\(tan(\theta)=\cfrac{x_{d1}+x_{d2}}{D+D_o}\)
Now consider a short array of such valence electrons on the surface of a material, that is part of the material structural lattice.
The optical path difference between two adjacent deflections is given by
\(\Delta OP=a_icos(\theta)\)
where \(a_i\) is the atomic distance of the material. If the deflected paths are out of phase,
\(\Delta OP=a_icos(\theta_d)=(2n+1)\cfrac{\lambda}{2}\)
\(cos(\theta_d)=(2n+1)\cfrac{\lambda}{2a_i}\)
\(\theta_{d0}=cos^{-1}(\cfrac{\lambda}{2a_i})\), \(\theta_{d1}=cos^{-1}(\cfrac{3\lambda}{2a_i})\), \(\theta_{d2}=cos^{-1}(\cfrac{5\lambda}{2a_i})\)...
we will see destructive interference on a screen in the path of the deflection.
If the deflected paths are in phase,
\(\Delta OP=a_icos(\theta)=n\lambda\)
\(cos(\theta)=\cfrac{n\lambda}{a_i}\)
\(\theta_{c0}=cos^{-1}(\cfrac{\lambda}{a_i})\), \(\theta_{c1}=cos^{-1}(\cfrac{2\lambda}{a_i})\), \(\theta_{c2}=cos^{-1}(\cfrac{3\lambda}{a_i})\)...
If \(\lambda\approx a_i\) then ,
\(\theta_{d0}=1.0472\,r = 60^o\)
and
\(\theta_{c0}=0^o\)
The two angles are sufficiently wide apart and will project as distinctive bright and dark bands.
In order to achieved a uniformly aligned array of electrons, the material dimension must be very short to be straight and flat along the path of the photons, otherwise other interference from different angles as a resulting of material non-uniformity will over lap and render a indistinguishable image.
So, what about diffraction? Diffraction is the above interference over a short dimension that is the thickness of a thin material, where photons run parallel along the short surface. This can happen at a flat sharp edge or over the thickness of a thin material, smooth and flat.
