Theoretical physics

Sizing up atoms

Niels Bohr's model of the structure of the atom raised the question of how large an atom can be. One hundred years on, the issue is still unresolved. Two physicists discuss the theoretical limits of atomic and nuclear size.

Orbital arguments

Paul Indelicato

Bohr's model of the atom1 (Fig. 1) provided a new way of thinking about atomic size. For example, it predicted that the radius of the smallest atom, hydrogen, in its ground state was 0.5 × 10−10 metres, 100,000 times larger than the size of the nucleus. This value, known as the Bohr radius (a0), was remarkably accurate and is now one of the fundamental constants of atomic physics. The model also proposed that the speed of an electron in the inner orbital of an atom was approximately Zcα (where Z is the proton number, c is the speed of light and α is the fine-structure constant, approximately 1/137). Intriguingly, this limits Z to a maximum of about 137, because, above this value, the electron's speed would be greater than the speed of light.

Figure 1: Atomic structure.

This cartoon of a bismuth-209 atom exemplifies the features of Niels Bohr's model of the atom: a nucleus, composed of protons (purple) and neutrons (yellow) is orbited by electrons (blue), which occupy distinct shells. The relative size of the nucleus and the electron shells are not shown to scale. Bismuth-209 can decay by emitting α-radiation, but the measured half-life13 for this process is 1.9 ± 0.2 × 1019 years, a billion times longer than the estimated age of the Universe. It is, therefore, essentially the heaviest naturally occurring stable atom.

Nowadays, atomic models are based on the Dirac equation, which combines relativity and quantum mechanics in a theory called quantum electrodynamics (QED). The Dirac equation for a point nucleus leads to the same limit: the electron-binding energy becomes complex when Z is greater than or equal to 1/α. But for an extended nucleus, the limit is around Z = 173. Above that value, the electron-binding energy is more than twice the electron's rest mass, a condition that allows the formation of electron–antielectron pairs, which would render the atom unstable.

The size of an atom can be defined in different ways2. If the mean spherical radius of the whole atom is considered, based on the total electron density, then the possible range of sizes is small: from 1.06a0 to 1.5a0. But if the size of the outermost orbital is considered, then atomic size ranges from a0 at Z = 1 to 8a0 at Z = 172 (refs 2,3). What happens above Z = 172 is still being investigated4 to study how the emission of real electron–antielectron pairs causes the breakdown of the quantum vacuum — a mysterious state predicted by QED, consisting of empty space in which virtual particles such as photons and electron–antielectron pairs are constantly created and annihilated.

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The heaviest nucleus to have been identified5 has Z = 118. Nuclei with higher numbers of protons can be studied only by creating them temporarily during collisions of two lower-charged nuclei. This was attempted in the 1980s, but the accelerators of the time could not produce bare nuclei (or nuclei with single electrons) that had large enough Z to succeed. Today, high-quality beams of bare nuclei of heavy elements can be produced at energies that could allow binary nuclear systems to be prepared for approximately 10−21 seconds. Projects to study the quasi-molecular state created in such collisions, and to investigate the properties of the resulting quasi-atoms, have been proposed.

But it is not only large atoms that can be made — smaller, exotic atoms can also be created by replacing electrons with heavier particles, such as muons, pions or antiprotons. The resulting systems are 207 to 1,836 times smaller than the corresponding 'normal' atoms, and are thus close to the size of a nucleus. Such atoms have been used to study nuclear properties, such as the size of a proton6.

The nuclear question

Alexander Karpov

The maximum size of an atomic nucleus is determined by its stability towards decay. In general, only a few isotopes of each element are stable, the heaviest of which is bismuth-209 (83 protons and 126 neutrons; Fig. 1). All elements heavier than this are radioactive, although two of them (thorium and uranium) have tremendously long half-lives and are found in large amounts in nature. In some respects, such long-lived radioactive elements can be thought of as 'stable'.

If we enlarged a nucleus by adding neutrons, then it would become increasingly short-lived, eventually reaching the border of neutron stability. Beyond this border, the nuclear system is unbound and spontaneously emits neutrons. The number of neutrons that can be added to a stable nucleus depends mainly on Z: the larger Z is, the further away is the border of neutron stability. The border has already been reached experimentally in elements up to oxygen (and probably up to aluminium, although some dispute this). But for heavy elements, only theoretical estimates of the border's position are available. For example, the heaviest uranium atom is predicted to bind 92 protons and about 208 neutrons, a total mass number of around 300; by comparison, the heaviest naturally occurring uranium nucleus has a mass number of 238.

If we added protons to uranium, the heaviest naturally occurring element, then we would produce new elements. (In fact, we would need to add protons and neutrons, to avoid reaching the border of proton stability). The resulting nuclei would be progressively less stable to spontaneous fission because of Coulomb repulsion in their interiors. Nuclei become totally unstable towards fission at about Z = 106, in the absence of quantum effects.

But nuclei consisting of certain 'magic' numbers of protons and neutrons are especially stable by comparison with their neighbours. Superheavy nuclei that have nearly magic numbers of protons and neutrons form islands of relatively long-lived nuclei surrounded by a sea of short-lived nuclei. A pair of magic numbers in the superheavy region (114 protons and 184 neutrons) was predicted7,8,9,10 in the 1960s. The centre of this island has not been reached experimentally, and the ways to reach it are debated11. However, elements up to Z =118 have been synthesized5,12. The existence of the island unambiguously follows from these results, but the data do not indicate where the top of the island is, nor how long-lived the nuclei at the top would be. No consensus on this topic has been reached from theoretical considerations.

Are there other islands of stability? Probably, yes. But different theories of nuclear stability diverge from each other when extrapolated into remote domains of nuclei, so the opposite answer cannot be excluded. One hypothesis proposes that very heavy nuclei do not have a 'normal', nearly uniform distribution of nuclear matter, but a bubble-like distribution. This should substantially suppress the Coulomb forces and increase nuclear stability. Some theories predict bubble-like structures in the vicinity of the first island of stability of superheavy nuclei — in which case, massive, long-lived nuclei might have rather exotic structures.


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Indelicato, P., Karpov, A. Sizing up atoms. Nature 498, 40–41 (2013).

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