Chapter 16 in Elements of X-ray diffraction. by B. D. Cullity

Shortly explained, caused by stress the lattice becomes distorted which changes the lattice parameters and therefore the Bragg position of the reflections. However, you need a stress gradient, i.e. under hydrostatic pressure you wont be able to measure no strain and therefore no stress (except by comparison of the same material under non-stressed conditions). The standard sin²psi-technique also only works for polycrystalline materials of small grain size since you need millions of crystals which give you the required information. However, it is better to read a book or some fundamental  papers published in the 70th to 90th.

You can find other good introductions to this topic in:

Springer Handbook of Experimental Solid Mechanics (Springer 2008), Chapter 28 "X-Ray Stress Analysis" by J. D. Almer and R. A. Winholtz

and

Residual Stress - Measurement by Diffraction and Interpretation by I. C. Noyan and J. B. Cohen (Springer, 1987)

Deformation stacking fault shifts (111) line towards larger angle, and (200) line towards the smaller angles in FCC structure.  Instead of trying to measure the displacement of one reflection, it is better to combine a neighbouring pair, which have opposite displacements, and use the following formula given by E.E. Warren in Prog.  Met. Phys. V. 8, p. 178.  You can get directly the stacking fault probability., as I  did in  my early work in 1975-1979.  

But then your material is no more cubic, or a mixture of phases, right? I think Braggs equation clearly expresses that if one reflection shifts into positive direction all other have to do the same. Or did you measured the interferences under different sample tilt as used for sin²psi technique?

Dear Dr. Noize,   you are over simplifying the problems related to the effects of deformation on the XRD spectrum obtained from powder specimes of metals and alloys.  Any one dealing with the imperfections in deformed metals and alloys, and if he/she wishes  to get some quantitative  information from  XRD -power analysis then he/she should go over the review article by  B. E. Warren of MIT, which was  published in Progress in Metal Physics Vol. 8 (1959) pp. 147-202. The formulation here was due to the method introduced by Stokes and Wilson, which  is not limited only for {00L}  reflections in cubic system but also for  {00L} reflection in in terms of orthorhombic axes. 

I was working  in  alpha-Cu-Zn alloys( f.c.c) having various compositions using the samples prepared by  fillings to generate severe  plastic deformation.  I had done in situ experiments under the argon atmosphere using various isothermal annealing  scenarios to study segregation kinetics of Zn on the stacking faults. I  published two papers in Metal. Transaction i 1975 and 1979, and you may find in my publ. list in RG. As a by product in addition to the stacking fault probability variations with  alloying, I obtained diffusion activation energy of super saturated vacancies and  dislocation densities using the stacking fault width measurements by TM, These papers were heavily involved theoretically by solving BVP related to the  segregation on 2D systems (stacking faults, interfaces and grain boundaries) as well as experimentally. 

I had a special set-up, which had atmosphere and temperature control  facilities obtained from MRC-USA.  

You can also look at the Good Practice Guide for X-ray Residual Stress Measurement:

http://www.npl.co.uk/upload/pdf/Determination_of_Residual_Stresses_by_X-ray_Diffraction_-_Issue_2.pdf

The most powerful analytic  technique to extract information deal with XRD powder diffraction pattern is the Fourier analysis.  Where thecosine coefficients with respect mode numbers gives you the  product  of  a particle size coefficient and the distortion coefficient, which can be separated by a special method. The slope of particle size versus mode number results the average domain size, and the second derivation gives you domain size distribution probability. The expectation value of  Sine Fourier coefficients  can be affected only by the existence of stacking and twin faults otherwise they are all zero. According to Warren reprentation  from the semi logarithmic plot of cosine coefficient with respect  square of  L  gives you straight line, where  {00L}. The slope in this plot gives directly the expectation value of square of distortion (strain) and the intersection is the Ln of  size coeffient for given Fourier mode!!!

As a summary: 

i. The initial slope of cosine coefficients gives a combination of an average domain size and the faulting probability (stacking fault + twin fault).

ii. The peak displacement  gives the deformation(stacking faults) fault probability. 

iii. The magnitude of the sine coefficients gives the twin probability.

There are specific formulas for FCC, HCP and BBC structure in above cited reference.

This was my term paper project  in Advanced XRD course at Stanford in 1962, and it was specially assigned for me because of my background in applied mathematics and the computer programming. 

Well, I am certainly not that specialized in this field as you Tarik, but from my simple understanding of the translation-symmetric lattices one can only get a proportional shift of peaks with theta if the phase is cubic. I am not talking about anomalous peak broadening or peak shape analysis which certainly enables the extraction of crystal defects, e.g. as mentioned by you. Moreover, I do not exclude an apparent shift of a peak caused by the described lattice defects, but isn't this the result of a superimposition of "defect-free" and "non-defect-free"  regions of your sample?

Best greetings

Gert

Dear Gert; when we deal with a powder specimen we talking about statistical averages,  and relying implicitly on the principle superposition. Otherwise all these sophisticated mathematical tools such convolutions as such should go to the garbage cane. In actual practice why we are preferring to use Fourier space rather then the other tool of  convolutions is the fact that  multiple convolution integrals appears to be additive algebraically in Fourier  representation. This way you can organized  experiments related to the various - independent- error sources separately to be used in the final stage of the analysis. Actually, I had an alpha brass specimens with various compositions prepared in England.  I took two measurements after cold -worked  by fillings using the same mesh. The first  XRD experiment done on the cold-worked specimen, and the second one on the same sample after prolong isothermal annealing at the recrystallization temperature range, where substantial decrease takes place in the line defects as well as in the stacking fault defects concentrations. Then I did the folding of cold-worked  results over the annealed final state (reference) to extract what was solely due to  dislocations and stacking faults generated during the cold-worked stage, at any given temperature and the isothermal annealing time. This way I  was able to track down the kinetics of segregation of Zn at the stacking faults  at various isothermal annealing scenario.  Best regards.

Note: I haven't done and supervise any  experimental work in XRD, since I joined Max Planck Institute für Physik  (Stuttgart :1980-1988) but doing only theoretical work and computer simulations. 

XRD  Studies of Stacking Fault Segregation in Alpha Brass

Dear Tarik, this is really interesting stuff, but I guess the question of A. Rahdar is more related to the classical interpretation of the shift of the peaks caused by resdual stress measurements. Your work I interpret actually as important extension to this classical approach. Whereas the classical approach does not look for dislocations or stacking faults but interpretes the shift by an elastic deformation of the lattice, you try to interprete the peak broadening and peak shape in oder to find out which possible lattice defect may cause these effects. You are absolutely right to mention that the interpretation of an only elastic deformation of the lattice is practically redicules since we know that the material is full of lattice defects, but until now this gives a "simple" number for a technologist which tries to find a simple correlation between this number and the respecive property he is interested in. Nevertheless, nature is much more complex and complicated as you already showed in your paper from 1975. Many thanks!

Dear Gert, what you have written right now,  I am completely in agreeing   with your comments.  A rule of thumb;  any things changes the lattice parameters and/or the structure of a crystalline material affect its  observed XRD  spectrum in terms of  its line positions and their intensities.  Best Regards