Vector fields symmetry properties

The electric-dipole transition is determined by the symmetry properties of the initial-state and the final-state wave functions, i.e., their irreducible representations. In the case of electric-dipole transitions, the selection rules shown in table 7 hold true (n and a represent the polarizations where the electric field vector of the incident light is parallel and perpendicular to the crystal c axis, respectively. Forbidden transitions are denoted by the x sign). In the relativistic DVME method, the Slater determinants are symmetrized according to the Clebsch-Gordan coefficients and the symmetry-adapted Slater determinants are used as the basis functions. Therefore, the diagonalization of the many-electron Dirac Hamiltonian is performed separately for each irreducible representation. 

Further, one must pay attention to Eq. (5.10) for /mm - In isotropic collisions, when we have Tmm> — ram=m-M i and in the absence of an external field, when we have jkjJmm1 — 0) linearly polarized excitation yields /mm i = -f-M i-M- And since /mm i and f-M i-M enter into the sum (5.41) with equal coefficients, the aforesaid implies the absence of transversal orientation f v More precisely, the contention concerning the antisymmetry of the respective density matrix elements /mm i = —f-M i-M follows from the explicit form of the cyclic components of the polarization vector (see Appendix A), and from the symmetry properties of the Clebsch-Gordan coefficients (see Appendix C). 

Closed orbits from symmetry arguments) Give a simple proof that orbits are closed for the simple harmonic oscillator x = v, v = -x, using only the symmetry properties of the vector field. (Hint Consider a trajectory that starts on the v-axis at (0, ), and suppose that the trajectory intersects the x-axis at (x,0). Then use symmetry arguments to find the subsequent intersections with the v-axis and x-axis.)... 

Finally, the reader should appreciate a significant difference between the way in which a and r were introduced. In Section 1.2.1 the strain tensor was defined on purely mathematical grounds, whereas the conjugate stress tensor was introduced by purely physical reasoning (i.e., force balance). However, both elastic body arc explicitly excluded. As far as rr is concerned, this is effected by introducing the displacement of mass elements (r) as a vector field and by defining displacement tensor V (r) (see Ekjs. (1.1) and (1-6)] for the stress tensor r, the symmetry property stated in Eki. (1-14) serves to eliminate rotations. 

For instance, at the present, by optical measurements (by in situ electroreflectance) it was shown that the symmetry properties of the crystal faces were not perturbed by contact with aqueous solutions. For copper, all faces studied present their azimuthal anisotropy with respect to the plane of polarization of the incident light, except for (100), (111), (211), and some neighboring faces of (111) for which no azimuthal anisotropy is observed. " For silver, the twofold symmetry of the (110) face was observed the electroreflectance spectra at normal incidence differ markedly when the electric field vector of light is parallel or perpendicular to the surface atomic rails. This is not the case for the (111) and (100) faces which have higher symmetry. For gold, no azimuthal anisotropy was observed for the (111) and (100) faces with respect to the plane of polarization of the incident light, while the (110) and (311) faces present an azimuthal anisotropy. It does not necessarily mean that the outermost layer of surface atoms is 1 x 1, because a surface reconstruction such as 1 x2 for Au (110) would have the same symmetry order as 1x1. These are already studies of the metal surface in the presence of the electrochemical dl. 

The expression for Po oM in the case of mechanical excitons has the same form (2.57), but the functions u ),(()) must be replaced by u (0), obtained by neglecting the effects of the long-wavelength field. Since the operator P° is transformed like a polar vector, and the wavefunction To is invariant under all crystal symmetry transformations, the matrix element (2.57) will be nonzero only for those excitonic states whose wavefunctions are transformed like the components of a polar vector. If, for example, the function ToM transforms like the x-component of a polar vector, the vector Po o will be parallel to the x-axis. Thus the symmetry properties of the excitonic wavefunctions determine the polarization of a light wave which can create a given type of exciton. In the above example only a light wave polarized in the x-direction will be absorbed, obviously, if we restrict the consideration to dipole-type absorption. In a similar way, for example, the quadrupole absorption in the excitonic region of the spectrum can be discussed (for details see, for example, 8 in (12)). 

