Problem 24.6
\(The\ parameter, σ\_A, 
    is\ determined\ by\ the\ self - 
        consistent\ \(equation : \ \ 
ArcTanh[\(Q + 1\)\/2] + ArcTanh[\(3  Q - 1\)\/2]\)\  = \ α\ Q, 
with\ \ Q = \(\(-1\) - 2  σ\_A\ \ and\ \ α\  = 
        8  β | 
          J | \(\(.\)\(\ \)\(
\)\(The\)\)\ \((relative\ )\)\ order\ parameter\ is\ Q\), \ 
Q\  = \ 
      1\ when\ the\ system\ is\ completely\ ordered\ \((σ\_A\  = \ \(-1\))\)\)
Q  =  
      0 when the system is completely disordered 
\(\(\((σ\_A\  = \ \(3\/4\ \((\(-1\))\) + \(1\/4\) \((\(+1\))\) = \ \(-\(1\/2\)\)\))\)\(.\)\)\)

f[Q_] = ArcTanh[((Q + 1)/2)] + ArcTanh[((3  Q - 1)/2)];   g1 = 
      Plot[f[Q], {Q, 0.99, -0.33}]

FindRoot[f[Q ] - Q f'[Q] == 0, {Q, 0.5}]

(Subscript[Q, c]) = Q  /. %; α = f'[(Subscript[Q, c])]

g2 = Plot[{α  Q, f[Q]}, {Q, 0.8, -0.3}]

\(The\ order\ parameter\ Q\ as\ a\ function\ of\ temperature\ t = \(1/α\  = \ \(\(k\_B\) T/8 | 
            J | \(\(.\)\(
\)\(The\)\)\ transition\ is\ of\ first\ order\ and\ \ happens\ at\ 1/α\  = \(1/α\_c = 
            0.4143\)\)\), 
    where\ Q = \(Q\_c = \(0.3455\ as\ determined\ by\ the\ condition\ f[
              Q\_c] = Q\_c\ \(f'\)[Q\_c]\)\), \ \(f'\)[
        Q\_c] = \(\(α\_c\)\(.\)\)\)

ListPlot[m, PlotJoined -> True, 
      AxesLabel -> {"Temperature (t)", "Order parameter (Q)"}, 
      PlotRange -> {0, 1.1}, AspectRatio -> 1, 
      TextStyle -> {FontFamily -> "Times", 
          FontSize -> 13}, PlotStyle -> {Thickness[0.011]}]

Clear[g, α]

α = 1/0.405; 
    FindRoot[f[Q ] - Q α == 0, {Q, 0.9}]

m = {{1/f'[(Subscript[Q, c])], (Subscript[Q, c])}, {0.413, 0.410562}, {0.412, 
            0.432453}, {0.41, 0.46458}, {0.405, 0.5201560}, {0.4, 
            0.56113}, {0.39, 0.623796}, {0.38, 0.672618}, {0.37, 
            0.713134}, {0.36, 0.747856}, {0.35, 0.77818}, {0.34, 
            0.804982}, {0.33, 0.828856}, {0.32, 0.850226}, {0.31, 
            0.86941}, {0.3, 0.886656}, {0.29, 0.902156}, {0.28, 
            0.916074}, {0.27, 0.928538}, {0.26, 0.939666}, {0.24, 
            0.95829}, {0.22, 0.972616}, {0.2, 0.98322}, {0.18, 
            0.99064}, {0.16, 0.995423}, {0.14, 0.998152}, {0, 
            1}};