In about 1985, while considering a banal problem of scattering form atoms, I occasionally derived a positive charge (“small” or a “second”) atomic form-factor
describing effects of electrostatic interaction of a charged projectile with a point-like (structureless) atomic nucleus in atom :
To my surprise, it was unknown and it was absent in textbooks despite easiness of its derivation and a transparent physical sense. Namely, for elastic scattering describes the positive charge distribution in atom (not within the nucleus) and for inelastic processes the form-factors describe the atom excitation amplitudes due to transmitting an essential momentum q to the atomic nucleus while scattering (exciting atom due to pushing its nucleus).
In Fig. 1 one can see cross-sections (cuts) of the atomic (Hydrogen) wave functions squared describing the relative electron-nucleus motion in the Hydrogen atom or a Hydrogen-like ion. (A nice applet to visualize and rotate with your mouse 3D and 2D images of Hydrogen configurations is here.) Strictly speaking, they are the relative distance probabilities although they are often erroneously called “negative charge clouds”:
Fig. 1. Probability density 2D plots (electron-nucleus relative distance probability).
I say “erroneously” because in absolute coordinates quite similar, “positive charge clouds”, exist at much shorter distances due to nucleus motion around the atomic center of inertia (CI). The nucleus does not stay at the atomic CI, but moves around it. So the charge density pictures “seen” by a fast projectile include the positive clouds too. This fact is contained in formulas (1-3), both charge clouds being expressed via the same atomic wave function ! If we make a zoom in, we will see a picture similar to Fig. 1 since
Fig. 2 represents qualitatively such a picture for a particular state of a Hydrogen (or a Hydrogen-like ion).
Fig. 2. Qualitative image of atomic charge density, 2D plot of for the state |3,2,2>.
Two dots in the middle of Fig. 2 are “positive charge clouds”. In other words, Fig. 1, without scale indicated in it, describes the negative charge density 2D plots at the “atomic distances” equally well as it does the positive charge density at much shorter distances. This is the true, elastic (non destructive) physics of short distances. Unexpected?
Another beautiful qualitative picture, this time for the state |4,3,1>:
I would like to underline that the nucleus bound in atom is not seen as a point-like in elastic processes. Its charge and probability are quantum-mechanically smeared. And the most surprising thing about it is the smear size dependence on the electron configuration. The farther electron “orbit”, the larger the positive cloud. It looks counter-intuitively first, but it is so! The atomic form-factors (2) and (3) are just the Fourier images of the cloud densities. The corresponding elastic cross section (1) can be written also via an effective projectile-atomic potential which is softer than the Coulomb “singularity” at short distances . By the way, in a solid state the positive charge “clouds” are rather large and comparable with the lattice step. So in elastic picture the positive charge in a solid is distributed as in a “plum pudding” model! And in molecules too – the positive charge clouds are very large, but nobody pictures them either!
I can even mention here the fact that each “sub-cloud” contains a fractional charge😉 , which is never observed as such outside of atom, separately, like quarks. And the cloud configuration is related to groups of symmetries. Thus, let me call those clouds with fractional charges “shmarks“. They are “visible” in elastic picture with respect to atomic excitations, but in a deeply inelastic (inclusive) picture with respect to soft photon emissions.
Smearing is always the case for bound states. But one should not confuse how the atomic electron “sees” the nucleus and how a fast projectile sees the atom (Fig. 2, 3). The atomic electron does not see the positive clouds pictured in Fig. 2 and Fig. 3 because the latter is a result of mutual (not independent) motion.
In scattering experiments, however, it is extremely difficult to observe these pictures because of difficulties in preparing the target atoms in certain |n,l,m> states and in selecting the true elastic events. Normally it is impossible to distinguish (resolve in energy) an elastically scattered projectile from inelastically scattered one (the energy loss difference is relatively small). So experimentally, when all projectiles scattered in a given solid angle dΩ are counted, one observes an inclusive cross section. It is the inclusive cross section that corresponds to the Rutherford formula! So the point-like “free” nucleus in the Rutherford cross section is an inclusive (rich) picture, a sum of an elastic and all inelastic atomic cross sections.
Similarly, the “free point-like electron” in QED is not really free, but is permanently coupled to the quantized electromagnetic field. So its charge is also smeared quantum-mechanically and its smear size is state-dependent. This explains what elastic (non destructive) physics occurs at short distances: there is no Coulomb singularity, as a matter of fact. But the sum of an elastic and all inelastic cross sections results in a Rutherford-like cross section as if the electron were point-like, free, and “long-handed” (i.e., with long-range Coulomb elastic potential for mechanical problems).
(By the way, in this approximation the Positronium looks as a rather neutral system: its positive and negative clouds coincide so due to cancellation of positive and negative charge form-factors. Inelastic channels are open instead: )
The Born approximation is very important: if you scatter a too slow particle from an atom, the atom will get polarised and its clouds will essentially change in course of interaction.
Some more advanced results are given here.
 V. Kalitvianski, Atom as a “Dressed” Nucleus, CEJP, V. 7, N. 1, pp. 1-11 (2009), http://arxiv.org/abs/0806.2635 .