Mass defect and nuclear binding energy

1. Overview

The fundamental principle of nuclear physics is the equivalence of mass and energy. Einstein’s mass-energy relation, $E = mc^2$, implies that mass is a highly concentrated form of energy. In the context of the nucleus, this principle explains why the mass of a stable nucleus is always less than the sum of the masses of its constituent protons and neutrons. This difference, known as the mass defect, represents the energy released when the nucleus was formed, or conversely, the binding energy required to dismantle it. Nuclear reactions, including fission and fusion, occur because nuclei transition from states of lower stability to higher stability, releasing the difference in binding energy as kinetic energy or gamma radiation.

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Key Definitions

• Mass Defect ($\Delta m$): The difference between the total mass of the individual, separate nucleons (protons and neutrons) and the rest mass of the nucleus.
• Binding Energy ($E_B$): The minimum external energy required to completely separate the nucleons in a nucleus to infinity.
• Binding Energy per Nucleon ($E_B/A$): The total binding energy of a nucleus divided by its nucleon number ($A$). It represents the average energy required to remove a single nucleon from the nucleus and is the true measure of nuclear stability.
• Nuclear Fusion: A process where two light nuclei combine to form a heavier, more stable nucleus. This process occurs for nuclei with low nucleon numbers (typically $A < 56$).
• Nuclear Fission: A process where a heavy, unstable nucleus (such as Uranium-235) splits into two lighter, more stable daughter nuclei and neutrons. This occurs for nuclei with high nucleon numbers (typically $A > 56$).
• Unified Atomic Mass Unit (u): Defined as exactly one-twelfth of the mass of an isolated, ground-state carbon-12 atom. ($1 \text{ u} \approx 1.6605 \times 10^{-27} \text{ kg}$).

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Content

3.1 Mass-Energy Equivalence

Einstein’s theory of special relativity states that mass and energy are interchangeable. A change in the energy of a system results in a proportional change in its mass.

$E = mc^2$

Where:

• $E$ is the energy equivalent of the mass (Joules, J).
• $m$ is the rest mass (kilograms, kg).
• $c$ is the speed of light in a vacuum ($\approx 3.00 \times 10^8 \text{ m s}^{-1}$).

In nuclear physics, we often deal with the change in mass ($\Delta m$) to find the change in energy ($\Delta E$):

$\Delta E = \Delta m c^2$

Because $c^2$ is a very large constant ($\approx 9 \times 10^{16} \text{ m}^2 \text{ s}^{-2}$), a tiny change in mass results in a massive release of energy. This is why nuclear reactions are millions of times more energetic than chemical reactions (which involve electron transitions and much smaller mass changes).

3.2 Representing Nuclear Reactions

Nuclear reactions are represented by equations where the identity of the elements changes. To be valid, these equations must satisfy two conservation laws:

1. Conservation of Nucleon Number ($A$): The sum of the top numbers (protons + neutrons) must be equal on both sides.
2. Conservation of Proton Number ($Z$): The sum of the bottom numbers (charge) must be equal on both sides.

General Notation: ${}^A_Z\text{X}$

Example: Alpha Decay of Radium-226 $${}^{226}_{88}\text{Ra} \rightarrow {}^{222}_{86}\text{Rn} + {}^{4}_{2}\text{He}$$

• $A$: $226 = 222 + 4$ (Conserved)
• $Z$: $88 = 86 + 2$ (Conserved)

3.3 Mass Defect and Binding Energy

When nucleons are brought together to form a nucleus, they are bound by the strong nuclear force. As they move closer, they lose potential energy. According to $E=mc^2$, this loss of energy manifests as a loss of mass.

Calculating Mass Defect ($\Delta m$):
To find the mass defect, compare the mass of the "parts" to the mass of the "whole":
$\Delta m = [Z m_p + (A - Z) m_n] - M_{\text{nucleus}}$

Where:

• $Z$ = number of protons.
• $(A-Z)$ = number of neutrons.
• $m_p$ = mass of a free proton.
• $m_n$ = mass of a free neutron.
• $M_{\text{nucleus}}$ = mass of the bound nucleus.

