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Fundamentals of Nuclear Physics

CSCA Fundamentals of Nuclear Physics study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Modern Physics: Fundamentals of Nuclear Physics

1. Composition of the Nucleus

The nucleus consists of **Nucleons**, which are **Protons** and **Neutrons**.

**Notation**:

$$_Z^A\text{X}$$

* **$X$**: Element symbol.

* **$Z$**: **Proton Number** (Atomic Number). Determines chemical properties.

* **$A$**: **Mass Number**. $A = Z + N$ (where $N$ is Neutron Number).

**Isotopes**: Nuclei with same $Z$ but different $N$ (e.g., $^{1}\text{H}, ^{2}\text{H}, ^{3}\text{H}$).

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2. Radioactive Decay

Exams often test the properties of three types of rays and their deflection in magnetic fields.

(1) Types of Decay

| Type | Symbol | Nature | Charge | Penetration | Equation Example |

| :--- | :--- | :--- | :--- | :--- | :--- |

| **$\alpha$ Decay** | $_2^4\text{He}$ | Helium Nuclei | $+2e$ | Weak (Paper) | $_Z^A\text{X} \rightarrow _{Z-2}^{A-4}\text{Y} + _2^4\text{He}$ |

| **$\beta$ Decay** | $_{-1}^0\text{e}$ | Fast Electrons | $-1e$ | Medium (Aluminum) | $_Z^A\text{X} \rightarrow _{Z+1}^{A}\text{Y} + _{-1}^0\text{e}$ |

| **$\gamma$ Decay** | $\gamma$ | High Energy Photon | $0$ | Strong (Lead) | Accompanies $\alpha$ or $\beta$, no change in $A, Z$ |

**Notes**:

* The mechanism of $\beta$ decay is a **neutron** converting into a **proton** and an **electron**: $_0^1\text{n} \rightarrow _1^1\text{p} + _{-1}^0\text{e}$.

* In a magnetic field, $\alpha$ and $\beta$ deflect in opposite directions (Lorentz force), while $\gamma$ goes straight. $\beta$ usually has a smaller radius of curvature due to its tiny mass.

(2) Half-life ($T$)

The time required for half the nuclei to decay. **Independent of temperature, pressure, or chemical state**.

**Formula**:

$$ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} $$

Or using mass: $m = m_0 (1/2)^{t/T}$.

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3. Nuclear Reactions & Conservation Laws

All nuclear reactions (decay, fission, fusion) must obey two conservation laws:

1. **Conservation of Mass Number** (Sum of superscripts stays constant).

2. **Conservation of Charge Number** (Sum of subscripts stays constant).

**Exam Tip**:

Set the unknown particle as $_Z^A\text{X}$ and solve simple linear equations for $A$ and $Z$.

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4. Nuclear Energy & Mass-Energy Equation

(1) Mass-Energy Equivalence

$$ E = mc^2 $$

Change in energy: $\Delta E = \Delta m c^2$.

(2) Mass Defect ($\Delta m$)

The mass of a nucleus is always **less** than the sum of its constituent nucleons. The difference becomes **Binding Energy**.

$$ \Delta m = (Z m_p + N m_n) - m_{\text{nucleus}} $$

(3) Specific Binding Energy ($\epsilon$)

Also called **Average Binding Energy per Nucleon**: $\epsilon = \frac{\Delta E}{A}$.

* Higher $\epsilon$ means the nucleus is more **stable**.

* Nuclei near **Iron ($^{56}\text{Fe}$)** have the highest $\epsilon$ (most stable).

* **Fission**: Heavy nuclei (U) split into medium nuclei. $\epsilon$ increases $\rightarrow$ **Energy Released**.

* **Fusion**: Light nuclei (H) combine into heavier nuclei. $\epsilon$ increases drastically $\rightarrow$ **Energy Released**.

(4) Calculation Shortcut (Crucial for CSCA)

When mass defect is given in **atomic mass units (u)**, use:

$$ \Delta E (\text{MeV}) = \Delta m (\text{u}) \times 931.5 $$

This is much faster than using kg and $c^2$.