Component out-of-phase

This behavior is usually analy2ed by setting up what are known as complex variables to represent stress and strain. These variables, complex stress and complex strain, ie, T and y, respectively, are vectors in complex planes. They can be resolved into real (in phase) and imaginary (90° out of phase) components similar to those for complex modulus shown in Figure 18. 

G is called the loss modulus. It arises from the out-of-phase components of y and T and is associated with viscous energy dissipation, ie, damping. The ratio of G and G gives another measure of damping, the dissipation factor or loss tangent (often just called tan 5), which is the ratio of energy dissipated to energy stored (eq. 16). 

Some gadolinium-based MOF have also revealed the peculiar dynamics of the magnetization with the appearance of the out-of-phase component of the dynamic magnetization in presence of an applied static magnetic field. This behaviour, which is reminiscent of the field-induced slow relaxation that characterize many lanthanide-based SMMs, is however not related to the magnetic anisotropy, which is vanishingly small in most Gd3+ systems. 

Note, also, that the presence of a circular-difference effect is not a necessary requirement for chirality. The presence of an intensity difference effect depends on the exact phase relationships between the chiral and achiral expansion coefficients in-phase coefficients will produce no circular-difference effect (see Figures 9.9b and e), while out-of-phase components will... 

Other cw methods of obtaining Ti, generally applicable only to liquids, include measurement of the in and out of phase components of the disper-... 

Adsorbed water was observed to have a large effect on the F spin-lattice relaxation time for fluorine-doped aluminas in the dilute and intermediate concentration range of fluorine (0.3-8 wt. % F). An increase in Ti by a factor of 2 to 3 was observed in these samples when adsorbed water was removed from the solid by heating between 200-300°. The effect was completely reversible addition of oxygen-free water back to the solid resulted in recovery of the original (shorter) relaxation time. This effect was observed by the measurement of the in phase and ir/2 out of phase components of the dispersion derivative at resonance dx /6Ho at high rf power, from which effective values of Ti may be calculated 46). Values of Ti were also obtained by saturation of the resonance absorption derivative. 

The scattering of the X-ray beam in the forward direction adds out-of-phase components to the propagating beam, as the classical electron scattering has a negative sign. This implies that the X-ray refractive index n differs from unity. The reduction in n leads to total reflection at very small angles, which is applied in the design of X-ray mirrors. 

Where D" is the loss compliance or the out-of-phase component of the dynamic compliance. The rate is given by... 

The piezoelectricity of polymeric materials has in general a relax-ational nature and the piezoelectric stress constant e is a function of the frequency of the applied strain in a similar way to the elastic modulus and dielectric constant. The induced polarization has in-phase and out-of-phase components to the strain and the e-constant is expressed as a complex quantity, as in Eq. (32). 

Dynamic mechanical analysers, as discussed in chapter 9, can be constructed so that they can be used with unvulcanised materials and, hence, the in phase and out of phase components of modulus and the loss angle measured. The usual test piece geometries for cured rubbers are not convenient for the uncured materials where some form of oscillating shear is probably the best approach. This is the geometry used in cure meters discussed in the next section and such instruments have formed the basis for apparatus which measures dynamic properties from before and through the curing process. 

The equation for Qfv is derived in Section V.C together with the scheme of its solution down to the third order in From the solutions obtained, the terms yielding the linear and cubic low-frequency responses to the probing magnetic field H(t) = H cos cot are extracted. In terms of linear and cubic susceptibilities those quantities evaluated numerically are compared in Figures 4.31 and 4.32 primes and double primes there denote, as usual, the in-phase and out-of-phase components of the dynamic susceptibilities. 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

The storage modulus (G ) is derived from the in-phase component of the stress while the loss modulus (G") is derived from the out-of-phase component. As stated above, the moduli depend on the phase angle obtained from the following relationships ... 

Look at this carefully it has an in-phase component (the term in sinot) and an out-of-phase component (the term in coscot). This can be used to define the relationship between stress and strain in terms of two moduli. First writing Equation 13-78 ... 

The in-phase component, G, is called the storage modulus while the out-of-phase component, G", is called the loss modulus. It also follows that (Equation 13-81) ... 

Figure 1. TICA temperature scan of the AT quinoxallne resin in (a) nitrogen environment and in (b) air. The in-phase component is labelled as curve a, and the out-of-phase component as b.

Figure 9.14 Thermal dependence at different fields of xiviTfor [Sm(H20)4(bpy)( x-CN)2Cr(CN)4]n-3.5nH201.5nbpy. Inset plot in the in-phase and out of phase components of the AC susceptibility at 1500, 1000, 750, 500, 100, and 50 Hz [93]. (Reprinted with permission from M. Estrader, et al., Crystal structure, and magnetic studies of one-dimensional cyano-bridged Ln — Cr complexes with bpy as a blocking ligand," Inorganic Chemistry, 45, 8239-8250, 2006. 2006 American Chemical Society.)...

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