KAHIBARO
Table of Contents
What gamma-ray emission is
Gamma-ray emission is the process in which an atomic nucleus releases excess energy by emitting a high-energy photon, called a gamma ray. This usually happens after the nucleus has already been left in an excited state by some other process, such as alpha decay, beta decay, or a nuclear reaction. The number of protons and neutrons does not change during pure gamma emission. Only the energy of the nucleus changes.
A useful way to picture this is to think of the nucleus as having allowed energy levels. If the nucleus is in a higher energy state and moves to a lower one, the energy difference is carried away by a gamma photon.
In gamma-ray emission, the nucleus changes from a higher energy state to a lower energy state without changing its atomic number $Z$ or mass number $A$.
$$
{}^A_ZX^* \to {}^A_ZX + \gamma
$$
Here, $X^*$ means the nucleus is in an excited state.
Nuclear energy levels
Just as electrons in atoms occupy discrete energy levels, nuclei also have discrete energy levels. However, nuclear energy differences are usually much larger than atomic ones, so the emitted photons are much more energetic. That is why gamma rays have much higher energies than visible light or ordinary X rays.
If the excited nucleus has energy $E_i$ and the lower state has energy $E_f$, then the emitted gamma ray has energy approximately
$$
E_\gamma = E_i - E_f
$$
This relation is very important, although in a more precise treatment one also includes a very small recoil energy of the nucleus.
The gamma-ray energy is set by the difference between nuclear energy levels:
$$
E_\gamma \approx E_i - E_f
$$
Gamma rays therefore appear at specific, discrete energies.
How gamma emission differs from alpha and beta decay
Gamma decay is different from alpha decay and beta decay because it does not change the identity of the nuclide. In alpha decay, the nucleus loses two protons and two neutrons. In beta decay, a neutron changes into a proton or a proton changes into a neutron. In gamma emission, the same nucleus remains, but with less internal energy.
This means gamma emission is often a follow-up process. A nucleus may first undergo alpha or beta decay and be produced in an excited daughter state. That daughter then emits one or more gamma rays to reach its ground state.
A simple example is
$$
{}^A_ZX \to {}^A_{Z+1}Y^* + \beta^- + \bar{\nu}
$$
followed by
$$
{}^A_{Z+1}Y^* \to {}^A_{Z+1}Y + \gamma
$$
Energy and frequency of gamma rays
Because a gamma ray is a photon, its energy is related to its frequency by Planck's relation:
$$
E_\gamma = hf
$$
where $h$ is Planck's constant and $f$ is the frequency. Since gamma-ray energies are very large, their frequencies are extremely high and their wavelengths are very short.
Using the photon relation,
$$
E_\gamma = \frac{hc}{\lambda}
$$
where $\lambda$ is the wavelength and $c$ is the speed of light.
Recoil of the nucleus
When the nucleus emits a gamma photon, momentum must also be conserved. The photon carries momentum, so the nucleus recoils slightly in the opposite direction. Because of this, a tiny part of the transition energy goes into nuclear recoil rather than photon energy.
If the photon momentum is $p_\gamma$, then
$$
p_\gamma = \frac{E_\gamma}{c}
$$
and the recoiling nucleus gets equal and opposite momentum.
For most beginner calculations, the recoil correction is very small and is often neglected.
Single gamma rays and gamma cascades
Sometimes an excited nucleus reaches the ground state in one step and emits one gamma ray. In other cases, it passes through several intermediate excited states, emitting a sequence of gamma rays. This is called a gamma cascade.
For example,
$$
E_3 \to E_2 + \gamma_1
$$
then
$$
E_2 \to E_1 + \gamma_2
$$
and finally
$$
E_1 \to E_0 + \gamma_3
$$
The total emitted energy equals the total drop in nuclear energy:
$$
E_{\gamma_1} + E_{\gamma_2} + E_{\gamma_3} = E_3 - E_0
$$
Gamma cascades are very useful in identifying nuclear level structures.
Why gamma rays are discrete
A radioactive source often emits gamma rays only at certain specific energies. This happens because the nucleus can only occupy particular energy levels. A transition can occur only between allowed levels, so the emitted photons form a line spectrum rather than a continuous spread of energies.
This is one of the key signatures of nuclear structure.
| Quantity | Meaning |
|---|---|
| $E_i$ | Initial excited nuclear energy |
| $E_f$ | Final lower nuclear energy |
| $E_\gamma$ | Gamma-ray energy |
| $f$ | Gamma-ray frequency |
| $\lambda$ | Gamma-ray wavelength |
Example of a gamma transition
Suppose an excited nucleus is at an energy level $1.20\ \text{MeV}$ above the ground state and drops directly to the ground state. Then the emitted gamma ray has energy
$$
E_\gamma \approx 1.20\ \text{MeV}
$$
If instead it first drops to an intermediate level at $0.45\ \text{MeV}$ and then to the ground state, the two gamma rays have energies
$$
E_{\gamma_1} = 1.20 - 0.45 = 0.75\ \text{MeV}
$$
$$
E_{\gamma_2} = 0.45\ \text{MeV}
$$
The total emitted gamma energy is still
$$
0.75 + 0.45 = 1.20\ \text{MeV}
$$
Visualizing gamma emission
Typical features of gamma radiation
Gamma radiation has no electric charge and no rest mass because it consists of photons. It travels at the speed of light in vacuum. Compared with alpha and beta radiation, gamma rays are usually much more penetrating in matter. Their strong penetrating ability comes from the fact that they are high-energy photons, not massive charged particles.
However, the detailed way gamma rays interact with matter belongs to a later topic. Here the key point is that gamma-ray emission is an electromagnetic release of nuclear excitation energy.
Gamma rays are high-energy photons emitted by nuclei during transitions between nuclear energy levels.
They do not by themselves change $A$ or $Z$.
They carry away energy and momentum.
Relationship to nuclear spectroscopy
Because each nucleus has its own set of energy levels, the gamma rays it emits act like a fingerprint. By measuring gamma-ray energies, physicists can infer the spacing of nuclear levels and learn about nuclear structure. This is the basis of gamma-ray spectroscopy.
For beginners, the essential idea is simple. If you know the emitted gamma energies, you can work backward to the differences between nuclear energy states.
Summary equations
The most important formulas for gamma-ray emission are collected here.
Nuclear gamma transition:
$$
{}^A_ZX^* \to {}^A_ZX + \gamma
$$
Gamma-ray energy from level difference:
$$
E_\gamma \approx E_i - E_f
$$
Photon energy relation:
$$
E_\gamma = hf = \frac{hc}{\lambda}
$$
Photon momentum:
$$
p_\gamma = \frac{E_\gamma}{c}
$$
Gamma-ray emission is therefore the release of nuclear excitation energy in the form of a photon, usually after the nucleus has been created in an excited state. It is one of the clearest pieces of evidence that nuclei, like atoms, have discrete energy levels.