8.3.6.2. Inelastic Scattering

Table of Contents

What Inelastic Scattering Means

Inelastic scattering of neutrons happens when a neutron collides with a nucleus and the total kinetic energy of the particles is not the same before and after the collision. Momentum is still conserved, but part of the neutron's incoming kinetic energy is transferred into internal energy of the nucleus.

In a typical inelastic scattering event, the neutron approaches a nucleus, interacts with it, and leaves with lower kinetic energy. The nucleus is left in an excited state. Afterward, the nucleus often returns to a lower energy state by emitting one or more gamma rays.

A simple reaction notation is

$$
n + A \rightarrow n + A^*
$$

followed by

$$
A^* \rightarrow A + \gamma
$$

Here, $A$ is the target nucleus and $A^*$ is the same nucleus in an excited state.

How It Differs from Elastic Scattering

In elastic scattering, the nucleus is not left internally excited. Only the motion of the neutron and nucleus changes. In inelastic scattering, some of the neutron's kinetic energy goes into nuclear excitation.

This difference is very important. Elastic scattering is mainly a slowing down process through energy transfer to nuclear recoil. Inelastic scattering removes energy in larger steps because energy can be absorbed by the nucleus as excitation energy.

In inelastic scattering, kinetic energy is not conserved by itself, but total energy is conserved. The missing kinetic energy appears as internal excitation energy of the nucleus, and often later as gamma radiation.

Energy Balance

Suppose a neutron with initial kinetic energy $E_i$ strikes a nucleus. After the interaction, the outgoing neutron has kinetic energy $E_f$, the nucleus may recoil, and the nucleus may be excited by an amount $E_{\text{exc}}$.

The energy balance is

$$
E_i + Q = E_f + E_{\text{recoil}} + E_{\text{exc}}
$$

For inelastic scattering, the reaction effectively has a negative $Q$ value when excitation is produced, because energy must be supplied to excite the nucleus.

If the nucleus is excited by energy $E_{\text{exc}}$, then the incoming neutron must have enough energy for this process to occur. This means inelastic scattering usually has a threshold energy.

Inelastic scattering generally requires the neutron energy to be high enough to excite the nucleus. Below this threshold, the process cannot occur.

Threshold Energy

Because the nucleus must be excited, a neutron cannot undergo inelastic scattering unless it brings enough kinetic energy into the collision. If the excitation energy is $E_{\text{exc}}$, then the threshold neutron energy is at least of that order, with a correction due to conservation of momentum and recoil.

So, unlike elastic scattering, inelastic scattering is not usually important for very low energy neutrons. It becomes more important for fast neutrons, especially in the MeV range for many nuclei.

Physical Picture

When a neutron enters the region of the nucleus, the strong nuclear interaction can transfer energy to the nucleus. The nucleus behaves like a quantum system with discrete energy levels. If the neutron has the right energy, it can promote the nucleus from its ground state to one of these excited states.

After the collision, the neutron leaves with less energy, and the excited nucleus later emits gamma radiation. This makes inelastic scattering both a neutron energy loss mechanism and a source of gamma rays.

Rendering scientific sketch…

Neutron inelastic scattering from a nucleus

Role in Neutron Moderation

Inelastic scattering helps reduce neutron energy, especially for fast neutrons in heavy materials. In some reactor and shielding materials, it is an important way to remove energy from neutrons quickly.

Heavy nuclei are often more effective for inelastic scattering than for elastic slowing down alone, because a heavy nucleus does not gain much recoil energy in elastic collisions. But if that heavy nucleus can be excited, the neutron can lose a significant chunk of energy in one event.

This is why inelastic scattering is useful in fast neutron shielding and in reactor physics.

Gamma Emission After Inelastic Scattering

The excited nucleus produced in inelastic scattering is usually unstable in that excited state. It typically de-excites by emitting gamma rays. These gamma rays can then interact with matter by other processes.

This means inelastic neutron scattering can create secondary radiation. A shield that slows neutrons through inelastic scattering may also need to absorb the resulting gamma rays.

Inelastic neutron scattering often produces prompt gamma rays. Neutron shielding design must account for both the scattered neutrons and the secondary gamma radiation.

Where It Is Important

Inelastic scattering is especially important for fast neutrons and medium to heavy nuclei. It is less important for thermal neutrons because their energies are usually too small to excite nuclear levels.

Examples of situations where it matters include reactor cores, biological shielding, neutron spectroscopy, and nuclear structure studies.

Comparison Table

Feature	Elastic Scattering	Inelastic Scattering
Nucleus excited	No	Yes
Neutron kinetic energy after collision	Reduced or changed	Reduced more strongly
Internal nuclear energy change	None	Increases
Gamma emission afterward	No direct nuclear de-excitation gamma	Often yes
Threshold energy	Usually no	Usually yes
Importance for thermal neutrons	Very important	Usually small
Importance for fast neutrons	Important	Often important

Reaction Cross Section and Energy Dependence

The probability of inelastic scattering is described by its cross section. This cross section depends strongly on neutron energy and on the structure of the target nucleus. Near threshold, the cross section is small or zero. As neutron energy increases above the excitation threshold, the cross section often rises.

Because nuclei have discrete excited states, different inelastic channels may open as the neutron energy becomes large enough to excite higher levels.

So the neutron does not lose arbitrary amounts of energy. The allowed energy losses are connected to the nuclear energy level structure.

Summary of the Main Idea

Inelastic scattering is a neutron interaction in which the neutron collides with a nucleus and leaves behind some of its kinetic energy as nuclear excitation. The nucleus is then often left in an excited state and later emits gamma rays. This process requires sufficient neutron energy, so it is mainly important for fast neutrons. It is a key mechanism in neutron slowing down in some materials and in the production of secondary gamma radiation.

Key idea:
$$
n + A \rightarrow n + A^*
$$
The outgoing neutron has less kinetic energy because some energy has gone into exciting the nucleus.

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