Comprehensive Study Notes on Radioactivity and Modern Physics

Structure of the Atom and Nucleus

• Historical Discovery and Foundation:

  • The structure of the atom and nucleus was established towards the end of the nineteenth century following the discovery of electrons by Sir J.J. Thomson and alpha particle scattering experiments performed by Rutherford and others.

• Atomic Constituents:

  • An atom consists of three primary subatomic particles: electrons, protons, and neutrons.

  • Protons and neutrons reside inside the centrally located nucleus.

  • Electrons revolve around the nucleus in specific paths called stationary orbits (or stationary shells).

  • Electrons moving in stationary orbits do not radiate energy. Every stationary shell corresponds to a definite electron energy level.

• Electron Shell Configuration:

  • The total number of shells in an atom depends on its total electron count, with each shell having a strict capacity limit.

  • The maximum number of electrons a shell can accommodate is given by the formula $2n^2$, where $n$ represents the shell number.

  • Shell designations, corresponding values of $n$, and their maximum electron capacities are:

    • $n = 1$: K shell, maximum $2$ electrons

    • $n = 2$: L shell, maximum $8$ electrons

    • $n = 3$: M shell, maximum $18$ electrons

    • $n = 4$: N shell, maximum $32$ electrons

    • $n = 5$: O shell, maximum $50$ electrons

    • $n = 6$: P shell, maximum $72$ electrons

    • $n = 7$: Q shell, maximum $98$ electrons

  • Electron energy increases progressively as the shell number $n$ increases.

• Dimensions of Atom and Nucleus:

  • The overall size of an atom is determined by the radius of its outermost electron shell and is of the order of $10^{-10}\,\text{m}$.

  • The nucleus is positioned at the center of the atom with a size of the order of $10^{-15}\,\text{m}$ to $10^{-14}\,\text{m}$ (which is $10^{-5}$ to $10^{-4}$ times the overall size of the atom).

• Properties of Subatomic Particles:

  • Electron:

    • Symbol: $e$

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

    • Mass: $9.1 \times 10^{-31}\,\text{kg}$, which is approximately $\frac{1}{1840}$ times the mass of a proton.

  • Proton:

    • Symbol: $p$

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

    • Mass: $1.67 \times 10^{-27}\,\text{kg}$ (more precisely $m_p = 1.6726 \times 10^{-27}\,\text{kg}$).

  • Neutron:

    • Symbol: $n$

    • Charge: Zero (electrically neutral particle)

    • Mass: $1.67 \times 10^{-27}\,\text{kg}$ (more precisely $m_n = 1.6749 \times 10^{-27}\,\text{kg}$).

• Nucleons, Atomic Mass, and Charge:

  • Protons and neutrons, being the main constituent particles of the nucleus, are collectively termed nucleons.

  • Atomic Number ($Z$): The number of protons in the nucleus of an atom. In a neutral atom, it is equal to the number of orbital electrons. It determines the position of the element in the periodic table.

    • $\text{Atomic Number } (Z) = \text{number of protons in the nucleus}$

  • Mass Number ($A$): The total number of nucleons (protons plus neutrons) in the nucleus.

    • $\text{Mass Number } (A) = \text{number of protons} + \text{number of neutrons}$

  • Nuclear Charge and Mass:

    • The nucleus carries a total positive charge of $+Ze$.

    • The mass of the nucleus is approximately equal to the mass of a proton or neutron multiplied by mass number $A$.

    • Because the mass of an electron is negligible relative to protons or neutrons, the total mass of an atom is considered identical to the mass of its nucleus.

• Atomic Symbol Specification and Models:

  • An atom is specified symbolically as ${}_Z^A \text{X}$, where $\text{X}$ is the chemical symbol of the element.

  • For any specified nuclide ${}_Z^A \text{X}$:

    • $\text{Number of electrons} = Z$

    • $\text{Number of protons} = Z$

    • $\text{Number of neutrons} = A - Z$

  • Specific Atomic Examples:

    • Hydrogen (${}_1^1 \text{H}$): Lightest atom; $A = 1$, $Z = 1$. Nucleus contains $1$ proton and $0$ neutrons; $1$ electron revolves in the K shell.

    • Helium (${}_2^4 \text{He}$): $A = 4$, $Z = 2$. Nucleus contains $2$ protons and $2$ neutrons ($A - Z = 4 - 2 = 2$); $2$ electrons revolve in the K shell.

    • Sodium (${}_{11}^{23} \text{Na}$): $A = 23$, $Z = 11$. Nucleus contains $11$ protons and $12$ neutrons ($23 - 11 = 12$); $11$ electrons distributed as $2$ in K shell, $8$ in L shell, and $1$ in M shell.

