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The single configurational coordinate model (SCCM) is a very useful model for analysing and interpreting the transitions within transition metal ions. Consider an optically active ion (A) in a transparent host material consisting of ions (B). The A ion will be surrounded by a number of B ions belonging to the host material. This environment is dynamic because the A and B ions form part of a vibrating lattice. Also consider that the optically active ion A is coupled to the vibrating lattice. This means that neighbouring B ions can vibrate about some average point and this affects the electronic states of the A ion.[19] The SCCM is dependent on two main approximations:

• The ions move very slowly in comparison to the valence electrons, this approximation is reasonable because the nuclei are much heavier than electrons and therefore move on a much slower timescale.

• The movement of the ligand B ions is considered as a single symmetrical ‘breathing’ mode. In this case only one nuclear coordinate, which corresponds to the distance A-B, is needed to describe the position of all the ligands. This coordinate is called the configurational coordinate Q,

The potential energy curves for the ground state (electronic state a) and an excited state (electronic state b) for the one-coordinate dynamic centre A are represented diagrammatically in the SCCM in figure 2.7. These potential energy curves are approximated by parabolas according to the harmonic oscillator approximation. In this approximation the B ions pulsate in harmonic oscillation around the equilibrium positions.[19] The horizontal lines on each potential energy curve represent the allowed vibration modes or phonon levels. For the harmonic oscillator of electronic state a at frequency ω, the permitted phonon energies En are given by

ω

Where n = 0, 1, 2… and so on. Similarly for electronic state b, which may have a different harmonic oscillator frequency, the allowed phonon levels are characterised by m = 0, 1, 2… and so on. The probability distribution in each of these phonon levels is given by the square of its oscillator function. These probability distributions are represented very approximately by the red curves on each of the phonon levels. These distributions show that in the lowest phonon level the probability distribution is centred around the equilibrium position and in higher order phonon levels the maximum amplitude probability occurs where the phonon levels cross the potential energy curves. This has a strong influence on determining the line shapes of absorption and emission spectra. The Frank-Condon principle states that electronic transitions are most likely to occur when two vibrational wavefunctions overlap and that they are very fast in comparison to the motion of the lattice. This implies that electronic transitions can be represented by vertical lines, as in figure 2.7.

FIGURE 2.7 The single configurational coordinate model, showing how phonon assisted absorption gives rise to absorption line shapes and the mechanisms for phonon assisted non-radiative decay.

The peak in the absorption band occurs at an energy where the overlap between the probability distribution in the phonon levels is at a maximum, which is illustrated in figure 2.7 by the transition Ea. Similarly the peak in the emission band corresponds to

the transition Ee. The difference between the absorption and emission peaks (Ea - Ee) is

known as the Stokes shift (SS), i.e. SS = Ea - Ee. It should be noted that the equilibrium

position coordinates Q0a and Q0b are different for the electronic states a and b. This

reflects the difference in electron-phonon coupling between the two states. The dimensionless Huang-Rhys parameter (S) quantifies this difference in electron-phonon coupling and is given by

ω

h

dis E

S = (2.12)

Where Edis is defined in figure 2.7 and ħω is the energy of the breathing mode vibration.

The Huang-Rhys parameter is related to the Stokes shift by ω h ) 1 2 ( − = − =E E S SS _{a e} (2.13)

The absorption band shape, induced from phenomena illustrated in the SCCM in figure 2.7, is due to overlapping occurring between the vibrational m states and the n = 0 phonon level, which would occur at very low temperatures (~0 K). The transitions n = 0

↔ m = 0 are termed zero phonon lines (ZPL) as they occur without the participation of phonons. ZPL are characterised by relatively narrow line-widths which, disregarding the effect of the host, is the natural linewidth discussed in section 2.6.1. These transitions can commonly be observed in the low temperature absorption and emission spectra of transition metal doped crystals, as in V2+ doped ZnSe for example.[61] However they are rarely observed in transition metal doped glasses because of the greater inhomogeneous broadening in these hosts. It can be seen from figure 2.7 that for sufficient S no ZPL will be observed.

Once in an excited state, the ion A can reach its ground state through the emission of a photon (radiative decay) or through the emission of phonons (non-radiative decay). Non-radiative decay can be accounted for by the SCCM, illustrated in figure 2.7. For sufficiently large S, excitation from the ground state results in the population of higher order phonon modes in the excited state. These higher order phonon modes can coincide with the crossing of the potential energy curves of excited state a and b and therefore the system can relax through the phonon levels of excited state a. If the populated higher order phonon modes coincides with the proximity of the potential energy curves of excited state a and b then the same process may occur by tunnelling.