Example:Particle spacing from peak position

Consider the case of trying to measure the particle-particle spacing from the q-value of a particular peak. The interpretation of the q value of course depends upon the packing of the particles; i.e. the unit cell. Consider a cubic unit cell (SC, BCC, FCC). Note that in general:

${\begin{alignedat}{2}\left({\frac {1}{d_{hkl}}}\right)^{2}=\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}\end{alignedat}}$

Since $q=2\pi /d$, and since $a=b=c$, the realspace spacing of planes is:

${\begin{alignedat}{2}d_{hkl}&={\frac {a}{\sqrt {h^{2}+k^{2}+l^{2}}}}\end{alignedat}}$

BCC 110

${\begin{alignedat}{2}d_{110}&={\frac {a}{\sqrt {1^{2}+1^{2}+0^{2}}}}\\&={\frac {a}{\sqrt {2}}}\end{alignedat}}$

Note that for BCC, the particle-particle distance is given by:

$d_{nn}={\frac {{\sqrt {3}}a}{2}}$

So we expect:

${\begin{alignedat}{2}d_{nn}&={\frac {{\sqrt {3}}a}{2}}\\&={\frac {{\sqrt {3}}d_{110}{\sqrt {2}}}{2}}\\&={\frac {{\sqrt {6}}d_{110}}{2}}\\&={\frac {{\sqrt {6}}(2\pi /q_{110}}{2}}\\&={\frac {\pi {\sqrt {6}}}{q_{110}}}\\\end{alignedat}}$