Energy Dependence of Profile Functions in rm{pbar p} and rm pp Scattering

We construct analytical forms in the impact parameter $b$-space for the real and imaginary amplitudes describing elastic pp and $\rm p \bar p $ scattering. The amplitudes converted analytically to the momentum transfer $t$-space have magnitudes, slopes, curvatures, zeros, signs, obeying phenomenological and theoretical expectations, and describe with high precision all details of the data, in the full-t range, and for energies from 20 GeV to 7 TeV. The connection of forward with large-$|t|$ behavior allows precise determination of total cross-sections, slopes and other scattering parameters. We study the properties and the energy dependence of the $b$-space profile functions, observing that the real part has fundamental influence in the structure of $d\sigma/dt$ at intermediate and large $|t|$ values. We discuss the 540/546 GeV and 1.8/1.96 TeV data from CERN SPS and Fermilab TEVATRON and the 7 TeV results from TOTEM measurements at LHC, and investigate the extrapolation to 14 TeV and higher energies.