8.6.4.2. Coulomb Barrier

Table of Contents

Electric Repulsion Before Fusion

For two atomic nuclei to fuse, they must get extremely close to each other. Fusion happens only when the strong nuclear force can act effectively, and that force works over a very short distance, about the size of a nucleus. Before the nuclei get that close, they feel a strong electric repulsion because both nuclei carry positive charge. This repulsive effect is called the Coulomb barrier.

The Coulomb barrier is the energy obstacle that nuclei must overcome, or somehow pass through, in order to fuse.

Why the Barrier Exists

Each proton has positive electric charge. A nucleus with atomic number $Z$ contains $Z$ protons, so it has charge $+Ze$. If two nuclei approach each other, the electric force between them is repulsive. According to Coulomb's law, the electric potential energy between two nuclei is

$$
U(r) = \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{r}
$$

where $Z_1$ and $Z_2$ are the proton numbers of the two nuclei, $e$ is the elementary charge, and $r$ is the distance between their centers.

As $r$ becomes smaller, $U(r)$ becomes larger. This means the nuclei need more and more energy to get closer together.

For two positively charged nuclei, the Coulomb potential energy is
$$
U(r) = \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{r}
$$
This energy increases sharply as the separation $r$ decreases.

The Meaning of "Barrier"

The word barrier does not mean a physical wall. It means that, classically, the nuclei need enough kinetic energy to climb the electric potential energy hill. If their kinetic energy is too small, they would be pushed apart before they get close enough for the strong nuclear force to bind them.

A simple picture is this. Far apart, the nuclei move toward each other. As they approach, their kinetic energy is converted into electric potential energy. If all of the kinetic energy is used up before they reach nuclear distances, they stop approaching and then move apart again.

Competition Between Forces

At larger distances, electric repulsion dominates. At very short distances, the strong nuclear force becomes attractive and much stronger than the electric force. So fusion requires the nuclei to enter the tiny distance range where the strong force can take over.

This creates a basic situation:

Distance between nuclei	Dominant effect
Large distance	Very weak interaction
Intermediate distance	Coulomb repulsion dominates
Very short distance	Strong nuclear attraction dominates

The Coulomb barrier is therefore the region of repulsion that must be crossed before the attractive nuclear interaction can bind the nuclei.

Barrier Height

A rough estimate of the barrier height can be found by evaluating the Coulomb potential energy when the nuclei are nearly touching. If the nuclear radii are $R_1$ and $R_2$, then the closest classical approach is roughly at

$$
r \approx R_1 + R_2
$$

so the barrier energy is approximately

$$
U_{\text{barrier}} \approx \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{R_1 + R_2}
$$

This is only an estimate, but it shows the key dependence. The barrier becomes larger when the nuclear charges are larger.

A useful estimate for the Coulomb barrier is
$$
U_{\text{barrier}} \approx \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{R_1 + R_2}
$$
Larger values of $Z_1 Z_2$ mean a higher barrier and more difficult fusion.

Why Light Nuclei Fuse More Easily

Fusion is easiest for light nuclei because their charges are small. For example, hydrogen isotopes have $Z=1$, so the electric repulsion is much smaller than for heavier nuclei such as carbon or oxygen. This is one reason why fusion in stars begins with hydrogen and why controlled fusion on Earth focuses on light nuclei like deuterium and tritium.

Here is the trend:

Nuclei	Product $Z_1 Z_2$	Relative barrier difficulty
Proton and proton	1	Low
Deuterium and tritium	1	Low
Deuterium and helium nucleus	2	Higher
Carbon and carbon	36	Much higher

The exact barrier also depends on nuclear size, but the charge product $Z_1 Z_2$ is the main reason heavier nuclei are harder to fuse.

Thermal Energy and Temperature

In a hot gas or plasma, nuclei move randomly with a range of speeds. Higher temperature means higher average kinetic energy. To approach each other closely, nuclei need enough kinetic energy to fight the Coulomb repulsion.

This is why fusion requires extremely high temperatures. At high temperature, some nuclei move fast enough to get close to the barrier region.

For a gas, a typical thermal energy scale is about

$$
E \sim k_B T
$$

where $k_B$ is Boltzmann's constant and $T$ is temperature.

However, even in very hot plasmas, the average thermal energy is often smaller than the full classical barrier height. This leads to a crucial quantum effect.

Quantum Tunneling Through the Barrier

Classically, a nucleus with energy below the barrier cannot cross it. Quantum mechanics changes this result. A nucleus has a finite probability to tunnel through the Coulomb barrier even if its kinetic energy is less than the barrier height.

This is essential for fusion in stars. The temperature in stellar cores is high, but not high enough for most nuclei to classically overcome the barrier. Fusion still occurs because a small fraction of collisions succeed by quantum tunneling.

Fusion often occurs not because nuclei classically go over the Coulomb barrier, but because they quantum mechanically tunnel through it.

The tunneling probability increases when the barrier is lower and narrower, and when the nuclei have higher kinetic energy.

The Gamow Idea

The probability of penetrating the Coulomb barrier is often described using the Gamow factor. For beginners, the most important idea is qualitative. The tunneling probability depends very strongly on charge and energy. Small increases in particle energy can greatly increase fusion probability, while larger nuclear charges greatly reduce it.

This is why fusion rates are extremely sensitive to temperature.

Classical Turning Point

If two nuclei approach each other classically with center-of-mass kinetic energy $E$, the closest distance they can reach without tunneling is found by setting

$$
E = \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{r_{\min}}
$$

which gives

$$
r_{\min} = \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{E}
$$

If $r_{\min}$ is still larger than the nuclear interaction range, fusion will not happen classically.

Visual Picture

A simple energy diagram helps show the idea. The potential energy rises as the nuclei approach, then at very short range the strong force creates an attractive region.

Rendering scientific sketch…

Schematic Coulomb barrier for two nuclei

In this sketch, the dashed line represents the kinetic energy level of the approaching nuclei. If the line lies below the peak, classical motion cannot pass the barrier. Quantum tunneling allows some probability of getting through anyway.

Astrophysical Importance

The Coulomb barrier is one of the main reasons stars need high core temperatures and pressures. It controls how often nuclei can fuse and therefore strongly affects stellar energy production. Even though protons repel each other electrically, the enormous temperature, density, and long timescales inside stars allow fusion to occur.

The same barrier is a central challenge in human attempts to produce controlled fusion energy. Engineers must create conditions where enough nuclei come close enough, often with the help of very high temperatures.

Main Lessons

The Coulomb barrier is the electric repulsion barrier between two positively charged nuclei. It comes from Coulomb potential energy and grows with nuclear charge. Fusion requires nuclei to reach very short distances where the strong nuclear force can bind them. High temperature helps nuclei approach each other, but in many important cases, especially in stars, quantum tunneling is what makes fusion possible.

Key ideas about the Coulomb barrier:
$$
U(r) = \frac{1}{4\pi \varepsilon_0}\frac{Z_1 Z_2 e^2}{r}
$$
A larger charge product $Z_1 Z_2$ means a higher barrier.
Fusion requires nuclei to get close enough for the strong nuclear force to act.
In many real fusion processes, especially stellar fusion, quantum tunneling through the Coulomb barrier is essential.

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