Dear Tanbir Ahmed Billah

the key point here is that in x-ray tube based diffractometry one uses the K-alpha characteristic lines of the x-ray tube spectrum, which is a doublet line described by K-alpha1 and K-alpha2. The wavelength of K-alpha1 is a bit lower than that of K-alpha2. The intensity ratio is nearly K-alpha1 : K-alpha2 ~ 2 : 1.

Let us calculate the difference delta_theta of the angular positions of the diffraction peaks*), which are caused by the wavelength difference delta_lambda of the two K-alpha lines, as a function of the Bragg angle theta.

For doing that, let us play around with the Bragg law, which relates the x-ray wavelength lambda of the diffracted beam to the interplanar spacing d of the 'reflecting' lattice planes and the Bragg angle theta:

(1) lambda = 2*d*sin(theta)

Differentiating this equation will end up with:

(2) delta_lambda = 2*d*cos(theta)*delta_theta

Dividing that result by the Bragg equation (1) will give you:

(3) delta_lambda/lamba = cos(theta)/sin(theta)*delta_theta =1/tan(theta)*delta_theta

Resolving (3) for delta_theta will end up with:

(4) delta_theta = delta_lambda/lambda*tan(theta)

From eq. (4) you see, that (i) delta_theta is proportional to tan(theta) and (ii) delta_lambda/lambda is constant scaling factor.

That's it...

For low angles in your case the peak broadening is too large; so one is not able to resolve the two diffraction peaks caused by K-alpha1 and K-alpha2; but due to the tan(theta) function the peak distance increases with theta and two peaks can be resolved in many cases.

*) in your case the 'diffraction peaks' are equal to the profile(s) of your circular diffraction lines.

Best regards

G.M.

One of the important points in regard to the measurement of linear distances and then their conversion to θ values, is to distinguish the low angle diffraction lines from the high-angle ones. This can be achieved by observing the following two characteristic features on the film. One of these relates to the background intensity which arises mainly due to the scattering of radiation from the air in the camera. It is known that this scattering is maximum near θ=0. Thus one expects more of background scattering and hence more blackening of the film near the low angle side around the exit hole whose centre corresponds to 0°. The second feature which helps in distinguishing the low angle side from the high angle one, is in regard to the resolution of lines corresponding to Ka1 and Ka2 components of the Ka, doublet. For CuKa, which is the usual radiation employed for taking powder photographs, this doublet corresponds to CuKa1 (A 1.54050 A) and CuKa2(A-1.54434 Å). Thus every diffraction line should be a doublet, the two components arising from the Ka1 and Ka2 wavelengths. The two components, however, get resolved only at the high angle side where the separation between the two Bragg angles is significant and hence the doublet can be recognized. In practice the doublet is easily discernible for higher camera diameters where the separation between the two com- ponents is appreciable. For the usual camera diameter (57.3 mm), the doublet appears as a pair of two closely-spaced lines merging with each other and giving the appearance of the presence of a rather thick line.