Example:Particle spacing from peak position

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


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

BCC 110


\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:

d_{nn} = \sqrt{3}a /2

So we expect:


\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:


\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


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

And:

d_{nn}=\sqrt{2}a/2

So:


\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:


\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}

Comparison

Notice that the BCC 110 and FCC 111 have the same 'conversion factor' from q-position into particle-particle distance. Thus, even without knowing which unit cell the particles are packing into, one can perform the conversion, provided one is confident about the peak being either the BCC 110 or the FCC 111.

See Also