8.2.4.5. Energy Spectrum

Table of Contents

Continuous Beta Spectrum

In beta decay, the emitted electron or positron does not come out with just one fixed energy. Instead, it can have many possible energies, from nearly zero up to some maximum value. This spread of possible energies is called the beta decay energy spectrum.

This behavior is very different from alpha decay, where the emitted alpha particle usually has a sharply defined energy. The key reason is that beta decay involves more than two particles in the final state. In beta minus decay, a neutron in the nucleus changes into a proton while emitting an electron and an antineutrino. In beta plus decay, a proton changes into a neutron while emitting a positron and a neutrino. Because the decay energy must be shared among the daughter nucleus, the beta particle, and the neutrino or antineutrino, the beta particle can carry different amounts of energy in different decay events.

Why the Spectrum Is Continuous

Suppose a nucleus undergoes beta minus decay:

$$
{}^A_ZX \to {}^A_{Z+1}Y + e^- + \bar{\nu}_e
$$

The total decay energy, often called the $Q$ value, is fixed for a given nuclear transition. But that fixed energy is divided among the emitted particles. Since the neutrino can take away different amounts of energy, the electron energy is not fixed.

If the daughter nucleus is much heavier than the electron, its recoil energy is usually small, though not exactly zero. So, to a good approximation, the available decay energy is shared mainly between the beta particle and the neutrino.

This gives a continuous distribution of beta energies.

In beta decay, the beta particle does not have a single energy. Its kinetic energy ranges continuously from near zero up to a maximum endpoint energy.

Endpoint Energy

The largest possible beta particle energy is called the endpoint energy, or maximum beta energy. It occurs when the neutrino carries away as little energy as possible and the daughter nucleus recoils only slightly.

If $K_\beta$ is the beta particle kinetic energy, then

$$
0 \le K_\beta \le K_{\beta,\max}
$$

The endpoint energy is close to, but not always exactly equal to, the total decay energy available to the leptons, because a small amount goes into nuclear recoil.

A typical beta spectrum therefore starts at low energy, rises to some peak, and then falls to zero at the endpoint.

Shape of the Spectrum

The beta spectrum is not flat. Some energies are more likely than others. In many cases, the number of beta particles emitted with kinetic energy between $K$ and $K + dK$ is written as

$$
N(K)\,dK
$$

where $N(K)$ is the spectral distribution function.

A simplified form of the allowed beta spectrum is

$$
N(K) \propto pE(E_0 - K)^2
$$

where $p$ is the beta particle momentum, $E$ is its total energy, and $E_0$ is the endpoint kinetic energy in a simplified treatment.

More carefully, the total electron energy is

$$
E = K + m_ec^2
$$

and the momentum is related by

$$
E^2 = p^2c^2 + m_e^2c^4
$$

The factor $(E_0 - K)^2$ appears because the neutrino also carries energy and momentum, and the number of possible neutrino states changes with the leftover energy.

For real nuclei, the spectrum is also modified by the electric field of the nucleus, which attracts electrons in beta minus decay and repels positrons in beta plus decay. This changes the detailed shape, especially at low energies.

A beta spectrum has a characteristic endpoint. The count rate goes to zero at the maximum beta energy because no more energy is available for the beta particle beyond that point.

Comparison of Beta Minus and Beta Plus Spectra

Both beta minus and beta plus decays produce continuous spectra, but the detailed shapes differ because of the charge of the emitted particle and the Coulomb interaction with the nucleus.

Decay type	Emitted particle	Neutral lepton	Spectrum type	Nuclear effect
Beta minus	$e^-$	$\bar{\nu}_e$	Continuous	Electron is attracted by nucleus
Beta plus	$e^+$	$\nu_e$	Continuous	Positron is repelled by nucleus

In beta plus decay, some decay energy is also needed to create the positron mass, so the available kinetic energy is reduced compared with a similar beta minus transition.

Historical Importance

The continuous beta spectrum was one of the most important clues in nuclear physics. Early physicists expected the emitted electron to have a fixed energy if only the daughter nucleus and electron were produced. But experiments showed a continuous distribution instead.

This seemed to threaten energy conservation, until the neutrino was proposed. The missing energy was understood to be carried away by an unseen particle. Once the neutrino was included, conservation of energy and momentum were restored.

The continuous beta spectrum provided strong evidence that a third particle, the neutrino, is emitted in beta decay.

Typical Spectrum Sketch

A beta spectrum is often shown as the number of particles detected versus beta particle kinetic energy.

Rendering scientific sketch…

Typical beta decay energy spectrum

The curve rises from low energy, reaches a maximum, and then falls to zero at the endpoint. Real measured spectra may look less smooth because of detector effects, background, and finite resolution.

Role of Nuclear Recoil and Excited States

The daughter nucleus recoils slightly, so not all the decay energy goes to the beta particle and neutrino. Also, if the daughter nucleus is produced in an excited state, some energy is used to excite the nucleus. Then the beta endpoint is lower than for decay to the ground state.

This means one radioactive nucleus can sometimes produce several beta spectral branches, each with its own endpoint energy, corresponding to different final nuclear states.

Final state of daughter nucleus	Energy available to beta particle and neutrino	Endpoint energy
Ground state	Largest	Highest
Excited state	Reduced by excitation energy	Lower
Higher excited state	Even smaller	Even lower

Fermi-Kurie View

To analyze beta spectra, physicists often transform the data into a form that should become approximately linear for an allowed transition. This is called a Fermi-Kurie plot. The endpoint is then easier to determine from the intercept.

The detailed theory belongs to more advanced nuclear physics, but the key beginner idea is simple. A transformed beta spectrum can help reveal the maximum energy and whether the decay follows the expected spectral shape.

What Experiments Measure

In experiments, a detector records how many beta particles arrive with different energies. The observed spectrum is influenced by several effects. Low-energy beta particles may be absorbed in the source material itself. Detector resolution can blur the endpoint. Background radiation can add extra counts. Even so, the essential continuous form remains clear.

Because the endpoint energy depends on the nuclear transition, measuring the beta spectrum gives information about the decay process and the energy released.

Essential Takeaway

The energy spectrum in beta decay is continuous because the available decay energy is shared among the beta particle, the neutrino, and the recoiling daughter nucleus. The beta particle energy therefore varies from event to event, up to a definite maximum called the endpoint energy.

Key facts about beta decay energy spectrum:
$$
0 \le K_\beta \le K_{\beta,\max}
$$
The spectrum is continuous, not discrete.
The endpoint energy is the maximum beta kinetic energy.
The neutrino is essential for explaining the continuous spectrum and for conserving energy and momentum.

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