Examination of the phenomenological scaling functions for critical scattering

In the scaling limit $k\to 0$, $\xi \to \infty$ such that $y=k\xi$ is fixed, the $k$-dependent susceptibility $\chi (\overrightarrow{\mathrm{k}},T)$ can, according to the scaling hypotheses of Kadanoff and Fisher, be written as $\chi (\overrightarrow{\mathrm{k}},T)={\xi }^{\frac{\gamma }{\nu }}X{}_{\pm }{}^{}{}_{}{}^{}\mathrm{}\left(y\right)\mathrm{}+o\left({\xi }^{\frac{\gamma }{\nu }}\right)$. We exactly compute the scale functions $X{}_{\pm }{}^{}{}_{}{}^{}\mathrm{}\left(y\right)\mathrm{}$ for the two-dimensional Ising model in zero magnetic field. We then compare the various phenomenological scale functions (Ornstein-Zernike pole approximate, Fisher approximate, Fisher-Burford approximate, Tarko-Fisher approximates, etc.) with the exact $X{}_{\pm }{}^{}{}_{}{}^{}\mathrm{}\left(y\right)\mathrm{}$ for the two-dimensional Ising model. This comparison provides insight into those regions of $y=k\xi$ where these phenomenological scale functions are applicable. Such insight is important since the region of experimentally accessible $y$ is rather limited. We then use these results to examine the method of data analysis used in critical scattering experiments. We conclude that no experiment to date unambiguously and directly establishes that the critical exponent $\eta$ is greater than zero.