The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter $x$ is defined such that the change in critical temperature from the "pure" or unrenormalized system is $\left|T_c\mathrm{}\right(x)-T_c^0|=\frac{x{T}_c}{f}$, where $f=0.6-0.9\mathrm{}$, then we conclude that the effective exponents ${\beta }_{\mathrm{fit}}\mathrm{}\left(x\right)\mathrm{}$ and ${\gamma }_{\mathrm{fit}}\mathrm{}\left(x\right)\mathrm{}$ which will be observed experimentally, vary roughly as ${\beta }_{\mathrm{fit}}\approx \beta +x\Delta {\beta }_X$ and ${\gamma }_{\mathrm{fit}}\simeq \gamma +\frac{1}{5}x\times (1+2\mathrm{}{x}^2)\Delta {\gamma }_X$. Here $\beta$ and $\gamma$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\Delta {\beta }_X={\beta }_X-\beta$ and $\Delta {\gamma }_X={\gamma }_X-\gamma$, in which ${\beta }_X=\frac{\beta }{(1-{\alpha }^{\prime })}$ and ${\gamma }_X=\frac{\gamma }{(1-\alpha )}$ are the fully renormalized exponents, while $\alpha$ and ${\alpha }^{\prime }$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\beta \mathrm{}\left(x\right)={\beta }_X$ and $\gamma \mathrm{}\left(x\right)={\gamma }_X$ for all $x>\mathrm{}0\mathrm{}$.] The renormalized specific heats are found to be sensitive to $x$ but their true renormalized behavior is not evident until $x\gtrsim 0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.

• Received 4 March 1970

DOI:https://doi.org/10.1103/PhysRevA.2.825