Building up a face centered cubic lattice: Difference between revisions
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{{Jmol_general|Face_centered_cubic_lattice.xyz|A face centered cubic lattice}} | |||
* Consider: | * Consider: | ||
# a | # a cubic simulation box whose sides are of length <math>\left. L \right. </math> | ||
# a number of lattice positions, <math> \left. M \right. </math> given by | # a number of lattice positions, <math> \left. M \right. </math> given by <math> \left. M = 4 m^3 \right. </math>, | ||
with <math> m </math> being a positive integer | |||
* The <math> \left. M \right. </math> positions are those given by: | * The <math> \left. M \right. </math> positions are those given by: | ||
<math> | :<math> | ||
\left\{ \begin{array}{l} | \left\{ \begin{array}{l} | ||
x_a = i_a \times (\delta l) \\ | x_a = i_a \times (\delta l) \\ | ||
| Line 18: | Line 16: | ||
</math> | </math> | ||
where the indices of a given valid site are integer | where the indices of a given valid site are integer numbers that must fulfill the following criteria | ||
* <math> 0 \le i_a < | * <math> 0 \le i_a < 2m </math> | ||
* <math> 0 \le j_a < | * <math> 0 \le j_a < 2m </math> | ||
* <math> 0 \le k_a < | * <math> 0 \le k_a < 2m </math>, | ||
* | * the sum of <math> \left. i_a + j_a + k_a \right. </math> must be, for instance, an even number. | ||
with | with | ||
| Line 31: | Line 29: | ||
\right. | \right. | ||
</math> | </math> | ||
== Atomic position(s) on a cubic cell == | |||
* Number of atoms per cell: 4 | |||
* Coordinates: | |||
Atom 1: <math> \left( x_1, y_1, z_1 \right) = \left( 0, 0, 0 \right) </math> | |||
Atom 2: <math> \left( x_2, y_2, z_2 \right) = \left( 0 , \frac{l}{2}, \frac{l}{2}\right) </math> | |||
Atom 3: <math> \left( x_3, y_3, z_2 \right) = \left( \frac{l}{2}, 0, \frac{l}{2} \right) </math> | |||
Atom 4: <math> \left( x_4, y_4, z_2 \right) = \left( \frac{l}{2}, \frac{l}{2}, 0 \right) </math> | |||
Cell dimensions: | |||
*<math> a=b=c = l </math> | |||
*<math> \alpha = \beta = \gamma = 90^0 </math> | |||
[[category: computer simulation techniques]] | |||
[[category: Contains Jmol]] | |||
Latest revision as of 09:58, 25 June 2012
<jmol> <jmolApplet> <script>set spin X 10; spin on</script> <size>200</size> <color>lightgrey</color> <wikiPageContents>Face_centered_cubic_lattice.xyz</wikiPageContents> </jmolApplet></jmol> |
- Consider:
- a cubic simulation box whose sides are of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. L \right. }
- a number of lattice positions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. M \right. } given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. M = 4 m^3 \right. } ,
with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m } being a positive integer
- The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. M \right. } positions are those given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ \begin{array}{l} x_a = i_a \times (\delta l) \\ y_a = j_a \times (\delta l) \\ z_a = k_a \times (\delta l) \end{array} \right\} }
where the indices of a given valid site are integer numbers that must fulfill the following criteria
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le i_a < 2m }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le j_a < 2m }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le k_a < 2m } ,
- the sum of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. i_a + j_a + k_a \right. } must be, for instance, an even number.
with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \delta l = L/(2m) \right. }
Atomic position(s) on a cubic cell[edit]
- Number of atoms per cell: 4
- Coordinates:
Atom 1: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x_1, y_1, z_1 \right) = \left( 0, 0, 0 \right) }
Atom 2: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x_2, y_2, z_2 \right) = \left( 0 , \frac{l}{2}, \frac{l}{2}\right) }
Atom 3: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x_3, y_3, z_2 \right) = \left( \frac{l}{2}, 0, \frac{l}{2} \right) }
Atom 4: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x_4, y_4, z_2 \right) = \left( \frac{l}{2}, \frac{l}{2}, 0 \right) }
Cell dimensions:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=b=c = l }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \beta = \gamma = 90^0 }