Eigenvalue zero-field

Figure 8. Eigenvalues of a 3D mapping system with zero field. Top to bottom piston and tilts measured by the LGSs, piston not measured and neither piston nor tilts measured. Curves are normalized to 1, 10 and 100 from bottom to top.

For zero field of view (Fig. 8), as expeeted, there are respeetively 2, 4 and 8 singular modes when the piston and the tilts are measured from the LGSs, when the piston is not but the tilt is (ease of the polychromatie LGS, see 15.3), and when neither the piston nor the tilts are measured (monoehromatie LGS ease). Even and odd modes eorrespond respeetively to the high and low eigenvalues (lowest and highest modes). 

For axial symmetry (i.e., / = 0) this is a diagonal matrix with only two eigenvalues D and -D and a zero-field splitting AE = D —D = 2D as we have previously noted in... 

If we are only interested in the frequency of the modulations in the vicinity of the zero field limit we may employ a different approach, used by Freeman et al,6,7 and Rau10. They used the fact that the motion in the direction is bound and found the energy separation between successive eigenvalues. Specifically, they used Eq. (8.8), the WKB quantization condition for the bound motion in the direction, and differentiated it to find the energy spacing between states of adjacent n1 or, equivalently, between the oscillations observed in the cross sections. Differentiating Eq. (8.8) with respect to energy yields... 

In the presence of an external magnetic field, the eigenfunctions ( w >) and eigenvalues ( ) are changed from those in zero field... 

The zero-field resonances can be identified with respect to the system energy levels and the field frequency when the field is off. They are usually one- or two-photon resonances. The one-photon resonance is of first order with respect to the field amplitude in the sense that the degeneracy of the eigenvalues is lifted linearly with the field amplitude. The two-photon resonance is of second order since the degeneracy of the eigenvalues is lifted quadratically with the field amplitude. Multiphoton resonances (more than two-photon) are more complicated since they are generally accompanied by dynamical shifts of second order... 

As we have stated, the Floquet Hamiltonian (113) has no terms that are resonant if we take small enough e, and the iteration of the KAM procedure converges. However, if we take e large enough, we encounter new resonances that are not present at zero or small fields that is, they are not related to degeneracies of the unperturbed eigenvalues of Kq that lead to the zero-field resonances we have discussed in the previous subsection. These new resonances are related to degeneracies of the new effective unperturbed operator K 0(e), which appear at some specific finite values of e. These are the dynamical resonances. 

Although the determination of the zero field eigenvalues and wavefunctions is described in many standard texts on rotational spectroscopy, we will briefly recall the principles and give some results for later reference. From Eq. (IV. 59 a) and IV.59b), the zero field Hamiltonian of an asymmetric top molecule is given by (from now on quantum mechanical operators will be denoted by underlining) ... 

Analytical expressions for the exact eigenvalues of zero-field splitting systems [14, 17]... 

The resulting zero-field eigenvalues are listed in Table 8.40 and plotted in Fig. 

In the case of a weak exchange limit (17 1 is low) the field-dependent molecular states are far from the zero-field states the eigenvectors of 52 and Sz do not give a proper description of the spin states and correct eigenvalues and eigenvectors should be obtained by diagonalisation of the full Hamiltonian matrix. The off-diagonal Hamiltonian contains only the tensor... 

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