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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=2 \pi / d} , and since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_{nn}=\frac{ \sqrt{3}a }{2}}

So we expect:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }