Bethe phase variation due to a non-exponential nuclear amplitude and the possibility of using a $t$-dependent phase to determine the $\rho$-parameter from elastic scattering data

We evaluate the possible deviation (from the conventional Cahn's result) of the phase between the one-photon-exchange and the `nuclear' high energy $pp$ scattering amplitudes in a small $t\to 0$ region caused by a more complicated (not just $exp(Bt)$) behaviour of the nuclear amplitude. Furthermore we look at the possible role of the $t$-dependence of the $\rho(t) \equiv$ Real/Imaginary amplitude ratio. It turns out that both effects are rather small - much smaller than to have any influence on the experimental accuracy of $\rho(t=0)$ extracted from the elastic proton-proton scattering data.