Quadrupole-Dipole Transform

There are several physical implementation methods that can achieve such a transform using optical near-field interactions. One is based on optical excitation transfer between quantum dots via optical near-field interactions [23, 24]. For instance, assume two cubic quantum dots whose side lengths L are a and -Jla, which we call QDa and QDb, respectively. 

Suppose that the energy eigenvalues for the quantized exciton energy level specified by quantum numbers ( x z) in a QD with side length L are given by 

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Fig. 2.2 A quadrupole-dipole transform in the transition from the (2,1,1) level to the (1,1,1) level in QDb- Such a transform is unachievable without the optical near-field interactions between QDa and QDb, which allow the (2,1,1) level in QDb to be populated with excitons...

Fig. 2.3 Schematic diagram of nanophotonic matching system. The function is based on a quadrupole-dipole transform via optical near-fleld interactions, tind it is achieved through shape-engineered nanostructures and their associated optical near-field interactions...

To verify this quadrupole-dipole transform mechanism brought about by shape-engineered nanostructures, we numerically calculated the surface charge distributions induced in the nanostructures and their associated far-field radiation based on a finite-difference time-domain (FDTD) electromagnetic simulator (Poyntingfor Optics, a product of Fujitsu, Japan). Figure 2.4a schematically represents the design... 

Fig. 2.4 (a) Specifications of the three types of nanostructures used in numerical evaluation of the conversion eflSdency based on the FDTD method, and corresponding surface charge density distributions induced in each nanostmctuie. (1) Shape A only, (2) Shape B only, and (3) a stacked stiucture of Shapes A and B. (b) Calculated performance figure of the quadrupole-dipole transform, namely, polarization conversion elfidency, with the three types of nanostmctures. (c) Selective comparison at a wavelength of 690 nm... 

Here, we make a few remarks regarding the quadrupole-dipole transform demonstrated in this study. We can engineer many more degrees-of-freedom on the nanometer-scale while using far-field radiation for straightforward characterization. 

Fig. 2.7 (a) Horizontal and vertical misalignments between Shape A and Shape B denoted by Ax and Ay, and (b), (c), their relations to conversion efficiency, (d) Schematic cross-sectional profiles of samples with different gaps. The thickness of the Si02 gap layer between the first and second layers was set in three steps, (e) The conversion efficiency decreased as the gap increased. The result validates the principle of the quadrupole-dipole transform that requires optical near-field interactions between closely arranged nanostructures... 

From a system perspective, the quadrupole-dipole transform can be regarded as a kind of mutual authentication or certification function of two devices, meaning that the authentication of Device A (with Shape A) and Device B (with Shape... 

B) is achieved through the quadrupole-dipole transform. Because such fine nanostructures are difficult to falsify, the vulnerability of a security system based on this technology is expected to be extremely low. 

Quadrupole-Dipole Transform based on Optical Near-Field Interactions in Engineered Nanostructures, Optics Express, Vol. 17,11113,2009... 

Accordingly, the transition cannot be vibrationally induced in the usual manner. The transition is not magnetic dipole allowed because A[ does not transform like a rotation. Nor is it allowed in electric quadrupole radiation since the matrix elements of the quadrupole moment transform like squared polar vectors, and E X E = a x +, 4 -H E . Since the transition is apparently observed, a most reasonable mechanism would involve a combination of two vibrations, say E and A z. The combination band symmetry is E", and A l X E" = E which is the rep of a polar vector. 

Because of difficulties in calculating the non-adiabatic conpling terms, this method did not become very popular. Nevertheless, this approach, was employed extensively in particular to simulate spectroscopic measurements, with a modification introduced by Macias and Riera [47,48]. They suggested looking for a symmetric operator that behaves violently at the vicinity of the conical intersection and use it, instead of the non-adiabatic coupling term, as the integrand to calculate the adiabatic-to-diabatic transformation. Consequently, a series of operators such as the electronic dipole moment operator, the transition dipole moment operator, the quadrupole moment operator, and so on, were employed for this purpose [49,52,53,105]. However, it has to be emphasized that immaterial to the success of this approach, it is still an ad hoc procedure. 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

The details of electron distribution in atoms characterized by higher multipoles (dipoles and quadrupoles) are defined by the deviations S P . As it has been already mentioned the vector parts of the HOs centered at each given atom vA transform as 3-vectors under the molecule/space rotations, and the hybrid densities sAvA transform as 3-vectors as well. On the other hand, the diadic products v,A vA under 3-dimensional rotations transform as a sum of a scalar and of the rank two tensor of the 3-dimensional space. The values of these momenta are obtained by averaging their standard definitions ... 

Transformation of coordinates for the nuclear magnetic dipole and electric quadrupole terms... 

The dynamic coupling mechanism predicts that hypersensitivity should be observed when the point group of the lanthanide complex contains Y3m spherical harmonics in the expansion of the point potential. The good agreement between the calculated and observed values for Tj or Q.2 parameters shows that the dynamic coupling mechanism makes a significant contribution to the intensities of the quadrupole allowed f-electron transitions in lanthanide complexes. Qualitatively, the mechanism is allowed for all lanthanide group symmetries in which the electric quadrupole component 6,a,fi and the electric dipole moment p, a transform under a common representation. 

In the molecular point groups, the three trace elements of / always transform under the totaUy symmetric representation. The symmetry behaviour of the three components of (i corresponds with that of the components of the magnetic dipole operator, which transform like the rotations Ry, R. The components of 0(2) transform like those of the quadrupole moment operator, that is, like the five d orbitals. 

The determination of polymer structure at the atomic level is possible by analyzing orientation-dependent NMR interactions such as dipole-dipole, quadrupole and chemical shielding anisotropy as mentioned above. The outline of the atomic coordinate determination for oriented protein fibers used here is described more fully in Ref. [30]. The chemical shielding anisotropy (CSA) interaction for N nucleus in an amide (peptide) plane can be interpreted with the chemical shielding tensor transformation as shown in Fig. 8.3. [31, 32]. 

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)). 

All cfv reflections transform the dipole-quadrupole to its enantiomeric form the simplest choices of reflection plane are those transforming either or Qy°z into its negative. 

Third, the quadrupole moments are referred to the principal inertia axis system, i.e., to the center of mass. Since the quadrupole moments are origin dependent for molecules which have nonzero electric dipole moments, the appropriate transformation to a common origin must be performed, if quadrupole moments of different molecules or if quadrupole moments determined by different methods are to be compared. For instance, if the reference system is shifted parallel to the a-axis by an amount Aa, av=av + Aa, a =a -[- Aa, b y—bp,...) the new primed quadrupole moments are ... 

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