Introduction to the Physics of Matter by Nicola Manini

Author:Nicola Manini
Language: eng
Format: epub
ISBN: 9783030572433
Publisher: Springer International Publishing

(4.124)

The comparison of the two explicit expressions, which must coincide for any temperature, requires the following identities for the coefficients:

(4.125)

known as Einstein relations. Relations (4.125), which can be reformulated as

(4.126)

were derived under the assumption of equilibrium. Since the coefficients depend only on electromechanical properties of the particles, these relations hold unchanged for arbitrary field conditions. Once one of these coefficients is evaluated, either experimentally or, e.g., by of Eq. (2.​45), the two others are fixed by Eq. (4.126).

The first equality expresses the symmetric role of the initial and final states in QM, implying that a radiation field induces equal rates of excitation and of stimulated emission . For a strong radiation intensity, such that , the spontaneous emission rate becomes negligible, and implies in particular that , thus rapidly also (saturated transition). This result clarifies the role of the intense pump beam in the experiment of Fig. 2.​11: it saturates the individual components of the transition of hydrogen.

The second relation implies that, for a given spectral energy density (independent of ), the ratio of spontaneous emission to stimulated emission varies with . Accordingly, in a low-energy (microwave, IR) transition, stimulated emission tends to prevail, while at higher energy (UV, X-rays) spontaneous emission usually dominates. Even when equilibrium radiation is considered, the ratio of spontaneous to stimulated emission

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