Polarization moments

Poiar stationary phases which have a polar moment. These phases interact with the dipoiar moments of poiar components themselves and those components capable of induced polarization such as aromatics. 

Component reliability will vary as a function of the power of a dimensional variable in a stress function. Powers of dimensional variables greater than unity magnify the effect. For example, the equation for the polar moment of area for a circular shaft varies as the fourth power of the diameter. Other similar cases liable to dimensional variation effects include the radius of gyration, cross-sectional area and moment of inertia properties. Such variations affect stability, deflection, strains and angular twists as well as stresses levels (Haugen, 1980). It can be seen that variations in tolerance may be of importance for critical components which need to be designed to a high reliability (Bury, 1974). 

Fig. 3-15 Polar moments of inertia for common cross-sections.

That way, the Distributed Electrostatic Moments based on the ELF Partition (DE-MEP) allows computing of local moments located at non-atomic centres such as lone pairs, a bonds and n systems. Local dipole contributions have been shown to be useful to rationalize inductive polarization effects and typical hydrogen bond interactions. Moreover, bond quadrupole polarization moments being related to a n character enable to discuss bond multiplicities, and to sort families of molecules according to their bond order. 

Table 9. Conduction electron polarization effect in uranium ferromagnetic monocompounds conduction electron polarization moment Psai saturation moment from magnetization studies ordered magnetic moment as determined by neutron scattering...

For lateral force measurements, the constant of the cantilever torsion can be calculated from k=GJ/l, where G is the shear modulus and J is the polar moment of inertia ... 

The dipole moment of a compound is a function of the distribution of charge within the molecule, and appears to be a sensitive test for the accuracy of the compound s molecular wave functions. The dipole moment of a molecule can be approximated for a given direction as the sum of two components, /iq, the contribution from net charge densities on the atoms, and for each atom A, ftsp (A), an atomic polarization moment produced by the distortion of the electronic cloud around the atom. The atomic polarization moment results essentially from the mixture of s and p orbitals and, for a heteroatom, includes mv, the lone pair moment. 

The expansion coefficients Pq are called polarization moments or multipole moments. The expansion (2.14) may also be carried out by slightly alternative methods which are presented in Appendix D and differ from the above one by the normalization and by the phase of the complex coefficients Pq. The normalization used in (2.14) agrees with [19]. Considering the formula (B.2) from Appendix B of the complex conjugation for the spherical function Ykq(0, [Pg.30]

Another advantage of the polarization moments is the possibility of describing relaxation processes in the most rational way. Thus, if the relaxation process is isotropic, then the various moments and their components relax independently for details see Section 5.8. All multipole components of certain rank K relax at one and the same rate constant... 

If excitation takes place by intensive light, then, naturally, there may emerge in the molecular ensemble polarization moments of rank higher than 2. This may take place both in excited and in ground states. In this case the set Pq of odd rank characterizes orientation, whilst that of even rank characterizes alignment of the angular momenta of the molecules. 

The multipole (or polarization) moments introduced according to (2.14) present a classical analogue of quantum mechanical polarization moments [6, 73, 96,133, 304]. They are obtained by expanding the quantum density matrix [73, 139] over irreducible tensor operators [136, 140, 379] and will be discussed in Chapters 3 and 5. 

We will now analyze in greater detail the connection between the polarization moments of an excited molecular state and the characteristics of the emitted light. 

From the form of Eq. (2.33) it becomes understandable why the anisotropy of polarization 7Z is sometimes called the degree of alignment. From the point of view of the determination of the magnitude of the polarization moments bPo the measurement of 71 is preferable, as compared with that of V, all the more so if one bears in mind that the population bPo appears only as a normalizing factor for all other bPQ and does not influence the shape of the probability density p(B,multipole moment dependence of V and 71 for various types of radiational transition (A = 0, 1) can be obtained using the numerical values of the Clebsch-Gordan coefficient from Table C.l, Appendix C. 

The simplest method consists of investigating the collisional depopulation of a laser excited rovibronic level, i.e. of measuring the rates and cross-sections of the collisional relaxation of its population bPo- The relaxation rate Tk of polarization moments bPQ of various rank K may be represented, in the case of isotropic collisions, as follows ... 

Following the general approach, as presented in Chapter 2, let us expand the solution (3.5) over spherical functions (2.14) in order to pass from pa(9,(p) to classic ground state polarization moments aPq (2.16). It is important to stress that, since absorption is non-linear with respect to the exciting light, here, unlike in Section 2.3, we obtain polarization moments aPq of rank k > 2 in the ground state. We will denote the rank and the projection by k and q respectively (unlike K and Q for the excited state). We can, however, state that all the produced polarization... 

Fig. 3.3. Relative polarization moment value in ground (initial) state at linearly polarized Q-absorption.

Table 3.1. Analytical expressions for classic polarization moments apQ (k < 4) describing optical polarization of angular momenta in the ground (initial) state via light absorption...

Table 3.2. Values of polarization moments apo describing optical alignment in the ground (initial) state under linearly polarized Q-type absorption. It is assumed that the pumping parameter y = 1...

The quantitative characteristic of the alignment created is given, as already stated, by multipole moments of even rank. A more rigorous treatment of the expansion of the quantum mechanical density matrix over irreducible tensorial operators will be performed later, in Chapter 5 and in Appendix D. As an example we will write the zero, second and fourth rank polarization moments and [Pg.62]

Here the first term in the righthand side determines the rate of creation of polarization moments in the spontaneous transition process, whilst the second term describes their relaxation. In writing Eq. (3.27) it is assumed that collisions do not lead to population of the state J", since it lies sufficiently high and is surrounded by non-populated rovibronic levels within the range of thermal energy kT. 

In particular, in the case of linearly polarized excitation through bPo> bPo-> and according to (3.19), three ground state polarization moments may emerge with k = 0,2 and 4. Indeed, substituting coefficients from Table 2.1 and Clebsch-Gordan coefficients from Table C.l into (3.19) we obtain... 

It is simple to arrange the registration of the transient process in the case of fluorescent population considered in Section 3.4. For this purpose it is, for instance, sufficient that the first beam, which causes a transition J" — J (Fig. 3.14) should act in short pulses Tp = GS(t). Then in the time dependence of the fluorescence produced by the continuous probe beam (cycle J — J[ — J f dotted line in Fig. 3.14) we observe the relaxation of the polarization moments of level J" directly. 

Firstly, there may be a difference in the decay rates of the polarization moments bpQ of various ranks K, which is frequently the case with atoms, as was demonstrated for instance in [6, 45, 96, 135]. In order to avoid this effect, one uses well-known methods of separating the moment po, as described in Sections 2.4 and 2.5, namely using the magic angle Oo = 54.7°, or convenient linear combinations I +2I which follow from (2.29), (2.30) and the corresponding discussion. In this way we can obtain signals depending only on bPo-... 

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