Grade 12 Physics Revision: Quantum and Nuclear Physics

General Physics Revision: Origins of Quantum Theory

• Failure of Classical Physics: Prior to Max Planck, physicists could not explain emission spectra because classical electromagnetic wave theory predicted continuous spectra, whereas observations showed discrete lines.
• Planck’s Quantum Hypothesis:
  • Energy changes in atoms are not continuous but are quantized (energy exists in specific bundles or packets).
  • Planck’s Equation: $E = nhf$. In this formula, $n$ is the principal quantum number (an integer), $h$ is Planck’s constant, and $f$ is the frequency of vibration.
  • Planck’s Constant ($h$): Approximately $6.63 \times 10^{-34}\,J \cdot s$.
  • Energy States: Energy values must be whole-number multiples of $hf$ (e.g., $1hf$, $2hf$, $3hf$). Fractions like $\frac{3}{2}hf$ are not allowed.
  • Vibrational Change: Atoms emit radiation only during changes in vibrational energy. If an atom transitions from $3hf$ to $2hf$, the energy emitted equals $hf$.
  • Quantization in Everyday Life: The quantized nature of energy is not noticed in daily life because Planck’s constant ($h$) is extremely small.
• Scientific Impact: Planck’s quantum theory was troubling to classical physicists because it suggested that classical physics was limited. It eventually led to the development of modern technologies like solar cells and computers.

The Photoelectric Effect

• Definition: The emission of electrons from a metal surface when it is exposed to electromagnetic radiation.
• Threshold Frequency ($f_0$): The minimum frequency of incident light required to eject electrons from a specific metal. If the incident frequency is below $f_0$, no electrons are emitted regardless of light intensity.
• Materials and Thresholds:
  • Zinc: Discharges when hit by ultraviolet (UV) light but fails to discharge with visible light because visible light's frequency is below the threshold for zinc.
  • Cesium: Usually used to coat cathodes in photocells because it can eject electrons with most visible light frequencies.
• The Photocell:
  • Consists of a large electrode called the Cathode (negative) and a smaller, thin electrode called the Anode (positive).
  • The tube is evacuated to prevent oxidation and collisions between emitted electrons (photoelectrons) and gas particles.
  • The anode is made thin to minimize the blocking of incident radiation.
• The Photocurrent:
  • Current flows only when the incident radiation frequency exceeds the threshold frequency ($f > f_0$).
  • Increasing light intensity (brightness) above the threshold frequency increases the photocurrent (more photons strike the surface per second, ejecting more electrons).
  • Classical wave theory incorrectly predicted that low-frequency light would eventually eject electrons if given enough time for the electrons to absorb energy; in reality, emission is immediate if $f \ge f_0$.

Photon Theory and Einstein’s Explanation

• Photons: Einstein proposed that light consists of discrete bundles of energy called photons. Light behaves as particles carrying energy packets.
• Photon Energy Equation: $E = hf$ or $E = \frac{hc}{\lambda}$.
• The Electron Volt ($eV$): The energy gained by an electron moving across a potential difference of $1\,V$. Small atomic-scale energies are measured in $eV$ because Joules are too large ($1\,eV = 1.6 \times 10^{-19}\,J$).
• Work Function ($\phi$): The minimum energy required to free an electron from a specific metal ($\phi = hf_0$).
• Conservation of Energy: Excess energy from a photon (the energy remaining after overcoming the work function) becomes the kinetic energy ($KE$) of the emitted electron.
  • $KE_{max} = hf - \phi$
• Stopping Potential ($V_0$): The negative potential applied to the anode at which the current becomes zero because no photoelectrons can reach it.
  • $KE_{max} = e \Delta V_0$
• Spectral Trends: Within the visible spectrum, Violet light has the highest photon energy, while Red light has the lowest. As wavelength ($\lambda$) increases, photon energy ($E$) decreases.

Wave-Particle Duality and Quantum Mechanics

• De Broglie Wavelength: Louis de Broglie proposed that matter particles (like electrons) exhibit wave properties.
  • $\lambda = \frac{h}{mv}$
  • Everyday objects (like cars or bowling balls) do not show wave behavior because their mass is large, making their de Broglie wavelength extremely small.
• Heisenberg Uncertainty Principle: It is impossible to know exactly both the position and the momentum (velocity) of a particle simultaneously. Measuring one accurately introduces uncertainty into the other.
• Quantum Model of the Atom:
  • Developed by Erwin Schrödinger using wave properties.
  • Replaces defined orbits with an "Electron Cloud," representing the probability region for finding an electron.

