23. Mass Defect and Nuclear Binding Energy

Mass-Energy Equivalence

Equivalence of Mass and Energy

In 1905, Albert Einstein published his research showing that matter can be considered a form of energy. He proposed that mass and energy are entirely interchangeable.

• Mass can be converted into energy (e.g., nuclear fusion in the sun, nuclear power plants, particle accelerators).
• Energy can be converted into mass.

According to Einstein, since the decay of any radioactive nucleus releases energy, the total mass before the decay MUST exceed the total mass after the decay. The difference in mass before and after the decay is entirely equivalent to the total kinetic and electromagnetic energy of the decay products.

Mass-Energy Equivalence Equation

$\Delta E=\Delta m{c}^2$ Where:

• $E$ = Energy (J)
• $m$ = Mass (kg)
• $c$ = Speed of light ($3.00\times 10^8{\text{ m s}}^{-1}$)

Atomic Mass Unit Conversion: Atomic, nuclear, and particle masses are usually expressed in unified atomic mass units (u). $1\text{ u}=1.6605\times 10^{-27}\text{ kg}$ The energy equivalent of $1\text{ u}$ is exactly $931\text{ MeV}$. This enables a direct conversion from mass in u to energy in MeV without needing to use $E=m{c}^2$ in kg and Joules first.

Mass Defect

Definition: Mass Defect

The difference between the mass of a nucleus and the total mass of its constituent nucleons when they are infinitely separated

Calculating Mass Defect

$\Delta m=Z{m}_p+(A-Z)m_n-m_{total}$ Where:

• $Z$ = proton number
• $A$ = nucleon number
• $m_p$ = mass of a proton
• $m_n$ = mass of a neutron
• $m_{total}$ = measured mass of the intact nucleus

Nuclear Binding Energy

But why is Infinitely Separated Nucleons > Mass of Nucleus

Mass of nucleus compared to mass of infinitely separated nucleons

• In the nucleus, the nucleons are held together by the strong nuclear force.
• In order to infinitely separate the nucleons, energy must be supplied to the system to overcome the strong nuclear force.
• This supplied energy is known as the Nuclear Binding Energy.
• Since energy is supplied to the system, the total energy of the separated nucleons will be larger than the total energy of the nucleons when they are in the nucleus.
• Since $\Delta E=\Delta m{c}^2$, the infinitely separated nucleons must have a larger total mass than the nucleus.

Energy change when nucleons are separated to infinity

Definition: Nuclear Binding Energy

The minimum energy required to separate all nucleons in a nucleus to infinity.

Binding Energy per Nucleon

Binding Energy per Nucleon

$\text{Binding Energy Per Nucleon}=\frac{\text{Binding Energy}}{\text{Number of Nucleons}}$

Meaning of Binding Energy Per Nucleon

A higher binding energy per nucleon indicates a higher stability (more energy is required to separate the nucleons).

Binding Energy per Nucleon Graph

Graph of binding energy per nucleon against nucleon number showing the peak at Iron-56

Key features of the graph:

1. Low Values of A

• Light nuclei have lower binding energies per nucleus.
• They have weaker electrostatic forces and will undergo nuclear fusion to become more stable.
• Helium-4 (${}_2^4\text{He}$), Carbon-12 (${}_6^{12}\text{C}$), and Oxygen-16 (${}_8^{16}\text{O}$) are anomalies; they do not fit the smooth trend and have exceptionally high BE/nucleon for their size.

2. Peak at A $\approx$ 56

• Iron-56 has the highest binding energy per nucleon ($\approx 8.8\text{ MeV}$) making it the most stable of all elements.

3. High Values of A

• The heaviest elements unstable and will undergo nuclear fission.
• This instability is the result of electrostatic repulsion between protons starting to challenge the strong nuclear force.

4. Average Binding Energy per Nucleus

• The average binding energy per nucleus is approximately $8\text{ MeV}$.

Examiner Advice for Graph Sketching

If asked to sketch the Binding Energy per Nucleon graph:

• Start the curve at A = 1 (do not start at A = 0, as $A=0$ is not a nucleus!).
• Draw a steep initial positive gradient on the left.
• Create a rounded peak around A = 56 (Iron).
• Draw a shallower, continually decreasing negative gradient for higher A values.
• The line must not drop back down to 0 binding energy.

