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8.5.3.3. Energy Dependence

Table of Contents

Why cross section depends on energy

In nuclear reactions, the cross section is not usually a fixed number. It often changes strongly with the kinetic energy of the incoming particle. This energy dependence tells us how likely a reaction is at different projectile energies, and it is one of the most important ideas in reaction physics.

A projectile approaching a nucleus does not simply hit a hard sphere. The interaction depends on wave behavior, forces between the particles, and the internal structure of the nucleus. Because of this, the reaction probability can rise, fall, or show sharp peaks as energy changes.

The reaction cross section is generally a function of energy:
$$\sigma = \sigma(E)$$
This means that quoting a cross section without specifying the projectile energy is often incomplete.

Physical reasons for energy dependence

Several effects make the cross section vary with energy. At low energies, the projectile may not have enough energy to overcome repulsion or to excite the nucleus. At higher energies, new reaction channels can open. At certain special energies, the projectile and target can form an intermediate state especially efficiently, causing a resonance.

For charged particles, the Coulomb repulsion between the positively charged projectile and the nucleus is often very important. A proton or alpha particle can be pushed away before it gets close enough for the strong nuclear force to act. As the projectile energy increases, it can approach the nucleus more easily, so the cross section may increase.

For neutrons, there is no electric repulsion, so even very low energy neutrons can interact strongly. This is one reason neutron cross sections can be large at low energies.

Threshold behavior

Some reactions can happen only if the projectile has at least a certain minimum energy. This is called the threshold energy. Below this energy, the cross section is zero because the reaction is energetically impossible. Just above threshold, the cross section usually starts small and then grows.

For example, if a reaction requires energy to produce an outgoing particle or to leave the nucleus in an excited state, the incoming particle must supply enough kinetic energy.

For a reaction with threshold energy $E_{\text{th}}$,
$$\sigma(E) = 0 \quad \text{for} \quad E < E_{\text{th}}$$
Only for energies above threshold can the reaction occur.

Low energy behavior

At low projectile energy, the cross section may behave very differently for neutrons and charged particles.

For neutrons, many capture and absorption processes become more probable as the neutron slows down. In many cases, especially for thermal neutrons, the cross section approximately follows the inverse velocity law. Since kinetic energy is related to speed, this means the cross section can increase as energy decreases.

$$\sigma \propto \frac{1}{v}$$

or, using $E \propto v^2$ for nonrelativistic motion,

$$\sigma \propto \frac{1}{\sqrt{E}}$$

This behavior is common in slow neutron reactions, though not universal.

For charged particles, low energy usually means strong suppression because of the Coulomb barrier. Even if the reaction is energetically allowed, the cross section may remain very small until the projectile energy becomes high enough for close approach, or until quantum tunneling gives a small probability of penetration.

Coulomb barrier effects

If the projectile and target are both positively charged, electrostatic repulsion creates a barrier. The projectile must either go over this barrier or tunnel through it. At low energies, this makes the cross section very small. As energy increases, penetration becomes easier and the cross section often rises rapidly.

This is especially important in reactions involving protons, alpha particles, or heavier ions. In contrast, neutron-induced reactions avoid this barrier.

A simple picture is that the projectile must come close enough to the nucleus for the strong force to act. The Coulomb force resists this approach, so the energy dependence can be dramatic.

Charged particle approaching a nucleus

In this sketch, a projectile with energy $E_1$ is much less likely to reach the nucleus than one with energy $E_2$.

Resonances

One of the most striking forms of energy dependence is a resonance. A resonance happens when the projectile and target form an intermediate compound system with an energy close to one of its allowed excited states. At that energy, the reaction probability can become much larger.

Instead of changing smoothly, the cross section shows a sharp peak at the resonance energy. This is similar to resonance in ordinary waves or oscillations, where a system responds strongly at certain frequencies.

Near a resonance energy $E_r$, the cross section often has a peak shape. A common approximate description is the Breit-Wigner form:

$$\sigma(E) \propto \frac{1}{(E - E_r)^2 + (\Gamma/2)^2}$$

Here $E_r$ is the resonance energy, and $\Gamma$ is the width of the resonance. A small $\Gamma$ means a narrow peak. A large $\Gamma$ means a broad peak.

Resonances cause sharp increases in cross section at special energies.
A resonance peak indicates that the nuclear system can form an intermediate state efficiently at that energy.

Typical resonance peak in cross section

Opening of reaction channels

As projectile energy increases, more outcomes can become possible. At low energy, maybe only elastic scattering occurs. At higher energy, the nucleus might be excited, emit a neutron, emit a proton, or split into other products. Each newly allowed process is called an open reaction channel.

When a new channel opens, the total cross section and the individual reaction cross sections can change noticeably. Some probability that previously went into one process may now be shared among several processes.

This means the energy dependence is often not smooth over a wide energy range. Instead, it may show threshold onsets, gradual changes, and resonance peaks.

Smooth trends and complicated structure

In some energy regions, the cross section changes smoothly. In others, especially where many nuclear energy levels are involved, the dependence can be irregular and full of narrow peaks. The exact pattern depends on the target nucleus, the projectile type, and the reaction being studied.

A simple way to think about this is shown below.

Energy regionTypical behavior
Below thresholdNo reaction, $\sigma = 0$
Just above thresholdSmall cross section, often rising with energy
Low energy neutronsOften large cross sections, sometimes $\sigma \propto 1/v$
Low energy charged particlesOften very small because of Coulomb repulsion
Resonance regionSharp peaks at special energies
Higher energy regionMore channels open, behavior may become smoother or more complex

Elastic scattering compared with reaction cross sections

Energy dependence appears not only in reactions that transform the nucleus, but also in scattering. The probability for elastic scattering can vary with energy because the projectile wavelength changes and because the interaction with the nuclear potential changes.

At some energies, scattering and true reaction processes compete strongly. Therefore, when measuring cross sections, physicists often examine how both the scattering cross section and reaction cross section vary with energy.

Why energy dependence matters

The energy dependence of cross section is essential in reactors, particle detectors, astrophysics, and nuclear experiments. A material that strongly absorbs slow neutrons may be almost transparent to fast neutrons. A reaction used to produce isotopes may require a carefully chosen beam energy. In stars, fusion rates depend very sensitively on energy because of the Coulomb barrier.

By measuring $\sigma(E)$, physicists learn not only how likely a reaction is, but also details about nuclear structure and nuclear forces.

The graph of cross section versus energy, $\sigma(E)$, is a key experimental tool.
It reveals threshold energies, resonance states, barrier effects, and the opening of new reaction channels.

A visual summary

Qualitative energy dependence of a nuclear reaction cross section

This sketch shows a common pattern. Below the threshold energy, the reaction cannot occur. Above threshold, the cross section rises. Then resonances may appear as peaks. At still higher energies, the behavior may continue to change as other channels become possible.

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