    • Strontium Observation: A single atom of Strontium trapped in an electric field has been photographed. Its nucleus contains $38$ protons, and its atomic diameter measures a few millionths of a millimeter.

Chemical versus Nuclear Changes

• Fundamental Mechanisms:

  • Chemical Change: Occurs at low energy and involves changes only in the number or configuration of orbital electrons.

  • Nuclear Change: Occurs at high energy and involves changes in the number of nucleons inside the atomic nucleus.

• Energy Requirements:

  • A chemical change requires low energy, on the order of a few $\text{eV}$.

  • A nuclear change requires significantly higher energy, on the order of a few $\text{MeV}$.

  • The energy required for a nuclear change is approximately $10^6$ times higher than that required for a chemical change.

• Conservation Laws in Nuclear Reactions:

  • Both Atomic Number ($Z$) and Mass Number ($A$) are strictly conserved during nuclear reactions.

  • $\sum Z_{\text{reactants}} = \sum Z_{\text{products}}$

  • $\sum A_{\text{reactants}} = \sum A_{\text{products}}$

Classification of Nuclides

• Isotopes:

  • Definition: Atoms of the same element having the same atomic number $Z$ but different mass numbers $A$.

  • Properties: Isotopes occupy the exact same position in the periodic table. They possess the same number of protons ($Z$) and orbital electrons ($Z$), giving them identical chemical properties, but contain different numbers of neutrons ($A - Z$).

  • Abundance: Different isotopes of an element occur in varying proportions in nature. Usually, one isotope has a dominant relative abundance, and the element is commonly specified by the mass number of that most abundant isotope.

  • Examples of Isotopes:

    • Hydrogen: Protium (${}_1^1 \text{H}$, ordinary hydrogen; $1$ proton, $0$ neutrons), Deuterium (${}_1^2 \text{H}$ or $\text{H}$, heavy hydrogen; $1$ proton, $1$ neutron), and Tritium (${}_1^3 \text{H}$; $1$ proton, $2$ neutrons). Each has $1$ orbital electron.

    • Carbon: ${}_6^{12} \text{C}$, ${}_6^{13} \text{C}$, and ${}_6^{14} \text{C}$. Each contains $6$ protons and $6$ electrons. ${}_6^{12} \text{C}$ (most abundant) has $6$ neutrons, ${}_6^{13} \text{C}$ has $7$ neutrons, and ${}_6^{14} \text{C}$ has $8$ neutrons.

    • Chlorine: ${}_{17}^{35} \text{Cl}$ and ${}_{17}^{37} \text{Cl}$, existing in nature in a relative abundance ratio of $3 : 1$. Each has $17$ protons and $17$ electrons; ${}_{17}^{35} \text{Cl}$ contains $18$ neutrons, while ${}_{17}^{37} \text{Cl}$ contains $20$ neutrons.

  • Types of Isotopes:

    • Stable Isotopes: Possess a neutron count nearly equal to their proton count ($N \approx Z$).

    • Unstable or Radioactive Isotopes (Radioisotopes): Possess a neutron count significantly greater than their proton count ($N \gg Z$), driving them to undergo spontaneous radioactive decay. Used extensively in medical and industrial applications.

    • Examples of Radioisotopes: ${}_{92}^{235} \text{U}$ ($92$ protons, $143$ neutrons) and ${}_{92}^{238} \text{U}$ ($92$ protons, $146$ neutrons); ${}_6^{14} \text{C}$ is also a radioisotope.

  • Note: Tin ($\text{Sn}$) possesses the largest number of isotopes ($10$ isotopes).

• Isobars:

  • Definition: Atoms of different elements having the same mass number $A$ but different atomic numbers $Z$.

  • Properties: Isobars contain the same total number of nucleons ($A$), but different numbers of protons ($Z$), neutrons ($A - Z$), and orbital electrons ($Z$). Consequently, they exhibit different chemical properties.

  • Examples of Isobars:

    • ${}_{11}^{23} \text{Na}$ ($11$ protons, $12$ neutrons, $11$ electrons) and ${}_{12}^{23} \text{Mg}$ ($12$ protons, $11$ neutrons, $12$ electrons). Total nucleons = $23$ each.

    • ${}_6^{14} \text{C}$ ($6$ protons, $8$ neutrons, $6$ electrons) and ${}_7^{14} \text{N}$ ($7$ protons, $7$ neutrons, $7$ electrons). Total nucleons = $14$ each.

  • Mirror Isobars: Isobars in which the numeric values of protons and neutrons are mutually interchanged, such as ${}_{11}^{23} \text{Na}$ ($11$ p, $12$ n) and ${}_{12}^{23} \text{Mg}$ ($12$ p, $11$ n).

