Level repulsion

Figure 22. Shown in panel (a) is the relation between the bare energy difference e between frozen-in structural states in a glass and the effective splitting e that is smaller due the level repulsion in the tunnehng center. Panel (b) depicts schematically the derivative of e with respect to e, which is used to compute the new effective distribution P(e) of the transition energies.

These results are very similar to the normal two-level problem, except (i) the real parts of the energies exhibit level repulsion, but this repulsion is reduced (but never reversed) by the <5r2/2<5e term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (h) the imaginary parts of the energies exhibit level attraction (the interaction causes the widths of the mixed states to become more similar), and this attraction is reduced (but never reversed) by the 48e2/8Y term in the denominator of Eqs. (9.3.12a) and (9.3.12b) (Hi) when <5T — 0, the level widths are unaffected by the interaction (iv) when Se = 0 but <5Tj >> V, the level positions are unaffected by the interaction (because the narrower level is symmetrically surrounded by the much broader level and the usual level shift by V is suppressed). This amounts to an intuitively-sensible extension of the normal (real E, Hermitian H) two-level problem. 

In the weak coupling limit, the real part of AE differs from the familiar JT = 0 two-level perturbation result by a small reduction in level repulsion due to the <5r2 term and the imaginary part shows that, as intuitively expected, the difference in zero-order level widths is reduced by the interaction, except that, when > V2, a narrrow level can tune through resonance (Se = 0) with a broad level without any significant change in width ... 

The levels of a GOE ensemble (and hence those of irregular systems) have a NNSD distribution very close to the Wigner one. Notice that while the Poisson distribution is maximal at S = 0 (no level repulsion), the Wigner distribution vanishes at S = 0 (there is level repulsion). An intermediate behavior can be described in different ways. Considering also a non-linear repulsion, P2 S) oc S , Brod3 ° computed the distribution... 

Fig, 17,34. Level repulsion of different excitation modes in the ferromagnetic phase of TbAlj as seen by inelastic neutron scattering. The intensity was measured at the (111) reciprocal lattice point corresponding to q = 0. ha is the energy transfer and k is the wave vector of the scattered neutrons (from Purwins et al., 1973). 

---