Huang-Rhys parameter

Thus, the Huang-Rhys parameter is a measure of the Stokes shift (or the displacement between the ground and excited parabolas). In fact, from Figure 5.12 it can be shown that... 

At this point, we are able to predict the low-temperature optical (absorption and emission) band shapes for different coupling strengths (i.e., different Huang-Rhys parameters), as shown in the next example. 

EXAMPLE 5.5 Sketch the absorption and emission spectra at OK for bands with zero-phonon line at 600 nm, a coupling with an unique breathing mode of energy 200 cm and a Huang-Rhys parameter ofS = l. 

Figure 5.14 The low-temperature band shape, plotted as intensity versus m (final state), expected for different coupling (Huang-Rhys) parameters.

A certain transition metal ion presents two optical absorption bands in a host crystal whose zero-phonon lines are at 600 nm and 700 nm, respectively. The former band has a Huang-Rhys parameter 5 = 4, while for the latter 5 = 0. Assuming coupling with a phonon of 300 cm for the two bands (a) display the 0 K absorption spectrum (absorption versus wavelength) for such a transition metal ion (b) display the emission spectra that you expect to obtain nnder excitation in both absorption bands and (c) explain how you expect these two bands to be affected by a temperature increase. 

A host material is activated with a certain concentration of Ti + ions. The Huang-Rhys parameter for the absorption band of these ions is 5 = 3 and the electronic levels couple with phonons of 150 cm . (a) If the zero-phonon line is at 522 nm, display the 0 K absorption spectrum (optical density versus wavelength) for a sample with an optical density of 0.3 at this wavelength, (b) If this sample is illuminated with the 514 nm line of a 1 mW Ar+ CW laser, estimate the laser power after the beam has crossed the sample, (c) Determine the peak wavelength of the 0 K emission spectrum, (d) If the quantum efficiency is 0.8, determine the power emitted as spontaneons emission. 

Finally, it is important to recall that the simple nonradiative rate law described by Equations (6.1) and (6.2) is only vaUd for (RE) + ions. This is a consequence of the weak ion-lattice interactions for these ions, that leads to a Huang-Rhys parameter of... 

The F center in sodium fluoride (NaF) shows a broad absorption band that peaks at 335 nm (77 K). The shape of this absorption band fits a Huang-Rhys parameter of 28 and coupling with a phonon mode of 0.0369 eV. Estimate the peak position of the F center emission in NaF. 

HS redox couple, 33 91-92 HSe/HSe redox couple, 33 98 HSO," /HSO4 redox couple, 33 96 HSO5/HSO5" redox couple, 33 96 [HTcCpJ, 41 29 Huang-Rhys parameter, 35 325 optical centers, interaction with surroundings, 35 380-381... 

This Huang-Rhys parameter S is related to the Franck-Condon factor for the corresponding transition. It is possible to derive a simple expression for the low-... 

In summary, it should be kept in mind that a Huang-Rhys parameter of 0.3 must still be regarded as very small. This becomes obvious when this value is compared to the observed range of S values. These are frequently larger than one and one also finds S values greater than ten for other compounds (e.g. compare Refs. [96-101]). Therefore, the results foimd for Pd(2-thpy)2 show that the nuclear equilibrium positions of the triplet state Tj and of the ground state Sq are rather similar. 

In a situation, when the purely electronic transition between an excited state and the ground state is allowed or not totally forbidden, a different vibrational activity, the Franck-Condon activity, can become dominant. This can lead, for example, to the occurrence of progressions. Although, the Franck-Condon effect is well known, it is appropriate to summarize briefly the background and to introduce the useful Huang-Rhys parameter. This summary is largely based on the descriptions found in the Refs. [96-100,154], in particular it is referred to Ref. [99, p. 200]. 

Fig. 18. Franck-Condon progressions, (a) The equilibrium positions of the potential surfaces of the excited state II and the ground state 0 are shifted by AQ. This leads to a progression, if the dipole moment of the transition between the states II and 0 is non-zero. The progression is characterized by vertical transitions that are depicted for a low-temperature emission, (b) The intensity distribution of a progression of vibrational satellites depends on the Huang-Rhys parameter S which is proportional to (AQ) (see Eq. (12)). The examples given in (b) are calculated according to Eq. (13). The peaks of highest intensity are normalized for the different diagrams. It is marked that the maximum Huang-Rhys parameter for Pd(2-thpy)2 and Pt(2-thpy)2 have been determined to = 0.3 and = 0.08, respectively. (Compare also Sect. 4.2.4)...

The Franck-Condon factor is given by the squared overlap integral of displaced harmonic oscillator functions (Hermite functions). It can be related [154, p. 113] to the so-called Huang-Rhys parameter (or factor) S according to... 

This Huang-Rhys parameter is directly connected with the shift AQ of the equilibrium positions of the involved electronic states. (Fig. 18a) According to the Refs. [96] and [154, p. 112],for example,one obtains... 

This relation has already been appHed to determine the maximum Huang-Rhys parameter for the T, —> Sq transition of Pd(2-thpy)2 (see Eq. (1)). 

Intrastate conversion time in a substate of T, for an excited vibrational mode of = 10 cm Maximal Huang-Rhys parameter for the T, So transition. 

Compound Lowest triplet sublevel 1 [cm ] Total zfs [cm ] Blue-shift [cm" ] Blue-shift per proton [cm" ] Huang-Rhys parameter Assignments and references... 

Referring to the number of protons involved in the spatial region, where the electronic transition occurs, j Maximum Huang-Rhys parameter for the T, S transition (compare Fig. 18 and Eqs. (12), (13)). 

The emission transition will usually be situated at lower energy than the absorption transition. This phenomenon is known as the Stokes shift. Only the zero-vibrational transition is expected to occur at the same energy in the absorption and emission spectra. The Stokes shift is a direct consequence of the relaxation processes that occur after the optical transitions. It is obvious that the larger Q a Qo is, the larger the Stokes shift will be. If the two parabolas have the same shape and vibrational frequency, it is possible to define a parameter S (the so-called Huang-Rhys parameter) as follows... 

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