Atomic Models and Spectroscopy

• Thomson Model: Electrons distributed in a positively charged sphere ("Raisins in a muffin").
• Rutherford Model:
  • Gold Foil Experiment: Alpha particles (massive positively charged particles) were fired at gold foil.
  • Findings: Most particles passed through (atom is mostly empty space), but a few bounced back (dense, positive nucleus).
• Bohr’s Model of the Atom:
  • Electrons move in specific, quantized energy levels called "Stationary States."
  • Electrons in stable orbits do not radiate energy.
  • Ground State: The lowest possible energy level ($n=1$).
  • Excited State: Any energy level above the ground state. Atoms absorb energy to move electrons to higher levels and emit photons to move to lower levels.
  • Hydrogen Energy Levels: $E_n = \frac{-13.6}{n^2}\,eV$.
  • Hydrogen radius: $r_n \propto n^2$.
• Spectroscopes: Instruments used to study spectra.
  • Emission Spectrum: Bright lines on a dark background; unique to each element gas-discharge tube.
  • Absorption Spectrum: Dark lines (Fraunhofer lines) in a continuous spectrum produced when white light passes through a cooler gas.
  • Hydrogen Series:
    • Lyman Series: Transitions to $n=1$ (Ultraviolet light).
    • Balmer Series: Transitions to $n=2$ (Visible light: red, green, blue, violet).
    • Paschen Series: Transitions to $n=3$ (Infrared radiation).

Lasers and Stimulated Emission

• LASER: Light Amplification by Stimulated Emission of Radiation.
• Spontaneous Emission: An excited atom naturally drops to a lower state, releasing a random photon.
• Stimulated Emission: An excited atom is struck by a photon of the exact matching energy, causing it to emit a photon that is in phase and identical to the first. These photons are coherent (in-phase) and monochromatic (single wavelength).
• Parallel Mirrors: Used to maintain directionality and stimulate further emission. One mirror is partially reflective for the laser beam to exit.

Nuclear Physics: Structure and Stability

• Nucleons: Protons and Neutrons.
• Atomic Number ($Z$): Number of protons.
• Mass Number ($A$): Total nucleons (Protons + Neutrons).
• Isotopes: Atoms with the same number of protons but different numbers of neutrons (e.g., Helium-3 vs Helium-4).
• Atomic Mass Unit ($u$): Defined as $\frac{1}{12}$ the mass of a Carbon-12 atom ($1\,u \approx 1.66 \times 10^{-27}\,kg$).
• Strong Nuclear Force: The powerful, short-range ($1.4 \times 10^{-15}\,m$) attractive force that holds nucleons together, overcoming the electromagnetic repulsion between protons.
• Mass Defect ($\Delta m$): The difference between the sum of individual nucleon masses and the actual mass of the nucleus. This "lost" mass is converted into binding energy.
• Binding Energy ($BE$): The energy required to separate a nucleus into its nucleons ($BE = \Delta m c^2$).
  • $1\,u = 931.5\,MeV$.
  • Iron-56 is the most stable nucleus.

Radioactivity and Half-Life

• Discovery: Henri Becquerel (1896) discovered radioactivity using photographic plates.
• Types of Radiation:
  1. Alpha ($\alpha$): Helium-4 nuclei. High mass, low penetration (stopped by paper). Mass number decreases by 4, Atomic number decreases by 2.
  2. Beta ($\beta$): High-speed electrons ($\beta^-$) or positrons ($\beta^+$). Medium penetration (stopped by aluminium). In $\beta^-$ decay, a neutron becomes a proton; atomic number increases by 1.
  3. Gamma ($\gamma$): High-energy photons. High penetration (stopped by thick lead). No change in mass or atomic number.
• Half-Life ($T_{1/2}$): The time required for half the atoms in a radioactive sample to decay.
  • Remaining mass/fraction after $n$ half-lives: $(\frac{1}{2})^n$.
  • Example: Tritium ($T_{1/2} = 12.3$ years). After 24.6 years (2 half-lives), $\frac{1}{4}$ of the sample remains.
• Nuclear Fission: Splitting a heavy nucleus (like Uranium-235) into smaller nuclei, releasing energy and neutrons (Chain Reaction). Moderated by materials like water (to slow neutrons) and cadmium rods (to absorb neutrons).
• Nuclear Fusion: Combining light nuclei (like Hydrogen) to form a heavier nucleus (Helium). Occurs in the Sun (Proton-Proton chain) at temperatures near $2 \times 10^7\,K$.

Key Physical Formulas

• Planck’s Equation: $E = hf$
• Photon Wavelength: $E = \frac{hc}{\lambda}$
• De Broglie Wavelength: $\lambda = \frac{h}{mv}$
• Bohr Energy Level: $E_n = \frac{-13.6}{n^2}\,eV$
• Bohr Orbit Radius: $r_n = n^2 a_0$
• Mass-Energy Equivalence: $E = mc^2$
• Binding Energy: $BE = \Delta m \times 931.5\,MeV/u$
• Decay Formula: $N = N_0 (\frac{1}{2})^{t/T_{1/2}}$
• Work Function: $\phi = hf_0$

Important Constants and Conversions

• Speed of light ($c$): $3.0 \times 10^8\,m/s$
• Planck constant ($h$): $6.63 \times 10^{-34}\,J \cdot s$
• Elementary charge ($e$): $1.6 \times 10^{-19}\,C$
• Atomic mass unit ($u$): $1.66 \times 10^{-27}\,kg$
• Electron mass ($m_e$): $9.11 \times 10^{-31}\,kg$
• Proton mass ($m_p$): $1.673 \times 10^{-27}\,kg$
• Neutron mass ($m_n$): $1.675 \times 10^{-27}\,kg$
• Energy Conversion: $1\,eV = 1.6 \times 10^{-19}\,J$