File:Nearly free electron model Brilouin zone.webm
Nearly_free_electron_model_Brilouin_zone.webm (WebM audio/video file, VP9, length 17 s, 710 × 292 pixels, 869 kbps overall, file size: 1.76 MB)
Captions
Summary[edit]
DescriptionNearly free electron model Brilouin zone.webm |
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 |
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]
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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
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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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