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Background

Defects in materials can be investigated by the means of diffraction line broadening. Anisotropic X-ray line broadening means that neither the FWHM nor the integral breadth nor the Fourier coefficients of diffraction profiles are monotonous functions of the diffraction vector or its square, g or g2, respectively (Williamson & Smallman 1955; Caglioti et al., 1958; Warren, Averbach, 1952, 1950; Warren, 1959). It has been shown that the strain anisotropy in powder diffraction can be well accounted using the contrast factors C of dislocations related to particular Burgers-, line- and diffraction vectors or to the average contrast factors . The contrast factors of dislocation characterize the "visibility" of dislocations in the diffraction experiments.

This parameter enable the evaluation of dislocation densities, arrangement, the determination of active slip systems as well as the size distributions of crystallites.

The dislocation contrast factors, C, in elastically anisotropic crystals can be obtained by numerical methods taking into account the elastic constants of the material, the lattice parameters and the relative orientations of the diffraction vector and the line- and Burgers vectors of the dislocations (Wilkens, 1970; Klimanek, Kuzel, 1988, 1989; Groma et al. 1988)

Since the displacement fields of dislocations superimpose linearly the average contrast factors corresponding to a specific slip system can be obtained by the weighted linear combination of the individual contrast factors (Wilkens, 1970).

Contrast factors of dislocations in cubic and hexagonal crystals

It can be shown that if the specimen is either untextured or if all possible slip systems are equally populated the dislocation contrast factor for a specific diffraction vector, g, in cubic crystals will have a simple form:

hkl=h00(1-qH2) (1a),

where H2=(h2k2+ h2l2+ k2l2)/(h2+k2+l2)2, and h, k and l are the Miller indices of the diffraction vector, g. In hexagonal crystals

hk.l = hk.0 [1 + q1x + q2 x2 ] (1b),

where x =(2/3)(l/ga)2 and and h, k and l are the indices of the diffraction vector, g, (Ungár &Tichy, 1999; Ungár et al. 1999; Dragomir & Ungár 2001). In the cubic system two parameters are sufficient to fully characterize the average contrast factors of dislocation in the selected slip system. h00 is the average contrast factor corresponding to the h00 reflections and q is a parameter depending on the dislocation type and the elastic constants of the material. The hexagonal crystal system is by far more complex than the cubic one. Here three numbers are needed for the parameterization of average contrast factors: hk.0, the average contrast factor corresponding to the hk.0 reflection, q1 and q2 parameters which depends on the character of dislocations and the elastic constants. Therefore these parameters allow the determination of the prevailing dislocation type in the sample ( Ungár et al. 1999; Dragomir & Ungár 2001).

The ANIZC program enables the calculation of the average contrast factors of dislocations for the must common dislocation types in cubic and hexagonal crystals, but also the determination of the individual contrast factors for any other dislocation.

In both cases one has to input the elastic constants of the investigated material: cij or sij and the hkl indices of diffraction vector, g.

The calculation of the individual contrast factors

For the evaluation of the individual contrast factor of a single dislocation the input of Burger vector, b, line vector, l and slip plan, n, is required and also the elastic constants of the material in question. Please click here to see an example.

The calculation of the average contrast factors

For the evaluation of the average contrast factors the elastic constants and the lattice ratio c/a in the case of hexagonal crystals are needed. Please click example to see how these data have to be inputted.

By using this web-site the evaluation of average contrast factor can be done for the most common type of dislocations observed in the cubic and hexagonal crystals. These are listed in the follows:

Dislocation types in cubic crystals: Edge <110>{111}

<111>{110}

<111>{211}

Screw <110>

<111>
 
 

The following slip systems are considered for fcc and bcc lattices:
 
fcc lattice
    
bcc lattice, {110} slip
    
bcc lattice, {211} slip
  
slip system no. slip plane slip direction slip system no. slip plane slip direction slip system no. slip plane slip direction
1 11-1 011 1 011 11-1 1 2-11 11-1
2 11-1 101 2 101 11-1 2 1-2-1 11-1
3 11-1 1-10 3 1-10 11-1 3 112 11-1
4 1-1-1 01-1 4 01-1 1-1-1 4 211 1-1-1
5 1-1-1 101 5 101 1-1-1 5 12-1 1-1-1
6 1-1-1 110 6 110 1-1-1 6 1-12 1-1-1
7 1-11 011 7 011 1-11 7 21-1 1-11
8 1-11 10-1 8 10-1 1-11 8 121 1-11
9 1-11 110 9 110 1-11 9 1-1-2 1-11
10 111 01-1 10 01-1 111 10 2-1-1 111
11 111 10-1 11 10-1 111 11 1-21 111
12 111 1-10 12 1-10 111 12 11-2 111
  Dislocation types in hexagonal crystals: Edge <2-1-10>{0001}

<-2110>{01-10}

<0001>{01-10}

<-2113>{01-10}

<-12-10>{10-11}

<-2113>{2-1-12}

<-2113>{11-21}

<-2113>{10-11}

Screw <2-1-10>

<-2113>

<0001>

In this case you have to choose from the above list the slip system for which you would like to evaluate the average contrast factor of dislocation. For this the program will access a batch file, which calculates the individual contrast factors for all equivalent dislocations of this kind. With the view to obtain the average contrast factor of dislocation the program average the individual contrast factors as it was shown by Wilkens, 1970. The user does not have to make this calculation. The number received at the end is the average contrast factor of the chosen dislocation type and the given diffraction vector, g.

Further more from eq. (1a) and (1b) the q parameters for the different dislocation type can be estimated.
 
 

Methodology to evaluate the main dislocation character

The value of q parameters cumulate the dependence of average contrast factor on the elastic constants, dislocation type and c/a ratio. Comparing the numerically evaluated q factors with the ones obtained experimentally by X-ray profile analysis the dominant dislocation character in the sample can be determinate.

The average contrast factor which characterize the investigated material can be written as a weighted combination of the average contrast factor for pure edge, e, and pure screw dislocation, s:

m = f e+(1-f) s ,

where f is the fraction of edge dislocations.

The measured q factors, qm, can be given as the linear combinations of the numerically calculated q factors:

qm = hqe+ (1-h)qs ,

where qe and qs is the numerically evaluated q factors for edge and screw dislocations, respectively.

Than the fraction of the edge dislocations is:

f = h/[(1-h)+h] .

The same procedure can be applied for hexagonal crystal also. However, in the case of hexagonal crystal the evaluation of the active dislocation type is more complex. Instead of one major type of slip system there are at least three fundamentally different slip systems (Jones & Hutchinson, 1981; Castelnau et al., 2001; Yadav & Ramesh, 1997).

For a more detailed description of this procedure applied for determination of activ dislocation type in hexagonal crystals please see Dragomir & Ungár 2001.

For additional information please see the article.