The g Factor

Since the classical equation of motion equates the rate of change of angular momentum J to the applied torque Mj X H, the equation of motion for the magnetization of a solid is (damping terms neglected) 

The spectroscopic splitting factor g that is determined from resonance experiments must be distinguished from the g factor determined by gyromagnetic experiments (347,526,632). In a gyromagnetic experiment (Einstein-DeHaas (164) or Barnett (40) methods), what is measured is the magnetomechanical ratio (see eq. 73) 

If the orbital angular momentum is quenched by the crystalline fields, so that e is small, then it is possible to show (347) that for a resonance experiment the spectroscopic splitting factor g is given by 

In practice it is found that the values of e determined from resonance experiments are appreciably ( factor of two) higher than those found from gyromagnetic measurements, and this discrepancy is attributed by Kittel and Mitchell (350) to an apparent tendency of the g values to decrease as the resonant frequency is increased. 

Smit and Wijn (590) have pointed out that if the atomic moments are oriented purely parallel or antiparallel to the magnetic field, i.e., the orbital angular momentum is quenched, in first approximation, by the crystalline fields (c C 1), then 

ESR spectra are characterised by three parameters, the g factor, the hyperfine coupling, and the line width. It is these parameters that allow identification of the nature and environment of radicals. 

If eqn. (5) represented the resonance condition for all electrons, one would predict that they would all resonate at the same applied field for a given applied frequency. Consequently, the g factor would always be equal to that of the free electron. However, as well as possessing spin angular momentum, the unpaired electron also possesses electronic orbital angular momentum. The interaction between the two (via spin-orbit coupling) ensures that the electron has an effective magnetic moment different from that of the free electron. Hence, the resonance condition for an unpaired electron in a References pp. 349-352 

As already mentioned G is purely an instrumental factor which must be known in order to perform correct FA measurements. Its determination is quite simple thanks to the fact that when excitation is performed with horizontally polarized light (instead than vertical) the vertical and horizontal component of the emitted light are expected to be equal, independently on the properties of the sample. A difference between the measured intensities is henee, in this exeitation condition, due exclusively to instrumental effect and it can be used to obtain the G factor. 

It is hence possible to calculate the G factor in a given spectral range by recording the fluorescence spectra of a sample which emits in that spectral window keeping the excitation polarizer in the horizontal position and the emission one first horizontal and then vertical. Indeed it is usually convenient to measure the G factor using the same sample under investigation. In this case it is necessary to record four different luminescence spectra exactly in the same experimental conditions changing the orientation of the two polarizer in all the possible combination. Two of the four spectra are used to calculate the G factor (Eq. 6.21) and the other two to calculate the anisotropy. In alternative it is possible to express the anisotropy as a function of the four spectra. 

Afree radical (with just one unpaired electron) is described as an electronic doublet because, in an external magnetic field, the electron can only exist in one of two possible spin states ( up or down ). By contrast, a pair or radicals, or a biradical (a species with two unpaired electrons in the same molecule) can exist in either of two electronic states singlet or triplet. In the singlet state the electrons are paired (opposite spin) a singlet radical pair (or biradical) is thus diamagnetic (W5 = I -1 = 0) and not observable by EPR. The radical pair above is shown in the singlet state. 

How is the Ms = 0 combination of the triplet different from a singlet The answer lies in the precessional phase of the two 

The precessional frequency (v) of an unpaired electron is directly proportional to the applied magnetic field strength (B) and can be expressed by the equation 

TABLE 11.1 Values for g Factors of Several Common Organic Radicals 1 

Especially notable are nitroxide radicals such as 11-1. Molecules of this class are sufficiently stable to survive a variety of chemical reactions, allowing them to be covalently bonded to other molecules (e.g., biopolymers such as proteins and nucleic acids). They can then serve as spin labels, transmitting information (via their EPR spectra) about the molecules to which they are attached. 

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The calculations show that for both considered types of signals the CCF are very sensitive to initial phases of current and reference pulses and difference in their carrier frequencies. The CCF is also sensitive to the g-factor of loaded vibrators of probes and number of periods in the signal. 

In equation (bl. 15.24), r is the vector coimecting the electron spin with the nuclear spin, r is the length of this vector and g and are the g-factor and the Boln- magneton of the nucleus, respectively. The dipolar coupling is purely anisotropic, arising from the spin density of the impaired electron in an orbital of non-... 

The g-factors of radicals 101 and 102 are 2.0065 and 2.0059, respectively. The pyrazolylnitroxides have effective magnetic moments at room temperature corresponding to the standard values for one unpaired electron per molecule (1.71 0.05 B.M.). The values of effective magnetic moments of the nitroxyls practically do not change in the temperature range 5-300K. 

Pu(C5H5)3 and (C5H5)3Pu C=NC6Hi1 (Fig. 7) show a similar temperature dependence to that of PUCI3. The lower magnetic moments indicate some reduction of the g-factor (21,31). 

The values for the atomic saturation magnetization at the absolute zero, ferromagnetic metals iron, cobalt, and nickel are 2.22, 1.71, and 0.61 Bohr magnetons per atom, respectively.9 These numbers are the average numbers of unpaired electron spins in the metals (the approximation of the g factor to 2 found in gyromagnetic experiments shows that the orbital moment is nearly completely quenched, as in complex ions containing the transition elements). 

