The size of atoms, uncertainty and relativity

Busch Stadium: Source: Ryan Pastorino’s flickr

You may have read somewhere that atoms are actually made up mostly of empty space. The usual description goes more or less as follows. At the centre of each atom there is a very tiny nucleus, which carries all of the positive charge and nearly all of the mass of the atom. Surrounding the nucleus are a number of electrons, equal to the positive charge of the nucleus. The electrons are arranged on various quantum orbits. The lowest one is called the “ground state.” But even the orbit of the ground state is still far away from the nucleus. In most atoms, the radius of the ground state is about 10-10 m, while the radius of the nucleus is only about 10-14 m. This means that a nucleus occupies only a tiny fraction of the space inside an atom; the rest is empty (except for a few point-sized electrons.). This is usually accompanied by an analogy the layperson can easily visualize, something like “If the nucleus were the size of a peanut, the atom would be about the size of a baseball stadium”.

Well, strictly speaking, this is simply not true. We have known for more than a century that in this universe there are three kind of physical realities: matter (in all its varieties), movement (and everything associated with it) and fields. According to quantum electrodynamics, the space is filled by an electron field around the nucleus which neutralizes its charge and fills the space defining the atom size. The electrons are therefore rather like a very low-density glue-like viscous fluid surrounding the nuclei and making up the spatial extent of the atom, transparent for neutrons but not for other electrons. Chemists usually draw the shape of these fluid clouds (more precisely, the electron density) as orbitals. Electrons show up as particles, then, only under particular circumstances.

This seems very complicated and not very easy to explain or understand. Still, we can grasp why the atoms have the size they have using another, simpler, approach. One with no mathematics and just a general basic knowledge of quantum physics and, wait for it,…relativity.

In the Rutherford’s model, the atom should collapse into the nucleus, because the electrons should radiate away their energy and spiral inward. This was a huge problem for the model, so Bohr attempted to account for why this does not happen by postulating the existence of stationary quantum states, while quantum mechanics associated such states with standing waves, the ground state having the smallest standing electron wave in that orbit. But this explains nothing regarding the size of the atom or, put in another way, why the stationary states are where they are and not any closer to the nucleus.

It is the uncertainty principle that explains why we can’t have any lower states, and why the negative electrons cannot exist within or on the positive nucleus. Let’s recall what the principle says: It is impossible to measure both the position and the momentum of a subatomic particle, in the same instant to unlimited accuracy. The more accurate is the measurement of the momentum, the less accurate is the measurement of the position in that instant, and vice versa.

A simple application of the uncertainty relation for position and momentum shows that if an electron is confined to a space of 10-10 m, the size of an average atom, then the uncertainty in its speed is less than the speed of light. But if it is confined to a much smaller space, or even down to the size of the nucleus, 10-14 m, the uncertainty in its speed would exceed the speed of light.

Enter relativity. As you know, no material particle may exceed the speed of light not even equal it. So the space inside the atom between the nucleus and the first quantum state must be left “empty”.


Arnold Neumaier (2004-2016) A theoretical physics FAQ. University of Vienna

Cassidy, D. et al (2002) Understanding Physics Springer Verlag New York

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

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