Absorption, Induced and Spontaneous Emission

Assume that molecules with the energy levels E and E2 have been brought into the thermal radiation field of Sect. 2.2. If a molecule absorbs a photon of energy hv = E2 — E, it is excited from the lower energy level E into 

The constant factor B 2 is the Einstein coefficient of induced absorption. Each absorbed photon of energy hv decreases the number of photons in one mode of the radiation field by one. 

The radiation field can also induce molecules in the excited state 2 to make a transition to the lower state E] with simultaneous emission of a photon of energy hv. This process is called induced or stimulated) emission. The induced photon of energy hv is emitted into the same mode that caused the emission. This means that the number of photons in this mode is increased by one. The probability d P2i/dt that one molecule emits one induced photon per second is in analogy to (2.15) 

The constant factor 21 is the Einstein coefficient of induced emission. 

An excited molecule in the state 2 rnfiy also spontaneously convert its excitation energy into an emitted photon hv. This spontaneous radiation can be emitted in the arbitrary direction k and increases the number of photons in the mode with frequency v and wave vector k by one. In the case of isotropic emission, the probability of gaining a spontaneous photon is equal for all modes with the same frequency v but different directions k. 

It is therefore possible to determine p(v) experimentally by measuring the spectral distribution of the radiation penetrating through a small hole in the walls of the cavity. If the hole is sufficiently small, the energy loss through this hole is negligibly small and does not disturb the thermal equilibriiun inside the cavity. 

The probability d /dt per second that a photon ht = Ej-Ej is spontaneously emitted by a molecule, depends on the structure of the molecule and the selected transition 2) -+ l) but it is independent of the external radiation field. 

Assume that molecules with energy levels and l ve been brought into the thermal radiation field of Sect.2.2. If a molecule absorbs a photon of energy hv with hv = E - it is excited from the lower energy level E into the higher level E (see Fig.2.4). This process is called induced absorption. The probability per second that a molecule will absorb a photon, is proportional to the number of photons of energy hv per unit volume, and can be expressed in terms of the spectral energy density p(v) of the radiation field as 

The constant factor 12 is the Einstein coefficient of induced absorption. It depends on the electronic structure of the atom, i.e. on its electronic wave functions in the 

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From these equations one also finds the rate coefficient matrix for themial radiative transitions including absorption, induced and spontaneous emission in a themial radiation field following Planck s law [35] ... 

FIGURE 18.35 Black-body (BB)-induced and spontaneous emission rates within the v = 0 rotational manifold of HD" ". Time values are the natural lifetimes. Rate values are for absorption of BB radiation and stimulated emission by BB radiation. BB temperature is 300 K. 

The book begins with a discussion of the fundamental definitions and concepts of classical spectroscopy, such as thermal radiation, induced and spontaneous emission, radiation power and intensity, transition probabilities and oscillator strengths, linear and nonlinear absorption and dispersion, and coherent and incoherent radiation fields. In order to understand the theoretical limitations of spectral resolution in classical spectroscopy, the next chapter treats the different causes of the broadening of spectral lines. Numerical examples at the end of each section illustrate the order of magnitude of the different effects. 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

We now consider the transition rates for absorption of and stimulated emission induced by the black body radiation and compare these rates to the spontaneous... 

Einstein obtained coefficients for induced absorption B , induced emission Bu i, and spontaneous emission Au, of light by the following thermodynamic arguments, based on Arrhenius 116 law. 

The transfer of momentum from the beam to an absorbing particle is therefore straightforward compared to other cases, such as transparent particles and atoms. For transparent particles, refraction and induced polarisation must be taken into account. For an atom, the frequency dependence of the absorption and spontaneous emission must be considered, while for an absorbing particle, the absorption can be assumed to be independent of frequency, and inter-atomic collision rates within the particle can be assumed to be high enough to cause deexcitation without re-emission. 

Einstein coefficients Coefficients used in the quantum theory of radiation, related to the probability of a transition occurring between the ground state and an excited state (or vice versa) in the processes of induced emission and spontaneous emission. For an atom exposed to electromagnetic radiation, the rate of absorption is given by... 

The optical cooling techniques discussed so far are restricted to true two-level systems because the cooling cycle of induced absorption and spontaneous emission has to be performed many times before the atoms come to rest. In molecules the fluorescence from the upper excited level generally ends in many rotational-vibrational levels in the electronic ground state that differ Irom the initial level. Therefore most of the molecules cannot be excited again with the same laser. They are lost for further cooling cycles. 

Since we are interested mainly in the roots of spontaneous emission, we shall quantize the electromagnetic field because we know that the semi-classical description where the atom is quantized and the field is classical, does not provide aity spontaneous emission it is introduced phenomenologically by a detailed balance of the population of the two-states atom and comparison with Planck s law. This procedure introduced by Einstein gave the well-known relationship between induced absorption (or emission), and spontaneous emission probabilities, the B12, B21 and A21 coefficients, respectively, but caimot produce the coherent aspect and its link with spontaneous emission. 

Let us consider a molecule and two of its energy levels E) and f 2- The Einstein coefficients are defined as follows (Scheme B2.2) Bn is the induced absorption coefficient, B2i is the induced emission coefficient and A21 is the spontaneous emission coefficient. 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

In addition to absorption and stimulated emission, a third process, spontaneous emission, is required in the theory of radiation. In this process, an excited species may lose energy in the absence of a radiation field to reach a lower energy state. Spontaneous emission is a random process, and the rate of loss of excited species by spontaneous emission (from a statistically large number of excited species) is kinetically first-order. A first-order rate constant may therefore be used to describe the intensity of spontaneous emission this constant is the Einstein A factor, Ami, which corresponds for the spontaneous process to the second-order B constant of the induced processes. The rate of spontaneous emission is equal to Aminm, and intensities of spontaneous emission can be used to calculate nm if Am is known. Most of the emission phenomena with which we are concerned in photochemistry—fluorescence, phosphorescence, and chemiluminescence—are spontaneous, and the descriptive adjective will be dropped henceforth. Where emission is stimulated, the fact will be stated. 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

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