Anti-crossing

Molecular Crystals as a Consequence of the Non-Crossing Rule (Level Anti-Crossing). 

The level anti-crossings due to the off-diagonal elements can give rise to line shifts and to sharp frequency responses, the so-called rotor-RF lines in the spectrum. These effects are observed in many spectra and are discussed extensively elsewhere [52]. The positions of these sharp features, ujrrfl, are... 

Use To improve the aging and service life of rubbers anti-cross-linking agent for SBR (styrene-butadi-ene-rubber). 

Figure 5.25 Zeeman splitting of an 5 = 2 paramagnet with D = -0.5 cm" and E = 0.01 cm" with the applied field along the molecular z axis. Inset expansion of level anti-crossing between Ms = +2 at zero field...

Bu4N)[Y(Pc)2l at 0.04 K and with apphed field (various sweep rates) along the easy axis, (b, lower) Expansion of low field area, (b, upper) Zeeman diagram for the M = 6 doublet of Tb including nuclear hyperfine and quadrupole interaction with the I = 3/2 b nucleus. The states are labelled by their Mj, Mi quantum numbers and correspondence between level anti-crossings and QTM steps are shown. Reprinted with permission from Ishikawa et al., 2005 [89]. Copyright (2005) Wiley-VCH... 

In the general case, a transition involving degenerate levels is split into two or more components under an external perturbation. One usually follows the spectral positions of the component as a function of the amplitude of the perturbation, and this result in fan charts of the kind shown in Fig. 8.8. In such charts, there are values of the perturbation for which two components from different transitions can, in principle, intersect. As a function of the symmetries of these components with respect to the perturbation, these components can cross without interaction, but there are cases where the two components interact, giving rise to an anti-crossing or avoided crossing configuration which can be properly dealt with by an appropriate perturbation calculation. 

Fig. 8.9. Stress dependence of line I s (To) of Se° in silicon at LHeT. The lines are drawn using parameters obtained to fit the Is (Ai) —> 2p i transition. For F// [110], the anti-crossing of the h-e component of Is (T2) near 200MPa is due to its interaction with the h-e component of the ls(E) level, undetectable at zero stress. The anti-crossing on the low-energy component is due to s-o interaction with the h-e component of the Is (3T2) line located at 2146.4 cm-1 at zero stress. The spectral range is 260.4-291.4 meV [14], Copyright 1989 by the American Physical Society...

Figure 8.9 shows for F// [110], the anti-crossing of the h-e component of Is (T2) (Se°) with the h-e component of Is (E) as these components have the same symmetry (see Table 8.5). This interaction allows one to extrapolate the position of the Is (E) component at zero stress to 2.1 meV ( 25cm 1) above the Is (T2) line and to obtain a value of the position of this symmetry- and parity-forbidden line in good agreement with the one deduced from the Fano resonances [14]. 

The DPs bi and di for the irs+ ground state and for the ITg and 2Tg excited states pertaining to lines 1 and 2 were determined for the B, A/, and In acceptors in silicon by [32] for stresses up to 140 MPa along <100>, <110>, and < 111 >. Such stresses allowed the study of the anti-crossing behaviour of components of different lines with the same symmetry. The results of the detailed piezospectroscopic measurements on B in silicon performed for lower values of the stress (up to 40 MPa) have also been reported by Lewis et al. [97], They provide values of the DPs b and d for the 1 /g1 ground state and several excited states. In Table 8.8 are presented experimental values of the uniaxial DPs 5j and di of the first acceptor levels in silicon, where they are compared with calculated values. 

At a difference with line 2p, the stress splitting observed for lines 1 and 2 is a combination of the ground and excited state splitting. Anti-crossing behaviour between components of different zero-stress lines can be observed when they have the same symmetry This is the case for component 1.4 of line 1 and component 2.1 of line 2 of B in silicon (see Fig. 7 of the paper by Chandrasekhar et al. [32]). 

Fine structures in the magnetic-field-tuned PTIS spectrum of P donors in germanium have been related to the spin splitting of the Is (Ai) and Is (T2) levels [50]. Spin splitting is also involved in the anti-crossing of Zeeman components with opposite spin of Is (T2) states of As donors in germanium measured by four-wave spectroscopy, and the effect is more marked when a fixed uniaxial stress along the [110] direction is used to split the Is (T2) state into three sublevels ([151], and references therein). 

Experimentally this effect has been found in dhcp Pr (Houmann et al. 1979) and in PrAlj (Purwins et al. 1976). The latter has cubic site symmetry and the ground and first excited states are Fj (OK) and 7 (27.4K). It orders ferromag-netically at = 33 K. At low temperatures (F T ) only three of the field-split F3-F4 magnetic excitons are seen (fig. 30) and those with 7+ polarization show a strong anti-crossing effect with a transverse acoustic phonon mode in the [001] direction. Detailed model calculations for this mixed-mode spectrum were performed by Aksenov et al. (1981) using an equation of motion approach. 

An external electric field may also prevent level crossings caused by the magnetic field. Such anti-crossing effects in Rydberg states of Li atoms in the... 

Because of the selection rule AM = 0, 1 we find that Kj 0 only for M-M = 0, 1 or 2. For M-M = 0 a level crossing cannot occur because of the non-crossing rule. Instead a related phenomenon, "anti-crossing" can be observed under certain circumstances [7.29]. For the arrangement shown in Fig.7.13 it can be shown that Kj = 0 for M-M = 1. If we now consider the special case where the sublevels of the excited state are linear Zeeman levels we have Ij m m I = 2/iggB/h for M-M = 2 yielding... 

Figure 7. Zeeman dependence of ENDOR transitions in a S=l/2,1=1/2 spin system. As the nuclear Zeeman energy is varied, the peak positions shift and there exists a critical point at which the nuclear Zeeman energy equals half the nuclear hyperfme energy (dashed vertical line). The critical point corresponds to the energy level (anti-)crossing situation.

Figure 8. Level (anti-)crossing of nuclear sublevels S=l/2, 1=1 by adjusting the nuclear Zeeman splitting. Only the m=l transitions are shown in the diagram. As in Figure 7, the ENDOR transitions are mobile and subject to a critical point as the magnitude of the nuclear zeeman energy assumes a value that leads to the crossing condititm in (me electrcm spin...

If ESEEM is indeed a variation of level (anti-)crossing spectroscopy, then is should be possible to perform ZF-NQR studies of any species in the region of the... 

Figure 14. The peak positions of the ESEEM spectra of Figure 13 plotted as Zeeman dependent trajectories. When a P, the trajectories of the peaks associated with the (anti-) crossing energy levels are parabolas from whose minima one can directly read the ZF-NQR parameters. The peaks from non-crossing levels trace linear diverging trajectOTies. Note that the sum/harmonic (i.e. combination) lines mirror the fundamentals and can be distinguished from lines of other origin based on their behavior.

Figure 9 Factor group analysis of the symmetry of the guest vibrations and the schematic plot illustrating symmetry forhidden anti-crossing hetween localized guest vibrations and framework acoustic transverse (TA) and longimdinal (LA) vibrations.

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