Unit 2: Thermal Radiation and Blackbody Radiation

1. Properties of Thermal Radiation

Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation.

Key Properties:

• It travels with the speed of light in vacuum.
• It travels in straight lines and can be reflected, refracted, and polarized just like light.
• It can travel through vacuum; it does not require a material medium for propagation.
• The wavelength of emitted radiation depends on the temperature of the body.

2. Kirchhoff's Law

Kirchhoff's Law of thermal radiation states that for an arbitrary body in thermal equilibrium with its surroundings, its emissivity is equal to its absorptivity.

e_lambda / a_lambda = E_lambda (Blackbody)

In simpler terms, good absorbers of a particular wavelength are also good emitters of that same wavelength at the same temperature.

3. Blackbody Radiation and Spectral Distribution

A perfect blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

Spectral Distribution:

When a blackbody is heated, it emits radiation. The spectral distribution of this radiation (how much energy is emitted at each wavelength) was experimentally determined by Lummer and Pringsheim.

• As temperature increases, the total energy emitted (area under the curve) increases rapidly.
• As temperature increases, the peak of the curve shifts toward shorter wavelengths.

4. Wien's Law and Rayleigh-Jeans Law

Classical physics attempted to explain the blackbody spectrum using two different approaches:

Wien's Distribution Law:

Derived using thermodynamics, it worked well for short wavelengths (high frequencies) but failed to match the experimental data at long wavelengths.

Rayleigh-Jeans Law:

Based on the principle of equipartition of energy, it assumed radiation in a cavity consists of standing waves. It matched experimental results at long wavelengths but failed spectacularly at short wavelengths.

5. The Ultraviolet Catastrophe

The Ultraviolet Catastrophe was a prediction of the Rayleigh-Jeans law that an ideal blackbody at thermal equilibrium would emit radiation with infinite power as the wavelength approaches zero (the ultraviolet range).

This was a major failure of classical physics because it contradicted the Law of Conservation of Energy and experimental observations, which showed that emission power actually drops to zero at very short wavelengths.

6. Planck's Quantum Postulates

In 1900, Max Planck solved the blackbody radiation problem by introducing a revolutionary hypothesis:

1. The atoms in the walls of the blackbody cavity act as atomic oscillators.
2. These oscillators do not emit or absorb energy continuously. Instead, energy is exchanged in discrete packets called quanta.
3. The energy (E) of a quantum is proportional to the frequency (f) of the radiation.

Where h is Planck's constant (6.626 x 10^-34 J·s) and n is an integer (1, 2, 3...).

7. Planck's Law of Blackbody Radiation

By applying statistical mechanics to these quantized oscillators, Planck derived the correct formula for the energy density of blackbody radiation:

This formula perfectly matches the experimental blackbody spectrum across all wavelengths.

8. Deductions from Planck's Law

Planck's Law is the "master equation" from which all other radiation laws can be derived:

• Wien's Distribution Law: Derived when the wavelength (lambda) is very small (h*c / lambda*k*T >> 1).
• Rayleigh-Jeans Law: Derived when the wavelength (lambda) is very large (h*c / lambda*k*T << 1), using the Taylor expansion of the exponential term.
• Stefan-Boltzmann Law: Obtained by integrating Planck's energy density over all wavelengths from 0 to infinity. It shows that total energy E is proportional to T^4.
• Wien's Displacement Law: Obtained by differentiating Planck's Law with respect to wavelength and finding the maximum. It proves that lambda_max * T = constant.

9. Saha's Ionization Formula

Developed by Meghnad Saha, this formula relates the ionization state of a gas to its temperature and pressure.

It is used in astrophysics to determine the temperatures and chemical compositions of stellar atmospheres by analyzing the absorption lines in stellar spectra. It provides a qualitative idea of how thermal ionization occurs in high-temperature environments like stars.