Nuclear Fusion

Definition: Nuclear Fusion

• Two small nuclei join together to form one large nucleus.
• Total binding energy increases.

Example Nuclear Fusion Reaction

${}_1^2\mathrm{H}+{}_1^2\mathrm{H}\to {}_2^3\mathrm{He}+{}_0^1\mathrm{n}+\text{energy}$

The Mechanics of Nuclear Fusion

• Fusion requires extremely high temperatures (like the core of a star) because both nuclei are positively charge and electrostatically repel each other strongly.
• High kinetic energy is needed to force the nuclei close enough together (within $10^{-15}\text{ m}$) so that the short-range attractive strong nuclear force can overcome the electrostatic repulsion and bind them.

Diagram showing deuterium and tritium fusing to form helium, releasing a neutron and energy

Why Total Binding Energy Increases

Increase in strong nuclear force interactions during Deuterium-Deuterium fusion

1. Fundamental Strengths

• The strong nuclear force is the strongest of the four fundamental interactions.
• It is roughly 100 times stronger than the electrostatic repulsion between protons at subatomic distances.

2. Range Limitation

• Unlike electromagnetism which has an infinite range, the strong nuclear force has an extremely short range ($\approx 1-2\text{ fm}$).
• It effectively acts only between immediate neighbours in the nucleus.

3. Nuclear Coordination in Fusion

• When small nuclei fuse into a larger nucleus, each nucleon is surrounded by more direct neighbours.

4. More Attractive Bonds

• The total number of strong nuclear interactions increases.

5. The Iron-56 Peak

• Up to Iron-56, the increase in strong nuclear interactions is larger than the increase in electrostatic repulsion between protons.

6. Resulting Binding Energy

• More energy is needed to separate the nucleons to infinity, therefore binding energy increases.

Nuclear Fission

Definition: Nuclear Fission

• A single large nucleus divides to form smaller nuclei
• Total binding energy increases

Example Nuclear Fission Reaction

${}_{92}^{236}\mathrm{U}\to {}_{56}^{141}\mathrm{Ba}+{}_{36}^{92}\mathrm{Kr}+3{}_0^1\mathrm{n}+\text{energy}$

The Mechanics of Nuclear Fission

• Usually initiated by neutron bombardment.
• An unstable heavy nucleus absorbs a slow-moving neutron, becomes highly unstable, and splits into fission fragments (daughter nuclei).
• Energy and more neutrons are releases (which can trigger a chain reaction)

Diagram showing a neutron striking a target nucleus causing it to split into two fission products and three neutrons

Why Total Binding Energy Increases

Splitting of a large nucleus during nuclear fission

Cumulative repulsion from protons in large vs small nuclei

1. Saturation of the Strong Nuclear Force

• A proton only feels attraction from its 3 or 4 immediate neighbours.

• The attractive strong force per nucleon is roughly the same in both large and small nuclei.

2. Long Range Electrostatic Repulsion

• Electromagnetism has an infinite range.

• In the smaller nucleus, there are fewer total protons.

• Proton-Proton repulsion is less in smaller nuclei.

3. Effect on Binding Energy

• In smaller nuclei, there is less repulsion fighting against the strong force.
• Binding energy per nucleon increases (since they are held more tightly together)
• Total binding energy increases.

Energy Released during Nuclear Reactions

Calculating Energy Released

The energy released in a nuclear reaction can be calculated using the mass-energy equivalence formula: $E=c^2\Delta m$

Steps for calculation:

1. Calculate the total mass of reactants.
2. Calculate the total mass of products.
3. Find the mass difference ($\Delta m={\text{mass}}_{reactants}-{\text{mass}}_{products}$).
4. Convert $\Delta m$ from unified atomic mass units (u) into kg (multiply by $1.66\times 10^{-27}$).
5. Multiply by $c^2$ to find Energy in Joules. (Alternatively, multiply mass difference in ‘u’ directly by 931 to get energy in MeV).