7.5 Quantum Physics, Photoelectric Effect, Atomic Spectra & Nuclear Physics

7.5 Quantum Physics, Photoelectric Effect, Atomic Spectra & Nuclear Physics

Modern Physics represents a cornerstone of the FSc Class 12 curriculum and is heavily featured in the AMC Physics test. This section covers quantum postulates, wave-particle duality, atomic structure, X-rays, radioactivity, and nuclear reactions.

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1. Planck's Quantum Theory & Photoelectric Effect

Planck's Quantum Postulate

Classical electromagnetic theory failed to explain blackbody radiation curves (leading to the ultraviolet catastrophe). In 1900, Max Planck proposed that electromagnetic radiation is emitted or absorbed in discrete packets of energy called quanta or photons:

$E = h f = \frac{h c}{\lambda}$

where Planck's constant $h \approx 6.63 \times 10^{-34} \text{ J}\cdot\text{s} = 4.14 \times 10^{-15} \text{ eV}\cdot\text{s}$. Photons travel at speed $c = 3 \times 10^8 \text{ m/s}$ in vacuum and possess momentum $p = \frac{E}{c} = \frac{h}{\lambda}$.

Photoelectric Effect & Einstein's Explanation

When light of sufficiently high frequency falls on a clean metallic surface, electrons are ejected instantaneously. Albert Einstein explained this using photon theory:

$h f = \Phi + K.E._{max}$

where:

• $h f$ is incident photon energy.
• $\Phi = h f_0$ is the metal work function (minimum energy required to liberate an electron).
• $f_0$ is the threshold frequency.
• $K.E._{max} = e V_s$ is maximum kinetic energy of ejected photoelectrons ($V_s$ is stopping potential).

Fundamental Photoelectric Laws for AMC:

1. Emission occurs only if incident frequency $f \ge f_0$ (or wavelength $\lambda \le \lambda_0$).
2. $K.E._{max}$ depends only on light frequency $f$, NOT on light intensity.
3. Photoelectric current (number of emitted electrons per second) is directly proportional to light intensity.
4. Emission process is instantaneous ($< 10^{-9} \text{ s}$ delay).

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2. De Broglie Matter Waves & Compton Effect

Compton Effect

Compton scattering occurs when a high-energy X-ray photon collides with a free or loosely bound electron, undergoing a wavelength increase $\Delta \lambda$:

$\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos\theta)$

This experiment decisively confirmed the particle nature of electromagnetic radiation.

De Broglie Matter Wave Hypothesis

Louis de Broglie proposed that moving material particles exhibit wave-like properties with a characteristic matter wavelength $\lambda$:

$\lambda = \frac{h}{p} = \frac{h}{m v}$

For an electron accelerated through a potential difference $V$:

$\lambda = \frac{h}{\sqrt{2 m_e e V}} = \frac{1.227}{\sqrt{V}} \text{ nm}$

The Davisson-Germer experiment confirmed electron wave character via nickel crystal diffraction.

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3. Bohr's Model of Hydrogen Atom & Line Spectra

Niels Bohr formulated three postulates for the hydrogen atom:

1. Stationary Orbits: Electrons revolve in non-radiating circular orbits where electrostatic attraction provides centripetal force.
2. Quantized Angular Momentum: Angular momentum $L$ is an integral multiple of $\frac{h}{2\pi}$: $L = m_e v r = n \frac{h}{2\pi} \quad (n = 1, 2, 3, \dots)$
3. Frequency Condition: Emission/absorption of a photon occurs when an electron transitions between energy states $E_2$ and $E_1$: $\Delta E = E_2 - E_1 = h f$

Key Quantized Formulas:

• Bohr Radius: $r_n = n^2 r_1$ (where $r_1 = 0.053 \text{ nm} = 0.53 \text{ \AA}$).
• Quantized Energy Levels: $E_n = -\frac{13.6}{n^2} \text{ eV}$. Ground state energy $E_1 = -13.6 \text{ eV}$.

