Magnetocaloric effect
for a af spin dimer

©  Apl. Prof. Dr. Jürgen Schnack, University of Osnbrueck, Germany

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<< Graphics`Legend` 
<< Graphics`ContourPlot3D`

Constants

J = -1    ;

    gmuBDurchkB = 1;

Energy eigenvalues without field

E0 = {3*J/2, -J/2, -J/2, -J/2}

Magnetic quantum numbers

M = {0, 1, 0, -1}

Energy eigenvalues with field

EB[B_] = E0 + M*gmuBDurchkB*B

Partition function

Z[T_, B_] = 
      Sum[(E^(-EB[B][[m]]/T)), {m, 1, 
          4}]

Free energy

F[T_, B_] = -T*Log[Z[T, B]]

Entropy

S[T_, B_] = -Simplify[(∂_T) F[T, B]]

Graphical representation of the entropy for a paramagnet

\(\(PISENPara = 
        ContourPlot[
          T/B, {B, 0.01, 3.0}, {T, 0.01, 
            2.0}, 
PlotPoints -> 200, 
          ColorFunction -> 
            Hue, 
ContourShading -> 
            False, 
Contours -> {0.1, 0.65, 3}, 
          ContourStyle -> {Thickness[0.005], 
              RGBColor[0, 0, 
                1]}, 
AxesLabel -> {"B", "T"}, 
PlotLabel -> "T-B isentropes", 
TextStyle -> {FontFamily -> Helvetica, 
              FontSize -> 18}];\)\)

Graphical representation of the entropy for the dimer

\(\(PISEN = 
        ContourPlot[
          S[T, B], {B, 0, 3.0}, {T, 0.01, 
            2.0}, 
PlotPoints -> 200, 
          ColorFunction -> 
            Hue, 
ContourShading -> 
            False, 
Contours -> 20, 
          ContourStyle -> {Thickness[0.005], 
              RGBColor[1, 0, 
                0]}, 
AxesLabel -> {"B", "T"}, 
PlotLabel -> "T-B isentropes", 
TextStyle -> {FontFamily -> Helvetica, 
              FontSize -> 18}];\)\)

Graphical representation of the entropy for the dimer 
with highlighted regions

K1 = 
        Graphics[{Thickness[0.01], Hue[0.3], 
            Circle[{0.6, 1.85}, 0.15]}];
K2 = 
        Graphics[{Thickness[0.01], Hue[0.3], 
            Circle[{2.6, 1.7}, 0.15]}];
K3 = 
        Graphics[{Thickness[0.01], Hue[0.3], 
            Circle[{2.20, 0.22}, 0.15]}];
K4 = 
        Graphics[{Thickness[0.01], Hue[0.3], 
            Circle[{1.85, 0.22}, 0.15]}];
T1 = Graphics[Text[a, {0.85, 1.75}]];
T2 = Graphics[Text[b, {2.35, 1.7}]];
T3 = Graphics[Text[c, {2.20, 0.430}]];
T6 = Graphics[Text[d, {1.85, 0.430}]];
T4 = Graphics[Text[T, {0.051, 1.95}]];
T5 = Graphics[Text[B, {2.90, 0.03}]];
PISENTR = 
        Show[PISEN, PISENPara, 
K1, K2, K3, K4, T1, T2, T3, 
          T4, T5, T6
];
Export["dimer-isentrops.eps", PISENTR];

Graphical representation of the entropy for the dimer (color coded)

\(\(\(DimerS = 
        Plot3D[S[T, B], {B, 0, 3.0}, {T, 0.01, 
            2.0}, 
PlotPoints -> 400, 
          PlotRange -> {0, 
              1.5}, 
ColorFunction -> \((Hue[
                  0.7 - 0.9*#] &)\), 
Mesh -> 
            False, 
AxesLabel -> {"B", "T", "S"}, 
PlotLabel -> "S as function of T and B", 
TextStyle -> {FontFamily -> Helvetica, 
              FontSize -> 18}];\)\(
\)
    \)\)
Export["dimer-S.eps", DimerS];

Internal energy U

U[T_, B_] = (T^2)*Simplify[(∂_T) Log[Z[T, B]]]

Heat capacity

SH[T_, B_] = Simplify[(∂_T) U[T, B]]

Graphical representation of the heat capacity

\(\(\(DimerC = 
        Plot3D[SH[T, B], {B, 0, 3.0}, {T, 0.01, 
            2.0}, 
PlotPoints -> 200, 
          PlotRange -> {0, 
              1.2}, 
ColorFunction -> \((Hue[
                  0.7 - 0.9*#] &)\), 
Mesh -> 
            False, 
AxesLabel -> {"B", "T", "C"}, 
PlotLabel -> "C as function of T and B", 
TextStyle -> {FontFamily -> Helvetica, 
              FontSize -> 18}];\)\(
\)
    \)\)
Export["dimer-C.eps", DimerC];

Absolute adiabatic temperature change

AMCE[T_, B_] = -T*
        Simplify[(∂_B) S[T, B]]/SH[T, B]

Relative adiabatic temperature change
(compared to a paramagnet)

RMCE[T_, B_] = AMCE[T, B]*B/T

Graphical representation

\(\(\(DimerRMCE = 
        Plot3D[RMCE[T, B], {B, 0, 3.0}, {T, 0.01, 
            2.0}, 
PlotPoints -> 200, 
          PlotRange -> {\(-15\), 
              15}, 
ColorFunction -> \((Hue[
                  0.7 - 0.7*#] &)\), 
Mesh -> 
            False, 
AxesLabel -> {"B", "T", " "}, 
PlotLabel -> "relative magnetocaloric effect", 
TextStyle -> {FontFamily -> 
                Helvetica, FontSize -> 18}];\)\(
\)
    \)\)
Export["dimer-RMCE.eps", DimerRMCE];