Note on Atomic vs. Nuclear Mass: Most data tables provide atomic masses (which include electrons). To find the nuclear mass, you must subtract the mass of the $Z$ electrons from the atomic mass. In a reaction equation, the electron masses usually cancel if you use atomic masses on both sides. But when finding a mass defect from the masses of free protons and neutrons, they do not cancel: use the nuclear mass, or use hydrogen-atom masses in place of proton masses.

Calculating Binding Energy ($E_B$):
Once $\Delta m$ is found in kg, use:
$E_B = \Delta m c^2$

3.4 The Binding Energy per Nucleon Curve

The stability of a nucleus is not determined by its total binding energy, but by its binding energy per nucleon.

Key Features and Interpretations:

1. Maximum Stability: Nuclei near iron-56 have the highest $E_B/A$. They are the most tightly bound nuclei.
2. Fusion (Light Nuclei): For $A < 56$, joining two light nuclei to form a heavier one increases the $E_B/A$. The nucleons become more tightly bound, and the "missing" energy is released.
3. Fission (Heavy Nuclei): For $A > 56$, splitting a heavy nucleus into two medium-sized nuclei increases the $E_B/A$. The daughter nuclei are more stable than the parent, and energy is released.
4. The Drop at High A: The curve drops for heavy nuclei because the electrostatic repulsion between the large number of protons starts to counteract the short-range strong nuclear force.

3.5 Fission vs. Fusion: Energetics

Energy is released in a nuclear reaction if the total mass of the products is less than the total mass of the reactants.

• Nuclear Fusion:
  • Occurs when light nuclei (e.g., Hydrogen isotopes) fuse.
  • Requires extremely high temperatures and pressures to overcome the Coulomb repulsion (electrostatic force) between positively charged nuclei.
  • Releases significantly more energy per unit mass than fission.
• Nuclear Fission:
  • Occurs when a heavy nucleus (e.g., U-235) captures a neutron and becomes unstable.
  • The nucleus splits into two smaller "fission fragments" and several neutrons.
  • The energy released is primarily the kinetic energy of the fragments.

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Key Equations

Equation	Meaning	Status
$E = mc^2$	Mass-energy equivalence	Memorise
$\Delta E = c^2 \Delta m$	Energy released from mass defect	Memorise
$\Delta m = (Z m_p + (A-Z) m_n) - M_{\text{nucleus}}$	Calculation of mass defect	Memorise
$E_{B/A} = \frac{E_B}{A}$	Binding energy per nucleon	Memorise
$1 \text{ u} = 1.66 \times 10^{-27} \text{ kg}$	Mass conversion factor	Data Sheet
$1 \text{ u} = 931.5 \text{ MeV}$	Energy conversion shortcut	Not on data sheet
$1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$	Energy unit conversion	Data Sheet

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5. Worked Examples

Worked Example 1 — Binding Energy of Helium-4

Calculate the binding energy per nucleon of a Helium-4 nucleus (${}^4_2\text{He}$) in MeV.
Data:
Mass of ${}^4_2\text{He}$ nucleus = $4.001506 \text{ u}$
Mass of proton ($m_p$) = $1.007276 \text{ u}$
Mass of neutron ($m_n$) = $1.008665 \text{ u}$
$1 \text{ u} = 931.5 \text{ MeV}$

Step 1: Identify the constituents.
Helium-4 has $Z=2$ protons and $A-Z = 4-2 = 2$ neutrons.

Step 2: Calculate the mass of the individual nucleons.
$m_{\text{parts}} = (2 \times 1.007276) + (2 \times 1.008665)$
$m_{\text{parts}} = 2.014552 + 2.017330 = 4.031882 \text{ u}$

Step 3: Calculate the mass defect ($\Delta m$).
$\Delta m = m_{\text{parts}} - M_{\text{nucleus}}$
$\Delta m = 4.031882 - 4.001506 = 0.030376 \text{ u}$

Step 4: Convert mass defect to Binding Energy ($E_B$).
Using the shortcut: $E_B = 0.030376 \times 931.5 \text{ MeV}$
$E_B = 28.295 \text{ MeV}$

Step 5: Calculate Binding Energy per nucleon.
$E_{B/A} = \frac{28.295}{4} = 7.07 \text{ MeV}$

Final Answer: $7.07 \text{ MeV}$

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Worked Example 2 — Energy Release in Fusion