Vectors which transform according to equation (4.5) are called polar. The vector product of two polar vectors p and q, r = p x q, is equally considered as a vector. The transformation properties of r, however, differ from those of p and q. r is invariant with respect to a mirror plane perpendicular to the vector and inverted by a mirror plane parallel to the vector. The symmetry of p and q is oom, that of r is oo/m. r is an axial vector. As an example, we cite the magnetic field H. If the components of p and q are p and respectively, the components of r are ... 

The time-reversal symmetry may be understood as a property of the vector field. This engenders a symmetry at the level of the trajectories of the system. 

The second and third sums will vanish due to the symmetry of second partial deriva tives and because the commutator of the two vector fields is again a vector field, that is, a first-(but not second-) order differential operator. Eventually, property 4 follows from the fact that... 

Taking into account that the field amplitudes E, E2 are vectors and that the second-order susceptibility is a tensor of rank 3 with components Xijk depending on the symmetry properties of the nonlinear crystal [5.217], we can write the second-order term in the explicit form... 

TheEinEqs. 17.9 and 17.10 is an electric field vector. As such, the symmetry properties of the material will be important. In particular, it can be shown that the odd-order terms of these equations are independent of any symmetry considerations, but the even-order terms vanish in a centrosymmetric environment. That is, x is zero for any sample that has a center of inversion. This is always the case for bulk liquids, and it is true for most solids. Thus, symmetry and orientation effects should be crucial for discussions of SHG and related phenomena. We have noted earlier that crystal engineering has significant implications for many types of applications, and NLO is definitely one of them. The ability to rationally design or predict non-centrosymmetric crystals would be very valuable. Similarly, at the molecular level, p will be zero for molecules with a center of symmetry. No such restrictions apply to the third-order terms (x and y). 

Cubane is a Platonic hydrocarbon with chemical formula CgHg, possessing octahedral Oh symmetry. Its special magnetic properties have been investigated by some authors, whose work was reviewed in a recent paper [116]. A compact spatial model for the electronic current density vector field induced in the cubane molecule by a magnetic field perpendicular to a face has been proposed to interpret its magnetic response. 

Although Eqs. (13-7) and (13-8) determine the spatial dependence of e((x, y), they give no information about its polarization, i.e. its vector field direction. This direction must be determined from the polarization properties of the waveguide, or by its symmetry properties. 

This is equivalent to the statement that any two-terminal transport phenomenon can only have an even magnetic field dependence. In chiral systems, symmetry allows all microscopic properties to have in principle an odd dependence on the wave vector k of the moving particles. As the wave vector is also odd under time-reversal, from Eq. (33) it follows that... 

The dielectric properties of a material are properly specified by a symmetric second-rank tensor relating the three components of the electrical displacement vector D to those of the field E. By choosing axes naturally related to the crystal structure the six independent components of this tensor can be reduced to three and, taking account of the hexagonal symmetry of the ice crystal, only two independent components remain. These are the relative permittivities parallel and perpendicular to the unique c-axis direction and we shall denote them by e, and e. We shall discuss the experimental determination of these quantities when we come to consider dielectric relaxation, since some difficulties are involved. For the present we simply note the results which are shown in fig. 9.2. The often-quoted careful measurements of Auty Cole (1952) were made with polycrystalline samples and removed many of the uncertainties in earlier work. They represent, however, a weighted mean of the values of e, and Humbel et al. (1953 [Pg.201]

The importance of the hyperpolarizability and susceptibility values relates to the fact that, provided these values are sufficiently large, a material exposed to a high-intensity laser beam exhibits nonlinear optical (NLO) properties. Remarkably, the optical properties of the material are altered by the light itself, although neither physical nor chemical alterations remain after the light is switched off. The quahty of nonlinear optical effects is cmciaUy determined by symmetry parameters. With respect to the electric field dependence of the vector P given by Eq. (3-4), second- and third-order NLO processes may be discriminated, depending on whether or determines the process. The discrimination between second- and third-order effects stems from the fact that second-order NLO processes are forbidden in centrosymmetric materials, a restriction that does not hold for third-order NLO processes. In the case of centrosymmetric materials, x is equal to zero, and the nonhnear dependence of the vector P is solely determined by Consequently, third-order NLO processes can occur with all materials, whereas second-order optical nonlinearity requires non-centrosymmetric materials. 

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