 Solution

A further derivative gives the magnetic susceptibility within a selected pure thermodynamic phase:Assuming a finite nonzero zero-field amplitude on the branch considered, comparison with $\chi (t,0)\asymp |t{|}^{-\gamma }$ givesTogether with part (a), this proves the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the Widom scaling relation $\gamma =\beta (\delta -1)$. Below the transition, use the one-sided response of an ordered phase; differentiating through the cusp of the equilibrium free energy at $h=0$ would include phase switching rather than this critical response.