A fusion reaction occurs between Deuterium (${}^2_1\text{H}$) and Tritium (${}^3_1\text{H}$): $${}^2_1\text{H} + {}^3_1\text{H} \rightarrow {}^4_2\text{He} + {}^1_0\text{n}$$ Calculate the energy released in Joules.
Data:
Mass of ${}^2_1\text{H} = 2.014102 \text{ u}$
Mass of ${}^3_1\text{H} = 3.016049 \text{ u}$
Mass of ${}^4_2\text{He} = 4.002603 \text{ u}$
Mass of ${}^1_0\text{n} = 1.008665 \text{ u}$

Step 1: Calculate total mass of reactants.
$m_{\text{reactants}} = 2.014102 + 3.016049 = 5.030151 \text{ u}$

Step 2: Calculate total mass of products.
$m_{\text{products}} = 4.002603 + 1.008665 = 5.011268 \text{ u}$

Step 3: Calculate the mass change ($\Delta m$).
$\Delta m = 5.030151 - 5.011268 = 0.018883 \text{ u}$

Step 4: Convert $\Delta m$ to kg.
$\Delta m = 0.018883 \times 1.66 \times 10^{-27} \text{ kg} = 3.134578 \times 10^{-29} \text{ kg}$

Step 5: Calculate energy released ($\Delta E$).
$\Delta E = \Delta m c^2$
$\Delta E = (3.134578 \times 10^{-29}) \times (3.00 \times 10^8)^2$
$\Delta E = 2.82 \times 10^{-12} \text{ J}$

Final Answer: $2.82 \times 10^{-12} \text{ J}$

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Common Mistakes to Avoid

• ❌ Wrong: Rounding mass values to 2 or 3 decimal places mid-calculation.
• ✅ Right: Keep all decimal places provided in the data (usually 5 or 6). The mass defect is a tiny difference between two large numbers; rounding early will result in a zero or highly inaccurate energy value.
• ❌ Wrong: Using $E = mc^2$ with mass in atomic units (u) and expecting Joules.
• ✅ Right: If you use $c = 3.00 \times 10^8 \text{ m s}^{-1}$, your mass must be in kg to get energy in Joules.
• ❌ Wrong: Confusing Binding Energy with Binding Energy per Nucleon.
• ✅ Right: Total Binding Energy is the energy to dismantle the whole nucleus. Binding Energy per Nucleon is the measure of stability. A large nucleus like Uranium has a higher total $E_B$ than Helium, but it is less stable because its $E_B/A$ is lower.
• ❌ Wrong: Thinking energy is "stored" in the mass defect.
• ✅ Right: Mass defect is the mass that is no longer there because it was released as energy.

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Exam Tips

1. The "Separated to Infinity" Keyword: When defining Binding Energy, you must mention that the nucleons are separated to infinity. This ensures there is no residual strong force interaction between them.
2. Energy from B.E. per Nucleon: If a question gives you the $E_B/A$ of reactants and products:
   • $\text{Energy Released} = (\text{Total } E_B \text{ of Products}) - (\text{Total } E_B \text{ of Reactants})$.
   • Remember to multiply the $E_B/A$ by the nucleon number $A$ for each nucleus involved.
3. Graph Accuracy: When sketching the $E_B/A$ vs $A$ graph:
   • Start the curve at $(1, 0)$.
   • The peak is at $A \approx 56$ (Iron).
   • The peak value is approx $8.8 \text{ MeV}$.
   • The curve for heavy nuclei ($A > 200$) should still be around $7.5 \text{ - } 8.0 \text{ MeV}$—do not let it drop too low.
4. Unit Conversions: Practice moving between $\text{u} \rightarrow \text{kg} \rightarrow \text{J} \rightarrow \text{eV} \rightarrow \text{MeV}$ fluently.
   • $1 \text{ u} = 931.5 \text{ MeV}$ is a massive time-saver for multiple-choice questions.
5. Explain Questions: If asked why energy is released in a reaction:
   • State that the products have a higher binding energy per nucleon than the reactants.
   • State that the total rest mass of the products is less than the total rest mass of the reactants.
   • Mention $\Delta E = \Delta mc^2$.