Spectral Distribution of Blackbody Radiation

What is being plotted

The radiant energy emitted per unit area per unit time by a blackbody depends on its temperature. Hot objects radiate electromagnetic radiation over a large range of frequencies, so the rate of emission per unit area, or power per unit area of the surface, is defined as a function of the wavelength $\lambda$ of the emitted radiation. Spectral distribution of the power radiated by a body means the power radiated at different wavelengths.

At low temperature the power radiated is small and lies primarily in the long-wavelength region. As the temperature is increased the rate of emission increases fast. At each temperature the radiant energy contains a mixture of different wavelengths, and at higher temperatures both the total energy radiated per unit time and the proportion of energy emitted at higher frequencies (shorter wavelengths) increase.

The experiment

Lummer and Pringsheim studied the energy distribution of blackbody radiation as a function of wavelength. They kept the source of radiation — a cavity radiator — at different fixed temperatures and measured the radiant power corresponding to different wavelengths. The measurements were represented graphically as curves showing the variation of radiant power per unit area with wavelength $\lambda$ at different constant temperatures. Experimental observations showed that the spectral distribution depended only on the absolute temperature $T$ of the blackbody and was independent of the material.

Diagram in words

Radiant power per unit area per unit wavelength range, $R_{\lambda}$, is plotted on the vertical axis against wavelength $\lambda$ on the horizontal axis. There is one continuous hump-shaped curve for each temperature. Each curve starts from zero at $\lambda = 0$, rises steeply to a maximum at a wavelength $\lambda_{max}$, and then falls off more slowly, approaching zero asymptotically at large $\lambda$. A curve for a higher temperature lies entirely above the curves for lower temperatures, and its peak is displaced towards shorter wavelengths, that is, to the left. The dotted line joining the peaks of successive curves therefore slopes to the left as $T$ increases.

Observations from the curves

1. At a given temperature, the energy is not uniformly distributed in the spectrum of the blackbody, that is, as a function of wavelength.

2. At a given temperature, the radiant power emitted initially increases with increase of wavelength, reaches a maximum, and then decreases.

3. The wavelength corresponding to the radiation of maximum intensity, $\lambda_{max}$, is characteristic of the temperature of the radiating body. It is not the maximum wavelength emitted by the object.

4. The area under a curve represents the total energy emitted per unit time per unit area by the blackbody at all wavelengths.

5. The peaks of the curves shift towards the left, to shorter wavelengths: the value of $\lambda_{max}$ decreases with increase in temperature.

6. At higher temperatures, the radiant power or total energy emitted per unit time per unit area corresponding to all wavelengths — the area under the curve — increases.

7. At a temperature of 300 K, around room temperature, the most intense of these waves has a wavelength of about $5 \times 10^{-5}$ m, and the radiant power is smaller for other wavelengths. Practically all the radiant energy at this temperature is carried by waves longer than those of red light; these are infrared radiations.

Wien's displacement law

It is observed that the wavelength for which the emissive power of a blackbody is maximum is inversely proportional to the absolute temperature of the blackbody. This is Wien's displacement law.

$$\lambda_{max} \propto \dfrac{1}{T} \qquad \Rightarrow \qquad \lambda_{max} = \dfrac{b}{T} \qquad \Rightarrow \qquad \lambda_{max}T = b$$

where $b$ is Wien's constant, of value $b = 2.897 \times 10^{-3}$ m K.

$\lambda_{max}$ indicates the wavelength at which the blackbody dominantly radiates, so it corresponds to the dominant colour of the radiating body and is a function of its temperature. White dwarfs are hot stars with surface temperature about 10000 K, while red giants are cooler, with surface temperature about 3000 K. The law is used to determine the temperatures of distant stars, the Sun and the Moon.

Worked value: for the Sun, taking the surface temperature as 5800 K,

$$\lambda_{max} = \dfrac{2.897 \times 10^{-3}}{5800} = 4.995 \times 10^{-7}\ \text{m} = 4995\ \mathring{\text{A}}$$

which lies in the visible region of the electromagnetic spectrum.

The theoretical background

A theoretical explanation of these observations could not be given by the theories then existing.

• Wien gave an expression for the spectral distribution from the laws of thermodynamics, which fitted the experimental observations only for short wavelengths.

• Lord Rayleigh and Sir James Jeans gave a formula from the equipartition of energy, which fits well in the long-wavelength region but tends to infinity at short wavelengths.

• It was therefore essential to propose a new model. Planck, aware of the shortcomings of both, combined them using an empirical formula and could describe the observed spectrum quite well.

Planck's quantum hypothesis: he considered the atoms of the walls of the cavity as tiny electromagnetic oscillators with characteristic frequencies that exchange energy with the cavity, and supposed this energy could have only the specific values

$$E = nh\nu$$

where $\nu$ is the frequency of the oscillator, $h = 6.626 \times 10^{-34}$ J s is a universal constant, and $n$ takes only positive integral values. The oscillators do not radiate energy continuously but only in jumps or quanta, corresponding to transitions from one quantised energy level to another of lower energy; while in one of the quantised states the oscillator neither emits nor absorbs energy.

Standard results

$$\lambda_{max}T = b, \qquad b = 2.897 \times 10^{-3}\ \text{m K}$$

• $\lambda_{max}$ is the wavelength of maximum intensity, not the longest wavelength emitted.

• Area under the $R_{\lambda}$ versus $\lambda$ curve = total power radiated per unit area at that temperature.

• The distribution depends only on the absolute temperature of the blackbody, not on its material.