Extra degeneracy

Find the spinor representations of the space group 219 (F4ic or T ) at A and at X. Comment on whether time-reversal symmetry introduces any extra degeneracy. 

Each DCT(q) block appears ca times along the diagonal. The eigenvalues of DCT(q) are and their degeneracy is 1(a), the dimension of the IR a. This completes the solution to the problem of finding the frequencies and the eigenvectors of the dynamical matrix D(q), except for a consideration of extra degeneracies that may arise from time-reversal symmetry. 

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

Let us now consider an extra degeneracy i i(O) = 0 and i7 (0) 0. In order to study bifurcations in this case one should introduce a new independent governing parameter which is the constant term of the Taylor expansion of Hi. If we set / = 1 according to relation (C.4.22), then system (C.4.20) reduces to the following asymptotic form ... 

Because the rotational energies now depend on K (as well as on J), the degeneracies are lower than for spherical tops. In particular, because the energies do not depend on M and depend on the square of K, the degeneracies are (2J+1) for states with K=0 and 2(2J+1) for states with K > 0 the extra factor of 2 arises for K > 0 states because pairs of states with K = K and K = -K are degenerate. 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

We went from the one-dimensional velocity distribution to the three-dimensional speed distribution by adding in an extra v2 factor to account for the added degeneracy in velocity space. The one-dimensional diffusion equation 7.36 has a form which is mathematically very similar to the one-dimensional velocity distribution... 

More Complicated Molecules.—Calculations have been reported on a number of more complicated molecules, as indicated in Table 4. The work on BFa and SOa, and on NH3 and NF3, is of particular interest since these are the simplest symmetric top structures for which the calculation is practical, and for which there exist sufficient spectroscopic data to make it worthwhile for symmetric top molecules there are extra observable vibration-rotation interaction constants associated with the vibrational degeneracy that provide further information on the force field (see Table 3). Formaldehyde and ethylene, and their simple halogen derivatives, and also methane, are obvious candidates for further work. 

For shallow multiple acceptors associated with the Fs+VB, the fourfold degeneracy allows one to accommodate a maximum of four holes. Photoconductivity measurements in the very-far IR at LHeT and down to 1.2 K have indeed shown that group-II neutral acceptors and Cu° could bind an extra... 

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