Spin vector

If the angular momentum of a free electron is represented by a spin vector S=(S, S, S ), the magnetic moment... 

In terms of the magnitudes of the spin vectors, thismeans that... 

Fig. 12-5. The Magnetic Unit Cell of MnFa Showing the Spin Orientation of the Magnetic Ions. The fourfold axis is in the direction of the spin vectors. The nonmagnetic ions are not shown.

This simplified treatment does not account for the fine-structure of the hydrogen spectrum. It has been shown by Dirac (22) that the assumption that the system conform to the principles of the quantum mechanics and of the theory of relativity leads to results which are to a first approximation equivalent to attributing to each electron a spin that is, a mechanical moment and a magnetic moment, and to assuming that the spin vector can take either one of two possible orientations in space. The existence of this spin of the electron had been previously deduced by Uhlenbeck and Goudsmit (23) from the empirical study of line spectra. This result is of particular importance for the problems of chemistry. 

Tin has fourteen electrons outside of its krypton-like core. These may occupy the nine orbitals in the following three most stable ways (atomic electrons are indicated by spin vectors, bonding electrons by dots, the metallic orbitals by open circles) ... 

Determine the angle between the spin vector S and the z-axis for an electron in spin state a). 

An unpaired electron executes a spin about its own axis. The mechanical spin momentum is related to a spin vector which specifies the direction of the rotation axis and the magnitude of the momentum. The spin vector s of an electron has an exactly defined magnitude ... 

Table 19.3 Coupling of the spin vectors related to cooperative magnetic effects...

Unlike these conventional techniques, NSE measures the neutron velocities of the incident and scattered neutrons using the Larmor precession of the neutron spin in an external magnetic field, whereby, the neutron spin vector acts like the hand of an internal clock, which is linked to each neutron and stores the result of the velocity measurement at the neutron itself. The velocity measurement is thus performed individually for each neutron. For this reason, the... 

Whatever the other aspects of its state are, an electron itself also has a spin vector. That is, it has a self-state with a quantum number of either +1/2, or -1/2. 

Before going on to calculate the energy levels it is necessary to digress and briefly describe the wavefunction. The spin Hamiltonian only operates on the spin part of the wavefunction. Every unpaired electron has a spin vector /S = with spin quantum numbers ms = + and mB = — f. The wavefunctions for these two spin states are denoted by ae) and d ), respectively. The proton likewise has I = with spin wavefunctions an) and dn)- In the present example these will be used as the basis functions in our calculation of energy levels, although it is sometimes convenient to use a linear combination of these spin states. 

In the absence of an applied magnetic field, the spin vector of the electron can be in any direction. The magnitude of the spin vector is ... 

Figure 2.93 The precession of the spin vector S around a magnetic field. The component of the vector in the r direction, the direction of the field, is Ms.

In the presence of the applied field, B, the spin vector precesses around the direction of the field, taken by convention as the z-direction (see Figure 2.93). This precession is quantised such that the component of S in the z-direction, Sz, can only take one of two values, + 1/2 or — 1/2, and the quantum number Ms is used to label the allowed values of Sz. Thus ... 

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

The torque is simply the rate of change of angular momentum and since magnetic moment is related to angular momentum by Equation (1), one may solve Equation (6) to find the motion of the spin vector. The spin precesses about H and the angular frequency of this precession, known as the Larmor precession, is yH. This situation is illustrated in Fig. 2. 

Empirically it was found that the field independent fine structure superimposed on the chemical shifts of NMR spectroscopy could be described by an interaction energy, ENN/, between nuclear spin vectors IN and In- of the form (46, 48)... 

If a nucleus with a non-zero spin number, which can be compared to a small magnet, is exposed to a magnetic field 7i0, with an angle 6 with the spin vector, fl and Bq will become coupled. This coupling modifies the potential energy E of the nucleus. However, fl will not necessarily align itself in the direction of the external field, in contrast to the action of a compass. 

According to quantum mechanics, //y for a nucleus can only take 21 + 1 different values. This means that in a magnetic field 7 0, the potential energy E can also only take 21 + 1 values. The value of /iz, the projection of the spin vector on the Oz axis, is related to the values determined by m. These values (given in h/2n units) are given by the following series ... 

The projection of the spin vector onto the Oz axis essentially traces the surface of a cone of revolution with an angle between the axis and atom that can be calculated knowing fl and flz. This precession around an axis parallel to that of the magnetic field (Fig. 9.2 and 9.4a) is characterised by a frequency that increases with the field intensity. 

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