File:Nearly free electron model Brilouin zone.webm

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English

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

Summary[edit]

Description	English: Dispersion relation for the "nearly free electron model" (in 2D) as a function of the underlying crystalline structure.
Date	22 February 2022
Source	https://twitter.com/j_bertolotti/status/1496091333508972544
Author	Jacopo Bertolotti
Permission (Reusing this file)	https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 13.0 code[edit]

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[edit]

I, the copyright holder of this work, hereby publish it under the following license:

	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. CC0falsefalse

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	12:37, 23 February 2022		17 s, 710 × 292 (1.76 MB)	Berto (talk \| contribs)	Uploaded own work with UploadWizard

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Format	Bitrate	Download	Status	Encode time
VP9 360P	436 kbps		Completed 12:38, 23 February 2022	10 s
Streaming 360p (VP9)		Not ready	Unknown status
VP9 240P	206 kbps		Completed 12:37, 23 February 2022	8.0 s
Streaming 240p (VP9)	206 kbps		Completed 21:27, 5 February 2024	1.0 s
WebM 360P	336 kbps		Completed 12:38, 23 February 2022	4.0 s
Streaming 144p (MJPEG)	834 kbps		Completed 17:40, 15 November 2023	1.0 s

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User comments	Created with the Wolfram Language : www.wolfram.com
Software used	Lavf58.76.100 Lavc58.134.100 libvpx-vp9

File:Nearly free electron model Brilouin zone.webm

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Mathematica 13.0 code[edit]

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crystal structure

dispersion relation

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inception

22 February 2022

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File:Nearly free electron model Brilouin zone.webm

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Mathematica 13.0 code[edit]

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