Hydrogen Spectral Series

Transition wavelengths follow the Rydberg formula $\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$, where $R_H \approx 1.097 \times 10^7 \text{ m}^{-1}$:

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4. X-Rays & Production

X-rays are short-wavelength electromagnetic radiation ($0.01 \text{ nm} - 10 \text{ nm}$) produced in a Coolidge tube when fast-moving electrons strike a heavy metal target (tungsten).

• Continuous X-rays (Bremsstrahlung): Produced by deceleration of electrons in target metal. Minimum cutoff wavelength is $\lambda_{min} = \frac{h c}{e V}$.
• Characteristic X-rays ($K_\alpha, K_\beta$): Produced when incident electrons ionize inner-shell ($K$-shell) electrons, causing outer-shell cascades.

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5. Nuclear Physics & Radioactivity

Nuclear Structure, Mass Defect & Binding Energy

A nucleus $^{A}_{Z}X$ contains $Z$ protons and $N = A - Z$ neutrons ($A$ is mass number).

• Mass Defect ($\Delta m$): Difference between sum of constituent nucleon masses and total bound nuclear mass: $\Delta m = [Z m_p + (A - Z) m_n] - m_{nucleus}$
• Binding Energy ($E_b$): Energy holding nucleons together: $E_b = \Delta m c^2$. ($1 \text{ u} = 931.5 \text{ MeV}$).
• Binding Energy per Nucleon ($E_b / A$): Reaches a peak of $\approx 8.8 \text{ MeV/nucleon}$ around Iron-56 ($^{56}\text{Fe}$), representing maximum nuclear stability.

Radioactive Decay Law & Half-Life

Radioactive decay is a random, spontaneous process obeying:

$N(t) = N_0 e^{-\lambda t}$

where $\lambda$ is the decay constant. Half-life ($T_{1/2}$) is time required for half of active nuclei to decay:

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

Fraction remaining after $n$ half-lives is $\left(\frac{1}{2}\right)^n$.

Radioactive Decays:

1. Alpha ($\alpha$) Decay: $^{A}{Z}X \to ^{A-4}{Z-2}Y + ^{4}_{2}\text{He}$.
2. Beta ($\beta^-$) Decay: $^{A}{Z}X \to ^{A}{Z+1}Y + ^{0}_{-1}e + \bar{\nu}$.
3. Gamma ($\gamma$) Decay: Excited nucleus drops state by emitting high-energy photon ($A, Z$ unchanged).

Fission vs Fusion

• Nuclear Fission: Heavy nucleus splits into medium-mass fragments (e.g., $^{235}\text{U} + n \to ^{141}\text{Ba} + ^{92}\text{Kr} + 3n + 200 \text{ MeV}$). Used in nuclear reactors.
• Nuclear Fusion: Light nuclei combine to form heavier nucleus (e.g., proton-proton cycle in Sun releasing $\approx 26.7 \text{ MeV}$). Requires extreme temperatures.

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6. Worked Numerical Examples for AMC Candidates

Example 1: Photon Energy & Frequency

Problem: Calculate the energy in electron-volts of a photon with frequency $f = 6 \times 10^{14} \text{ Hz}$.

Solution:

1. Calculate energy in Joules: $E = h f = (6.63 \times 10^{-34} \text{ J}\cdot\text{s}) \times (6 \times 10^{14} \text{ Hz}) = 3.978 \times 10^{-19} \text{ J}$
2. Convert to eV: $E_{eV} = \frac{3.978 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 2.48 \text{ eV}$

Example 2: Radioactive Decay Fraction

Problem: A radioactive isotope has a half-life of $8 \text{ days}$. What fraction of original active sample remains after $24 \text{ days}$?

Solution:

1. Find number of half-lives $n$: $n = \frac{\text{Total Time}}{\text{Half-life}} = \frac{24 \text{ days}}{8 \text{ days}} = 3$
2. Fraction remaining: $\text{Fraction} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} = 0.125 \text{ or } 12.5\%$