File:Nearly free electron model Brilouin zone.webm

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Nearly_free_electron_model_Brilouin_zone.webm (file size: 1.76 MB, MIME type: video/webm)

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Summary

Description
English: Dispersion relation for the "nearly free electron model" (in 2D) as a function of the underlying crystalline structure.
Date
Source https://twitter.com/j_bertolotti/status/1496091333508972544
Author Jacopo Bertolotti
Permission
(Reusing this file)		 https://twitter.com/j_bertolotti/status/1030470604418428929
WEBM development
InfoField
 This diagram was created with Mathematica.

Mathematica 13.0 code

v3 = {0, 0, 1};
sinstep[t_] := Sin[\[Pi]/2 t]^2;
frames1 = Table[
   v1 = {0.25 + sinstep[t], 0}; 
   v2 = {0, 0.25 + sinstep[t]}; (*Start at 0.25 stop at 1.25*)
   
   Print["v1=", v1, "  v2=", v2];
   b1 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[Join[v2, {0}], 
        v3])[[1 ;; 2]];
   b2 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[v3, 
        Join[v1, {0}]])[[1 ;; 2]];
   tmp = VoronoiMesh@
     Flatten[Table[n b1 + m b2, {n, -3, 3}, {m, -3, 3}], 1];
   pts = MeshCoordinates[tmp];
   polys = MeshCells[tmp, 2];
   
   firstBrillouin = 
    Position[
      Table[Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
        Dimensions[polys][[1]]}], 
      Min@Table[
        Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
         Dimensions[polys][[1]]}] ][[1, 1]];
   
   Grid[{{
      Graphics[{
        Table[Point[n v1 + m v2], {n, -100, 100}, {m, -100, 100}]
        }, PlotRange -> {{-10, 10}, {-10, 10}}, 
       PlotLabel -> Framed["Crystal", Background -> White], 
       LabelStyle -> {Black, Bold}]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, -10, 10}, {y, -10, 10}, 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(Extended zone scheme)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, y} \[Element] 
         Polygon[pts[[polys[[firstBrillouin, 1]] ]] ], 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(First Brillouin zone)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      }}]
   , {t, 0, 1, 1/30}];
Print["Done: 1"]

frames2 = Table[
   v1 = {1.25 - 0.25*sinstep[t], 0}; 
   v2 = {0, 1.25 + 0.25*sinstep[t]}; (*Start at 0.25 stop at 1.25*)
  
    Print["v1=", v1, "  v2=", v2];
   b1 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[Join[v2, {0}], 
        v3])[[1 ;; 2]];
   b2 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[v3, 
        Join[v1, {0}]])[[1 ;; 2]];
   tmp = VoronoiMesh@
     Flatten[Table[n b1 + m b2, {n, -3, 3}, {m, -3, 3}], 1];
   pts = MeshCoordinates[tmp];
   polys = MeshCells[tmp, 2];
   
   firstBrillouin = 
    Position[
      Table[Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
        Dimensions[polys][[1]]}], 
      Min@Table[
        Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
         Dimensions[polys][[1]]}] ][[1, 1]];
   
   Grid[{{
      Graphics[{
        Table[Point[n v1 + m v2], {n, -100, 100}, {m, -100, 100}]
        }, PlotRange -> {{-10, 10}, {-10, 10}}, 
       PlotLabel -> Framed["Crystal", Background -> White], 
       LabelStyle -> {Black, Bold}]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, -10, 10}, {y, -10, 10}, 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(Extended zone scheme)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, y} \[Element] 
         Polygon[pts[[polys[[firstBrillouin, 1]] ]] ], 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(First Brillouin zone)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      }}]
   , {t, 0, 1, 1/20}];
Print["Done: 2"]

frames3 = Table[
   v1 = {1.25, 1.25/2*sinstep[t]}; 
   v2 = {0, 1.25}; (*Start at 0.25 stop at 1.25*)
   
   Print["v1=", v1, "  v2=", v2];
   b1 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[Join[v2, {0}], 
        v3])[[1 ;; 2]];
   b2 = ((2 \[Pi])/
       v3 . Cross[Join[v1, {0}], Join[v2, {0}] ] Cross[v3, 
        Join[v1, {0}]])[[1 ;; 2]];
   tmp = VoronoiMesh@
     Flatten[Table[n b1 + m b2, {n, -3, 3}, {m, -3, 3}], 1];
   pts = MeshCoordinates[tmp];
   polys = MeshCells[tmp, 2];
   
   firstBrillouin = 
    Position[
      Table[Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
        Dimensions[polys][[1]]}], 
      Min@Table[
        Total[Norm /@ pts[[polys[[j, 1 ]] ]] ], {j, 1, 
         Dimensions[polys][[1]]}] ][[1, 1]];
   
   Grid[{{
      Graphics[{
        Table[Point[n v1 + m v2], {n, -100, 100}, {m, -100, 100}]
        }, PlotRange -> {{-10, 10}, {-10, 10}}, 
       PlotLabel -> Framed["Crystal", Background -> White], 
       LabelStyle -> {Black, Bold}]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, -10, 10}, {y, -10, 10}, 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(Extended zone scheme)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      ,
      Show[
       Plot3D[
        Table[Norm[{x, y} - n b1 - m b2 ]^2, {n, -3, 3}, {m, -3, 
          3}], {x, y} \[Element] 
         Polygon[pts[[polys[[firstBrillouin, 1]] ]] ], 
        PlotRange -> {{-10, 10}, {-10, 10}, {0, 20}}, 
        ClippingStyle -> None, PlotPoints -> 50, Mesh -> None, 
        PlotStyle -> Directive[Orange, Opacity[0.75]], 
        Lighting -> "Neutral", Boxed -> False, Axes -> False]
       ,
       Graphics3D[{Thick, Black, 
         Table[Line@
           Join[Join[#, {0}] & /@ 
             pts[[polys[[j, 1]] ]], {Join[
              pts[[polys[[j, 1]] ]][[1]], {0}]}], {j, 1, 
           Dimensions[polys][[1]]}]
         }], 
       PlotLabel -> 
        Framed["Dispersion relation\n(First Brillouin zone)", 
         Background -> White], LabelStyle -> {Black, Bold}
       ]
      }}]
   , {t, 0, 1, 1/20}];
Print["Done: 3"]

Export["/home/jb601/Documents/Physicsfactlet/BrillouinZone.mp4", 
 Join[
  Table[frames1[[1]], {5}],
  frames1,
  Table[frames1[[-1]], {5}],
  frames2,
  Table[frames2[[-1]], {5}],
  Reverse@frames2,
  Table[frames1[[-1]], {5}],
  frames3,
  Table[frames3[[-1]], {5}],
  Reverse@frames3,
  Table[frames1[[-1]], {5}],
  Reverse@frames1
  ]
 ]

Licensing

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.

Captions

Dispersion relation for the "nearly free electron model" (in 2D) as a function of the underlying crystalline structure.

inception

22 February 2022

media type

video/webm

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeDimensionsUserComment
current13:37, 23 February 2022 (1.76 MB)wikimediacommons>BertoUploaded own work with UploadWizard

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