1
Thermal Radiation

General objective

Gain knowledge on energy-related quantities and the laws of thermal radiation.

Specific objectives

On completing this chapter, the reader should be able to:

• - define thermal radiation;
• - know the origin of thermal radiation;
• - know the relations between energy, photon wavelength and frequency;
• - distinguish between black body and gray body;
• - define the energy-related quantities (flux, exitance, radiance and intensity);
• - apply the law of conservation of radiant flux;
• - state Lambert's, Kirchhoff's, Stefan-Boltzmann and Wien's laws;
• - provide an interpretation of Planck's law;
• - apply the laws of thermal radiation;
• - find the useful spectrum from a given isotherm of the black body.

Prerequisites

• - structure of matter;
• - modes of energy transfer;
• - range of electromagnetic waves.

1.1. Radiation

1.1.1. Definition

An object at temperature T can emit or absorb light waves of several frequencies [PÉR 86, PER 11, SAK 12]. The distribution of the energy exchanged by the object with its external environment depends on the temperature T. When two objects at different temperatures are in contact, thermal energy (heat) is transferred from the hot object to the cold object. By contrast, radiation is energy that is carried by an electromagnetic wave. In this case, energy is transferred by emission and absorption of light waves. Hence, by definition, thermal radiation is the electromagnetic radiation emitted by any object at non-zero temperature T.

1.1.2. Origin of radiation

In 1900, Max Planck laid the foundations of quantum physics by studying the black body emission spectrum within the theory of quanta [BRO 25, PAI 82, PLO 16]. He formulated the fundamental hypothesis according to which the energy generated by a periodic movement of frequency ? (rotation or vibration) has, similar to matter, a discontinuous structure. Consequently, radiant energy can only exist as bundles or quanta of energy h?. The number h is a universal constant known as the Planck constant. In 1905, Albert Einstein stated that light is made of particles subsequently called photons, each of which has an energy h?. Radiation results from electronic transitions between discrete levels of atomic or molecular systems. The energy exchanged during these transitions corresponds to photon absorption and emission processes. The energy E, angular frequency ?, frequency ? and wavelength ? of the photon are related by the following relations:

In relations [1.1], E is expressed in joules (J), ? in hertz (Hz), ? in radian per second (rad · s-1) and ? in meters (m). The quantity c designates the speed of light in a vacuum and h is the h-bar (or reduced) Planck constant.

Numerical expression:

The figures designate the absolute errors ?X (uncertainties) of the given values of the measured X quantity. For example, h = (6.626 068 96 ± 0.000 000 33) × 10-34 J · s.

This means an absolute error ?h = (0.000 000 33) × 10-34 J · s.

At the microscopic scale, it is convenient to use the electronvolt (eV) as a unit of energy:

Photon absorption and emission processes are illustrated in Figure 1.1.

Figure 1.1. Electronic transition between two discrete levels

APPLICATION 1.1.-

A He-Ne laser in a laboratory emits radiation whose wavelength is 633 nm. Calculate the energy E, frequency ? and angular frequency ? of a photon of this radiation. Express E in eV.

Given data. h = 6.63 × 10-34 J · s; c = 3.0 × 108 m · s-1; 1 eV = 1.60 × 10-19 J.

Solution. E = 1.96 eV; ? = 4.74 × 1014 Hz; ? = 2.98 × 1015 rad · s-1.

Box 1.1. Planck (1854-1947)

Max Planck, in full Max Karl Ernst Ludwig Planck, was a German physicist. He founded quantum physics in 1900 with his fundamental hypothesis on the theory of quanta. He was awarded the Nobel Prize for physics in 1918 for his essential contribution to the theory of quanta. Planck is also one of the founding fathers of quantum mechanics. He is also well known for his law giving monochromatic radiant exitance, which makes it possible to interpret the experimental observations related to black body isotherms.

1.1.3. Classification of objects

Objects susceptible to exchange energy are classified into three categories:

• - transparent objects that allow radiation to pass through without attenuation. This is the case with glass, transparent plastic material, etc.;
• - opaque objects that absorb radiation and get heated. This is the case with solid bodies (metals, rocks, etc.), cardboard and some viscous liquids, such as paint;
• - translucent objects that absorb a part of the radiation and allow the rest to pass through. For these objects, radiation propagation is accompanied by absorption that increases the energy of the medium. A familiar example is that of oil.

1.2. Radiant flux

1.2.1. Definition of radiant flux, coefficient of absorption

By definition, radiant flux denoted by F is the power emitted by a source throughout the space in which it can radiate. Radiant flux is expressed in Watts (W).

Let us consider an object receiving an incident energy flux Fi. The surface of the object is chosen to allow radiation reflection, absorption and transmission (Figure 1.2).

According to the law of conservation of energy, we have:

In this relation, Fr is the reflected radiant flux, Fa designates the absorbed radiant flux and Ft represents the transmitted radiant flux. Let us consider ?, a and t as the coefficients of reflection, absorption and transmission, respectively, of the radiant flux. Their expressions are given as:

Figure 1.2. Decomposition of an incident radiant flux Fi at the contact with the surface of an object

Implementing relations [1.3] in [1.2], the conservation of energy can be written as:

Coefficients ?, a and t characterize the behavior of an object that is subjected to radiation. It is worth noting that the absorption coefficient a is the most important parameter. This coefficient measures the proportion in which incident electromagnetic radiation is converted into thermal energy.

APPLICATION 1.2.-

Let us consider an arbitrary process of reflection, absorption and transmission of an incident radiant flux. Calculate the absorbed flux.

Given data. ? = 30 %; t = 20 %; transmitted flux: 200 W.

Solution. Fa = 500 W.

1.2.2. Black body and gray body

There are two types of bodies:

• - gray bodies for which a < 1;
• - black bodies for which a = 1.

By definition, a black body is an ideal (therefore fictitious) object that has the specific property of perfectly absorbing the radiations of the visible spectrum irrespective of their frequency. The adjective "black" highlights only the fact that the object absorbs all the radiations of the visible spectrum so that it appears to be black. A black body can be actually realized by piercing a small orifice in the wall of a temperature-controlled cavity (whose walls are brought to a given temperature T). No radiation entering this cavity can escape. Hence the orifice behaves as a black body. It is nevertheless worth noting that an insignificant amount of thermal radiation leaves the cavity, but it is not sufficient to perturb the thermal equilibrium established in the cavity (but it is sufficient to be studied experimentally). Black velvet and black ink are simple examples of black bodies. Let us finally note that a gray body is not necessarily gray. This term designates any object whose absorption coefficient is a < 1.

1.3. Black body emission spectrum

1.3.1. Isotherms of a black body: experimental facts

By definition, the electromagnetic energy density denoted by du in the band of angular frequency between ? and ? + d? (or...