Читать книгу Spectroscopy for Materials Characterization - Группа авторов - Страница 16

A simplified model of the atom constituted by two nondegenerate energy levels is assumed to evaluate the interaction with radiation. As reported in Figure 1.4, the lower energy level is E ₁ and the higher energy level is E ₂. This system interacts with the thermal equilibrium radiation field at temperature T.

According to Planck’s theory, the energy distribution of the radiation is given by [8, 11, 12]

(1.24)

where ρ(ν) is the density of energy for unit volume and unit frequency interval, h the Planck’s constant, ν the radiation frequency, c the speed of light, k the Boltzmann’s constant (1.38 ⋅ 10⁻²³ J K⁻¹), and T the absolute temperature.

Figure 1.4 Top: Schematic representation of two energy‐level atom. The arrows represent the absorption and emission transition processes whose probability is given by Einstein’s coefficients A ₂₁, B ₁₂, B ₂₁ and the density of radiation ρ(ν). Bottom: Schematic representation of two energy levels of an atom. The arrows represent the excitation and relaxation transitions. The absorption rate is R _abs, the radiative emission rate is k _r, and the non‐radiative transition rate is k _nr.

The interaction between a density N of atoms for unit of volume and the radiation field causes an exchange of energy with those electromagnetic waves’ modes having frequency related to the atom’s energy levels’ separation by the equation E ₂ − E ₁ = hν. As a consequence, based on Einstein’s treatment, the following processes can occur [8, 13]:

 Transition from the state E 1 to the state E 2, stimulated by the absorption of a photon; based on Einstein’s theory, the rate of this process is given by(1.25)

where N ₁ is the density of atoms (population) in the lower energy state.

 Transition from the state E 2 to the state E 1, stimulated by the emission of a photon; the rate is given by(1.26)

where N ₂ is the density of atoms in the upper energy state.

 Spontaneous transition from the state E 2 to the state E 1 with a rate(1.27)

The Einstein’s coefficients A ₂₁, B ₁₂, and B ₂₁ have been used. In particular, it is worth observing that B ₁₂ and B ₂₁ are related to the presence of the field (stimulated processes of absorption and emission, respectively), whereas A ₂₁ is present also without electromagnetic field and is related to spontaneous emission. This term is related to the radiative emission lifetime introduced in the previous paragraph and, in detail, it is the reciprocal of the lifetime at low temperature, A ₂₁ = 1/τ [13]. At thermal equilibrium, the population of atomic states should reach a stationary condition and it is expected that

(1.28)

and, based on the above reported processes, one obtains

(1.29)

and the relation

(1.30)

The Boltzmann distribution at thermal equilibrium in a two‐level system without degeneracy predicts that [14]

(1.31)

Equating (1.30) and (1.31) and solving with respect to ρ(ν), it is found that

(1.32)

This distribution of energy density in the electromagnetic field should coincide with the Planck’s law at thermal equilibrium. As a consequence, by equating (1.32) and (1.24), it is found that

(1.33)

and, finally

(1.34)

(1.35)

In the case of degenerate energy levels with degeneracy g ₁ and g ₂, it is shown that (1.35) transforms into

(1.36)

whereas (1.34) remains unchanged [8, 13].

Using the quantum mechanical treatment of the interaction between radiation and matter and, in particular, neglecting any magnetic contribution and considering the electric dipole approximation, the atom can be described by a dipole moment

(1.37)

where e is the electron charge (1.602 ⋅ 10⁻¹⁹ C) and r is its position vector with respect to the atomic nucleus. The time‐dependent perturbation theory enables to show that the probability to populate the higher energy level of the atom E ₂ (multiplied by unit frequency interval), starting from the level with energy E ₁, is given by [8, 9, 13]:

(1.38)

where V is the interaction energy between the electric field and the electric dipole moment:

(1.39)

and ℏ = h/2π. Considering a linearly polarized lightwave with electric field of amplitude ₀, wavevector k , and angular frequency ω = 2πν

(1.40)

the probability of population of the excited state per unit of time, coinciding with the transition rate, is then given by

(1.41)

in which the electric dipole matrix element μ ₁₂ relative to the considered atomic states in the direction of the external field has been introduced. Using (1.29), it is possible to find that

(1.42)

This result shows a connection between the macroscopic empiric quantities and the microscopic ones related to the quantum mechanical states of the electron in the atom. In particular, it is shown that the transition probability is related to the electric dipole matrix element μ ₁₂.

Considering that in vacuum [8, 13]

(1.43)

ε ₀ being the permittivity of free space (8.854 × 10⁻¹² kg⁻¹ m⁻³ s⁴ A²), it is possible to find that

(1.44)

Furthermore, since the intensity of radiation and the energy density are related by [1, 8, 13]

(1.45)

using (1.42), it is found that the transition rate between the atom’s energy states is given by

(1.46)

a connection with the intensity of radiation is made explicit now. The rate of energy absorbed per unit of volume by the atom from the electromagnetic field can then be written as

(1.47)

By assuming that all the atoms reside in the N ₁ state, this is the energy lost by the radiation field. If a sample of thickness dx is considered, the energy lost for unit area by the electromagnetic wave is then

(1.48)

By recalling the Lambert–Beer law in differential form from (1.1), it is shown that

(1.49)

where the frequency dependence has been inserted, and finally one obtains

(1.50)

This is a more direct connection between experimental parameters and the microscopic ones. In fact, by using (1.44), it is found that

(1.51)

Integrating (1.51) over the entire frequency range pertaining to the given atomic (or molecular) species gives

(1.52)

where the central absorption frequency has been introduced that is usually related to the maximum of the absorption band. This expression shows that absorption measurements give information on the electric dipole matrix element μ ₁₂ once the concentration of absorbing centers N ₁ is known. The connection reported in (1.41) with the electronic states’ wave functions enables to obtain information about them and vice versa, i.e. once the dipole matrix element is known, from the integral of the absorption band, the concentration of absorbing centers can be found.