Interaction with static magnetic fields

There is a general result from angular momentum theory in quantum mechanics, known as the Wigner-Eckart theorem [4], which allows the magnetic dipole moment to be directly related to the nuclear spin, according to  

Besides the magnetic dipole moment, nuclei with spin higher than 1 /2 also possesses an electric quadrupole moment In a semiclassical picture, the nuclear electric quadrupole moment informs about the deviation of the nuclear charge distribution from a spherical symmetry. Nuclei with spin 0 or 1/2 are therefore said to be spherical, with zero electric quadrupole moment. Quadrupolar nuclei, on the other hand, are not spherical, assuming cylindrically symmetrical shapes around the symmetry axis defined by the nuclear spin. Within the subspace I,m), the nuclear electric quadrupole moment operator is a traceless tensor operator of second rank, with Cartesian components written is terms of the nuclear spin  

The eigenvalues of the Zeeman Hamiltonian (2.2.2), which are clearly proportional to the eigenvalues of the operator 4, represent the energy levels of the nucleus, given by  

Therefore, for a nucleus with spin I, there are 2/ + 1 energy levels equally spaced by the amount hcoi. The lower energy states correspond to the higher (positive) m values. The ground state is thus the state with m = I, which means, in a semiclassical picture, that the nucleus is as aligned as possible with the direction of the field Bq. 

For an ensemble of identical nuclei in thermal equilibrium, the population of each energy level is given by the Boltzmann distribution [6]. In the case of / = 1/2, for example, one has a two-level system, with the populations and + of the m = —1/2 and m = +1/2 levels, respectively, related by the Boltzmann factor  

---

Fig. 2. Interaction of nucleus (electron) with static magnetic field, Bq, where the bulk magnetization, M, is (a) parallel to Bq and to the -axis, and (b), upon apphcation of a 90° radio frequency pulse along x, M perpendicular to Bq and to the -axis. See text.

The NMR phenomenon is based on the magnetic properties of nuclei and their interactions with applied magnetic fields either from static fields or alfemaling RF fields. Quanfum mechanically subatomic particles (protons and neutrons) have spin. In some nuclei these spins are paired and cancel each other out so that the nucleus of the atom has no overall spin that is, when the number of protons and neutrons is equal. However, in many cases the sum of the number of protons and neutrons is an odd number, giving rise to... 

As discussed above, the fraction of nuclei that are available for MR interactions increases with static magnetic field strength Bg- Noise in MR scans depends on the square root of the product of 4 times the bandwidth, temperature, Boltzmann s constant, and the resistance of the object to be imaged. 

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)... 

Fig. 10.2 Interactions of the local magnetic fields eDD(t) with EFSA(t) (see text). 80 s the static magnetic field. The CSA tensor a is displayed by...

Figure 6.39(a) shows the vs. T curve, normalized to the RT value, for a 100 nm thick a-/ -NPNN/glass film obtained from electron paramagnetic resonance (EPR) measurements with the static magnetic field applied perpendicular to the substrate plane. As previously shown in Fig. 3.19, the molecular a -planes are parallel to the substrate s surface. The data points closely follow the Curie-Weiss law = (T — w)/C, where C stands for the Curie constant. In this case w — —0.3 K, indicating that the net intermolecular interactions are weakly anfiferromagnetic. No hint of a transition at low temperature is observed. These results coincide with those derived from SQUID measurements on a single a-p-NPNN crystal (Tamura etal, 2003), where 0.5 < w < 0, which are displayed in Fig. 6.40. 

When a nucleus with spin I is placed in a static magnetic field, the energy splits into 21 + 1 equally spaced levels, and a 2J-fold degenerate resonance line can be observed in an NMR experiment (16). Nuclei with spin I > 1/2 possess an electric quadrupole moment Q which may interact with the gradient of the electric crystal field at the site of the nucleus. This field gradient is a traceless tensor (17). 

The electrons modify the magnetic field experienced by the nucleus. Chemical shift is caused by simultaneous interactions of a nucleus with surrounding electrons and of the electrons with the static magnetic field B0. The latter induces, via electronic polarization and circulation, a secondary local magnetic field which opposes B0 and therefore shields the nucleus under observation. Considering the nature of distribution of electrons in molecules, particularly in double bonds, it is apparent that this shielding will be spatially anisotropic. This effect is known as chemical shift anisotropy. The chemical shift interaction is described by the Hamiltonian... 

Now we can estimate the static magnetic field, where the electric field -induced spin-flips have a larger probability than the transitions due to the magnetic field of the EM wave. With the Hamiltonian of interaction of the spin... 

---