Operator electron-nuclear hyperfine

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)... 

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,... 

The nuclear spin-orbit operator or orbital hyperfine operator describes the coupling of the nuclear magnetic moments to the orbital of the electrons... 

This can be enriched by the operator of the spin-orbit coupling and eventually by the operator of the hyperfine (electron-nuclear) interaction... 

A/Mj = 0, whereas the nuclear spin transition operator connects states with Tx = and Airij 0. Pure electron spin and nuclear spin transitions can then be observed, as in ordinary high-field (hf) experiments. The probability of the former is considerably higher than that of the latter owing to differences in the magnitudes of the respective moments. However, as in zf experiments on doublet states (e.g., H atom), the mixing of the basis functions by off-diagonal hyperfine terms allows the observation in zf of simultaneous electron-nuclear transitions (i.e., Tx and Anij 0) and contributes additional oscillator strength to the pure nuclear spin transitions. The electron spin transition operator can be the major source of intensity in zf ENDOR experiments (Harris and Buckley, 1976). 

Previous experiments in this laboratory (Chen et ai, 1973) have shown that it is possible to resolve hyperfine structure in a hf ODMR spectrum providing the spectrometer is operated at low power in order to avoid the combined effects of microwave saturation and forbidden electron-nuclear transitions (as in the zf experiment). To illustrate this, we show in Fig. 9 first-derivative hf spectra of C-benzophenone-dio with 1 mW power. The lines shown are the Antg = 1 transitions with H x (low-field), H y (low-field), and H z (high-field). Each is split into a doublet by the first-order hyperfine interaction. Similar splittings are observed in the... 

Electron nuclear double resonance is a powerful tool for the study of the electronic structure of triplet states because of its high precision. ENDOR linewidths can be as narrow as 10 kHz, which represents an increase in resolution of better than six orders of magnitude over that which can be obtained optically. The technique is particularly useful when combined with hf methods owing to the first-order nature of the hyperfine interaction in the presence of a field. Although such experiments are difficult, the information obtained is unique. Accordingly, the hf EPR (or ODMR) spectrometer has been modified for ENDOR operation in several laboratories. In order to illustrate the power of the method, we discuss here some recent optically detected hf ENDOR experiments on (njr ) benzophenone and its iso-topically labeled derivatives (Brode and Pratt, 1977, 1978a,b). The results, although incomplete, show considerable promise for the ultimate determination of the complete spin distribution in this prototype triplet state. 

If the interaction of an electron with a magnetic field were the only effect operative, then all ESR spectra of free radicals would consist of one line. When the nuclear spin quantum number / is nonzero a nuclear hyperfine interaction A is observed. When several equivalent nuclei are present (e.g., -CHs, -C.Hs), the number of lines in the spectrum is given by n,(2n-,li + ). where n is the number of magnetically equivalent nuclei /. 

In Eq. (2) the summations are taken over all the nuclei in the molecular species. The new symbols in Eq. (2) are defined as follows g is the nuclear g factor, which is dimensionless is the nuclear magneton, having units of joules per gauss or per tesla the nuclear spin angular momentum operator I the electron-nucleus hyperfine tensor A the quadrupole interaction tensor Q and Planck s constant h. 

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

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