Dipoles energy

Fig. 4.17 The charge-charge energy decays much more slowly (cer ) than the dipole-dipole energy (ar ).

Fig. 2 Intramolecular twisting in DMABN. In the excited state, the molecule has a high propensity to rotate the dimethylamino group out of the planar ground-state configuration [3]. This process changes the dipole energy, and relaxation from...

Dipole-dipole energy transfer between an isolated... 

Fig. 2.12. Dependences of the dipole energies of various orientational states (see Fig. 2.11) on the rhombic angle a.

Like Eq. (2.1), the classical ion-dipole and dipole-dipole energy expressions (2.23) and (2.25) are engagingly simple, and can be rigorously justified at sufficiently large R. However, these approximations can be expected to fail at smaller R where quantal effects become appreciable. 

That such a matrix describes a metastable state when H0 Hl can easily be seen and will be discussed in the following section. Since / z is related to the mean magnetic moment and fiD to the mean dipole-dipole energy, this raises the question of the independence of these two invariants of the motion. It may be shown10 by introducing the Fourier transforms of the Iz(j)... 

The mean angular moment (in units of %) is fit = dqjdv and the mean dipole-dipole energy is ED — — dq/dp. In the limit of high temperatures we may expand the exponential and retain only the linear terms, v and /9 may then be considered as the coefficients of the expansion of p in orthogonal operators, since Tr(VmMa) — 0. In this limiting case it then becomes evident that v and / are independent. It is easily shown that the thermodynamical temperature, defined by... 

This constitutes a system with three quasi-invariants the two magnetic moments and the dipole-dipole energy. The corresponding density matrix is... 

Th. Forster Quantum mechanical theory of dipole-dipole energy transfer... 

An important parameter required for the calculation of R0 is the orientation factor k2 which takes into account the angular dependence of dipole-dipole energy transfer, as described by eq 22... 

Of all the systems where Forster dipole—dipole energy transfer has been identified, the most important is light harvesting by antennae chloro-phyll-b molecules and donation of singlet energy to the chlorophyll-a reaction centres in photosynthetic organisms. Typical values of R0 have been estimated to be 4—5nm. Further details of photosynthesis may be found in articles by Birks [6, 141,142], Berlman [127], Gregory [144], and Jortner [145]. 

DIPOLE-DIPOLE ENERGY TRANSFER IN VISCOUS MEDIA... 

If the donor and acceptor molecules are unable to rotate in the solvent during the donor fluorescence time, the value of R0 considered above is too large. Steinberg [139] has analysed Forster kinetics in this limit. Allinger and Blumen [153] have developed a more detailed analysis of dipole—dipole energy transfer from excited donors to acceptors in liquids and obtained essentially similar results to those of Yokota and Tanimoto. 

Kaplan and Jortner [164] have observed dipole—dipole energy transfer between the second excited state of rhodamine 6G and 2,5-bis(5 -f-butyl-2-benzoxazolyl)thiophene in ethanol. The donor excited state lifetime was estimated to be 0.19 ps based on energy transfer by Forster kinetics. 

Since much of the following statistical mechanical analysis follows the outlines developed above for dipole—dipole energy transfer and is similar to electron tunnelling (recently reviewed by Rice and Pilling [39]), the following discussion has been compressed a little. 

Triplet—triplet energy transfer from benzophenone to phenanthrene in polymethylmethacrylate at 77 and 298 K was studied by steady-state phosphorescence depolarisation techniques [182], They were unable to see any clear evidence for the orientational dependence of the transfer probability [eqn. (92)]. This may be due to the relative magnitude of the phosphorescence lifetime of benzophenone ( 5 ms) and the much shorter rotational relaxation time of benzophenone implied by the observation by Rice and Kenney-Wallace [250] that coumarin-2 and pyrene have rotational times of < 1 ns, and rhodamine 6G of 5.7 ns in polymethyl methacrylate at room temperature. Indeed, the latter system of rhodamine 6G in polymethyl methacrylate could provide an interesting donor (to rose bengal or some such acceptor) where the rotational time is comparable with the fluorescence time and hence to the dipole—dipole energy transfer time. In this case, the definition of R0 in eqn. (77) is incorrect, since k cannot now be averaged over all orientations. 

In many cases 3 possible positions exist differing by a solid angle of 120° (tetrahedron angle 109°). We compute the average difference of the dipole energy 2 i4diei in an electric field to be... 

The time Tev depends on the mechanism of spin evolution. For the dipole-dipole mechanism tcv = / gpD, where D is the dipole-dipole energy of interaction, p is the Bohr magneton. The values Tev=2.7T0"11 s, and TPaii=3.1 10 9 s were calculated for TBPDA(C6o)2-... 

Fig. 19. Room temperature q3+ Tm3+ transfer rate WDA in Cr3+ (0.7 at.%) Tm3+ (2 at.%) YAG as a function of pressure (from Shen et al. (2000)). W da was obtained from fits to die experimental decay curves at different pressures assuming dipole-dipole energy transfer. The solid line has been calculated as described in die text. (10 kbar = 1 GPa).

These equations describe only the geminate reaction, which is not followed by bimolecular reaction of products in the bulk. On the other hand, the Naumann theory was also developed for noncontact (dipole-dipole) energy transfer considered in Section III.B. 

Fig. 21. Total potential energy of glyoyl-L-alanine at intervals of 1 kcal mole-1 using the same values of the parameters as in Fig. 20, but ignoring the dipole-dipole energy (i.e., D= oo). This potential surface yields o/ j,l = 3-95 (Brant and Flory, 1965c).

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