Einstein radiative probabilities

The rationalization of absorption band intensities in lanthanide compounds (as developed since 1945) got in the Judd-Ofelt treatment a form in eq. (8) (see 2.5) directly involving the Einstein transition probabilities A J, / ). In the case of resonant absorption and emission being the only process occurring (like sodium atoms in yellow light), eq. (1) gives a direct relation between the (radiative) lifetime r and the oscillator strength P. As an important example of such a resonant situation can be mentioned r = 10.9 ms for the first excited state of gadolinium(III) aqua ions (Carnall, 1979) at 31200 cm of which three quarters... 

Einstein s A and B coefficients. Quantized systems, such as atoms and molecules, emit and absorb radiation of frequency vy = E — Ej /hc in transitions between states i) and j) of energy , Ej. Einstein assumed that the probability that a system in state i) will absorb a photon of energy hcvij is proportional to the density of radiative energy per frequency interval, u(vij) dv. The probability of absorbing a photon in the time interval dt is given by... 

All upward radiative transitions in Figure 3.23 are absorptions which can promote a molecule from the ground state to an excited state, or from an excited state to a higher excited state. We have seen that the probability of these transitions is related ultimately to the transition moment between the two states and thereby to the Einstein coefficient A. In practice two other related quantities are used to define the intensity5 of an absorption, the oscillator strength f and the molar decadic extinction coefficient e. 

Luminescence Kinetics, Luminescence Lifetimes. The Einstein coefficient A for spontaneous emission gives the probability of radiative transition. Since this probability is the same for all molecules of the same excited species, it follows that the decrease in the number of excited molecules within a differential time increment is simply proportional to the number of excited... 

The near equality of population in the two levels is an important factor in determining the intensity of the NMR signal. According to the Einstein formulation, the radiative transition probability between two levels is given by... 

Here, ijJ v)) is the mean and angle averaged value of the local radiation field, weighted with the profile function of the local absorption coefficient. The Aij and Bij are the Einstein coefficients for spontaneous and induced transitions, while denotes the probability for a collisional transition from state j —> i. Accordingly, the first row in eq. (10.20) accounts for spontaneous emission and collision of the molecule considered with H2, whereas in the second row induced emission processes are described. This system of rate equations has to be solved simultaneously with the generalized radiative transfer equation for every point in physical and velocity space. 

While the overall quantum yield is relatively easy to measure, Q, which is needed to evaluate /sens/ is quite difficult to determine experimentally in view of the weakness of the f-f transitions. One way to estimate it is by means of Eq. (10) where Tobs is the actual lifetime of the emitting excited state and Trad is its natural radiative lifetime which obey Einstein relation for spontaneous emission probability between two states with quantum numbers / and /, A( Pj, Pj ) (Gorller-Walrand and Binnemans, 1998) ... 

Using highly correlated MCSCF-Cl wave functions for the A rij and X states, the transition moment function for the A - X transition has been calculated which in turn allowed the evaluation of Einstein coefficients of spontaneous emission A, (v = 0,1 v" = 0,1,2), absorption oscillator strengths f v (v = 0,1 v" = 0,1), and radiative lifetimes for A Ili, v = 0,1 of PH and PD. The v = 0 lifetime Xrad = 399 ns for PH (390 ns for PD) is shorter than the experimental value, probably because the large correlation energy contributions to the transition moment have not been sufficiently accounted for in the calculation [32]. 

A simple relation will be derived between the probabilities for spontaneous and stimulated emission and absorption of radiation using well-known statistical distribution laws. Consider a system such as that illustrated in Fig.4.3 with two energy levels, E and E2, populated by and N2 atoms, respectively. Three radiative processes can occur between the levels, as discussed above. In the figure the processes are expressed using the so-called Einstein coefficients 6 2, B21 and A21, which are defined such that the rate of change in the population numbers is... 

In the case of emission, A(J,J ) is also known as Einstein s coefficient of spontaneous emission, and the sum of all probabilities for all radiative transitions is equal to the inverse of the radiative rate constant, Icr, in turn the reciprocal of the emissive state lifetime, Tr. 

---