Zeeman states

Fig. 4.10. The left side of the scheme represents the starting situation of pure Zeeman splitting, as described by (4.48) and shown before in Fig. 4.9. In this example, the field B = (0,0,B), which defines the quantization axis, is chosen as the z-direction. The additional quadrupole interaction, as shown on the right side of Fig. 4.10, leads to a pair-wise shift of the Zeeman states with mj = 3/2 and mi = 1/2 up- and down-wards in opposite sense. In first order, all lines are shifted by the same energy as expected from the m/-dependence of the electric...

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

The overlap between spherical and Zeeman states, was originally derived as a sum of the product of two vector coupling coefficients [3] ... 

Figure 8.2 The elastic scattering cross section for collisions of O2 molecules in the lowest energy Zeeman state Mg = = —1) as a function of the magnetic field. The collision energy is 10 ...

Similarly, the hyperfine transitions involving the spin-mixed Zeeman states within the Is level are expected to display frequency shifts due to CPT violation. In particular, the transition from the low field seeking Is — c states to the Is — d states could provide an unsupressed evidence of CPT violation that leads to differences between H and H A,h-h, -H fi... 

It should be apparent that the magnetic susceptibility x of a paramagnetic sample (corrected for diamagnetism) should be infinite at absolute zero (with perfect alignment of the magnetic dipoles, or 100% population of the lowest Zeeman state) and zero at infinite temperature (random alignment of dipoles, equal population of Zeeman states). An obvious expression which embodies these conditions is ... 

In a magnetically-dilute system, the distribution of paramagnetic centres over the available Zeeman states is determined by Boltzmann statistics, and the magnetic state adopted at any instant by one centre is in no way influenced by its neighbours. Deviations from such behaviour may be classified as ferromagnetic or antiferromagnetic. 

In the high field limit, where the quadrupole interaction acts as a perturbation of the Zeeman states, the terms of this Hamiltonian which commute with L lead to the perturbation of first-order... 

These perturbations then lead to a shift in the separation of the Zeeman states Em.m = Em Em- I which to first-order is... 

For cases (c ) and (d) one has to consider that the different possible Zeeman states m, for the I- nuclei differ in the respective sizes of their z-components, and hence generate dipolar fields of different magnitudes at the observe spins [29,30]. Therefore, the expression at Eq. (11) must be replaced by a superposition of 2I-I-1 individual de-phasing curves ... 

Figure 9b shows that for small Cq values, the two-component behavior, as predicted by Eq. (13), is readily apparent. With increasing magnitude of Cq, however, the REDOR response is successively attenuated, reflecting the fact that the spins in the outer 3/2> Zeeman states are less and less affected by the n(I) pulses. Note that the simulated curves approach the limiting cases (calculated via Eq. 13) of entirely non-selective irradiation of all the Zeeman states (upper dotted curve) and entirely selective irradiation of the l/2> -l/2> coherence (lower solid curve) of the I-nuclei. Extending now the initial curvature analysis approach discussed above to Fig. 9b, the early parts of these REDOR curves can be approximated by a sum of parabolic functions ... 

Because the internal interactions which characterize proton NMR are all small compared to the Zeeman interaction, perturbation theory to first order may be used, and ) is given by a superposition of orthonormal Zeeman states, I, k) = k), in an n-dimensional spin space ... 

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