All field emission models are wrong, $\ldots$ but are any of them useful?

Field emission data are often represented on a Fowler--Nordheim plot but a new empirical equation has been recently proposed to better analyze experiments. Such an equation is based on approximations of the Murphy and Good model and predicts that a constant parameter $\kappa$, depending only on the work function of the emitter, can be extracted from the data. We compared this empirical equation with simulations of the Murphy and Good model in order to determine the range of validity of the approximations and the robustness of the relationship between $\kappa$ and the work function. We found that $\kappa$ is constant only over a limited range of electric fields and so depends significantly on the field enhancement factor. This result calls into question the usefulness of the new empirical equation.