Planck's Quantum Theory & Wien's Displacement Law

 This is part of the HSC Physics course under the topic Light: Quantum Model.

HSC Physics Syllabus

  • analyse the experimental evidence gathered about black body radiation, including Wien's displacement Law related to Planck's contribution to a changed model of light (ACSPH137).
- λmax = b/T
 

Planck's Quantum Theory & Wien's Displacement Law

What is Black Body Radiation?

A black body is an idealistic object that absorbs all radiation falling within its vicinity.

A black body at thermal equilibrium and given temperature would also emit radiation of exactly the same amount as what it first absorbed.

  

Black bodies are not real, but scientists have been able to produce objects that possess similar properties.

Figure showing the theoretical path of radiation after it enters a black body-like cavity.
 
A large box with a small hole is a simple example of a near-perfect black body. Once radiation enters the box via the opening, it bounces around in the box, unable to exit.
 
In 1895, Wien and Lummer punched a small hole in the side of an otherwise completely closed oven. They began to measure the radiation coming out. The beam coming out of the hole was passed through a diffraction grating, which sent the different wavelengths/frequencies in different directions, all towards a screen. A detector was moved up and down along the screen to determine the intensity of each frequency of emitted radiation.

 

 

Classical Theory of Electromagnetic Radiation (Rayleigh-Jeans Law)

The classical theory states that the intensity of black body radiation is directly proportional to its frequency. That is, greater the frequency a radiation has, the more energy this is equivalent to. The theory always states that the radiation emitted by a black body is continuous and a function of temperature; greater a black body’s temperature, more energy it emits.   

The Rayleigh-Jeans Law is based on many assumptions and knowledge in classical physics. It assumes radiation is continuous and not discrete in energy. 

 

Ultraviolet Catastrophe

By this logic, the classical model proposes that a black body at thermal equilibrium would emit radiation at all frequencies, resulting in an infinite amount of energy. However, this disobeys the Law of Conservation of Energy as it is merely impossible to emit ‘infinite’ amount of energy.

 

Figure showing the ultraviolet catastrophe - inconsistent predications of classical and quantum theories of physics with regard to energy emitted by a black body radiation.

Experimentally, the classical theory of radiation is only valid to an extent. For radiation with frequencies below 105 GHz (1017 Hz), the energy measured is consistent with the model.

However, as the frequency becomes higher, the inconsistency between the classical theory and experimental results grows larger and larger. The divergent trend becomes most evident in the ultraviolet spectrum and points to flaws within the long accepted classical theory, hence the term ultraviolet catastrophe.

 

Planck’s quantum theory resolved the ultraviolet catastrophe

Planck suggested the idea that energy was not continuous but rather quantised. That is, energy of light was confined to discrete amounts equal to integral multiples of a smallest unit of energy (a quantum).

The energy of a quantum of light (which Einstein later termed a photon) is given by:


$$E=hf$$

 

The energy of total radiation emitted is therefore given by

 

$$E=nhf$$

 

where h is Planck’s constant: 6.626 × 10-34 Js (Joule second); n is an integer (now called a quantum number)

Since the energy contained within a quantum is finite, it means that energy emitted by a black body is also finite (rather than infinite as proposed by the classical theory). In particular, depending on a black body’s temperature, the intensity distribution of photons of various frequencies is different. See diagram above for reference.

Wien's Displacement Law

Wien’s displacement law builds upon Planck’s quantum theory

Wien’s displacement law states that the black-body radiation curve for different temperatures will peak at different frequencies/wavelengths. In specific, the wavelength that produces the highest intensity varies inversely with temperature; higher the temperature, shorter the wavelength according to:

$$\lambda_{max}=\frac{b}{T}$$

 

where b is Wien’s displacement constant: 2.898 × 10-3 metre Kelvin (m K) and T is the object’s temperature in Kelvins (K). The wavelength derived from this equation would have units in metres (m).

 

Examples of Wien’s Law

  • Stellar emission. Some stars radiate partially in the visible spectrum. The intensity distribution of a star’s emission depends on its surface temperature. 

Hertzsprung-Russell diagram shows different groups of stars, each classified according to various characteristics such as temperature. 

 

  • Mammals with a body/skin temperature of around 300 K emit infrared radiation. This can be observed with infrared detectors:

Mammals' skin temperature emits infrared radiation which can be captured by specialised detectors. 

 

  • Fire produced from burning wood has an average temperature of about 1500 K. This means majority of its radiation has a wavelength around 2000 nm.

 

However, the reason why we see fire is because it also emits a small portion of visible light. This is also the reason why wood fire can provide a large amount of heat but is not efficient at providing lighting.

   

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