Читать книгу Introduction to the Physics and Techniques of Remote Sensing - Jakob J. van Zyl - Страница 33

The interaction of electromagnetic waves with matter (e.g., molecular and atomic structures) calls into play a variety of mechanisms which are mainly dependent on the frequency of the wave (i.e., its photon energy) and the energy level structure of the matter. As the wave interacts with a certain material, be it gas, liquid, or solid, the electrons, molecules, and/or nuclei are put into motion (rotation, vibration, or displacement), which leads to exchange of energy between the wave and the material. This section gives a quick simplified overview of the interaction mechanisms between waves and matter. Detailed discussions are given later in the appropriate chapters throughout the text.

Atomic and molecular systems exist in certain stationary states with well‐defined energy levels. In the case of isolated atoms, the energy levels are related to the orbits and spins of the electrons. These are called the electronic levels. In the case of molecules, there are additional rotational and vibrational energy levels which correspond to the dynamics of the constituent atoms relative to each other. Rotational excitations occur in gases where molecules are free to rotate. The exact distribution of the energy levels depends on the exact atomic and molecular structure of the material. In the case of solids, the crystalline structure also affects the energy levels’ distribution.

In the case of thermal equilibrium, the density of population N_i at a certain level i is proportional to (Boltzmann’s law):

(2.39)

where E_i is the level energy, k is Boltzmann’s constant, and T is the absolute temperature. At absolute zero all the atoms will be in the ground state. Thermal equilibrium requires that a level with higher energy be less populated than a level of lower energy (Fig. 2.13).

To illustrate, for T = 300 K, the value for kT is 0.025 eV (one eV is 1.6 × 10⁻¹⁹ joules). This is small relative to the first excited energy level of most atoms and ions, which means that very few atoms will be in the excited states. However, in the case of molecules, some vibrational and many rotational energy levels could be even smaller than kT, thus allowing a relatively large population in the excited states.

Figure 2.13 Curve illustrating the exponential decrease of population as a function of the level energy for the case of thermal equilibrium.

Let us assume a wave of frequency ν is propagating in a material where two of the energy levels i and j are such that

(2.40)

This wave would then excite some of the population of level i to level j. In this process, the wave loses some of its energy and transfers it to the medium. The wave energy is absorbed. The rate p_ij of such an event happening is equal to:

(2.41)

where ℰ_ν is the wave energy density per unit frequency and B_ij is a constant determined by the atomic (or molecular) system. In many texts, p_ij is also called transition probability.

Once exited to a higher level by absorption, the atoms may return to the original lower level directly by spontaneous or stimulated emission, and in the process they emit a wave at frequency ν, or they could cascade down to intermediate levels and in the process emit waves at frequencies lower than ν (see Fig. 2.14). Spontaneous emission could occur any time an atom is at an excited state independent of the presence of an external field. The rate of downward transition from level j to level i is given by

(2.42)

where A_ji is characteristic of the pair of energy levels in question.

Stimulated emission corresponds to downward transition which occurs as a result of the presence of an external field with the appropriate frequency. In this case the emitted wave is in phase with the external field and will add energy to it coherently. This results in an amplification of the external field and energy transfer from the medium to the external field. The rate of downward transition is given by

(2.43)

Figure 2.14 An incident wave of frequency ν_ij is adsorbed due to population excitation from E_i to E_j. Spontaneous emission for the above system can occur at ν_ij as well as by cascading down via the intermediate levels l and k.

The relationships between A_ji, B_ji, and B_ij are known as the Einstein’s relations:

(2.44)

(2.45)

where n is the index of refraction of the medium.

Let us now consider a medium that is not necessarily in thermal equilibrium and where the two energy levels i and j are such that E_i < E_j. N_i and N_j are the population in the two levels, respectively. The number of downward transitions from level j to level i is (A_ji + B_jiℰ_ν)N_j. The number of upward transitions is B_ijℰ_νN_i = B_jiℰ_νN_i. The incident wave would then lose (N_i − N_j) B_ijℰ_ν quanta per second. The spontaneously emitted quanta will appear as scattered radiation which does not add coherently to the incident wave.

The wave absorption is a result of the fact that usually N_i > N_j. If this inequality can be reversed, the wave would be amplified. This requires that the population in the higher level is larger than the population in the lower energy level. This population inversion is the basis behind laser and maser operations. However, it is not usually encountered in the cases of natural matter/waves interactions which form the topic of this text. (Note: Natural maser effects have been observed in astronomical objects; however, these are beyond the scope of this text.)

The transition between different levels in usually characterized by the lifetime τ. The lifetime of an excited state i is equal to the time period after which the number of excited atoms in this state have been reduced by a factor e⁻¹. If the rate of transition out of the state i is A_i, the corresponding lifetime can be derived from the following relations:

(2.46)

(2.47)

(2.48)

If the transitions from i occur to a variety of lower levels j, then

(2.49)

(2.50)