Emission, spontaneous

If we start with excited molecules or atoms, we can observe spontaneous emission— generation of light as the atom or molecule drops down to a lower energy level. Equation 8.1 also dictates the possible frequencies in the emission spectrum. Spontaneous emission in general gives radiation in all directions, and the selection rules are almost the same as in Equation 8.4. The only difference is that we must have An 0 for the final state to be lower in energy than the initial state. 

A hydrogen atom with its electron in a 2p orbital will decay back down to the Is orbitral in approximately 1 nanosecond, giving off a photon with A = 121 nm (determined by the energy difference between the two states). On the other hand, an electron in a 2s state is stuck (we call 2s a metastable level), since emission to the only lower state (1. s ) is forbidden by the Al = 1 selection rule. On average, it takes about 100 ms for the electron to get back down to the ground state from 2s. 

As an application to the results obtained above we consider the spontaneous emission rate from our molecule after it is prepared in the excited state 2). In terms of zero-order states of the Hamiltonian Hq = + Hr the initial state is 2, 0 ) 

Care need to be taken in order to accommodate the vcctornalurc of /z and of the incident field. For spherically symmetric molecules, each of the two directions perpendicular to the wavevector k of a given mode contributes equally We can use for p any component, say /Zx, of the transition dipole, and the density of states used below takes an extra factor of 2 for the two possible directions of the polarization. 

Now consider the density p of 1-photon states. Because each of these states is characterized by one mode being in the first excited state while all others are in the ground state, the number of states is the same as the munber of modes and the required density of states per unit energy is given by pE of Eq. (3,20). Using this, together with (3.30) in (3.29) leads to 

Several observations should be made regarding this result. First, while regular chemical solvents are characterized by jh. 1, different solvents can differ considerably in their dielectric constants s, and Eq. (3.31) predicts the way in which the radiative lifetime changes with s. Note that a dependence on s may appear also in 6012 because different molecular states may respond differently to solvation, so a test of this prediction should be made by monitoring both and W2i as functions of the dielectric constant in different embedding solvents. 

Second, the dependence on a is a very significant property of the radiative decay rates. Assmning similar transition dipoles for allowed transitions, Eq. (3.31) predicts that lifetimes of electronically excited states ( x 2i of order lO cm ) are shorter by a factor of -lO than those of vibrational excitations ( 021 of order 10 cm ), while the latter are 10 shorter than those of rotational excitations (ct 2i of order lO cm ), as indeed observed. 

An excited atom (or molecule) can make a transition to a lower state and emit a photon with an energy equal to the transition energy. This radiative transition occurs spontaneously, as a quantum jump (Bohr jump), in a random direction, via the allowed decay channels (Fig. 2.1(a)). The instant of time when the spontaneous transition occurs is also random. Consequently complete information about the process can be obtained by averaging the results of a large number of independent measurements of the energy E = Hlv (or the frequency lv or the wavelength X = 2Trcju ) of the emitted photon and of the delay time between the excitation of the atom and its decay from the excited state e). 

Let the atom, at the instant of time t = 0, be in the upper state e). All the lower states will be designated as by i). The decay of the state e) to the lower states i) can be described by a decrease in the probability Ps t) that the atom is in the state e) at the instant t, which obeys the exponential law 

What is the spectrum of spontaneous emission for an individual transition e) i) For a finite lifetime of the excited state, this spectrum cannot be reduced to a spectrum only at the central frequency of the transition of the emitting atom (or molecule), but has a certain characteristic width that is usually defined as the full width at half maximum (FWHM). This radiative width is called the natural width. It is equal 

The function f Lv) in the above expression is normalized so as to make its integral over all frequencies equal to unity. Note that the natural width of the spontaneous-emission line for the transition between the levels indicated in Fig. 2.1 may be greater than 7s nt because of the decay of the excited state to other states via some channels other than the transition e) i). 

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A1.6.1.3 ABSORPTION, STIMULATED EMISSION AND SPONTANEOUS EMISSION OF LIGHT... 

For themial light, the iiumber of transitions per second induced by stimulated emission integrated over solid angles, is equal to The total emission, which is the sum of the stimulated and spontaneous emission, may be obtained by letting A A + 1 in the expression for stimulated emission, giving... 

As an example, consider the two-level system, with relaxation that arises from spontaneous emission. In this case there is just a single V. ... 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

The above fomuilae for the absorption spectrum can be applied, with minor modifications, to other one-photon spectroscopies, for example, emission spectroscopy, photoionization spectroscopy and photodetachment spectroscopy (photoionization of a negative ion). For stimulated emission spectroscopy, the factor of fflj is simply replaced by cOg, the stimulated light frequency however, for spontaneous emission... 

As described at the end of section Al.6.1. in nonlinear spectroscopy a polarization is created in the material which depends in a nonlinear way on the strength of the electric field. As we shall now see, the microscopic description of this nonlinear polarization involves multiple interactions of the material with the electric field. The multiple interactions in principle contain infomiation on both the ground electronic state and excited electronic state dynamics, and for a molecule in the presence of solvent, infomiation on the molecule-solvent interactions. Excellent general introductions to nonlinear spectroscopy may be found in [35, 36 and 37]. Raman spectroscopy, described at the end of the previous section, is also a nonlinear spectroscopy, in the sense that it involves more than one interaction of light with the material, but it is a pathological example since the second interaction is tlirough spontaneous emission and therefore not proportional to a driving field... 

