Related to error propagation in crystallite size calculation from Scherrer formula I am trying to calculate the error in crystallite size calculation from Scherrer formula $ t=kλ/β\cos\theta $ and I have calculated error propagation using the formula $\Delta t/t = \sqrt((\Delta\beta/\beta)^2 + (\Delta\theta \tan\theta)^2)$ ($\Delta\theta$ in radian). Using the above method I got $8$% error for a $37$ nm spherical crystallite i.e. ($37 \pm 3$) nm.
I know crystallite size calculation from Scherrer formula is a rough estimation but still, the error is quite high. Is this calculation right? Is this because the peak broadening is not due to crystallite size only? Please suggest me if there is another method.  Thank you.
 A: As you wrote the Scherrer formula is a rough estimation. The 8% error actually looks low. Usually I see people reporting the size without the error: the size calculated according to this method is 37nm and according to that method 52nm. Because the real error is hard to tell.
Instead of using the error propagation formula I'd calculate the size from each peak to get the idea how consistent the results are. You could then use the standard deviation as an error measure.
As you noted, the peak broadening is caused not only by the small size of crystallites.
All kinds of crystal defects broaden the peaks.
Traditionally, these defects (if you can call the grain boundaries a defect) are grouped into size and strain components; estimating the defects is called the size-strain analysis.
You also have instrumental broadening that you can measure on a standard sample in the same instrument.
The Scherrer equation ignores the strain broadening and this method is frowned upon nowadays. A few decades later, in 1950s, two other classical methods have been devised: Williamson-Hall and Warren-Averbach. They give two numbers: size and strain. But nowadays they are also regarded as too simplistic. In 1980s and '90s the double-Voigt method has been developed and it became one of the mainstream size-strain methods. It characterizes the size with two numbers (because normally you have a distribution of domain sizes, not a single size). More recently, the WPPM method became popular. I'm not in this field, but my impression is that the WPPM and perhaps also the double-Voigt method are nowadays considered the methods of choice.
