Hyperfine precession

Hyperfine structure arises through the interaction of the electron spin with a nuclear spin. Consider first the interaction of the electron spin with a single magnetic nucleus of spin , In an applied magnetic field the nuclear spin angular momentum vector, of magnitude (/ / -f l)]l/2, precesses around the direction of the field in an exactly analogous way to that of the electron spin. The orientations that the nuclear spin can take up are those for which the spin in the z-direction, M, has components of ... 

Fig. 1. Schematic for /zSR and fiLCR experiments. For pSR the muon spin polarization vector starts off in the x direction (open arrow). It then precesses about an effective field (the vector sum of the external field and the internal hyperfine field), which is normally approximately the z direction. The muons are detected in the M counter, and positrons from muon decay are detected in the L or R counters. For pLCR, the muon spin polarization is initially along the external field or t axis (solid arrow). The positron rates in the F and B counters are measured as a function of external field. A sharp decrease in the asymmetry of the F and B counting rates signifies a level crossing.

In this expression r is the inter-chain hopping time and ts is the phonon scattering time along a chain. The quantity s = (d2/a2) is the ratio of the anisotropic to isotropic contribution of the hyperfine interaction and /(cd) is the spectral density of the interaction, with coe and con being electron and nuclear precession frequencies respectively,... 

The forces which drive the intersystem crossing are the nuclear spin-dependent hyperfine interactions in the radicals and the electron Zeeman interactions. This becomes evident from the following after pair formation in the magnetic field of a NMR spectrometer, say, the two unpaired electron sfnns precess about the magnetic field axis starting from defined initial phase an es. These iititial phase angles are different for the four possible initial electronic states T Tq,... 

T-, and S. They are given in Fig. 3 for three of these states. During the lifetime of the pairs, the precession keeps phase provided the precession frequencies of the two electron spins are exactly equal. Now the precessioii frequency, or Lar-mor frequency, of a radical in a magnetic field is given for high fields by the g factor and the hyperfine interactions with the nuclei and is to first order (angular frequency units)... 

Low-spin iron(III) ions have an electron hole in the t2g orbitals. Therefore, these centers have S = 1/2 and spin-orbit interaction contribntes considerably to the magnetic hyperfine field. Low-spin iron(III) componnds in solution always show a rather complicated magnetic Mossbauer pattern at temperatures around 4.2 K and low external fields, which means that the relaxation rate of these centers is lower than the nnclear precession rate of 10 s. Sometimes a magnetic sphtting is observed even at 77 K. Therefore, in order to pin down 8 and A g, it is advisory to measure between 100 and... 

Figure 5 shows an example for a radical pair with one nucleus of spin 1/2, e.g. a proton, and a triplet precursor. Assume that radical 1 has the higher g-value, that the h)q5erfine coupling constant of the nucleus is positive, and that the sense of precession is clockwise. The action of the g-value difference is first shown separately. It is seen to cause a symmetrical displacement of the two vectors from their initial positions. Next, the effect of the h)q)erfine interaction for the nucleus having spin a and spin jS is superimposed on that. The resulting displacements are again symmetrical, but with respect to the positions of the vectors after rotation by the g-value difference. Obviously, the Zeeman and hyperfine terms add constructively... 

Figure 5. Explanation of an S- T0-type CIDNP net effect with vector models (left), resulting schematic population diagram (center), and NMR spectrum (right). The example describes a radical pair with one proton in radical 1, triplet precursor, product of the singlet exit channel, gq > g2, and positive hyperfine coupling constant. For the vector models, a clockwise sense of precession has been chosen, the labels 1 and 2 designate the radical and a> and /i) the nuclear spin state, and the dotted vertical lines in the projections give the amount of singlet character. For further details, see the text.

The collapse of a six-line hyperfine spectrum with decreasing relaxation time is illustrated by the calculated spectra in Fig. 3.8 [44] for a " Fe nucleus in a fluctuating magnetic field and a fixed electric field gradient for different values of the electronic relaxation time. If the fluctuation rate is very slow compared to the precession frequency of the nucleus in the field H, the full six-line hyperfine pattern is observed. If the fluctuation rate is extremely rapid the nucleus will see only the time-averaged field which is zero and a symmetric quadrupolar pattern will be seen. At intermediate frequencies the spectra reflect the fact that the 2> -> transitions which make up the low-velocity component of the quadrupole doublet relax at higher frequencies than do the I I i> and -]-> -> transitions which... 

Rotational motion exerts fluctuating magnetic fields on the electron spin and also averages the various tensor components (see Fig. 2). Nitro-xide EPR spectra are profoundly influenced by the rate of this motion relative to the range of electron spin precession frequencies within each hyperfine line. Thus, one can use EPR spectroscopy to determine tr accurately. There are distinct temporal regimes in which spin label spectra require different types of data collection methods and data analysis to achieve this. These regimes are discussed below. 

---