We have seen in section Al.6.2.4 that external fields alone caimot change the value of Tr(p ) Changes in the purity can arise only from the spontaneous emission, which is inlierently uncontrollable. Wliere then is the control ... 

Note that the differential equation obtained from this approaeh will never agree perfeetly with the results of a simulation. The above fomuilation is essentially an adiabatie fomuilation of die proeess the spontaneous emission is eonsidered to be slow eompared with the time seale for the purity-preserving transformations generated by the external field, whieh is what allows us to assume m the theory that the external field... 

Tannor D J and Bartana A 1999 On the interplay of control fields and spontaneous emission in laser cooling J. Phys. Chem. A 103 10 359-63... 

Classic examples are the spontaneous emission of light or spontaneous radioactive decay. In chemistry, an important class of monomolecular reactions is the predissociation of metastable (excited) species. An example is the fonnation of oxygen atoms in the upper atmosphere by predissociation of electronically excited O2 molecules [12, 13 and 14] ... 

The first mfonnation on the HE vibrational distribution was obtained in two landmark studies by Pimentel [39] and Polanyi [24] in 1969 both studies showed extensive vibrational excitation of the HE product. Pimental found that tire F + H2 reaction could pump an infrared chemical laser, i.e. the vibrational distribution was inverted, with the HF(u = 2) population higher than that for the HF(u = 1) level. A more complete picture was obtained by Polanyi by measuring and spectrally analysing tlie spontaneous emission from vibrationally excited HE produced by the reaction. This infrared chemiluminescence experiment yielded relative populations of 0.29, 1 and 0.47 for the HF(u =1,2 and 3)... 

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] ... 

The interpretation of emission spectra is somewhat different but similar to that of absorption spectra. The intensity observed m a typical emission spectrum is a complicated fiinction of the excitation conditions which detennine the number of excited states produced, quenching processes which compete with emission, and the efficiency of the detection system. The quantities of theoretical interest which replace the integrated intensity of absorption spectroscopy are the rate constant for spontaneous emission and the related excited-state lifetime. 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

For many reaction products and for the detection of molecules in their ground vibrational level, some laser-based spectroscopic method must be employed, rather than observation of spontaneous emission. The simplest spectroscopic method for detemiining concentrations of specified product internal states would involve the... 

Figure B2.3.8. Energy-level sehemes deseribing various optieal methods for state-seleetively deteeting ehemieal reaetion produets left-hand side, laser-indueed fluoreseenee (LIF) eentre, resonanee-enlianeed multiphoton ionization (REMPI) and right-hand side, eoherent anti-Stokes Raman speetroseopy (CARS). The ionization oontinuiim is denoted by a shaded area. The dashed lines indieate virtual eleetronie states. Straight arrows indieate eoherent radiation, while a wavy arrow denotes spontaneous emission.

Apart from the natural lifetime due to spontaneous emission, both uni- and bimolecular processes can contribute to the observed value of T. One important contribution comes from coiiisionai broadening, which can be distmguished by its pressure dependence (or dependence upon concentration [M] of tlie collision partner) ... 

Figure C 1.4.2. Spontaneous emission following absorjDtion occurs in random directions, but absorjDtion from a light beam occurs along only one direction.

In equation (Cl.4.14) the saturation parameter essentially defines a criterion to compare the time required for stimulated and spontaneous processes. If I then spontaneous coupling of the atom to the vacuum modes of the field is fast compared to the stimulated Rabi coupling and the field is considered weak. If s" 1 then the Rabi oscillation is fast compared to spontaneous emission and the field is said to be strong. Setting s equal to unity defines the saturation condition... 

Note that negative Acoj (red detuning) produces a force attracting the atom to the intensity maximum while positive (blue detuning) repels the atom away from the intensity maximum. The spontaneous force or cooling force can also be written in tenns of the saturation parameter and the spontaneous emission rate. 

Figure Cl.4.5. Population modulation as the atom moves through the standing wave in the Tin-periD-lin one dimensional optical molasses. The population lags the light shift such that kinetic is converted to potential energy then dissipated into the empty modes of the radiation field by spontaneous emission (after 1171).

The acronym LASER (Light Amplification via tire Stimulated Emission of Radiation) defines the process of amplification. For all intents and purjDoses tliis metliod was elegantly outlined by Einstein in 1917 [H] wherein he derived a treatment of the dynamic equilibrium of a material in a electromagnetic field absorbing and emitting photons. Key here is tire insight tliat, in addition to absorjDtion and spontaneous emission processes, in an excited system one can stimulate tire emission of a photon by interaction witli tire electromagnetic field. It is tliis stimulated emission process which lays tire conceptual foundation of tire laser. 

Returning to the kinetie equations that govern the time evolution of the populations of two levels eonneeted by photon absorption and emission, and adding in the term needed for spontaneous emission, one finds (with the initial level being of the lower energy) ... 

When the light souree s intensity is so large as to render gBf i Af i (i.e., when the rate of spontaneous emission is small eompared to the stimulated rate), this population ratio reaehes (Bi f/Bf i), whieh was shown earlier to equal (gf/gi). In this ease, one says that the populations have been saturated by the intense light souree. Any further inerease in light intensity will result in zero inerease in the rate at whieh photons are being absorbed. Transitions that have had their populations saturated by the applieation of intense light sourees are said to display optieal transparency because they are unable to absorb (or emit) any further photons because of their state of saturation. 

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