Difference between revisions of "Lattice:Hexagonal"

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(Created page with "===Reciprocal-Space Peaks=== * Allowed reflections: * Peak positions: *: <math>q_{hkl}=2\pi\left( \frac{(h^2 + hk + k^2)^2}{a^2} + \frac{l^2}{c^2} \right)</math> *: For ''a'' ...")
 
 
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===Reciprocal-Space Peaks===
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'''Hexagonal''' is a general class of [[lattice]] symmetries (i.e. how [[unit cell]]s can be arranged in space).
* Allowed reflections:
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===[[Reciprocal-space]] Peaks===
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* Forbidden reflections, when both:
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** <math>h + 2k = 3n</math>
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** <math>l</math> odd
 
* Peak positions:
 
* Peak positions:
*: <math>q_{hkl}=2\pi\left( \frac{(h^2 + hk + k^2)^2}{a^2} + \frac{l^2}{c^2} \right)</math>
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*: <math>q_{hkl}=2\pi\left( \frac{(h^2 + hk + k^2)^?}{a^2} + \frac{l^2}{c^2} \right)^{1/2}</math>
 
*: For ''a'' = ''b'' = 1.0, ''c'' = 1.0:
 
*: For ''a'' = ''b'' = 1.0, ''c'' = 1.0:
 
<pre>
 
<pre>
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</pre>
 
</pre>
 
[[Image:Lattice peaks-Hexagonal.png|450px]]
 
[[Image:Lattice peaks-Hexagonal.png|450px]]
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==Canonical 2D==
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A canonical 2D hexagonal lattice, such as formed by hexgonally-packed cylinders (where the ''c'' direction is conceptually infinite) has peak positions in the ratio:
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* <math>1:\sqrt{3}:2:\sqrt{7}:\sqrt{9}:\sqrt{12}:\sqrt{13}</math>
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==See Also==
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* [[Lattice:HCP|Hexagonal close-packed (HCP)]]
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* [[Lattice:Hexagonal diamond|Hexagonal diamond]]
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** [[Lattice:Wurtzite|Wurtzite]]
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* [[Lattice:AlB2|AlB<sub>2</sub>]]

Latest revision as of 09:23, 29 August 2019

Hexagonal is a general class of lattice symmetries (i.e. how unit cells can be arranged in space).

Reciprocal-space Peaks

  • Forbidden reflections, when both:
    • odd
  • Peak positions:
    For a = b = 1.0, c = 1.0:
peak    q value         h,k,l   m       f       intensity       intensity_scaled
1:      0.126933036509  1,0,0   2       1       2       0.192150
2:      0.146569645595  1,0,0   6       1       6       0.499222
3:      0.193893415997  1,1,0   12      1       12      0.754752
4:      0.253866073017  2,1,0   8       1       8       0.384301
5:      0.283830898224  2,1,1   12      1       12      0.515594
6:      0.293139291189  2,0,0   18      1       18      0.748832
7:      0.319441136669  2,1,0   12      1       12      0.458117
8:      0.359020843488  2,2,1   12      1       12      0.407613
9:      0.380799109526  3,0,0   2       1       2       0.064050
10:     0.387786831994  3,1,0   24      1       24      0.754752

Lattice peaks-Hexagonal.png

Canonical 2D

A canonical 2D hexagonal lattice, such as formed by hexgonally-packed cylinders (where the c direction is conceptually infinite) has peak positions in the ratio:

See Also