Blackbody Radiation and Planck's Hypothesis

A blackbody is an ideal absorber and emitter of radiation. Classical theory (Rayleigh-Jeans law) predicted that radiation intensity would become infinite at short wavelengths—the ultraviolet catastrophe—contradicting experimental observations.

Planck's Solution (1900): Energy is emitted in discrete packets called photons:

$E_n = nhf$

where $h = 6.63 \times 10^{-34}$ Js is Planck's constant, $f$ is frequency, and $n$ is a positive integer.

Planck's law for radiation intensity is:

$I(\lambda, T) = \frac{2\pi hc^2}{\lambda^5} \times \frac{1}{e^{hc/\lambda k_B T} - 1}$

This agrees perfectly with experimental data.

Experimental Observations

When light strikes a metal surface, electrons may be emitted. Key experimental facts:

• There exists a threshold frequency $f_0$ below which no electrons are emitted, regardless of intensity.
• For $f > f_0$, photocurrent is proportional to light intensity.
• Maximum kinetic energy of emitted electrons depends only on light frequency, not intensity.
• Emission is instantaneous (~10⁻⁹ s).

Failure of Wave Theory

Classical wave theory could not explain:

1. Why electron kinetic energy depends on frequency, not intensity
2. The existence of a threshold frequency
3. Instantaneous emission (classical theory predicted time delay)

Einstein's Photon Theory (1905)

Einstein proposed that light consists of particles called photons, each with energy:

$E = hf = \frac{hc}{\lambda}$

When a photon strikes a metal, it transfers all its energy to an electron. If this energy exceeds the metal's work function $\phi_0$, the electron is emitted with maximum kinetic energy:

$K_{max} = hf - \phi_0 = \frac{hc}{\lambda} - \phi_0$

The stopping potential $V_s$ satisfies:

$eV_s = K_{max} \Rightarrow V_s = \frac{hf}{e} - \frac{\phi_0}{e}$

Threshold wavelength: $\lambda_0 = \frac{hc}{\phi_0}$

Worked Example

Light of wavelength 400 nm falls on a metal with work function 1.25 eV. Find the stopping potential.

Solution

Energy of incident photon: $E = \frac{hc}{\lambda} = \frac{1240 \text{ eVnm}}{400 \text{ nm}} = 3.1 \text{ eV}$

Stopping potential: $V_s = \frac{E - \phi_0}{e} = \frac{3.1 - 1.25}{1} = 1.85 \text{ V}$

de Broglie Wavelength

Louis de Broglie (1924) proposed that particles of matter have wave properties. The de Broglie wavelength is:

$\lambda = \frac{h}{mv} = \frac{h}{p}$

This was experimentally confirmed by Davisson and Germer (1927) through electron diffraction.

Worked Example

Calculate the de Broglie wavelength for an electron moving at $10^5$ m/s.

Solution

$\lambda = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \times 10^5} = 7.3 \times 10^{-9} \text{ m} = 7.3 \text{ nm}$

Rutherford's Planetary Model (1911)

Alpha particle scattering experiments showed:

• Most alpha particles pass through undeflected (atom is mostly empty)
• Some deflect at large angles (positive charge concentrated in tiny nucleus)
• This led to the nuclear model: electrons orbit a central nucleus

Limitations:

• According to Maxwell's theory, accelerating charges emit radiation; electrons should spiral into the nucleus
• Could not explain atomic stability or line spectra

Bohr's Model of the Hydrogen Atom

Bohr postulates:

1. Electrons revolve in stationary orbits without radiating energy
2. Angular momentum is quantised: $mvr = \frac{nh}{2\pi}$
3. Energy is emitted/absorbed only during transitions between orbits

Radius of nth orbit: $r_n = \frac{n^2 h^2 \varepsilon_0}{Z\pi me^2}$

For hydrogen (Z = 1), the Bohr radius is $r_1 = 5.29 \times 10^{-11}$ m.

Energy of nth level: $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$

For hydrogen: $E_1 = -13.6$ eV (ground state), $E_2 = -3.39$ eV (first excited state).

Atomic Transitions and Spectral Series

When an electron jumps from outer orbit m to inner orbit n, radiation is emitted with energy:

$hf = E_m - E_n = 13.6 \text{ eV} \left( \frac{1}{n^2} - \frac{1}{m^2} \right)$

The Rydberg equation: $\frac{1}{\lambda} = R_H \left( \frac{1}{n^2} - \frac{1}{m^2} \right)$

where $R_H = 1.097 \times 10^7$ m⁻¹.

