Mini-excitons

Along the path from isolated, oriented molecules to excitons in non-doped crystals, we next take up the triplet states of oriented dimers, which we called mini-excitons in Sect. 6.4. Mini-triplet-excitons are excitations of triplet states which are spatially distributed over exactly two molecules, for example the two A and B molecules in the unit cell of Fig. 7.11, and they are localised there. In principle, mini-excitons can also be localised on a pair of molecules with the same orientation in crystals with a dimeric structure, e.g. in the a-perylene crystal (Fig. 2.12). We will however restrict ourselves in this section to the treatment of A - B mini-excitons. 

A - B mini-excitons exist in sufficiently concentrated mixed crystals, for example in perdeuteronaphthalene-naphthalene mixed crystals N-dg x N-hg, in statistical abimdance, because N-hg can be inserted at arbitrary concentration x into the N-dg crystal. There, nearest-neighbour N-hg pairs are formed, in whose immediate neighbourhood only N-dg molecules of the host crystal, and no other N-hg guest molecules are found, when the N-hg concentration x is not too high. If wab denotes 

In N-dg x-N-hg mixed crystals, one observes in the triplet ESR spectrum (Fig. 7.3) in addition to the ESR lines A+/A and /B from the isolated molecules also two additional lines, /M, whose intensity relative to the intensities of the lines from the individual molecules matches the ratio of their statistical probabilities, Wab. This result was found for N-hg concentrations between x = 0.2% and x=10% [4]. 

Quantitatively, the ESR spectra of the M lines are obtained from the spin Hamiltonian  

AB is thus formally identical to TLs (Eq. (7.7)). The values of D and for the mini-exciton as well as the orientations of x and y depend on the fine-structure constants D and E of the isolated molecules and on their orientations in the crystal. They are clearly derived uniquely through the averaging process. For naphthalene, a tedious but simple calculation according to Eq. (7.10) yields the values 

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In a first step, we consider a physical dimer. This is a pair of equivalent molecules, denoted by 1 and 2, whose distance and relative orientation are the same as in the crystal lattice. In the case of anthracene, let these be the two molecules in a unit cell. Such a configuration gives a mini-exciton [17] in an optically-excited state. This model is explained in Fig. 6.7. Later, we will expand it to include the whole crystal lattice, and will thus obtain the Frenkel excitons. 

Fig. 6.7 Left The splitting Aq and shift D - D of the excited term of a physical dimer or mini-exciton, relative to the monomer. The quantities Vi2, D , D and I12 are defined in the text. The energetic ordering of the terms E and E depends on the sign of I12 the case shown here is for I12 < 0. There are two optical transitions.

Right Monomer and pair emission (mini-exciton). The 0,0-transition of the phosphorescence of C- oHg as a guest molecule in a CioDg host crystal is shown. The spectra from crystals with 0.2, 2, 5, 10 and 20 mole % C- oHg are shifted vertically for... 

The simple dimer model as in Fig. 6.7 can be called a mini-exciton. It shows how intermolecular interactions can lead to shifts and spUttings in the spectra. An example is shown on the right-hand side of Fig. 6.7. D and D are based to first order on the molecular polarisability in the ground and excited states, and the resonance energy In in the singlet state is due to the resonance interaction between molecule 1 in an excited state and molecule 2 in its ground state or vice versa. In the triplet state, lu is determined in the main by the overlap of the orbitals of the two molecules, one of which is excited. 

Fig. 6.8 A sketch showing the orientation and magnitude of the transition dipole moments M for various mini-exciton geometries. When the two transition moments are parallel (beside one another or in a line), a blue or red shift occurs relative to the monomer. Only one optical transition is allowed. In the general...

There are thus two allowed transitions with the energy difference 2ii2. This holds equally well for excitons in the crystal and for the mini-exciton. In the mono-cHnic anthracene crystal lattice, the orientation 1 is converted into orientation 2 by a mirror-ghde operation (and vice versa). The mirror-ghde plane is the a-c plane (Fig. 2.10). M+ and M are therefore polarised parallel and perpendicular to the b axis. The two Davydov components thus differ in both energy and polarisation. 

Even with simple UV-VIS spectroscopy, the exchange interaction In of a naphthalene dimer in naphthalene, i.e. in a mini-exciton, could be detected as a spUtting of the 0,0 transition in the phosphorescence spectrum, yielding 1.2 0.2cm [18]. 

In Chap. 6, we discussed low-energy optical excitation states, the singlet and triplet excitons and energy transfer. The primary experimental method applied there was optical spectroscopy in the visible, in the near IR and in the UV spectral ranges. In the present chapter, we treat the structure and the dynamics of localised triplet states, of triplet mini-excitons, and of triplet excitons in molecular crystals. The primary experimental method for the investigation of the lowest-energy triplet level Ti is electron-spin resonance (ESR) (Fig. 7.1). 

Fig. 7.3 The ESR spectrum of a perdeuteronaphthalene-naphthalene (N-dg + 2% N-hg) mixed C7Stal at T = 4.2 K. A IA and 6 /6 are ESR transitions in isolated N-hg molecules with the two different orientations A and B (see Fig. 7.11) are transitions in N-hg A-B pairs (mini-excitons) the lines in the centre (g 2) are transitions in free radicals of the crystal mount. The relative concentrations of the T- states of the isolated guest molecules [A,B)...

X indicates a principal axis of the mini-excitons (cf Fig. 7.11). Open circles are the experimental values filled circles are calculated values 9.4 GHz. T = 4.2 K. After [4]. 

