Angular frequency tensor

The observed magnitude of the angular frequency, w of a spectral line of negligible width can be expressed by the Euler angles (a, / ), which relate the principal axes values of the angular frequency tensor (un, cj22, W33) to the laboratory frame as... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

The expressions for the various parts of the Hamiltonian (equation 1) are well documented and for our purpose and the following discussion it suffices to summarize the results for axially symmetric situations in angular frequency units with the equations 2-6, where and Ashielding tensor and the shielding anisotropy, respectively, D is the dipole coupling, eq or V is the electric field gradient at the nucleus, eQ is the nuclear quadrupole moment and the other symbols have their usual meaning ... 

This moment of inertia is essential for the analysis of rotational spectra of molecules. For anisotropic solids or for molecules, the moment of inertia I is a second-rank tensor, with three principal-axis components I, I2, and f3. This moment of inertia is important when the body rotates with angular frequency co radians per second (co FIz), or with v revolutions per second. 

Here, (> and /> are initial and final states of the supermolecular complex, 11 is the induced incremental polarizability (a second-rank tensor), fio and fij are unit vectors in the direction of the electric polarization of incident and scattered waves, which are often specified in the form of subscripts V and H, for vertical and horizontal, (aif = (Ej- — Ei)lh is the energy difference of initial and final state in units of angular frequency, Pi T) is the population of the initial state (a function of temperature), <5 (to) is Dirac s delta function, and the summation is over all initial and final states of the collisional complex. 

The molecular response tensors are characterized by peculiar properties and satisfy a series of very general quantum-mechanical relations. First we observe that the dynamic properties can be rewritten as a sum of the corresponding static property and a function multiplying the square of the angular frequency. Thus, for instance, in the case of dipole electric polarizability, using Eqn. (117),... 

If the hyperfine tensor is present as in the nitroxides spin probes, three transitions are seen when the angular frequency of the microwave field co equals the transition frequencies couin) = ( 1/2,M — -1/2,m)/ I. M = 1, 0 (see Fig. 2c) ... 

The subindices f, j, k indicate base directions (x,y,z) in Cartesian coordinates. Xfj i2co co,co) represents a second order nonlinear susceptibility tensor, a tensor or third rank, responsible for SHG process while Ej(a)) and Ej (co) are the laser field amplitudes in j and k directions. Pf 2coi) is the induced SHG nonlinear polarization in i direction at twice the fundamental frequency, which couples to and generates light field radiation Ei(2co) in twice the original angular frequency 2 CO. Summation over j and k is implied in Eq. (1) by the Einstein summation convention. Please notice that xfk(2(o cOy(o) vanishes for a medium with centro-sym-metry. For a non-centro-symmetry medium, not all the tensor elements exist. 

I = scattered light intensity K = defined by Equation [11] N = number of events NA = numerical aperture R = distance between scattering molecule and observer S = surface area S/N - signal to noise ratio a = half angle of light cone = isotropic Raman invariant aG = isotropic ROA invariant due to the optical activity tensor = anisotropic Raman invariant =j8(G )2 = anisotropic ROA invariant due to the optical activity tensor <5 = f A) = anisotropic ROA invariant due to the quadrupole tensor fiQ = permeability of the vacuum to = angular frequency. 

In order to understand some of the essential features of the RR experiment, it is convenient to assume negligible chemical shielding anisotropy. Under these conditions, o) and e CS-perturbed Zeeman angular frequencies, are independent of time. In addition, it is convenient to use the spherical tensor notation to describe the direct dipolar interaction. Thus, the total Hamiltonian is... 

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