Spectral series:

• Lyman series: transitions to n = 1 (ultraviolet)
• Balmer series: transitions to n = 2 (visible)
• Paschen series: transitions to n = 3 (infrared)

Production

X-rays are produced when high-speed electrons strike a heavy metal target (e.g., tungsten). The X-ray tube operates at high voltage (typically 30-100 kV).

Types of X-ray Spectra

1. Continuous spectrum (bremsstrahlung): Electron decelerates in the Coulomb field of nucleus, emitting radiation of varying wavelength.

Minimum wavelength: $\lambda_{min} = \frac{hc}{eV}$

2. Characteristic X-rays: Inner shell electrons are knocked out; electrons from higher levels drop down, emitting X-rays with specific energies.

Principle: Stimulated Emission

Einstein (1917) proposed that excited atoms can emit photons stimulated by incoming photons of the same frequency, phase, and direction.

Requirements for laser operation:

1. Metastable state: An excited state with relatively long lifetime (~10⁻³ s)
2. Population inversion: More atoms in excited state than in ground state
3. Optical cavity: Mirrors to amplify light through multiple reflections

Properties of Laser Light

• Monochromatic: Single wavelength
• Coherent: Waves in phase in space and time
• Collimated: Low divergence, travels in parallel rays

Types of Lasers

• Solid-state: Ruby laser (doped crystal)
• Gas: He-Ne laser, CO₂ laser
• Semiconductor: Laser diodes

Applications

• Medical surgery and diagnostics
• Cutting, welding, drilling
• CD/DVD players
• Barcode scanners
• Fiber optic communications
• Holography

Nuclear Structure

The nucleus contains:

• Protons: Z (atomic number)
• Neutrons: N
• Mass number: A = Z + N

Nuclides are written as $^A_ZX$. Isotopes have same Z but different N.

Mass Defect and Binding Energy

The measured mass of a nucleus is always less than the sum of its nucleon masses. This mass defect $\Delta m$ corresponds to binding energy:

$BE = \Delta m c^2 = [ZM_p + (A-Z)M_n - M(A,Z)]c^2$

Binding energy per nucleon determines nuclear stability. The curve peaks around $^{56}$Fe (8.8 MeV/nucleon).

Nuclear Stability

Stable nuclei have a specific neutron-to-proton ratio. Unstable nuclei undergo radioactive decay:

• Alpha decay: Emits $^4_2He$ nucleus (heavy nuclei, Z > 82)
• Beta-minus decay: Emits electron (neutron-rich nuclei)
• Beta-plus decay: Emits positron (proton-rich nuclei)
• Gamma decay: Emits high-energy photon

Radioactive Decay Law

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

where $\lambda$ is the decay constant.

Half-life: $t_{1/2} = \frac{0.693}{\lambda}$

Nuclear Reactions

Nuclear Fusion: Combining light nuclei to form heavier ones (e.g., in the Sun): $^2_1H + ^3_1H \rightarrow ^4_2He + ^1_0n + 17.6 \text{ MeV}$

Nuclear Fission: Splitting heavy nuclei: $^1_0n + ^{235}_{92}U \rightarrow ^{141}_{56}Ba + ^{92}_{36}Kr + 3^1_0n + 173 \text{ MeV}$

A chain reaction occurs when neutrons from fission induce further fissions.

Nuclear Reactor

Components:

• Fuel: Enriched uranium-235
• Moderator: Slows neutrons (water, graphite)
• Control rods: Absorb neutrons (cadmium, boron)
• Coolant: Removes heat (water, liquid sodium)
• Containment: Prevents radiation escape

A sample contains $10^{18}$ atoms with half-life 2000 days. Find the fraction remaining after 5000 days.

Solution

Number of half-lives: $n = \frac{5000}{2000} = 2.5$

Fraction remaining: $\frac{N}{N_0} = \left( \frac{1}{2} \right)^{2.5} = \frac{1}{5.66} \approx 0.177$

Alternatively, using $N = N_0 e^{-\lambda t}$ with $\lambda = \frac{0.693}{2000}$: $N = N_0 e^{-\frac{0.693}{2000} \times 5000} = N_0 e^{-1.7325} = 0.177 N_0$

In Tanzania, the principles of atomic physics are applied in medical imaging. At hospitals like Bugando Medical Centre in Mwanza or Muhimbili National Hospital in Dar es Salaam, X-ray machines and CT scanners use X-rays to image internal body structures. Similarly, radioisotopes are used in diagnosing thyroid conditions and in radiation therapy for cancer treatment. Understanding nuclear physics also helps Tanzania develop safe nuclear energy: the country is exploring nuclear power to address electricity shortages, where nuclear fission reactors would generate electricity for homes and industries across the nation.