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Spontaneous Magnetization

physicssolid state physicsmagnetismcetztikz

Spontaneous magnetization mm in an Ising ferromagnet appears below the critical temperature TcT_c. The derivative of mm diverges in the limit T→Tc−T \to T_c^-.


Spontaneous Magnetization

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  Code

  LaTeX

spontaneous-magnetization.tex (30 lines)

\documentclass{standalone}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}
\begin{tikzpicture}[
    declare function={
        % arcsinh
        arcsinh(\x) = ln(\x + sqrt(\x^2+1));
        % magnetization
        m(\x) = and(\x>=0,\x<1) * (1 - (sinh(arcsinh(1)/x))^(-4))^(1/8) + or(\x<0,\x>=1) * (0);
      }]

  \begin{axis}[
      xlabel = {$T/T_c$},
      ylabel = {$m(0,T)$},
      smooth,thick,
      domain=0:1.8,
      ymax=1.6,
      axis lines = center,
      every tick/.style = {thick}]

    \addplot[color=blue,very thick,samples=300]{m(x)};
    \addplot[color=black,dashed,domain=0:1]{x};

  \end{axis}
\end{tikzpicture}
\end{document}

  Typst

spontaneous-magnetization.typ (53 lines)

#import "@preview/cetz:0.3.2": canvas, draw
#import "@preview/cetz-plot:0.1.1": plot

#set page(width: auto, height: auto, margin: 8pt)

#let arcsinh(x) = calc.ln(x + calc.sqrt(calc.pow(x, 2) + 1))

// Magnetization function
#let magnetization(x) = {
  if x >= 0 and x < 1 {
    calc.pow(1 - calc.pow(calc.sinh(arcsinh(1) / x), -4), 1 / 8)
  } else {
    0
  }
}

#canvas({
  draw.set-style(
    axes: (
      y: (mark: (end: "stealth", fill: black), label: (anchor: "north-west", offset: -0.2)),
      x: (mark: (end: "stealth", fill: black), label: (anchor: "south-east", offset: -0.2)),
    ),
  )

  plot.plot(
    size: (8, 5),
    x-label: $T / T_c$,
    y-label: $m(0,T)$,
    y-max: 1.5,
    x-tick-step: 0.5,
    y-tick-step: 0.5,
    axis-style: "left",
    x-grid: true,
    y-grid: true,
    {
      // Main magnetization curve
      plot.add(
        style: (stroke: blue + 1.5pt),
        domain: (0.01, 2), // Avoid x=0 due to division
        samples: 300,
        magnetization,
      )

      // Dashed line y=x from 0 to 1
      plot.add(
        style: (stroke: (dash: "dashed", paint: black, thickness: 1pt)),
        domain: (0, 1),
        x => x,
      )
    },
  )
})