Coulomb barrier

The Coulomb barrier, named after Coulomb's law, which is in turn named after physicist Charles-Augustin de Coulomb, is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo a nuclear reaction.

Potential energy barrier

This energy barrier is given by the electric potential energy:

$U_{\text{coulomb}}={1 \over 4\pi \varepsilon _{0}}{q_{1}q_{2} \over r}$

where

ε₀ is the permittivity of free space;

q₁, q₂ are the charges of the interacting particles;

r is the interaction radius.

A positive value of U is due to a repulsive force, so interacting particles are at higher energy levels as they get closer. A negative potential energy indicates a bound state (due to an attractive force).

The Coulomb barrier increases with the atomic numbers (i.e. the number of protons) of the colliding nuclei:

$U_{\text{coulomb}}={{Z_{1}Z_{2}e^{2}} \over 4\pi \varepsilon _{0}r}$

where e is the elementary charge, and Z_i the corresponding atomic numbers.

To overcome this barrier, nuclei have to collide at high velocities, so their kinetic energies drive them close enough for the strong interaction to take place and bind them together.

According to the kinetic theory of gases, the temperature of a gas is just a measure of the average kinetic energy of the particles in that gas. For classical ideal gases the velocity distribution of the gas particles is given by Maxwell–Boltzmann. From this distribution, the fraction of particles with a velocity high enough to overcome the Coulomb barrier can be determined.

In practice, temperatures needed to overcome the Coulomb barrier turned out to be smaller than expected due to quantum mechanical tunnelling, as established by Gamow. The consideration of barrier-penetration through tunnelling and the speed distribution gives rise to a limited range of conditions where fusion can take place, known as the Gamow window.

The absence of the Coulomb barrier enabled the discovery of the neutron by James Chadwick in 1932.

Modeling a potential energy barrier

There is keen interest in the mechanics and parameters of nuclear fusion, including methods of modeling the Coulomb barrier for scientific and educational purposes. The Coulomb barrier is a type of potential energy barrier, and is central to nuclear fusion. It results from the interplay of two fundamental interactions: the strong interaction at close-range within ≈ 1 fm, and the electromagnetic interaction at far-range beyond the Coulomb barrier. The microscopic range of the strong interaction, on the order of one femtometre, makes it challenging to model and no classical examples exist on the human scale.  A visual and tactile classroom model of strong close-range attraction and far-range repulsion characteristic of the fusion potential curve is modeled in the magnetic “Coulomb” barrier apparatus. The apparatus won first place in the 2023 national apparatus competition of the American Academy of Physics Teachers in Sacramento, California. Essentially, a pair of opposing permanent magnet arrays generate asymmetric alternating N/S magnetic fields that result in repulsion at a distance and attraction within ≈ 1cm. A related patent method (US11,087,910 B2) further describes the apparatus and outlines criteria for more generally modeling an electromagnetic potential energy barrier. Magnetic and electric forces were unified within the electromagnetic fundamental force by James Clerk Maxwell in 1873 in A Treatise on Electricity and Magnetism. In the case of the magnetic “Coulomb” barrier, the patent describes alternating/unequal or asymmetric North and South magnetic poles but the patent method language is broad enough to include positive and negative electrostatic poles as well. The implication is that regularly spaced opposite and unequal electrostatic point charges possess the capacity to model an electrostatic potential energy barrier as well.