of dislocation line-broadening
N. Armstrong1, M. Leoni2 & P.
Scardi2
1 Department of Applied Physics,
University of Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia.
2 Departimento di Ingegneria dei Materiali
e Tecnologie Industriali, Università di Trento, 38050 Trento, Italy.
X-ray line
profile analysis is a potentially powerful non-destructive method for
characterising the microstructure of materials. In the past decade, the
technique has quickly evolved from a stage where "simple" assumptions
were made concerning the shape and breadth of line profiles to state-of-art
methods, allowing the synthesis of diffraction profiles directly from the
microstructural properties of materials (see [1] for state-of-art methods).
These microstructural properties include, lattice faulting/twinning,
dislocations and crystallite shape and size distributions (see [1] and
reference therein).
However, not
all available models have reached their maturity. In particular, the Wilkens
model [2-4] is commonly adopted for the (Fourier) description of X-ray
line-profile broadening caused by the presence of dislocations. In this
contribution, some insights concerning its physical significance and
interpretation will be presented.
The Wilkens model provides analytical expressions for the Fourier transform of a line profile broadened by dislocations in terms of two free parameters: the average dislocation density r and the effective outer cut-off radius Re of the strain field. This model was developed assuming a simple microstructure in which a restrictedly random distribution of dislocations is present. This means that equal numbers of parallel and antiparallel straight dislocations populate a single slip-system and are randomly distributed within a sub-area Fp of the total area F0 [2]. A number of important assumptions are made in this case: (i) the radius Rp of the sub-area is approximately equal to the outer cut-off radius Re; (ii) the total dislocation density r is uniform i.e. the ratio between the total number of dislocations N0 and the total cross-sectional area F0 and the ratio between the number Np of dislocations in the sub-area and the area of such region Fp are equal: r = N0/F0=Np/Fp. These assumptions are necessary to overcome the logarithmic divergence encountered by Krivoglaz & Ryaboshapka [5] in their alternative formulation of dislocation broadened line profiles.
It will be shown here that two formulations of the Wilkens model can be developed (see [2,3]); diffraction patterns for (real) elastically anisotropic materials will be simulated, showing qualitative and quantitative differences in the result of the two models.
The first and most widely used formulation [1] will be termed simplified Wilkens model. It assumes that the “average” Fourier term is independent of the slip system, the only hkl dependence being carried by the average contrast factor <Chkl> and by the reciprocal interplanar distance d*hkl [2,3].
The second formulation, called full Wilkens model, includes the slip-system and hkl dependencies into the Fourier coefficients, therefore expressing the resultant Fourier coefficients as the convolution product:
(1)
where 
is the Fourier
transform for the j-th slip-system, b is the magnitude of Burgers
vector, L is the Fourier length, 
defines the
slip-dependency of the contrast factors 
and 
is Wilkens function
(see [2]).
A set of simulated patterns for Cu, Ni and CeO2 will be employed to highlight the difference between the two models. Simulations are made assuming an ideal diffractometer with CuKa1 wavelength, dislocations density, r=2.0x1016 m-2 and an effective outer cut-off radius, Re=10.0 nm. These setting correspond to a Wilkens parameter M = Re r1/2 = 1.4, which is within the range of applicability of the theory [2-4]. These simulations highlight the slip-system and hkl dependency of the quantities in (1) and the influence they have on the profile for increasing d*hkl.
[1] E. J. Mittemeijer & P. Scardi, editors: Diffraction Analysis of the Microstructure of Materials, Springer Series in Materials Science, Vol. 68. (Springer-Verlag, Berlin, 2004).
[2] M. Wilkens (1970a), NBS Spec. Publ. 317, 2, 1195–1221. Proceedings of Fundamental aspects of dislocation theory, eds. J. A. Simmons, R. De Witt and R. Bullough.
[3] M. Wilkens (1970b), NBS Spec. Publ. 317, 2, 1191–1193. Proceedings of Fundamental aspects of dislocation theory, eds. J. A. Simmons, R. De Witt and R. Bullough.
[4] M. Wilkens (1970c), Phys. Stat. Sol. A, 2, 359–370
[5] A. Krivoglaz & K. P. Ryaboshapka, Fiz. Metal. Metalloved, 15(1), 18–31.