Zoom in on Atom or Unknown Physics of Short Distances

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 [1]:




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 f_n^ { n} describes the positive charge distribution in atom (not within the nucleus). Also the factor |f_n^ { n}|\le 1 being less than unity simply tells us that the amplitude of not exciting the atom is smaller than unity, if we push the atomic nucleus. Likewise, for inelastic processes the form-factors f_n^ {n^{\prime}} 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 \psi_{nlm}! 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 \rho(r) 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>:

Fig. 3. Qualitative image of atomic charge density, 2D plot of \rho(r) for the state |4,3,1>.

I would like to underline that the nucleus bound in atom is not seen “from the exterior” as a point-like in elastic scattering processes, i.e., in processes whose cross sections are expressed via non-perturbed atomic wave functions. The atomic 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 [1]. 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 it is a deeply inelastic (inclusive) picture with respect to soft photon emissions whose emission happens always.

Smearing is always the case for bound states. But one should not confuse how the atomic electron “sees” the nucleus with 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 only “seen” by a fast projectile in the elastic channel. (Generally, our mental pictures are only meaningful if they are encountered in our strict calculations.)

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 d\sigma_{\rm{incl}}=\sum_{n'}d\sigma_n^ {n^{\prime}}. 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 (sum of “pale” inelastic images).

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 d\sigma_{elastic}\approx 0 due to cancellation of positive and negative charge form-factors. Inelastic channels are open instead: d\sigma_n ^{n^{\prime}}>0 )

The Born approximation is very important: if you scatter a too slow particle from an atom, the atom will get polarized and its clouds will essentially change in course of interaction.

Some more advanced results are given here.


[1] V. Kalitvianski, Atom as a “Dressed” Nucleus, CEJP, V. 7, N. 1, pp. 1-11 (2009), http://arxiv.org/abs/0806.2635 .

10 Responses to “Zoom in on Atom or Unknown Physics of Short Distances”

  1. carlbrannen Says:

    Wonderful post, very understandable and well illustrated. I also think the subject is one of general interest and will attract readers for years. And it got me thinking.

    One of the things that has always bothered me about QM is the necessity for phase space and the fact that when one considers multiple particles, phase space does not correspond to regular spacetime.

    In this example, you’ve reduce the phase space to something that looks like regular space. I suppose it’s obvious that you can always do this with any two bodies. But I wonder if some similar technique could be used to give QM for multiple particles on a real space.

    It would need some sort of expansion in terms. And maybe mathematically, it would rely on taking objects two at a time and reducing them from Z^2 to Z where “Z” is the usual spacetime. So it would be iterative.

  2. Vladimir Kalitvianski Says:

    Hi Carl,

    Do you mean the configuration space {r1, r2, r3, …}?

    The number of r-s corresponds to the number of “degrees of freedom”. A compound system may have a lot of them, just like a multi-electron atom. Yet they all give the “charged clouds” in the 3D space because the form-factors F and f depend on the three-vector q and the cloud density is also a 3D function \rho(r).

  3. Carl Brannen Says:

    Yes, the configuration space (as I grow older I keep misusing language worse). In other words, I’m wondering whether your method gives a way to generalize “density functional theory” to higher energy states. For a reader unfamiliar with this see

    • Vladimir Kalitvianski Says:

      I am not aware of the “density functional theory”. I just would like to underline that even in case of two bodies, we deal already with quasi-particles. In particular, the relative distance r is determined with a reduced mass. In case of many-electron atoms, they often use the Slater determinant with some wave-functions (as ansatz) which I would also call a quasi-particle wave-functions, with necessarily now phenomenological (fitting) parameters. Collective motion excitations are always quasi-particles.

  4. Albert Pino Says:

    I would like to know if the electrons orbital speed around its nucleus varies from center. For example, in a Cupper particle that has 4 levels (orbits) the speed of the external electron is faster, equal or slower than the internal orbit, or the third orbit that has 18 electrons?

  5. Vladimir Kalitvianski Says:

    The electron velocity in the Hydrogen atom (Z=1) is estimated as e^2/h_bar where one of e-s comes from the nucleus charge (Ze). For an electron in a pure ion, it is Ze^2/h_bar or so. With presence of other electrons it is modifed due to collective effects but I believe in lower orbits the velocity is still higher.

  6. Quora Says:

    Do vibrational motion of a gas molecule affect its atomic radius (electronic cloud)?

    Yes, it does. Vibrational motion is much slower than the electron motion, so the electrons form “clouds” according to the instant distance [math]R(t)[/math] between the nuclei. But if one makes a quantum mechanical averaging over all kinds of motions…

  7. Christian Teske Says:

    If the positive charge of the nucleus smears out in an excited atom, shouldn’t the probability for fusion reactions with a positive projectile (deuteron) go up accordingly?

    • Vladimir Kalitvianski Says:

      No, it should, not. Because smearing out only happens in the elastic channel for the atom. In other words, the elastic form-factor is less than unity |f| < 1 simply to tell you that it is less probable to interact with the atomic nucleus and at the same time not to change the initial atomic state |n>. It means in practice a nearly obligatory change of the initial atomic state |n> –> |n'> during interaction with its nucleus. The sum of all possible atomic changes, i.e., an inclusive cross section, practically coincides with the interaction cross section for scattering from a "free" nucleus (where no information about atomic states is practically involved). Please read my paper in order to see the corresponding math.

  8. Sam Peppel Says:

    Very nice, thank you!

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