Nuclear Zeeman terms

Hfi includes a nuclear Zeeman term, a nuclear dipole-dipole term, an electron-nuclear dipole term and a term describing the interaction between the nuclear dipole and the electron orbital motion. 

The electron-electron dipolar term, Ho, equals S1.D.S2. The tensor D is completely anisotropic and only mixes T-states with one another. It is therefore dropped. The nuclear Zeeman term, tlzi =... 

The nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

For a hydrogen atom in an external field of 10,000 G, draw a figure that shows the effect on the original 1 s energy level of including first the electron Zeeman term, then the nuclear Zeeman term, and finally the hyperfine coupling term in the Hamiltonian. 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

First, the nuclear Zeeman term, using equation (8.5) ... 

We have made use of the results given in equation (8.181) in order to complete the last line of (8.184). Finally for the nuclear Zeeman term we have... 

The theoretical framework for a discussion of the hyperfine interactions in radicals is given by the so called spin Hamiltonian, which describes the interaction between the unpaired electrons and the magnetic nuclei (I>l/2) in the sample. When the radical is placed in a static magnetic field, the electrons and the magnetic nuclei will interact with the field. These interactions give rise to the electronic and nuclear Zeeman terms,... 

When EPR data are taken at X-band (9,500 MHz) the intensities of the so-called forbidden transitions, d here, are small and often neglected. This amounts to using only the first two terms in Eq. (18-1). However for small samples it is often necessary to use higher microwave frequencies (K-band, 24,000 MHz, Q-band, 35,000 MHz, or V-band, 75,000 MHz). In these experiments it is important to include the third term in Eq. (18-1), the nuclear Zeeman term. 

The nuclear Zeeman term in Eq. (4) can be omitted in the following presentation, since it does not change for levels involved in an ESR transition (AM/ = 0 if AM5 = 1). Thus, for example, the energy levels for an unpaired electron interacting with two different protons can be written as... 

We now turn to the last of the tree main magnetic interactions, the nuclear Zeeman term. The general approach (7) is to divide the spin Hamiltonian into two distinct parts ... 

The third term of the nuclear Hamiltonian contains two contributions. The nuclear Zeeman term couples the magnetic moment of the nucleus to the external magnetic field Bo. Furthermore, there is a term that describes the interaction of the nuclear spin with the internal magnetic hyperfine field. For paramagnetic samples this is often done in terms of the hyperfine coupling tensor, which multiplied by the spin... 

In Eq. (3-2), / is the value of the exchange integral between two electron spins (Si and S2) and the 1/2 term is put for convenience although it is not used in many textbooks. Eq. (3-3) is just the sum of spin Hamiltonian of one radical given by Eq. (2-22), but the nuclear Zeeman terms are omitted in Eq. (3-3) because their magnitude is much smaller than those of the electron Zeeman term and the HFC one. In Eq. (3-3), ga and gb are the isotropic g-values of two component radicals (radicals a and b) in a radical pair, respectively, and A, and A are the isotropic HFC constants with nuclear spins (/, and 4) in radicals a and b, respectively. 

The Nuclear Zeeman Interaction The nuclear Zeeman term represents the direct interaction between the external magnetic field and the nuclear magnetic moment. This is usually neglected since it cancels for transitions between states with identical values of m/. When forbidden transitions are being considered, however, it is sometimes necessary to take account of this effect. It is applicable to both anisotropic and isotropic spectra. 

For the hydrogen atom, where / = i, the nuclear Zeeman term in the Hamiltonian is by far the most important (FT He). The position... 

A well-known and important phenomenon in the area of nuclear-spin resonance (NMR) in gases, liquids, or solid samples is dynamic nuclear-spin polarisation (DNP) (see e.g. [M6]). This term refers to deviations of the nuclear magnetisation from its thermal-equilibrium value, thus a deviation from the Boltzmann distribution of the populations of the nuclear Zeeman terms, which is produced by optical pumping (Kastler [31]), by the Overhauser effect [32], or by the effet solide or solid-state effect [33]. In all these cases, the primary effect is a disturbance of the Boltzmann distribution in the electronic-spin system. In the Overhauser effect and the effet solide, this disturbance is caused for example by saturation of an ESR transition. Owing to the hyperfine coupling, a nuclear polarisation then results from coupled nuclear-electronic spin relaxation processes, whereby the polarisation of the electronic spins is transferred to the nuclear spins. 

E2.3 Explain using the example in Fig. 2.2 why the nuclear Zeeman term in the energy expression for a liquid E(nis, mi) = gi sBrns + amsmi - gNjJiNBmi must be taken into account in ENDOR, but can be neglected in ESR. 

This procedure (neglect of nuclear Zeeman term) is sometimes also adopted to obtain hyperfine coupling tensors from ESR measurements of free radicals. The method is, however, not suitable for the analysis of hyperfine structure due to a-H in jr-electron radicals of the type )Cc-H at X-band, and other cases where the anisotropic hyperfine coupling and the nuclear Zeeman energy are of comparable magnitudes, as discussed for case 3 below. 

As first pointed out by Minakata and Iwasaki [16] an asymmetry in the hyperfine pattern between the = 1 -o- 0 and ms = 0- -1 electronic transitions of a triplet state can be attributed to the influence of the nuclear Zeeman term. This makes it possible in principle to determine the relative signs of the hyperfine coupling and the zero-field splitting. An application to trimethylene methane is shown in Fig. 3.13. 

Az = -20.2 G typical of Jt-electron radicals of the type )C -H, (b) at 95 GHz (W-band) with A = 2.5, A = 27.6 G. Solid and dotted lines refer to spectra calculated with and without the nuclear Zeeman term... 

The quantity Gm contains contributions from the hyperfine tensor A and the nuclear Zeeman term i n- A quadrupole energy term Pm contributes when I > Vi. By applying 2nd order corrections frequencies can be obtained with better accuracy when the condition Gm Pm does not strictly apply. Equation (3.33) for the ENDOR transition probability first given by Toriyama et al. [45b] applies for species with small g anisotopy. 

Fig. 3.39 5 = 1/2 ESR line pattern for an / = Vi nucleus with an anisotropic hyperfine coupling and a nuclear Zeeman term of comparable magnitude. The intensities 1 and Ip depend on the values of Va and vp and the nuclear Zeeman energy vn in frequency units according to Eqs. (3.38) and (3.39)...

An analytical treatment applicable when the quadrupole interaction is small compared to the nuclear Zeeman term is given in [57]. A numerical method due to Hoffman et al. [58] is valid irrespective of the relative magnitudes of the hyperfine, nuclear Zeeman and quadrupole terms. 

Fig. 4.25 Effective fields B+ ms = +V2) and B. ms = -V2) acting on a nucleus with an anisotropic hfc and a nuclear Zeeman term of comparable magnitude. Ba is due to the hypetfine couphng, Bn to the applied magnetic field...

The quantities G contain contributions from the hyperfine tensor A and from the nuclear Zeeman term bi = B j li where lx, ly and 1 are the direction cosines of the applied magnetic field with respect to a suitably chosen coordinate system of a crystal or molecule. In the general case with non-negligible g-anisotropy the effective direction of the field is u = following the notation of Weil and Anderson [36]. The matrix notation (4.8b) is also employed, e.g. in [26] that also provides the 2nd order corrections reproduced in Chapter 3. 

Anisotropic A and g, and non-negligible b The general case with appreciable g- and hyperfine anisotropy and a nuclear Zeeman term that is comparable with the hyperfine coupling is rarely considered in practical apphcations. 

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