Inversion symmetry tensors

As mentioned earlier, heavy polar diatomic molecules, such as BaF, YbF, T1F, and PbO, are the prime experimental probes for the search of the violation of space inversion symmetry (P) and time reversal invariance (T). The experimental detection of these effects has important consequences [37, 38] for the theory of fundamental interactions or for physics beyond the standard model [39, 40]. For instance, a series of experiments on T1F [41] have already been reported, which provide the tightest limit available on the tensor coupling constant Cj, proton electric dipole moment (EDM) dp, and so on. Experiments on the YbF and BaF molecules are also of fundamental significance for the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDM de. Accurate theoretical calculations are also absolutely necessary to interpret these ongoing (and perhaps forthcoming) experimental outcomes. For example, knowledge of the effective electric field E (characterized by Wd) on the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron s EDM de in the ground (X2X /2) state of YbF and BaF. 

There are different paths to achieving surface specificity. One can exploit optical susceptibilities and resonances that are nonzero only at the surface or only for the molecular species of interest adsorbed on the surface. Examples include the use of second-order nonlinear mixing processes such as second harmonic generation7-9 for which the nonlinear susceptibility tensor is nonzero only where inversion symmetry is broken. Spectroscopic techniques with very high selectivity for molecular resonances such as surface-enhanced infrared or Raman spectroscopy10-12 may also be used. 

If the molecule or the solid has a center of inversion symmetry, then the permanent dipole moment m0 vanishes, as do the even-rank tensors j8 and 8 and the even-rank tensors y<2), y<4), and All matter, with or without a center of inversion symmetry, has nonzero values for the odd-rank molecular tensors a and y, and all odd-rank tensors ytensor components is vastly reduced. The components have values that depend seriously on the frequency of the electromagnetic radiation used to probe them. A practical application of nonlinear optics is frequency-doubling of the high-powered... 

Non-linear optical techniques, such as second harmonic generation (SHG), have recently been used as surface probes. Bulk materials with inversion symmetry do not generate second harmonic signals, while surfaces and interfaces cannot have inversion symmetry, so the total SHG signal will come from the surface region for many systems. The components of the non-linear polarizability tensor have been used to determine the orientation of chemisorbed molecules. 

Given the interest and importance of chiral molecules, there has been considerable activity in investigating the corresponding chiral surfaces [68, 69 and 70]. From the point of view of performing surface and interface spectroscopy with nonlinear optics, we must first examine the nonlinear response of the bulk liquid. Clearly, a chiral liquid lacks inversion symmetry. As such, it may be expected to have a strong (dipole-allowed) second-order nonlinear response. This is indeed true in the general case of SFG [71]. For SHG, however, the permutation symmetry for the last two indices of the nonlinear susceptibility tensor combined with the... 

The SFG technique probes the second-order nonhnear hyperpolarizability tensor this tensor includes the Raman and IR susceptibihty, which requires that the molecular vibrational modes are both Raman and IR active. Since Raman- and IR-dipole moment transition selection rules for molecules with a center of symmetry indicate that a vibrational mode is either Raman or IR active but not both, only molecules in a non-centrosymmetric environment on the surface interact with the electric fields molecules in the isotropic bulk phase show inversion symmetry where the third rank hyperpolarizability tensor goes to zero [25-27]. 

The second-order susceptibility ax, ox, m) is a third-rank tensor. For materials with inversion symmetry, in the electric dipole approximation for all i,j, k [78]. Ne-... 

We need to note that mathematically the splay and bending deformations are vectors, i.e., the flexoelectric coupling constant is a second-rank tensor. Accordingly, the flexoelectricity should not be confused with piezoelectricity, which is described by a third-rank tensor coupling constant. Piezoelectricity requires lack of inversion s)mtimetry of the phase, whereas flexoelectricity can exist in materials with inversion symmetry, such as in nematics with macroscopic symmetry. 

Although the order of frequencies in coqOJ — (o ) has no special meaning itself, owing to the intrinsic permutation symmetry, we will keep it fixed (Raman convention), using co and -co as second and third arguments. Besides the intrinsic permutation symmetry, the medium macroscopic symmetry imposes further restrictions on the tensor index of nonvanishing, independent components A very important result is that the nth-order susceptibility vanishes for even n in media showing inversion symmetry and contributes the low-est-order nonlinearity. For isotropic media, it can be shown that vanishes if some Cartesian index appears an odd number of times in the subscript. 

In Cartesian coordinates obviously there are altogether 3" elements in the third-order susceptibility X > ], a fourth-rank tensor, since i, j, k, /) each has three components 1, 2, 3. In an isotropic medium with inversion symmetry, however, it can be shown that there are only four different components, three of which are independent ... 

The application of perturbation theory to a nondegenerate state (for ko = 0) will involve no linear term in k (by symmetry) and the first significant term, which gives the curvature of the band, is given by second order and conventionally expressed as an effective (inverse) mass tensor. If the unperturbed state is denoted by 0> and all other k = 0 eigenstates by n>, the result for this is... 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

Symmetry is one of the most important issues in the field of second-order nonlinear optics. As an example, we will briefly demonstrate a method to determine the number of independent tensor components of a centrosymmetric medium. One of the symmetry elements present in such a system is a center of inversion that transforms the coordinates xyz as ... 

Any symmetry operation is required to leave the sign and magnitude of physical properties unchanged and therefore y xxx = 0. The same line of reasoning can be used to show that all tensor components will vanish under inversion. Hence, second-order nonlinear optical properties are not allowed in centrosymmetric media using the electric dipole approximation. The presence of noncentrosymmetry is one of the most stringent requirements in... 

The nonvanishing components of the tensors y a >--eem and ya >-mee can be determined by applying the symmetry elements of the medium to the respective tensors. However, in order to do so, one must take into account that there is a fundamental difference between the electric field vector and the magnetic field vector. The first is a polar vector whereas the latter is an axial vector. A polar vector transforms as the position vector for all spatial transformations. On the other hand, an axial vector transforms as the position vector for rotations, but transforms opposite to the position vector for reflections and inversions.9 Hence, electric quantities and magnetic quantities transform similarly under rotations, but differently under reflections and inversions. As a consequence, the nonvanishing tensor components of x(2),eem and can be different... 

The expansion (Equation 24.12) does not contain even powers of the field because of the spherical symmetry of an isolated atom. Indeed for an atom, the even derivatives in Equation 24.10 are zero as well as for any molecule having an inversion center. Note that a3 and a5 are, in fact, the components of tensors, respectively of the so-called second and fourth hyperpolarizabilities [4]. 

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