File:Crystal Structure as Lattice and Basis.gif

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Original file (923 × 431 pixels, file size: 3.47 MB, MIME type: image/gif, looped, 147 frames, 15 s)

Captions

Captions

Once we have the Bravais lattice, we can fully describe a crystal structure by specifying how the atoms are arranged around the lattice points (i.e. the "basis").

Summary

[edit]
Description
English: Once we have the Bravais lattice, we can fully describe a crystal structure by specifying how the atoms are arranged around the lattice points (i.e. the "basis").
Date
Source https://mathstodon.xyz/@j_bertolotti/112489920832771942
Author Berto
Permission
(Reusing this file)		 https://mathstodon.xyz/@j_bertolotti/111363365323269417

Mathematica 14.0 code

[edit]
sinstep[t_] := Sin[\[Pi]/2 t]^2
frames1 = Table[
   a1 = {1, 0}; a2 = {0, 1};
   basis1[point_] := point - {0.25*sinstep[t], 0}; 
   basis2[point_] := point + {0.25*sinstep[t], 0};  
   bravais = Flatten[Table[n1 a1 + n2 a2, {n1, -20, 20}, {n2, -20, 20}], 1];
   Grid[{{"Bravais Lattice", "", "Basis", "", "Crystal"}, {
      Graphics[{Gray, Table[Disk[n1 a1 + n2 a2, 0.05], {n1, -20, 20}, {n2, -20, 20}]}, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick], ImageSize -> 300 ]
      , Text[Style["\[Star]", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[{0, 0}, 0.05], Black, Disk[basis1[{0, 0}], 0.1], Disk[basis2[{0, 0}], 0.1]}, PlotRange -> 1.1*{{-1, 1}, {-1, 1}} , ImageSize -> 50]
      , Text[Style["=", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[#, 0.05] & /@ bravais, Black, Disk[basis1[#], 0.1] & /@ bravais, Disk[basis2[#], 0.1] & /@ bravais }, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick] , ImageSize -> 300]
      }}]
   , {t, 0, 1, 0.05}];
frames2 = Table[
   a1 = {1 + sinstep[t], 0}; 
   a2 = {sinstep[t], 1 + (Sqrt[3]/2 - 1)*sinstep[t]};
   basis1[point_] := point - {0.25, 0}; 
   basis2[point_] := point + {0.25, 0};  
   bravais = Flatten[Table[n1 a1 + n2 a2, {n1, -20, 20}, {n2, -20, 20}], 1];
   Grid[{{"Bravais Lattice", "", "Basis", "", "Crystal"}, {
      Graphics[{Gray, Table[Disk[n1 a1 + n2 a2, 0.05], {n1, -20, 20}, {n2, -20, 20}]}, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick], ImageSize -> 300 ]
      , Text[Style["\[Star]", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[{0, 0}, 0.05], Black, Disk[basis1[{0, 0}], 0.1], Disk[basis2[{0, 0}], 0.1]}, PlotRange -> 1.1*{{-1, 1}, {-1, 1}} , ImageSize -> 50]
      , Text[Style["=", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[#, 0.05] & /@ bravais, Black, Disk[basis1[#], 0.1] & /@ bravais,  Disk[basis2[#], 0.1] & /@ bravais }, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick] , ImageSize -> 300]
      }}]
   , {t, 0, 1, 0.05}];
frames3 = Table[
   a1 = {2, 0}; a2 = {1, Sqrt[3]/2};
   basis1[point_] := point - {0.25 Cos[2 \[Pi] sinstep[t]], 0.25 Sin[2 \[Pi] sinstep[t]]}; 
   basis2[point_] := point + {0.25 Cos[2 \[Pi] sinstep[t]], 0.25 Sin[2 \[Pi] sinstep[t]]};
   bravais = Flatten[Table[n1 a1 + n2 a2, {n1, -20, 20}, {n2, -20, 20}], 1];
   Grid[{{"Bravais Lattice", "", "Basis", "", "Crystal"}, {
      Graphics[{Gray, Table[Disk[n1 a1 + n2 a2, 0.05], {n1, -20, 20}, {n2, -20, 20}] }, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick], ImageSize -> 300 ]
      , Text[Style["\[Star]", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[{0, 0}, 0.05], Black, Disk[basis1[{0, 0}], 0.1], Disk[basis2[{0, 0}], 0.1]}, PlotRange -> 1.1*{{-1, 1}, {-1, 1}} , ImageSize -> 50]
      , Text[Style["=", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[#, 0.05] & /@ bravais, Black, Disk[basis1[#], 0.1] & /@ bravais,  Disk[basis2[#], 0.1] & /@ bravais}, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick] , ImageSize -> 300]
      }}]
   , {t, 0, 1, 0.05}];
frames4 = Table[
   a1 = {1, 0}; a2 = {0, 1};
   basis1[point_] := point - {0.25, 0}; 
   basis2[point_] := point + {0.25, 0};  
   basis3[point_] := point + {0.25 - 0.25*sinstep[t], 0.5*sinstep[t]};
   bravais = Flatten[Table[n1 a1 + n2 a2, {n1, -20, 20}, {n2, -20, 20}], 1];
   Grid[{{"Bravais Lattice", "", "Basis", "", "Crystal"}, {
      Graphics[{Gray, Table[Disk[n1 a1 + n2 a2, 0.05], {n1, -20, 20}, {n2, -20, 20}]}, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick], ImageSize -> 300 ]
      , Text[Style["\[Star]", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[{0, 0}, 0.05], Black, Disk[basis1[{0, 0}], 0.1], Disk[basis2[{0, 0}], 0.1], Disk[basis3[{0, 0}], 0.1]}, PlotRange -> 1.1*{{-1, 1}, {-1, 1}} , ImageSize -> 50]
      , Text[Style["=", Bold, Black, FontSize -> 20]],
      Graphics[{Gray, Disk[#, 0.05] & /@ bravais, Black, Disk[basis1[#], 0.1] & /@ bravais, Disk[basis2[#], 0.1] & /@ bravais, Disk[basis3[#], 0.1] & /@ bravais}, PlotRange -> {{-6, 6}, {-6, 6}}, Frame -> True, FrameTicks -> None, FrameStyle -> Directive[Black, Thick] , ImageSize -> 300]
      }}]
   , {t, 0, 1, 0.05}];
ListAnimate[Join[frames1, frames2, frames3, Reverse[frames2], frames4, Reverse[frames4], Reverse[frames1]]]

Licensing

[edit]
I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current13:54, 24 May 2024Thumbnail for version as of 13:54, 24 May 2024923 × 431 (3.47 MB)Berto (talk | contribs)Uploaded own work with UploadWizard

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