This introduction shows how useful the concept in the Rutherford scattering formula has been in the study of physics.
the massive target nucleus.
The initial discovery was made by Hans Geiger and Ernest Marsden in 1909 when they performed the gold foil experiment in collaboration with Rutherford, in which they fired a beam of alpha particles (helium nuclei) at foils of gold leaf only a few atoms thick. At the time of the experiment, the atom was thought to be analogous to a plum pudding (as proposed by J. J. Thomson), with the negatively-charged electrons (the plums) studded throughout a positive spherical matrix (the pudding). If the plum-pudding model were correct, the positive "pudding", being more spread out than in the correct model of a concentrated nucleus, would not be able to exert such large coulombic forces, and the alpha particles should only be deflected by small angles as they pass through.
In the progress of high energy physics experiments and their interpretation, Rutherford scattering has played the same role. The reason the nucleus of atoms was modeled with protons and neutrons was because in the experiments one could see that scattering particles off nuclei the atom had many hard cores,instead of the model at the time of spread out matter.
With higher energies and experiments of scattering on a nucleus , a single nucleus, again hard cores were found and the scattering could be described with the Rutherfored forumula.
This history is clarifying :
Drawing on Rutherford's groundbreaking experiments in the early years of the 20th century, ideas for detecting quarks were formulated. Rutherford had proven that atoms had a small, massive, charged nucleus at their centre by firing alpha particles at atoms of gold. Most had gone through with little or no deviation, but a few were deflected through large angles or came right back. This suggested that atoms had internal structure and a lot of empty space.
then one tried to see the internals of a proton scattering an electron at high energy:
The collision absorbs some kinetic energy, and as such it is inelastic. This is a contrast to Rutherford scattering, which is elastic: no loss of kinetic energy. The electron emerges from the nucleus, and its trajectory and velocity can be detected.
Analysis of the results led to the following conclusions:
The hadrons do have internal structure.
So at the level of quarks, it is only qualitatively that electrons scattering on quarks follow the Rutherford scattering formula, but the experiments showed that there were hard cores to the proton, the quarks expected from the quark model.
Quark- quark scattering can only happen with theoretical formulas, as there are no free quarks, and you are right, quantum mechanics in the form of QCD has a great role to describe and fit the data.
Quark-quark interactions are very complicated at the energies of the experiments we carry out now, and cannot be described by the formula of Rutherford scattering. Proton proton collisions at the LHC use the quark model and QCD to fit the data, and it is not a simple formula (for example using a single quark quark scattering formula). The plot can only be considered as an approximation to the QCD formulas used in fitting the data.
