File:Ferromagnet hysteresis.webm
Summary
| Description |
English: n a ferromagnet the equilibrium configuration is with all magnetic moments aligned with each other. If we want to flip them, we need to flip all of them at the same time, which requires a stronger field than if the moments were independent. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1671865946632253446 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 13.1 code
\[Gamma] = 1; \[Lambda] = 0.1; dt = 0.05; J = 0.2;
steps = 6000;
nx = 11; ny = 11; n = nx*ny;
adjacency = Normal@AdjacencyMatrix[TorusGraph[{nx, ny}] ];
M0 = (#/Norm[#] &) /@
Table[{RandomReal[{-0.1, 0.1}], RandomReal[{-0.1, 0.1}], 1}, {n}];
M = M0;(* M[[Floor[nx/2]*ny+Floor[ny/2] +1]]={1,1,0};*)
Hext[j_] := 2 {0, 0, (1 - 2 j/steps)};
evo = Reap[For[j = 1, j <= steps, j++,
M = (#/Norm[#] &) /@ M;
Heff = J*adjacency . M + Threaded[Hext[j] ]; (****)
M =
Table[M[[j]] +
dt*(-\[Gamma] Cross[M[[j]], Heff[[j]] ] - \[Lambda] Cross[
M[[j]], Cross[M[[j]], Heff[[j]] ]]), {j, 1,
n}] + (RandomReal[{-0.01, 0.01}, {n, 3}]);
M = (#/Norm[#] &) /@ M;
Sow[M];
]][[2, 1]];
Hext2[j_] := 2 {0, 0, (-1 + 2 j/steps)}(*8{0,0,(1-2j/steps)}*);
evo2 = Reap[For[j = 1, j <= steps, j++,
M = (#/Norm[#] &) /@ M;
Heff = J*adjacency . M + Threaded[Hext2[j] ]; (****)
M =
Table[M[[j]] +
dt*(-\[Gamma] Cross[M[[j]], Heff[[j]] ] - \[Lambda] Cross[
M[[j]], Cross[M[[j]], Heff[[j]] ]]), {j, 1,
n}] + (RandomReal[{-0.01, 0.01}, {n, 3}]);
M = (#/Norm[#] &) /@ M;
Sow[M];
]][[2, 1]];
frames1 = Table[
Grid[{{Graphics3D[{
Black,
Table[
Sphere[{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1, 0},
0.08], {j, 1, n}],
Table[
Sphere[{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + \[Alpha]*evo[[k, j]], 0.05], {j, 1, n}],
Thick,
Flatten@Table[
Line[{{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0}, {Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + \[Alpha]*evo[[k, j]]} ], {j, 1, n}],
Sphere[{-1.5, Sqrt[n]/2, 0}, 0.1], Arrowheads[.025],
Arrow[{{-1.5, Sqrt[n]/2, 0}, {-1.5, Sqrt[n]/2,
Hext[k][[3]]}}],
Text[Style["\!\(\*SubscriptBox[\(H\), \(ext\)]\)", Black,
Bold, FontSize -> 20], {-1.5, Sqrt[n]/2 + 1.5, 0}],
Red,
Table[
Line[({Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + # &) /@ (\[Alpha]*
evo[[Max[1, k - 50] ;; k, j]]) ], {j, 1, n}],
Opacity[0.5], Gray,
Cuboid[{0, 0, 0}, {Sqrt[n] + 1, Sqrt[n] + 1, -0.1}]
},
PlotRange -> {{-2, Sqrt[n] + 1}, {-1, Sqrt[n] + 1}, {-2, 2}},
ImageSize -> 500, Lighting -> "Neutral", Boxed -> False]
,
ListPlot[Table[{Hext[j][[3]], Mean[evo[[j]]][[3]]}, {j, 1, k}],
PlotRange -> {{-2, 2}, {-1.2, 1.2}}, Joined -> True,
PlotStyle -> {Thick, Black}, Ticks -> None,
AxesLabel -> {"\!\(\*SubscriptBox[\(H\), \(ext\)]\)",
"\!\(\*SubscriptBox[\(M\), \(z\)]\)"},
LabelStyle -> {Bold, Black}, ImageSize -> Medium]
}}]
, {k, 1, 6000, 10}];
frames2 = Table[
Grid[{{Graphics3D[{
Black,
Table[
Sphere[{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1, 0},
0.08], {j, 1, n}],
Table[
Sphere[{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + \[Alpha]*evo2[[k, j]], 0.05], {j, 1, n}],
Thick,
Flatten@Table[
Line[{{Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0}, {Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + \[Alpha]*evo2[[k, j]]} ], {j, 1, n}],
Sphere[{-1.5, Sqrt[n]/2, 0}, 0.1], Arrowheads[.025],
Arrow[{{-1.5, Sqrt[n]/2, 0}, {-1.5, Sqrt[n]/2,
Hext2[k][[3]]}}],
Text[Style["\!\(\*SubscriptBox[\(H\), \(ext\)]\)", Black,
Bold, FontSize -> 20], {-1.5, Sqrt[n]/2 + 1.5, 0}],
Red,
Table[
Line[({Mod[j, Sqrt[n], 1], Quotient[j, Sqrt[n], 1] + 1,
0} + # &) /@ (\[Alpha]*
evo2[[Max[1, k - 50] ;; k, j]]) ], {j, 1, n}],
Opacity[0.5], Gray,
Cuboid[{0, 0, 0}, {Sqrt[n] + 1, Sqrt[n] + 1, -0.1}]
},
PlotRange -> {{-2, Sqrt[n] + 1}, {-1, Sqrt[n] + 1}, {-2, 2}},
ImageSize -> 500, Lighting -> "Neutral", Boxed -> False]
,
ListPlot[{Table[{Hext[j][[3]], Mean[evo[[j]]][[3]]}, {j, 1,
steps}],
Table[{Hext2[j][[3]], Mean[evo2[[j]]][[3]]}, {j, 1, k}]},
PlotRange -> {{-2, 2}, {-1.2, 1.2}}, Joined -> True,
PlotStyle -> Directive[Thick, Black], Ticks -> None,
AxesLabel -> {"\!\(\*SubscriptBox[\(H\), \(ext\)]\)",
"\!\(\*SubscriptBox[\(M\), \(z\)]\)"},
LabelStyle -> {Bold, Black}, ImageSize -> Medium]
}}]
, {k, 1, 6000, 10}];
ListAnimate[Join[frames1, frames2]]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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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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