The energy spectrum of beta-particles [number N(E) as a function of beta-energy E] emitted from a ra

Q: Why isn't the energy of the $\beta$-particle discrete like alpha particles?

Alpha decay is a two-body process (daughter nucleus + alpha particle), so conservation of momentum and energy dictates a fixed energy for the alpha particle. Beta decay is a three-body process (daughter nucleus + electron + antineutrino), so the energy is shared continuously between the electron and the antineutrino.

Q: What exactly is $E_0$ on the graph?

$E_0$ is the maximum possible kinetic energy the $\beta$-particle can have. This happens in the extremely rare case where the antineutrino is emitted with zero kinetic energy. It is equal to the Q-value of the nuclear reaction.

Q: Why is the peak of the graph skewed to the left instead of being in the middle?

The skewness is caused by two main factors: the statistical phase space (which favors unequal sharing of energy) and the Coulomb attraction from the positively charged daughter nucleus, which slows down the escaping negative electron, shifting the average energy lower.

Q: Does the daughter nucleus take away any of the energy?

Yes, to conserve momentum, the daughter nucleus must recoil. However, because the nucleus is thousands of times more massive than the electron and antineutrino, its recoil kinetic energy is negligibly small and usually ignored in these calculations.

Q: What would the graph look like if the neutrino didn't exist?

If the neutrino didn't exist, beta decay would be a two-body process. The graph would not be a continuous curve at all; it would just be a single, sharp vertical line at $E = E_0$, indicating that every electron has the exact same energy.

Question

The energy spectrum of $\beta$-particles [number $N(E)$ as a function of $\beta$-energy $E$] emitted from a radioactive source is

Sigma Percentile · Accepted Answer

Have you ever looked at a graph and realized that it holds the secret to an entirely new, invisible particle? The energy spectrum of $\beta$-particles is not just a curve on a piece of paper; it is a historical map that led physicists to one of the greatest discoveries of the 20th century. Let's dive into the fascinating world of radioactivity and uncover the story behind this graph.

### The Setup: Beta Decay

Imagine a radioactive nucleus sitting quietly. Suddenly, it undergoes $\beta$-decay. For a long time, scientists believed that this process was simple: a neutron inside the nucleus turns into a proton, and an electron (the $\beta$-particle) is shot out.

If this were the whole story, the laws of physics would demand something very specific. According to the conservation of energy and momentum, a two-body decay (where a parent nucleus splits into a daughter nucleus and an electron) must result in the electron carrying away a fixed, discrete amount of kinetic energy.

Every single electron emitted from identical nuclei should have the exact same energy, let's call it $E_0$. If you were to plot the number of particles $N(E)$ against their energy $E$, you would expect to see a single, sharp spike at $E = E_0$.

### The Energy Crisis

But nature had a surprise in store. When experimentalists actually measured the kinetic energy of these emitted $\beta$-particles, they didn't see a sharp spike. Instead, they saw a broad, continuous curve!

The electrons were coming out with all sorts of energies, ranging from zero all the way up to the maximum energy $E_0$.

$$0 \le E_\beta \le E_0$$

This was a massive crisis in physics. Where was the missing energy going? Was the sacred law of conservation of energy being violated in the quantum realm? Niels Bohr famously suggested that maybe energy conservation was only a statistical law, not an absolute one.

### The Neutrino to the Rescue

Enter Wolfgang Pauli. In 1930, he proposed a desperate remedy. He suggested that the missing energy wasn't lost at all. Instead, it was being carried away by a third, invisible particle that was emitted alongside the electron.

This particle had to be electrically neutral (to conserve charge) and incredibly light (to explain the kinematics). Enrico Fermi later named it the "neutrino" (the little neutral one).

Because the decay actually produces three bodies—the daughter nucleus, the electron, and the antineutrino—the total disintegration energy $E_0$ is shared between the electron and the antineutrino.

$$E_\beta + E_
u = E_0$$

Since the energy is shared randomly, the electron can have any kinetic energy from zero to $E_0$, perfectly explaining the continuous spectrum!

### The Shape of the Spectrum

Now, let's look at the exact shape of this continuous curve. The number of particles $N(E)$ must be zero at $E=0$ (because it's highly unlikely for the electron to get absolutely no energy) and zero at $E=E_0$ (because it's equally unlikely for the electron to get all the energy while the antineutrino gets none).

But the curve isn't a perfectly symmetric bell shape. Due to the statistical nature of how the energy is shared (the phase space) and the attractive Coulomb force from the positively charged nucleus pulling on the escaping negative electron, the spectrum is skewed.

Most electrons end up with less than half of the total available energy. This means the peak of the curve shifts to the left, occurring at roughly $E_0/3$.

When we look at the options provided in the question, we are searching for a graph that is continuous, starts at the origin, ends at $E_0$, and has a peak skewed to the lower energy side. Option (c) perfectly captures this physical reality.

The next time you see this skewed curve, remember: you are looking at the exact mathematical footprint of the elusive neutrino!

Animated Solution for Physics - Atoms and Nuclei: The energy spectrum of $β$ -particles [number $N (E)$ as a function of $β$ -energy $E$ ] emitted from a radioactive source is

Select Answer:

Visualized Solution

$The β -Decay Process$

$Energy Conservation$

$Continuous Energy Spectrum$

$The Shape of the Spectrum$

$Conclusion$

The Sigma Insight: Radioactivity

The Setup

The Energy Crisis

The Neutrino to the Rescue

The Shape of the Spectrum

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Animated Solution for Physics - Atoms and Nuclei: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The energy spectrum of β-particles [number N(E) as a function of β-energy E] emitted from a radioactive source is