Building up a diamond lattice: Difference between revisions

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m (New page: [EN CONSTRUCCION] * Consider: # 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 ...)
 
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[EN CONSTRUCCION]
* Consider:
* Consider:
# a cubic simulation box whose sides are of length <math>\left. L  \right. </math>
# 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 <math> \left. M = 4 m^3    \right. </math>,
# a number of lattice positions, <math> \left. M \right. </math> given by <math> \left. M = 8 m^3    \right. </math>,
with <math> m </math> being a positive integer
with <math> m </math> being a positive integer


Line 19: Line 17:
where the indices of a given valid site are  integer numbers that must fulfill the following criteria
where the indices of a given valid site are  integer numbers that must fulfill the following criteria


* <math> 0 \le i_a < 2m </math>
* <math> 0 \le i_a < 4m </math>
* <math> 0 \le j_a < 2m </math>  
* <math> 0 \le j_a < 4m </math>  
* <math> 0 \le k_a < 2m </math>,
* <math> 0 \le k_a < 4m </math>,
* the sum of <math> \left. i_a + j_a + k_a \right. </math> must be, for instance, an even number.  
* the sum of <math> \left. i_a + j_a + k_a \right. </math> can have only the values: 0, 3, 4, 7,  8, 10, ...
 
i.e, <math> \left. i_a + j_a + k_a =  4 n \right. </math>; OR; <math> \left. i_a + j_a + k_a = 4 n + 3 \right. </math>, with <math> n </math> being
any integer number
* the indices <math> \left\{ i_a, j_a, k_a \right\} </math>must be either all even or all odd.
with
with
<math>
<math>
\left.
\left.
\delta l = L/(2m)
\delta l = L/(4m)
\right.
\right.
</math>
</math>
== Atomic position(s) on a cubic cell ==
* Number of atoms per cell: 8
* 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_3 \right) = \left( \frac{l}{2}, 0, \frac{l}{2} \right) </math>
Atom 4: <math> \left( x_4, y_4, z_4 \right) = \left( \frac{l}{2}, \frac{l}{2}, 0  \right) </math>
Atom 5: <math> \left( x_5, y_5, z_5 \right) = \left( \frac{l}{4}, \frac{l}{4}, \frac{l}{4}  \right) </math>
Atom 6: <math> \left( x_6, y_6, z_6 \right) = \left( \frac{l}{4}, \frac{3l}{4}, \frac{3l}{4}  \right) </math>
Atom 7: <math> \left( x_7, y_7, z_7 \right) = \left( \frac{3l}{4}, \frac{l}{4}, \frac{3l}{4}  \right) </math>
Atom 8: <math> \left( x_8, y_8, z_8 \right) = \left( \frac{3l}{4}, \frac{3l}{4}, \frac{l}{4}  \right) </math>
Cell dimensions:
*<math> a=b=c = l </math>
*<math> \alpha = \beta = \gamma = 90^0 </math>
[[category: computer simulation techniques]]

Latest revision as of 11:00, 13 February 2008

  • Consider:
  1. 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. }
  2. 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 = 8 m^3 \right. } ,

with 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 < 4m }
  • 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 < 4m }
  • 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 < 4m } ,
  • the sum of can have only the values: 0, 3, 4, 7, 8, 10, ...

i.e, 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 = 4 n \right. } ; OR; 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 = 4 n + 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 n } being any integer number

  • the indices 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 either all even or all odd.

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/(4m) \right. }

Atomic position(s) on a cubic cell[edit]

  • Number of atoms per cell: 8
  • Coordinates:

Atom 1:

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_3 \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_4 \right) = \left( \frac{l}{2}, \frac{l}{2}, 0 \right) }

Atom 5: 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_5, y_5, z_5 \right) = \left( \frac{l}{4}, \frac{l}{4}, \frac{l}{4} \right) }

Atom 6: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(x_{6},y_{6},z_{6}\right)=\left({\frac {l}{4}},{\frac {3l}{4}},{\frac {3l}{4}}\right)}

Atom 7: 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_7, y_7, z_7 \right) = \left( \frac{3l}{4}, \frac{l}{4}, \frac{3l}{4} \right) }

Atom 8: 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_8, y_8, z_8 \right) = \left( \frac{3l}{4}, \frac{3l}{4}, \frac{l}{4} \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 }
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