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:

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} \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{alignat} }

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:

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_{hkl} & = \frac{a}{\sqrt{ h^2 + k^2 + l^2 }} \end{alignat} }

BCC 110

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_{110} & = \frac{a}{\sqrt{ 1^2 + 1^2 + 0^2 }} \\ & = \frac{a}{\sqrt{ 2 }} \end{alignat} }

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} = \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} }

Of course, we could also have written:

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_{110} & = \frac{a}{\sqrt{ 2 }} \\ & = \frac{ 2 d_{nn} / \sqrt{3} }{\sqrt{ 2 }} \\ & = \frac{ 2 d_{nn} }{\sqrt{ 6 }} \\ \end{alignat} }

FCC 111

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_{111} & = \frac{a}{\sqrt{ 1^2 + 1^2 + 1^2 }} \\ & = \frac{a}{\sqrt{ 3 }} \end{alignat} }

And:

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}=\sqrt{2}a/2}

So:

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{2}a }{2} \\ & = \frac{ \sqrt{2} d_{111} \sqrt{3} }{2} \\ & = \frac{ \sqrt{6} d_{111} }{2} \\ & = \frac{ \pi \sqrt{6} }{q_{111}} \\ \end{alignat} }

Or:

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_{111} & = \frac{a}{\sqrt{ 3 }} \\ & = \frac{ 2 d_{nn} / \sqrt{2} }{\sqrt{ 3 }} \\ & = \frac{ 2 d_{nn} }{\sqrt{ 6 }} \\ \end{alignat} }