Difference between revisions of "Example:Particle spacing from peak position"

From GISAXS
Jump to: navigation, search
(→‎BCC 110)
(→‎BCC 110)
Line 23: Line 23:
 
</math>
 
</math>
 
[[Lattice:BCC|Note that for BCC]], the particle-particle distance is given by:
 
[[Lattice:BCC|Note that for BCC]], the particle-particle distance is given by:
−
:<math>d_{nn}=\frac{ \sqrt{3}a }{2}</math>
+
:<math>d_{nn} = \sqrt{3}a /2</math>
 
So we expect:
 
So we expect:
 
:<math>
 
:<math>
Line 31: Line 31:
 
   & = \frac{ \sqrt{3} d_{110} \sqrt{2} }{2} \\
 
   & = \frac{ \sqrt{3} d_{110} \sqrt{2} }{2} \\
 
   & = \frac{ \sqrt{6} d_{110}  }{2} \\
 
   & = \frac{ \sqrt{6} d_{110}  }{2} \\
−
   & = \frac{ \sqrt{6} (2 \pi / q_{110} }{2} \\
+
   & = \frac{ \sqrt{6} (2 \pi / q_{110} ) }{2} \\
 
   & = \frac{ \pi \sqrt{6}  }{q_{110}} \\
 
   & = \frac{ \pi \sqrt{6}  }{q_{110}} \\
 +
\end{alignat}
 +
</math>
 +
Of course, we could also have written:
 +
:<math>
 +
\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}
 +
</math>
 +
 +
===FCC 111===
 +
:<math>
 +
\begin{alignat}{2}
 +
d_{111}
 +
  & = \frac{a}{\sqrt{ 1^2 + 1^2 + 1^2 }} \\
 +
  & = \frac{a}{\sqrt{ 3 }}
 +
\end{alignat}
 +
</math>
 +
And:
 +
:<math>d_{nn}=\sqrt{2}a/2</math>
 +
So:
 +
:<math>
 +
\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}
 +
</math>
 +
Or:
 +
\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}
 
\end{alignat}
 
</math>
 
</math>

Revision as of 14:36, 2 September 2014

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: \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} </math>

Retrieved from "http://gisaxs.com/index.php?title=Example:Particle_spacing_from_peak_position&oldid=1484"