Sorry I miss-typed. It is difficult to write correctly the equations here. Replace lambda by lambda^2 in the above equations. Also B should be replaced 2B.

Here is the correct equation is for powder diffraction from a monatomic metal.

I = kASPf^2e^-2Bsin^2theta/lambda^2

I = observed intensity

k=scale factor

A = absorption factor

S= multiplicity factor

P= Lorentz-polarization factor

f=atomic scattering factor (For simplicity I assuime that you have only a single atom)

B= Debye-Waller factor

2theta=scattering angle

lambda=wavelength

Now take logarithmic from both sides and plot as I indicated above. This is the nasic principle. Now read text books.

When you refine a crystal structure then you refine also the Debye-Waller factor B. So you determine B from the measured diffraction intensities. Now B is also related with the mean square atomic displacements. So you can calculate the atomic displacements as well. If you have low symmetry then the displacement parameters will anisotropic and you can refine the symmetry allowed anisotropic displacement parameters. Have you never read books like "Thermal Vibrations in Crystallography" by B.T.M. Willis and A.W. Pryor. You can also consult X-ray diffraction by B.E. Warren. Please do some serious reading before you ask questions.

You can use Rietveld refinement method by application of various software such as GSAS or MAUD for calculation of B. in this method you should provide an initial guess for all parameters of applied models then after refinement you can find the parameters of applied models.

If you can measure the integrated intensities I of several reflections and plot the ln(I) as a function of sin^_theta/lambda then you will get a straight line. By fitting a straight line determine the slope. From this slope you get the Debye-Waller factor B. From B you can calculate mean square displacement .

The above method works because the intensity I can be written as I = Ae^(-Bsin^2theta/lambda). Taking ln from both sides you get

ln (I) = ln(A) -Bsin^2(theta)/lambda

So if you plot ln(I) vs. sin^2theta/lambda then the slope is B, the Debye waller factor.

Sorry I miss-typed. It is difficult to write correctly the equations here. Replace lambda by lambda^2 in the above equations. Also B should be replaced 2B.

Here is the correct equation is for powder diffraction from a monatomic metal.

I = kASPf^2e^-2Bsin^2theta/lambda^2

I = observed intensity

k=scale factor

A = absorption factor

S= multiplicity factor

P= Lorentz-polarization factor

f=atomic scattering factor (For simplicity I assuime that you have only a single atom)

B= Debye-Waller factor

2theta=scattering angle

lambda=wavelength

Now take logarithmic from both sides and plot as I indicated above. This is the nasic principle. Now read text books.

You have really helped me.I have calculated this but how to conform that these calculated values of Debye Waller parameter are correct or incorrect.what to do for error bar calculations?

What is the substance you are studying? Do you use Fullprof? If yes the program then calculates the error bars too. To estimate error bar of any measurement you have to know the statistical theory of error propagation. When you calculate a quantity that is comosed of several other quantities that you have really measured then the composite error depends on the individual errors of each measured quantities. Suppose if you have measured the cell parameters a and c of a tetragonal crystal and the errors of these measurements are Deltaa and Deltac. You now can calculte the volume of the unit cell V = a^2c. Now how to calculate error in volume Delta V? The DeltaV is simply related with Deltaa and Deltac by differentiating (partial):

DeltaV/V = 2cDeltaa/a +a^2 Deltac/c

For orthorhombic crystal the equations will be:

V = abc

and

DeltaV/V=bcDelta/a+acDeltab/b+abDeltac/c

For more complex equations consult book: Methods of experimental Physics, Academic Press vol. (I cannot remenber at the moment but I shall send letter) The article: Propagation of errors.

Exact reference is:

Evaluation of Measurements, by S. Reed in Methods of Expermental Physics, vol1., p. 1 , Academic Press.

This is a very nice article. The relevant equation is equatin 1.5.2 on p. 13

I have the reference of another nice pedagogic article "On the Propagation of Errors", Raymond T. Birge, The American Physics Teacher 7, 351-357 (1939). I am only afraid that you may not be able to get it at the BHU library. In case you don't find let me know. I can send you a scanned copy.

I dont have fullprof.I am struggling for these calculations.

But you have helped a lot.Thanks for it.

You should download the Fullprof program that is freely available.

I obtained all individual and overall B values as -ve for spinel structure refinement. However, refinement looks good.  Rp= 7.92%, Rwp =9.25%, Re=4.77% and chi2=3.75

Is this refinement wrong?