For a mixture of enantiomers it is thus possible to determine the ee-value without recourse to complicated calibration. The fact that the method is theoretically valid only if the g factor is independent of concentration and if it is linear with respect to ee has been emphasized repeatedly.84-89 However, it needs to be pointed out that these conditions may not hold if the chiral compounds form dimers or aggregates, because such enantiomeric or diastereomeric species would give rise to their own particular CD effects.88 Although such cases have yet to be reported, it is mandatory that this possibility be checked in each new system under study. 

For the hydrogen atom, two such resonance conditions occur, giving rise to two lines separated by 506 G, which is just the value of a for the hydrogen atom [Eq. (ID)]. The spectrum would look the same for a single crystal or for a polycrystalline sample because the g factor and the hyperfine constant are isotropic. 

The spin Hamiltonian for the hydrogen atom will be used to determine the energy levels in the presence of an external magnetic field. As indicated in Section II.A, the treatment may be simplified if it is recognized that the g factor and the hyperfine constant are essentially scalar quantities in this particular example. An additional simplification results if the z direction is defined as the direction of the magnetic field. For this case H = Hz and Hx = Hv = 0 hence,... 

It should be pointed out that a somewhat different expression has been given for the Knight shift [32] and used in the analysis of PbTe data that in addition to the g factor contains a factor A. The factor A corresponds to the I PF(0) I2 probability above except that it can be either positive or negative, depending upon which component of the Kramers-doublet wave function has s-character, as determined by the symmetry of the relevant states and the mixing of wavefunctions due to spin-orbit coupling. 

Three basic equations (3.10-3.12) are needed to describe the technique. In the equations, p is the magnetic moment of the electron, sometimes also written as pe, g is called the g factor or spectroscopic splitting factor, S is defined as the total spin associated with the electron (in bold type because it is considered as a vector), B is the imposed external magnetic field (also defined as a vector quantity), and... 

Smooth and uniform polymer surface after vacuum plays a key role to ensure good OFRR sensing performance. We have observed in experiments that toluene after vacuum is prone to leave a number of cavities of a few micrometers in diameter on the surface. These cavities will induce additional scattering loss for the WGMs in the OFRR, which greatly degrade the g-factor, and hence the detection limit of the OFRR vapor sensor. Moreover, these small cavities have different adsorption characteristics compared to smooth polymer surface. Vapor molecules may be retained for a longer time at the cavity, which increases the response time and recovery time. Acetone and methanol are found to be better candidates for solvents because they usually leave uniform and smooth surface after vacuum. 

The detection limit of the microresonator-based refractive index sensing device is directly related to the g-factor of the resonator and the sensitivity of the resonant mode discussed above. The g-factor of a microtube resonator is determined by the total loss of a resonant mode, including radiation loss, absorption loss, and surface roughness scattering loss. The overall g-factor can be expressed as... 

An experimental verification of Eqs. (65)-(67) has not yet been possible because the MALLS instrument that allows measurement of has only recently become available in laboratories. Furthermore, the combination of the MALLS with the Vise detector is not a trivial problem. Several experiments are, however, in progress. This work is of great importance since a quantitative estimation of the branching density would be possible if the Zimm-Stockmayer equations for the g-factor hold true. 

The relationships of Eqs. (65)-(67) are not the same as the g factors for regular stars. This is easily understood since Eqs. (65)-(67) refer to fractions which still remains an ensemble that contains a distribution of isomers of different architecture but the same molar mass. The equation for the stars can be derived from Eqs. (29) and (29 ) ... 

Fig. 26. Molar mass dependence of the g factor for three pregel and one postgel fraction of end linked PS stars. A good fit was obtained with the Zimm Stockmayer equation (Eq. 69) and an exponent in Eq. (70) of fi 0.63 [95] which agrees well with Kurata s estimation with b-0.6 [129]. Reprinted with permission from [129]. Copyright [1972] American Society...

A remark had previously been made in various places of the text that the parameter will increase with branching or more precisely with the g-factor. This -dependence must necessarily also result in a molar mass dependence of for which a power law behavior was tentatively assumed. From the KMHS-relation-ship one then finds [95]... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

According to convention we suppose that the g factors of a neutral diatomic molecule can be partitioned into a term depending on the electric dipolar moment d or its derivative AdJAR and an irreducible non-adiabatic contribution g, [19,28]... 

For a molecular ion with charge number Q a transformation between isotopic variants becomes complicated in that the g factors are related directly to the electric dipolar moment and irreducible quantities for only one particular isotopic variant taken as standard for this species these factors become partitioned into contributions for atomic centres A and B separately. For another isotopic variant the same parameters independent of mass are still applicable, but an extra term must be taken into account to obtain the g factor and electric dipolar moment of that variant [19]. The effective atomic mass of each isotopic variant other than that taken as standard includes another term [19]. In this way the relations between rotational and vibrational g factors and and its derivative, equations (9) and (10), are maintained as for neutral molecules. Apart from the qualification mentioned below, each of these formulae applies individually to each particular isotopic variant, but, because the electric dipolar moment, referred to the centre of molecular mass of each variant, varies from one cationic variant to another because the dipolar moment depends upon the origin of coordinates, the coefficients in the radial function apply rigorously to only the standard isotopic species for any isotopic variant the extra term is required to yield the correct value of either g factor from the value for that standard species [19]. 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

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