  We describe the main features of recent LHC data on elastic pp scattering
through a simple parametrization to the amplitude, inspired by a model proposed
by Barger and Phillips in 1973, comprising of two exponentials with a relative
phase. Despite its simplicity, this parameterization reproduces two essential
aspects of the elastic differential cross section, the well known precipitous
descent in the forward direction and a sharp `dip' structure. To include a
complete description of data sets near -t=0, we correct the original
parametrization. We examine two possibilities, the presence of the two-pion
threshold singularity or a multiplicative factor reflecting the proton form
factor. We find good descriptions of LHC7 and ISR data in either case. The form
factor model allows simple predictions for higher energies through asymptotic
theorems and asymptotic sum rules in impact parameter space. We present
predictions for this model at higher LHC energies, which can be used to test
whether asymptotia is reached. The black disk limit in this model is seen to be
reached only for energies near 10^6 TeV.
