If Strain is defined as $\frac{\partial d}{d}$ it is very easy to derive the enlargment of the peaks as function of $\theta$ by derivation of the Bragg formula $$d_{hkl}=\frac{n\lambda }{2sin(\theta )}$$ $$\partial d=-\frac{\lambda }{2tan(\theta )sin(\theta )}\partial \theta$$ $${ϵ}_{hkl}=\frac{\partial d_{hkl}}{d_{hkl}}=-\frac{1}{tan(\theta )}\partial \theta$$ $$\partial theta=-{ϵ}_{hkl}tan(\theta )$$

This size broadening is described by the Scherrer equation. We now reproduce the simple derivation following Klug and Alexander (1974) or Bilinge & Dinebier (2008). The derivation is build considering that for a crystal for small deviation from Bragg angle $\theta +ϵ$ there will be a plane for which the difference of path lenght $\mathrm{\Delta }$ will be $\frac{\lambda }{2}$ producing destructive interference. If the crystal is too small such plane will never arrive and intensity will be present.

The additional beam path between consecutive lattice planes at is: $$\mathrm{\Delta }=2dsin(\theta +ϵ)=2d(sin(\theta )cos(ϵ)+cos(\theta )sin(ϵ)$$ $$\mathrm{\Delta }\approx n\lambda +2dcos(\theta )ϵ$$ The corresponding phase difference is then: $$\delta \phi =2\pi \frac{\mathrm{\Delta }}{\lambda }=4\frac{4\pi ϵdcos(\theta )}{\lambda }$$ The phase difference between the top and the bottom layer, p is then: $$\delta \phi =4p\frac{4\pi ϵdcos(\theta )}{\lambda }=4\frac{4\pi ϵL_{hkl}cos(\theta )}{\lambda }$$ $$ϵ=\frac{\lambda \delta \phi }{4\pi L_{hkl}cos(\theta )}$$ where $L_{hkl}$ is the crystallite size. Using a $2\theta$ scale and the concept of integral breath: $${\beta }_{hkl}=4ϵ=\frac{\lambda }{L_{hkl}cos(\theta )}$$ generalizing for a spherical particle we obtain the Scherrer equation (K is function of grain shape, 0.89 for spheres) : $$\beta =\frac{K\lambda }{Lcos(\theta )}$$