Fig. 7.14 Schematic portion of the term diagram of the triplet states of the molecules A and B and the mini-excitons M in the naphthalene crystal for a selected orientation of the applied magnetic field Bq. 8B is the difference in the fine structure of the two ms = + ms = 0 ESR transitions A+ and B+ of the molecules A and B. The magnitude of 5B is independent of the magnitude Bol, if...

The characteristic property of the M lines is their spectral position they lie in the centre between the corresponding A and B lines, i.e. between A+ and and M between A" and B (Fig. 7.3). This result holds for all orientations of Bq (Fig. 7.4). Figure 7.14 gives an intuitively clear explanation for this mean value in this scheme, the Zeeman terms T+) and Tq) as well as the resonance fields of the corresponding Anis = l transitions of the two molecules A and B are shown. The diagram also shows the mean values M+ and Mq of the two molecular Zeeman terms and the resulting resonance field M+ for the mini-exciton. 

The orientations of the principal axes x, y and z are illustrated in Fig. 7.11, while Fig. 7.15 shows the term diagram of the mini-excitons in comparison to that of the isolated molecules in zero field. 

To answer this question, we consider in the following the experimentally determined line shape and the linewidth A of the mini-exciton ESR spectra, and then analyse them in terms of the theory of exchange or motional narrowing. 

The linewidths ABm of the two mini-exciton ESR lines M+ and M" depend in a very characteristic manner on the spectral spacing of the two associated lines - B+ and A - B, i.e. on the difference of the fine structures of the two inequivalendy-oriented molecules A and B ABm has a finite value when the A- B spacing goes to zero and then increases quadratically with increasing A-B spacing (Eig. 7.16). In frequency units, (Acom = giiBABulh), the empirical result is then... 

Fig. 7.16 The width ABm of the ESR lines from the mini-excitons in a naphthalene crystal as a function of the square of the line spacing (6/ + - Bb+) or (6 - 6g ) of the ESR lines of the isolated molecules. ABm S the total width of the derivative of the ESR lines at the points of their extreme values (distance between the...

Compared with the exchange frequency o>ab = 2 Iab/h (2 Iab is the Davydov splitting), the jump probability is smaller HP/IIab = 0.32. The energy and spin exchange in the mini-exciton must therefore be considered to be an incoherent process even at 4.2 K. At high temperatures (T 4.2 K), the mini-excitons are not observable, since they can then be thermally activated into the excitonic band of the host crystal. 

In the mini-exciton, the difference of the fine structure averages to an arbitrarily narrow value when it becomes very small, that is when the A - B spacing is neg-hgible. This process of motional or exchange narrowing of the fine structure was analysed above. If the two molecular partners A and B of all the mini-excitons had... 

The reduction of the inhomogeneous linewidth follows from Van Vleck s calculation of the second moment (AB ). It states that (AB > is proportional to N is here the number of nuclear spins. In an A - B pair, the number of nuclear spins is doubled in comparison to an isolated molecule. Therefore, the inhomogeneous linewidth is reduced by the factor 1 /-/z. The experimental values in naphthalene confirm this interpretation in the immediate neighbourhood of the A-M - B crossing points (a = 120° in Fig. 7.4), the experimental value for the ratio of the linewidths of the mini-exitons to those of the isolated molecules is ABm/ABab = 1/1.6. The width AB is thus an additional confirmation of the model for a mini-exciton it consists of the two molecules in a unit cell. At higher N-hg concentrations x = 10%), A-A-B and A-B-B mini-excitons have also been observed [4]. 

In non-doped anthracene crystals, as in non-doped naphthalene crystals at T = 300 K, two sharp Lorentzian ESR lines from excitons are observed (Figs. 7.18 and 7.21). They have, in complete analogy to the M lines of mini-excitons, a large and anisotropic fine structure (Fig. 7.19). The fine-structure constants of the triplet excitons in the anthracene crystal are D /he = -0.00575 cm and B /he = -1-0.0330 cm . The choice of axes is here the same as for the mini-excitons (cf. Fig. 7.11) z = b. From the measurements, the angle Z(x, a) = 27.25° is determined. VBth the methods treated in Sect. 7.4, this yields the fine-structure constants of the isolated anthracene molecule at room temperature, D/he = -1-0.0694 cm and E/hc = -0.00836 cm . (The small deviations from the values obtained for isolated anthracene molecules in a single-crystal matrix (see... 

Table 7.3) are not surprising, since the fine-structure constants are somewhat dependent on the local environment of the molecules.) In complete analogy to the mini-excitons, the width of the Lorentzian exciton ESR lines increases quadrati-cally with the spacing of the ESR lines of the associated isolated molecules A and B (Fig. 7.20). The latter can, to be sure, not be observed directly, since in the non-doped crystal only excitons but not isolated molecules can be excited however, their fine structure and thus the spacing of their ESR lines can be calculated unambiguously. The quantitative evaluation of the anisotropy of the ESR linewidth (Fig. 7.20) using the method described in Sect. 7.4 yields the correlation time tc,exc in a pure anthracene crystal at room temperature ... 

The large mean squared displacement can by the way explain the small residual linewidth of the triplet ESR lines in a natural way (for small A - B spacing) this was also observed for the triplet excitons in naphthalene crystals (Eig. 7.21). It is due to the complete averaging-out of the hyperfine structure, because the number N of molecules over which the average is carried out is not 2, as for the mini-excitons, but rather is very large (N 1). The second moment is then not reduced by the factor V2, as for the mini-excitons, but instead by the factor 1/N. The exciton ESR linewidth is therefore narrowed in relation to the linewidths of the isolated... 

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