First-order Zeeman splitting

For / = 0 there is obviously no first-order Zeeman splitting however, application of a magnetic field can result in second-order splitting. As such, it is necessary to evaluate the corresponding factor, which isg(j = 2 + 7,(2 + S). 

Figure 8.19. First-order Zeeman splitting ofthe N = 2, J = 1, 2, 3 fine-structure states for the c3nu state of H2.

In these expressions, w is a numerical coefficient which is a complicated function of both C and t. H is the field strength—that is, the first-order Zeeman splitting is proportional to the field strength—and P is the Bohr magneton. The first-order Zeeman effect has removed all degeneracies. Inclusion of the second-order Zeeman effect gives... 

Figure 27 Splitting of a free-ion state into 27+1 components by a magnetic field (a) for J even, (b) for J odd (i) for the first-order Zeeman effect (ii) including the second-order Zeeman effect...

At the present time it is unclear whether the spin echo decay spectroscopy approach successfully demonstrated for Na is generally transferable to other alkali ion nuclei such as Li and Cs. The latter two isotopes have moderately small electric quadrupole moments causing first order quadrupolar splittings that are comparable to the radio frequency excitation window. As a result, the 7i-pulse length is ill-defined in many situations and the contribution of dipolar coupling of the observed spins to nuclei in the outer Zeeman levels is difficult to quantify. Nevertheless, some promising initial results on cesium borate... 

Zeeman and first order quadrupole splitting for the nucleus (nuclear spin I = /2). A = Cq(3cos — 1)/8/ /(27 — 1).[ 1 The angle 0 defines the orientation of the principle axis of the (crystalline) sample with respect to the field direction. 

In an octahedral field, the ground state is split by spin-orbit coupling into a lower quartet Eg and a higher doublet f-. There is no first-order Zeeman effect for Eg and thus the g factor is zero. 

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...

The situation with a pair of spins is illustrated in Fig. 1, where the static field B0 is along the z-axis. To first order and neglecting terms in the expansion of Eq. (2) which lead to a change in nuclear quantum numbers of 1 or 2 (i.e. Am — 1 and + 2 transitions), the effect of is to split the Zeeman levels into many closely spaced energy levels, thereby causing a distribution of resonant frequencies and consequently a broad line. Eq. (2) has been simplified by the van Vleck formula 2... 

Figure 5.1 The effect of quadrupole interactions on an / = 3/2 nucleus in a magnetic field. The Zeeman interaction splits the levels by an equal amount, cot, (the Larmor frequency in frequency units). The central +1 /2 to -1 /2 transition is unaffected by first order coupling co j however, second order coupling, co , affects all transitions.

Another important interaction which contributes to broadening of resonance peaks is the quadurpolar interaction. When the nuclear spin, / > 1, the nuclei possess electric quadurpole moment, eQ. The quadurpole moment 0 is a measure of the distortion of spherical symmetry of the nuclear charge distribution. Quadurpole moment interacts with the electric field gradient, EFG, present at the nucleus. These are the fields caused by the electrons in the atom. Spherical distribution of electrons in closed shells do not produce any EFG at the nucleus but the bonding electrons do. Therefore the quadurpolar interaction can be used to obtain information on chemical bonding. Quadurpolar interaction shifts the Zeeman levels. The first order correction due to quadurpolar interaction to the Zeeman splitting between levels m and (w-1) is given by. 

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