Nuclear hyperfine terms

When a molecule contains an atom with a nuclear spin I 1/2, its energy levels acquire an additional (21 +1) degeneracy. This degeneracy is lifted in practice by magnetic and electric interactions which are called nuclear hyperfine interactions they have been described in detail in chapter 4. The magnitude of these interactions is usually relatively 

We recall from chapter 4 that there are many individual types of interaction which involve the nuclear magnetic dipole and electric quadrupole moments. Let us take just three of these to exemplify how the effective Hamiltonian is constructed. They are as follows  

In these expressions the index i runs over electrons and a runs over nuclei. The Fermi contact term describes the magnetic interaction between the electron spin and nuclear spin magnetic moments when there is electron spin density at the nucleus. This condition is imposed by the presence of the Dirac delta function S(rai) in the expression. The dipole-dipole coupling term describes the classical interaction between the magnetic dipole moments associated with the electron and nuclear spins. It depends on the relative orientations of the two moments described in equation (7.145) and falls off as the inverse cube of the separations of the two dipoles. The cartesian form of the dipole-dipole interaction to some extent masks the simplicity of this term. Using the results of spherical tensor algebra from the previous chapter, we can bring this into the open as 

In other words, it is a first-rank scalar product involving the nuclear spin angular momentum with an operator defined by 

The first-order contribution of these hyperfine interactions to the effective electronic Hamiltonian involves the diagonal matrix elements of the individual operator terms over the electronic wave function, see equation (7.43). As before, we factorise out those terms which involve the electronic spin and spatial coordinates. For example, for the Fermi contact term we need to evaluate matrix elements of the type  

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These definitions are consistent with those of Gallagher and Johnson [116]. This completes our calculation of the nuclear hyperfine terms in the primitive basis set. 

The Zeeman interaction and nuclear hyperfine terms have to be added to this effective Hamiltonian. The Zeeman terms are, as in previous examples,... 

A. Fine-Structure Terms Jfps Electron Zeeman Terms Electron-Nuclear Hyperfine Terms jP f Nuclear Quadrupole Terms J q Nuclear Zeeman Terms III. Zero-Field Experiments A. Selection Rules... 

The difficulty of this definition (or detection) of interactions is one of scale the perturbations that one is trying to observe are orders of magnitude smaller than actual measurements. Computationally, a better approach is to attempt evaluation of the individual perturbations directly, and then define the total interaction energy as a sum of the individual perturbation energies. In the case of EMR spectroscopy, this is exactly what we are doing by using ENDOR or ESEEM. We know that the effects that we expect to see will become manifest in the nuclear hyperfine terms, so rather than try to measure these from differences in the EMR spectrum, which includes the electron Zeeman term, we turn instead to the ENDOR and ESEEM, which detect the nuclear hyperfine interactions. 

The first term on the right-hand side arises from external eleetrie fields. The seeond (B) term arises from external magnetie induetions interaeting with elee-tronie orbital motion. The SL term arises from eleetron spin-orbital motion interaetions. The Z term arises from the Zeeman interaetion between eleetron spin and the external eleetrie field. Hss arises from eleetron spin-eleetron spin interaetions and includes all hyperfine terms arising from nuclear spins. 

Here and H describe radicals A and B of the radical pair and He the interaction of their electrons. The other terms in equation (15) are H g, the spin orbit coupling term, H g and Hgj, representing the interaction of the externally applied magnetic field with the electron spin and nuclear spin, respectively Hgg is the electron spin-spin interaction and Hgi the electron-nuclear hyperfine interaction. 

OIDEP usually results from Tq-S mixing in radical pairs, although T i-S mixing has also been considered (Atkins et al., 1971, 1973). The time development of electron-spin state populations is a function of the electron Zeeman interaction, the electron-nuclear hyperfine interaction, the electron-electron exchange interaction, together with spin-rotational and orientation dependent terms (Pedersen and Freed, 1972). Electron spin lattice relaxation Ti = 10 to 10 sec) is normally slower than the polarizing process. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

Although simple /rSR spectra that do not depend on the nuclear terms in the spin Hamiltonian are the easiest to observe, one loses valuable information on the electronic structure. Under certain circumstances it is possible to use conventional /rSR to obtain a limited amount of information on the largest nuclear hyperfine parameters. The trick is to find an intermediate field for which the muon is selectively coupled to only the nuclei with the largest nuclear hyperfine parameters. Then a relatively simple structure is observed that gives approximate nuclear hyperfine parameters. A good example of this is shown in Fig. 3a for one of the /xSR... 

The Mossbauer effect involves the resonance fluorescence of nuclear gamma radiation and can be observed during recoilless emission and absorption of radiation in solids. It can be exploited as a spectroscopic method by observing chemically dependent hyperfine interactions. The recent determination of the nuclear radius term in the isomer shift equation for shows that the isomer shift becomes more positive with increasing s electron density at the nucleus. Detailed studies of the temperature dependence of the recoil-free fraction in and labeled Sn/ show that the characteristic Mossbauer temperatures Om, are different for the two atoms. These results are typical of the kind of chemical information which can be obtained from Mossbauer spectra. 

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. 

If the unpaired electron is in a MO which includes atomic orbitals of ligand atoms, there is the possibility of a hyperfine interaction between the electron and the nuclear moment of the ligand atom. To see how these hyperfine terms can be calculated from the molecular orbitals we shall consider the dl case of Sec. V.C in which B is the ground state. 

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. 

Even in fairly small applied magnetic fields, say B = 20 mT, the terms of He are much larger than the hyperfine terms. This implies that the expectation value of the electronic spin, (S, alSIS, a) = (S), is determined by He. Under these circumstances, we can replace the spin operator S in the magnetic hyperfine term by its expectation value, (S), obtaining from Hm the nuclear Hamiltonian Hn... 

The resolution of the molecular beam experiments is high enough to observe even rather small nuclear hyperfine interactions such as the spin-spin and spin-rotation interactions as well as the larger quadrupole coupling interactions. The largest terms in the Hamiltonian for the hyperfine splittings are given below 66) ... 

Nuclear hyperfine splittings in the rotational spectra of dimers have been observed in the molecular beam electric resonance experiments and the Fourier transform microwave experiments. In most cases, the coupling constants are interpreted with the simplified expression given in Eqn. (6) for axially symmetric molecules in the K=0 rotational manifold. Thus both the nuclear quadrupole coupling term and the... 

The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

The nuclear hyperfine operators therefore have essentially the same form in the effective Hamiltonian as they do in the full Hamiltonian, certainly as far as the nuclear spin terms are concerned. Throughout our derivation, we have assumed that the electronic state r/, A) which is to be described by our effective Hamiltonian has a well-defined spin angular momentum S. It is therefore desirable to write the effective Hamiltonian in terms of the associated operator S rather than the individual spin angular momenta s,. We introduce the projection operators (P] for each electron i,... 

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