Coulomb barrier

The Coulomb barrier, named after Coulomb's law

Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

, which is named after physicist Charles-Augustin de Coulomb

Charles-Augustin de Coulomb

Charles-Augustin de Coulomb was a French physicist. He is best known for developing Coulomb's law, the definition of the electrostatic force of attraction and repulsion. The [SI unit] of charge, the coulomb, was named after him....

 (1736–1806), 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

Nuclear reaction

In nuclear physics and nuclear chemistry, a nuclear reaction is semantically considered to be the process in which two nuclei, or else a nucleus of an atom and a subatomic particle from outside the atom, collide to produce products different from the initial particles...

. This energy barrier is given by the electrostatic potential energy:

where

k is the Coulomb's constant = 8.9876×10⁹ N m² C⁻²;

ε₀ is the permittivity

Permittivity

In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...

 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).

Coulomb's barrier increases with the atomic number

Atomic number

In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...

s (i.e. the number of protons) of the colliding nuclei:


where e is the elementary charge

Elementary charge

The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...

, 1.602 176 53×10⁻¹⁹ C, 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

Strong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

 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 velocity of the particles in that gas. For normal gases, the Maxwell-Boltzmann distribution gives the fraction of particles moving at a given velocity as a function of gas temperature, and thus the fraction of particles moving at velocities high enough to overcome the Coulomb's barrier can be derived.

In practice, temperatures needed to overcome Coulomb's barrier turn out to be smaller than expected due to quantum-mechanical tunneling

Quantum tunnelling

Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the sun, and has important...

, as established by Gamow

George Gamow

George Gamow , born Georgiy Antonovich Gamov , was a Russian-born theoretical physicist and cosmologist. He discovered alpha decay via quantum tunneling and worked on radioactive decay of the atomic nucleus, star formation, stellar nucleosynthesis, Big Bang nucleosynthesis, cosmic microwave...

. The consideration of barrier-penetration through tunneling and the speed distribution gives rise to a limited range of conditions where the fusion can take place, known as the Gamow window.

It was the absence of a Coulomb barrier for the neutron that enabled James Chadwick

James Chadwick

Sir James Chadwick CH FRS was an English Nobel laureate in physics awarded for his discovery of the neutron....

to discover it.