• Isotones:

  • Definition: Atoms having different numbers of protons ($Z$) and different mass numbers ($A$), but the exact same number of neutrons ($A - Z$).

  • Properties: They belong to different elements and have different numbers of electrons.

  • Examples of Isotones:

    • ${}_{11}^{23} \text{Na}$ ($11$ protons) and ${}_{12}^{24} \text{Mg}$ ($12$ protons): Each contains exactly $12$ neutrons ($23 - 11 = 12$ and $24 - 12 = 12$).

    • ${}_{19}^{39} \text{K}$ ($19$ protons) and ${}_{20}^{40} \text{Ca}$ ($20$ protons): Each contains exactly $20$ neutrons ($39 - 19 = 20$ and $40 - 20 = 20$).

• Summary Comparison of Nuclide Classifications:

  • Isotope: Atomic Number $Z$ is Same; Mass Number $A$ is Different; Neutron Count $A - Z$ is Different.

  • Isobar: Atomic Number $Z$ is Different; Mass Number $A$ is Same; Neutron Count $A - Z$ is Different.

  • Isotone: Atomic Number $Z$ is Different; Mass Number $A$ is Different; Neutron Count $A - Z$ is Same.

Fundamentals of Radioactivity

• Discovery and History:

  • Discovered by Henry Becquerel in 1896.

  • Becquerel wrapped a uranium salt in black paper and placed it on a photographic plate inside a dark room. Upon later inspection, the photographic plate was found to be exposed.

  • Subsequent tests with other uranium salts yielded identical results, confirming that uranium and its salts spontaneously emit invisible radiations capable of penetrating paper, glass, or wood. These emissions were initially termed Becquerel rays.

• Definitions:

  • Radioactive Substances: Substances that disintegrate or decay by the spontaneous emission of radiation. Examples include uranium, radium, polonium, thorium, and actinium.

  • Natural Radioactivity: Isotopes of nearly all elements with atomic numbers higher than $82$ (elements positioned after lead in the periodic table) are naturally radioactive because their nuclei contain a massive excess of neutrons over protons.

• Radioactivity as a Nuclear Phenomenon:

  • Radioactivity is purely a nuclear process; it is the spontaneous emission of $\alpha$, $\beta$, or $\gamma$ radiations from atomic nuclei during their disintegration.

  • Proof of Nuclear Origin: No external physical change (such as variations in pressure or temperature) or chemical change (such as extreme heating, freezing, application of strong electric/magnetic fields, chemical reagents, or oxidation) alters either the nature of the radiation emitted or the rate of decay. Because orbital electrons are easily perturbed by such treatments without altering radiation properties, radioactivity cannot be attributed to orbital electrons.

• Random Nature of Decay:

  • In any radioactive sample containing a large number of atoms, it is impossible to predict precisely when or which individual nucleus will decay at any given moment.

  • Radioactive decay is fundamentally a random phenomenon governed by no deterministic law predicting individual nuclear disintegrations.

Radiation Analysis and Properties

• Rutherford's Magnetic Field Experiment (1903):

  • Rutherford analyzed emissions from a radioactive sample placed inside a lead box by subjecting them to a magnetic field oriented perpendicular to their path (directed perpendicularly into the plane of the paper).

  • The emissions separated into three distinct components:

    • Alpha ($\alpha$) particles: Deflected to the left (determined by Fleming's left-hand rule), indicating positive electric charge.

    • Beta ($\beta$) particles: Deflected to the right, indicating negative electric charge. Beta particles exhibit significantly greater trajectory curvature/deflection than alpha particles.

    • Gamma ($\gamma$) radiations: Deflected zero degrees, passing straight through. This confirms they are uncharged neutral electromagnetic waves similar to light.

• Electric Field Deflection Experiment:

  • When radioactive emissions pass through an electric field perpendicular to their direction of flight:

    • Alpha ($\alpha$) particles: Curve toward the negative plate, confirming positive charge.

    • Beta ($\beta$) particles: Curve toward the positive plate, confirming negative charge. Beta particles undergo greater deflection than alpha particles.

    • Gamma ($\gamma$) radiations: Pass completely undeviated, confirming neutrality.

• Properties of Alpha ($\alpha$) Particles:

  • An alpha particle consists of $2$ protons and $2$ neutrons bound together.

  • It is identical to a doubly ionised helium atom ($\text{He}^{2+}$), which is a helium nucleus that has lost both of its orbital electrons.

  • Represented symbolically as ${}_2^4 \text{He}$ or $\text{He}^{2+}$.