Configurational-coordinate diagram

It must be mentioned, however, that the one-dimensional model gives only a qualitative explanation of thermal quenching. A quantitatively valid explanation can by obtained only by a multidimensional model. 

Following the path of the optical absorption transition, presume that Hooke s law expresses the bonding force between the luminescent ion and a nearest-neighbor ion. The deviation from the equilibrium position of the ions is taken as the configurational coordinate denoted as Q. The total energies of the ground 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form  

Here Laguerre s polynomial functions are used. The quantity S can by expressed as shown below, with iC being the force constant of a harmonic oscillator and Qo the coordinate of the equilibrium position of the excited state. 

Physically, S is the number of emitted phonons accompanying the optical transition. It is commonly used as a measure of electron-phonon interaction and is called the Huang-Rhys factor. At m = 0, the transition probabihty is given by the simple relation  

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The Configuration Coordinate Model. To illustrate how the luminescent center in a phosphor works, a configurational coordinate diagram is used (2) in which the potential energy of the luminescent or activator center is plotted on the vertical axis and the value of a single parameter describing an effective displacement of the ions surrounding the activator, is plotted on the horizontal axis (Fig. 2). At low temperatures, near room... 

Fig. 2. General configurational—coordinate diagrams for (a) broad-band absorbers and emitters, and (b) narrow-band or line emitters. The ordinate represents the total energy of the activator center and the abscissa is a generalized coordinate representing the configuration of ions surrounding the...

Fig. 3. A configurational—coordinate diagram showing mechanisms of radiationless decay to the ground state. Nonradiative decay to the ground-state...

Figure 7-12. Configuration coordinate diagram of the four essential states showing the photophysical processes. Also shown is the calculated PA spectrum based on level energies from EA spectroscopy.

Fig. 12. Configurational coordinate diagram of Prussian blue. Curve g gives the ground state Fe(III)-NC-Fe(II) Curve e gives the MMCT state Fe(II)-NC-Fe(III). The optical transition is indicated by E p, whereas Eo gives the energy difference between the two states. See also text (after data in Ref. [66])...

There is no essential difference between quenching via a MMCT state or a LMCT state. The latter occurs, for example, in Eu(III) if the LMCT state is either at low energy or if this state shows a large offset in the configurational coordinate diagram [23, 35]. The latter occurs in glasses [123], certain cryptates [124] and lanthanum compounds [125]. 

Figure 5.10 The configurational coordinate diagram for the ABe center oscillating as a breathing mode. The broken curves are parabolas within the approximation of the harmonic oscillator. The horizontal full lines are phonon states.

Figure 5.12 A configurational coordinate diagram with which to analyze transitions between two electronic states. Harmonic oscillators at the same frequency Q are assumed for both states. The absorption and emission band profiles are sketched based on the 0 — m (absorption) and n <— 0 (emission) relative transition probabihties (see the text). For simphcity, the minima of these parabolas, Qo and Qg, are not represented.

Figure 5.16 Configurational coordinate diagrams to explain (a) radiative and (b) nonradiative (multiphonon emission) de-excitation process. The sinusoidal arrows indicate the nonradiative pathways.

Fig. 4. Configuration coordinate diagram of Pr + showing radiationless decay from the Po to the level via a c.t. state (virtual recharge mechanism). The relevant 4/ configuration levels have been drawn only. Note break in energy scale...

Fig. 5. Configuration coordinate diagram of Pr +. Drawn parabolas refer to levels of the 4/2 configuration, the upper broken parabola to the lowest level of the 4/5(i configuration of Pr3+ in CaZrOs and the lower broken parabola to the lowest 4/5d level of Pr + in Ca-stabiUzed Zr02 (schematically)...

Fig. 6. Configuration coordinate diagram of a luminescent centre. Non-radiative return from the excited state to the ground state is possible via the crossover S. This requires an activation energy AE which can be supplied at higher temperatures. Exc excitation, em emission...

Fig. 5.49. Configurational coordinate diagram with emission and excitation transitions in Bi ...

The energy relationships in a luminescence process are presented in a configurational coordinate diagram (Fig. 83). This illustrates the relationship between the potential energy E of the luminescence center (ordinate) and a space coordinate (abscissa), which gives the representative separation between the atom involved and its nearest neighbors or the deflection from its spatial equilibrium position. 

Fig. 1. The configurational coordinate diagram. The energy E is plotted versus a configurational coordinate Q. The offset between the parabolae is given by Qb0 - Q 0. The ground state a contains vibrational levels with quantum number n, the excited state b with quantum number ri...

In connection with our treatment of the configurational coordinate diagram (see above) it should be noticed that the occurrence of two progressions for Cr3 + in the elpasoiites indicates that a multi-configurational coordinate diagram must be used. 

Fig. 16. Schematic configurational coordinate diagram of the ground and 3LC and 3MLCT excited states of the Ir3 + complexes. The full and broken lines refer to the state energies and relaxation pathways of the complex in a crystal or in solution, respectively. Straight arrows correspond to radiative and curved arrows to nonradiative relaxation processes. The shaded area indicates the range, in which the 3MLCT state can be found, depending on the environment...

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