Nuclear Physics: Isotopes, Radioactivity, and Decay Laws

Basic Atomic and Nuclear Notation

• Element Representation: In nuclear physics, any element $x$ is represented using a specific notation where the symbol of the element is accompanied by its mass number and atomic number.

  • Mass Number ($A$): This is the number written at the top. It signifies the sum of protons and neutrons in the nucleus. These particles (protons and neutrons) are collectively known as nucleons (nucleus-residing particles).

    • Definition: $A = \text{protons} + \text{neutrons}$.

    • The mass number is always an integer because you cannot have a fraction of a proton or neutron.

  • Atomic Number ($Z$): This is the number written at the bottom. It represents the number of protons in the nucleus.

  • Neutron Number ($n$): This can be calculated by subtracting the atomic number from the mass number: $n = A - Z$.

• Atomic Number vs. Electrons: While in a neutral atom the number of protons equals the number of electrons, the definition of atomic number is strictly the number of protons.

  • In an Ion (like $Na^+$), the atomic number remains the same because the protons in the nucleus are untouched, but the electron count changes (e.g., $Na^+$ has 11 protons and 10 electrons).

Isotopes and Mass Spectrography

• General Definition of Isotopes: Atoms having the same atomic number ($Z$) but different mass numbers ($A$).

• Alternative Definition: Atoms having the same number of protons but a different number of neutrons.

• Properties:

  • Chemical Properties: These are the same because they depend on the atomic number and electron configuration.

  • Physical Properties: These differ due to the difference in mass.

• Hydrogen Isotopes:

  • Protium ($H_1^1$): 1 proton, 0 neutrons.

  • Deuterium ($H_1^2$): 1 proton, 1 neutron.

  • Tritium/Trityum ($H_1^3$): 1 proton, 2 neutrons. Tritium is a naturally occurring radioactive isotope of hydrogen.

• Atomic Mass vs. Mass Number: Atomic mass is the average mass of all the naturally occurring isotopes of an element. This is why atomic mass in the periodic table (like Chlorine 35.5) is often a decimal, whereas mass number is always an integer.

• Mass Spectrography: This is a technique used to segregate isotopes. It works on the principle that when a charged particle enters a magnetic field, it experiences a force ($f = qvB \times \text{sin}(\theta)$) causing it to move in a circular path.

  • Radius of path ($r$): $r = \frac{mv \times \text{sin}(\theta)}{qB}$.

  • Since different isotopes have different masses ($m$) but the same charge ($q$), they define different radii in the magnetic field, allowing them to be separated.

Radioactivity and Nuclear Stability

• Nature of the Nucleus: The nucleus is an extremely dense region with a radius of approximately $10^{-15}\text{m}$. To visualize this, if a whole room is an atom, the tip of a needle is the nucleus.

• Nuclear Forces:

  • Coulomb Force: The repulsive force between positively charged protons that tries to push the nucleus apart.

  • Strong Nuclear Force: An attractive force that exists between all nucleons (proton-proton, neutron-neutron, and proton-neutron). It is a short-range force effective only within the range of $10^{-15}\text{m}$.

• The Stability Criterion: Radioactivity occurs when the internal forces (Strong vs. Coulomb) are unbalanced, making the nucleus unstable.

• The Z-Threshold Myth: Many believe only elements with Z > 83 are radioactive. This is a guideline based on observation (heavy nuclei are generally unstable), not a strict condition. If any nucleus with a lower atomic number is unstable (e.g., $C^{14}$ or Tritium), it is radioactive.

Types of Nuclear Radiations

• Alpha ($\alpha) Particles:**\n * Resemblance: Identical to a **Helium nucleus** (He_2^4).\n * Charge: +2e$($3.2 imes 10^{-19} ext{C}).\n * Mass: Approximately 4 imes$the mass of a proton ($6.4 imes 10^{-27} ext{kg}).\n* **Beta ($\\beta) Particles:

  • Beta Negative ($\beta^-):** Fast-moving electrons.\n * **Beta Positive ($\\beta^+): Positrons (anti-particle of electrons).

