Splitting in an External Magnetic Field

So far we have been discussing magnetic (Zeeman) quantum beats taking place at frequencies of coherent sublevel splitting in an external magnetic field. 

The molecular level Eq with the total angular momentum J splits in an external magnetic field B into (2/ +1) Zeeman components. The sublevel with the magnetic quantum number M shifts from the energy Eq at zero field to... 

Recall that electron spin generates a tiny magnetic field, which causes a beam of H atoms to split in an external magnetic field (see Figure 8.1). Only a beam of a... 

Fig. 7. Zeeman splitting, in an external magnetic field of 10 gauss, of the four-wave mixing resonance in the geometry of Fig. 1, caused by collision-induced Zeeman coherences with Am = 1.

NMR spectroscopy (a commercial unit shown in Fig. 1.49) uses the fact that some atomic nuclei have a magnetic moment, e. g. very distinct in a proton, the nucleus of hydrogen, but also inl3C, 3IP, 14N, and 33S. In an external magnetic field the energy levels split, as described in quantum mechanics. The size and extend of the split is given by Eq. (9)... 

Example Problem Determine the temperature at which kBT is equal to the energy level splitting for the metastable state of 123Te at 247.6 keV in an external magnetic field of 4.0 tesla (T). 

S2 possesses the eigenvalues S(S + 1). For a spin-1/2 dimer (S can be 0 or 1), the same values are obtained from the derivation of the Bleaney-Bowers equation. For spin systems in an external magnetic field, the Zeeman operator Hmag = -g/ BB S accounts for Zeeman splitting. The isotropic Heisenberg Hamiltonian for multiple spin centers can be expanded by adding the individual coupling pairs ... 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

P and a D state, sketched in Figure 3. In an external magnetic field, an atomic D state splits into five components with... 

Fig. 2. Ground state Zeeman levels in an external magnetic field. The sum of the frequencies of the indicated transitions Aviz and Av i at the same magnetic field equals the zero field splitting Avhfs and their difference allows to determine the muon magnetic moment...

Fine-Structure Splittings in ESR Spectra of Triplet States. Consider the hamiltonian H for spin-spin (fine-structure) and Zeeman interactions of two spins Si and S2 a mutual distance r-12 apart, in an external magnetic field H0 ... 

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

ESR studies as applied to micellar systems rely on the sensitivity of a free radical probe to its microenvironment. Molecular species with a free electron posses labeled intrinsic angular momentum (spins), which in an external magnetic field undergoes Zeeman splitting. For a system with 5 = 1/2, two Zeeman energy levels are possible whose energy gap (AE) is given by ... 

In an external magnetic field, the Zeeman interaction splits the three triplet levels. As the inset of Figure 1 shows, this gives rise to two regions... 

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