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8.5.4.2. Inelastic Scattering

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What Inelastic Scattering Means

Inelastic scattering is a type of nuclear reaction in which an incoming particle collides with a nucleus and leaves again, but the nucleus is not left in the same internal state as before. Some of the kinetic energy of the incoming system is converted into internal energy of the nucleus. As a result, the nucleus becomes excited.

A common symbolic form is

$$
a + X \rightarrow a + X^*
$$

Here, $a$ is the incoming projectile, $X$ is the target nucleus, and $X^*$ means the same nucleus in an excited state. The important point is that the identity of the particles does not change, but the internal energy does.

In elastic scattering, kinetic energy is redistributed between projectile and target, but the total kinetic energy in the center-of-mass frame remains the same. In inelastic scattering, part of that kinetic energy is used to excite the nucleus, so the final kinetic energy is smaller.

In inelastic scattering, the target nucleus changes its energy state, even though its nuclear species remains the same.
$$
a + X \rightarrow a + X^*
$$
The final kinetic energy is less than the initial kinetic energy because energy has gone into nuclear excitation.

Energy Transfer in the Collision

When a projectile strikes a nucleus, conservation of energy still holds. However, some of the initial kinetic energy may be converted into excitation energy $E^*$ of the nucleus. If the nucleus is raised to an excited state, then

$$
E_{\text{initial}} = E_{\text{final}} + E^*
$$

where $E^*$ is the excitation energy.

This means that the scattered particle often emerges with reduced speed. Measuring this energy loss is one of the main ways physicists identify inelastic scattering experimentally.

If the excited nucleus later returns to a lower energy state, it often emits a gamma ray. So inelastic scattering is frequently followed by gamma emission:

$$
a + X \rightarrow a + X^*
$$

followed by

$$
X^* \rightarrow X + \gamma
$$

This gamma ray is evidence that the nucleus was excited during the collision.

Comparison with Elastic Scattering

The difference between elastic and inelastic scattering is easier to see in a direct comparison.

FeatureElastic ScatteringInelastic Scattering
Nuclear identityUnchangedUnchanged
Nuclear internal stateUnchangedChanged, excited
Kinetic energy in center-of-mass frameConservedDecreases
Excitation energy producedNoYes
Possible later gamma emissionNoOften yes

In both processes, the projectile and target can remain the same kinds of particles. The key difference is whether internal excitation occurs.

Do not confuse loss of kinetic energy with loss of total energy. In inelastic scattering, total energy is still conserved. The kinetic energy lost by the particles becomes internal excitation energy of the nucleus.

Excited Nuclear States

Nuclei can exist in discrete energy levels, somewhat like atoms. During inelastic scattering, the nucleus absorbs just enough energy to move into one of its allowed excited states.

If the excitation energy is $E^*$, then the scattered projectile must supply at least that energy, together with any energy needed by momentum conservation. Therefore, not every incoming energy can produce a given excitation. The projectile must have sufficient energy.

After excitation, the nucleus usually does not stay excited for long. It returns to a lower state by emitting radiation, most commonly gamma rays.

A simple picture is

Inelastic scattering and later gamma emission

Threshold Behavior

Because excitation requires energy, inelastic scattering often has a threshold. If the projectile energy is too low, the nucleus cannot be excited to the desired state, and only elastic scattering may occur.

Suppose the lowest excited state of a nucleus has energy $E_1$. Then the incoming projectile must have enough kinetic energy for that excitation to occur. In practice, the threshold is influenced not only by the excitation energy itself but also by momentum conservation.

This threshold behavior helps reveal the energy levels of nuclei. By varying the projectile energy and watching when new scattered-particle energies or gamma rays appear, physicists can determine excited nuclear states.

Inelastic scattering can occur only when the projectile has enough energy to excite the nucleus to an allowed state.

Common Projectiles

Many kinds of particles can produce inelastic scattering. Frequently used projectiles include neutrons, protons, alpha particles, and electrons. In nuclear physics, neutron and proton inelastic scattering are especially important.

For example,

$$
n + X \rightarrow n + X^*
$$

or

$$
p + X \rightarrow p + X^*
$$

Neutrons are especially useful because they have no electric charge, so they are not repelled by the positive nucleus by Coulomb forces. Charged particles such as protons must overcome electrical repulsion before reaching the nucleus.

Experimental Signatures

Inelastic scattering is identified by the measurable consequences of nuclear excitation. The main signatures are reduced outgoing kinetic energy and gamma radiation from de-excitation.

If a detector measures the energy of the outgoing projectile, several distinct groups may appear. One group can correspond to elastic scattering, where little or no energy is lost to excitation. Other groups can correspond to different excited states of the nucleus.

For example, if the projectile emerges with energy lower by $E^$, that suggests the nucleus absorbed excitation energy $E^$. If a gamma ray of the same energy is later detected, that provides strong confirmation.

A simplified energy-level picture is

Excitation of a nucleus in inelastic scattering

Why Inelastic Scattering Is Important

Inelastic scattering is a powerful tool for studying nuclear structure. Because nuclei have specific excited states, observing which excitations occur tells us about the allowed energy levels and properties of the nucleus.

By measuring scattering angles, outgoing energies, and emitted gamma rays, physicists can learn about nuclear shapes, collective motion, and internal arrangements of nucleons. The method acts like a probe of the nucleus.

Inelastic scattering is therefore not just a reaction type, it is also an experimental technique for exploring how nuclei are built.

Simple Summary

In inelastic scattering, the projectile collides with a nucleus and leaves as the same kind of particle, but the nucleus is excited. Some kinetic energy is converted into internal nuclear energy. The reaction is commonly written as

$$
a + X \rightarrow a + X^*
$$

and the excited nucleus often later emits a gamma ray:

$$
X^* \rightarrow X + \gamma
$$

This process differs from elastic scattering because the nucleus changes its internal state. Studying inelastic scattering helps reveal nuclear energy levels and nuclear structure.

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