  • Note: If only "Beta" is mentioned, it usually refers to $\\beta^-$ (electrons).

  • Charge: $e$ ($1.6 \times 10^{-19}\text{C}$).

  • Mass: Approximately $\frac{1}{1800}$ the mass of a proton.

• **Gamma ($\gamma) Rays:**\n * Nature: High-energy electromagnetic radiation (photons).\n * Charge: 0; Rest Mass: 0.\n * Note: Gamma rays are never emitted alone; they accompany alpha or beta emissions.\n\n# Properties: Ionization and Penetration\n\n* **Ionization Power:** The ability to strip electrons from atoms. It depends on charge.\n * Order: \alpha > \beta > \gamma.\n * Alpha has the highest charge (+2e), so it has maximum ionizing power.\n* **Penetration Power:** The ability to pass through matter. It depends on speed and lack of interaction.\n * Order: \gamma > \beta > \alpha.\n * Gamma rays travel at the speed of light (3 imes 10^8 ext{m/s}) and have no charge, allowing them to penetrate deeply.\n* **Deflection:** In electric/magnetic fields, Alpha and Beta particles deflect because they are charged. Gamma rays travel in a straight line (neutral).\n\n# Nuclear Decay Equations and Quark Theory\n\n* **Alpha Decay:** A nucleus emits an alpha particle (He_2^4).\n * The mass number decreases by 4 (A ightarrow A-4).\n * The atomic number decreases by 2 (Z ightarrow Z-2).\n* **Quarks:** Fundamental particles that make up nucleons.\n * **Up quark (u$):** Charge$+2/3e.\n * **Down quark (d$):** Charge$-1/3e.\n * **Proton Composition:** uud$(net charge$+e).\n * **Neutron Composition:** udd (net charge 0).\n* **Beta Decay Mechanism:**\n * **\beta^-$Decay:** A neutron converts into a proton ($udd ightarrow uud$). The atomic number increases by 1 ($Z ightarrow Z+1$), mass number remains the same ($A).\n * **\beta^+$Decay:** A proton converts into a neutron ($uud ightarrow udd$). The atomic number decreases by 1 ($Z ightarrow Z-1$), mass number remains the same ($A).\n* **Gamma Decay:** An excited nucleus releases energy as a photon; both A$and$Z remain unchanged.\n\n# Law of Radioactive Decay\n\n* **The Decay Law:** The rate of disintegration is directly proportional to the number of radioactive nuclei present.\n * \Delta N \propto -N_0 imes \Delta t.\n * **Decay Formula:** N = N_0 e^{-\lambda t}.\n * N_0$: Initial amount;$N$: Amount left after time$t.\n * \lambda$: Decay constant, measured in$ ext{s}^{-1}. It depends on the nature of the material.\n* **Half-Life (T_{1/2}):** The time required for half of the radioactive nuclei to decay.\n * T_{1/2} = \frac{0.693}{\lambda}.\n* **Activity (A$):** The number of disintegrations per unit time ($A = \lambda N).\n * **Units of Activity:**\n 1. **Becquerel (Bq):** 1 disintegration/second (SI Unit).\n 2. **Curie (Ci$):**$1 ext{Ci} = 3.7 imes 10^{10} ext{Bq}.\n 3. **Rutherford (Rd$):**$1 ext{Rd} = 10^6 ext{Bq}.\n\n# Calculation Method for Decay\n\n* **Number of Half-lives (n$):**$n = \frac{\text{Total Time}}{\text{Half-life}}.\n* **Amount Remaining:** N = \frac{N_0}{2^n}.\n* **Activity Rule:** After each half-life, the activity (A$) reduces by half, just like the mass. For example, if activity drops from$1600 ext{Bq}$to$100 ext{Bq}$, it has undergone 4 half-lives ($1600 ightarrow 800 ightarrow 400 ightarrow 200 ightarrow 100$$).