Incoherent limit

Let us now study the large energy (or incoherent) limit. We will actually find an analytical expression for the quantum corrections. Indeed if r 0 for e qqEt the parameter 9 vanishes like 1 /y/e, it is thus convenient to set up the following expansion ... 

Figure 2. Plots at 0.9Tc of JcJ(o/j/(0) for the isotropic s-wave (left)and extended-s-wave (right) OP s, respectively. The lines marked oo indicate the incoherent limit the other labels are as in the right panel of Fig. 1.[6]...

In the incoherent limiting case, the hopping time is proportional to (cf. Eq. (6.4)). Typical hopping times for singlet excitons in aromatic crystals lie as mentioned in the range of 10 s, while for triplet excitons, they are 10 to 10" s. From these times and from the lifetimes, the diffusion lengths can be computed (see Sect. 6.9.1). 

Let Q (t) denote the probability excitation energy resides at site id i i N) at time t, then in the incoherent limit, a valid approximation since both phase and spatial coherence time is considerably less than 10 sec for disordered systems ( ), the equations describing the evolution of Qi(t) are ... 

As an example, consider a planar wavefront from an incoherent source passing through an aberration-free circular lens. When the image is diffraction-limited, an Airy disc pattern is observed (Goodman, 1996). For an aperture of radius 1 / 2n) the pdf for photon arrival is given by... 

Just as above, we can derive expressions for any fluorescence lifetime for any number of pathways. In this chapter we limit our discussion to cases where the excited molecules have relaxed to their lowest excited-state vibrational level by internal conversion (ic) before pursuing any other de-excitation pathway (see the Perrin-Jablonski diagram in Fig. 1.4). This means we do not consider coherent effects whereby the molecule decays, or transfers energy, from a higher excited state, or from a non-Boltzmann distribution of vibrational levels, before coming to steady-state equilibrium in its ground electronic state (see Section 1.2.2). Internal conversion only takes a few picoseconds, or less [82-84, 106]. In the case of incoherent decay, the method of excitation does not play a role in the decay by any of the pathways from the excited state the excitation scheme is only peculiar to the method we choose to measure the fluorescence (Sections 1.7-1.11). 

Structural relaxation in glass-forming polymers has been studied for many years using chemically realistic simulations. Most of the early work that examined incoherent, as well as coherent scattering functions, is more of a qualitative nature because of the unsatisfactory quality of the force fields employed and the severe limitations on the length of the MD simulations performed. Roe studied the slowdown of structural relaxation in a PE-like model140,141 as well as for polystyrene.142 More recently Okada et al.143,144... 

The Krawiecka Goldberg Scale (or Manchester Scale) is a brief ten-item scale for assessment of changes in the clinical status of patients suffering from psychosis. The items include depression, anxiety, delusions and hallucinations, incoherence, flattened affect, poverty of speech and psychomotor retardation. The absence of items typical of schizoaffective and manic psychoses limits the use of this instrument. It is, however, useful for follow-up of inpatients and outpatients for longer periods of time (Krawiecka et al.y 1977). 

The relaxation rate R t) described by Eqs. (4.49)-(4.51) embodies our universal recipe for dynamically controlled relaxation [10, 21], which has the following merits (i) it holds for any bath and any type of interventions, that is, coherent modulations and incoherent interruptions/measurements alike (ii) it shows that in order to suppress relaxation, we need to minimize the spectral overlap of G( ), given to us by nature, and Ffo)), which we may design to some extent (iii) most importantly, it shows that in the short-time domain, only broad (coarse-grained) spectral features of G( ) and Ffa>) are important. The latter implies that, in contrast to the claim that correlations of the system with each individual bath mode must be accounted for, if we are to preserve coherence in the system, we actually only need to characterize and suppress (by means of Ffco)) the broad spectral features of G( ), the bath response function. The universality of Eqs. (4.49)-(4.51) will be elucidated in what follows, by focusing on several limits. 

The experiments just discussed made it clear that the motion of the hole on the series of As represents a different mechanism of transport than tunneling. Giese [13] and Bixon and Jortner [18] suggested that this mechanism is incoherent hopping of the hole between neighboring bases. This means that the hole wavefunction is Hmited to one base. The wavefunctions of the remaining electrons on that base would of course be distorted by the presence of the hole. Thus in this view of the transport process the base on which the hole sits could be called a molecular polaron, or a small polaron because it is limited to one site. 

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