Magnetocaloric effect for an af dimer of two spins s=1/2, Heisenberg model

©  Prof. Dr. Jürgen Schnack, Universität Bielefeld, 24. 09. 2013

Constants

J=-1  (* Kelvin (Subscript[k, B]) *);
gmuBDurchkB=1; (* normaly g 0.6717 K/T, but no absorbed in B *)

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}]

Gibbs Free Energy G/k

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

Entropy S/k

S[T_,B_]=-Simplify[(∂_T) G[T,B]]

Graphical representation of the entropy

PISENPara=ContourPlot[(T/B),{B,0.01`,3.`},{T,0.01`,2.`},PlotPoints->200,ContourShading->False,Contours->{0.1`,0.65`,3},
ContourStyle->{{Thick,Red},{Thick,Green},{Thick,Blue},{Thick,Cyan},{Thick,Magenta}},
AxesLabel->{"B","T"},PlotLabel->"T-B isentropes paramagnet",BaseStyle->{FontFamily->"Helvetica",FontSize->18}]

PISEN=ContourPlot[S[T,B],{B,0,3.`},{T,0.01`,2.`},PlotPoints->200,ColorFunction->Hue,ContourShading->False,Contours->20,ContourStyle->{Thickness[0.005`],RGBColor[1,0,0]},FrameLabel->{"B","T"},PlotLabel->"T-B isentropes",BaseStyle->{FontFamily->"Helvetica",FontSize->18}]

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[{2.2`,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,{2.2`,0.43`}]];
T6=Graphics[Text[d,{1.85`,0.43`}]];
PISENTR=Show[PISEN,PISENPara,K1,K2,K3,K4,T1,T2,T3,T6]
Export["dimer-isentrops.eps",PISENTR];

DimerS=Plot3D[S[T,B],{B,0,3.`},{T,0.01`,2.`},PlotPoints->100,
PlotRange->{0,1.5`},Mesh->False,
ColorFunction->Function[{x,y,z},Hue[.9*(1.5-z)]],
AxesLabel->{"B","T","S"},PlotLabel->"S as function of T and B",BaseStyle->{FontFamily->"Helvetica",FontSize->18}]

Tasks

- f coupling
- generalization to more than two spins