Sign ambiguity

Comparing the strain-induced anisotropic part of the effective g-value in Equation 9.24 with the linewidth Equation 9.18 for fully correlated g-strain reveals a remarkable correspondence they are equivalent (apart from a sign ambiguity) within the substitutions... 

In order to address polarization/rotation modulation—not just static polarization/rotation—an algebra is required which can reduce the ambiguity of a static representation. Such an algebra which is associated with 2, r, and that reduces the ambiguity up to a sign ambiguity, is available in the twistor formalism [28]. In this formalism, polarization/rotation modulation can be accomodated, and a spinor, K, can be represented not only by a null direction indicated by 2, q, or C, but also a real tangent vector L indicated in Fig. 4. 

Using this algebraic formalism, the Poincare vector—and its direction of change (up to sign ambiguity)—can be represented. A real tangent vector L of S+ at P is defined ... 

Note the inherent sign ambiguity in the matrices of SU (2) the positive and negative signs in the A matrices correspond to the same improper rotation in ft3. The choice of signs +,, ... 

BTaking the square root of a squared quantity Va = a, leads to a sign ambiguity which cannot be ignored. 

An important mathematical tool for structure determination was developed by A. L. Patterson, allowing both the length and the direction of the lines between various of the atoms (interatomic vectors) to be evaluated from the intensities of the reflections (for which there may be no sign ambiguities). In ideal cases, this method should result in a series of interatomic distances to which the atoms known to exist in the unit cell must be fit. For complicated structures, such fitting is itself a difficult puzzle, and almost always the three-dimensional vector problem is broken down to a series of two-dimensional vector problems. Projections of the interatomic vectors on the faces of the unit cell are obtained, and from such projections, a three-dimensional picture may often be reconstructed. 

Kaiser [90] showed how the sign ambiguity above can be resolved, and the absolute signs of the M matrix elements determined the interested reader is referred to his paper. By combining the M values with the theoretical vibrational wave functions, he was able to use equation (8.339) to derive values of the dipole moment function at a range of values of R R,.. He then fitted the dipole moment function to a sixth-order polynomial in R-Re, from which he was able to calculate the distance derivatives given in table 8.21. 

The radicals S3 and Se3 , which are isostructural with PF2, have been detected in alkali halides doped with sulfur or selenium, the identification being based upon the appearance of two different or 77Se coupling constants (Table VII). Because of sign ambiguity in the tensor, two sets of data result. However, we favor set (ii) because we expect the spin-density on the central atom to be less than that on the corresponding oxy-radicals SOo" and Se02". 

Therefore, we could define a useful surface of action by choosing a constant value of 1 = q f such that each trajectory passes it every time it undergoes an oscillation in the q direction (a good practical choice for q f would probably be the value of q for which the potential energy is a minimum). Each time a trajectory passes q f with pi > 0 (to remove the sign ambiguity), we record (p2, ) 100,122 Jhis means that all points on our Poincare map will lie on the surface... 

The sign ambiguity is related with the direction of propagation of the deformation. The rotational characteristic time tq is... 

The sign ambiguity in (2) can be resolved by noting that F < 1 (passivity requirement). From rectangular waveguide analysis, one can relate the relative (complex) constitutive parameters p and to the reflection and transmission coefficients ... 

It is seen fi-om relations (1.47) and (1.48) that absolute values of dipole moment derivatives can only be evaluated fi om experimental integrated intensities. Therefore, the directions of charge shifts accompanying particular vibrational distortions remain undetermined. This is a major difficulty in any further reduction of vibrational intensity data. For many years the sign ambiguity problem for the dipole moment derivatives has been a cause for the limited application of vibrational intensities in structural analysis. 

Existing perturbations in the spectra associated with F mi resonances, Cmiolis interactions and strong anharmonicity effects nu often hamper die interpretation of experimental intensity data. The sign ambiguity problem for dipole moment derivatives, as already discussed, is also present. 

As in the case of dipole moment derivatives widi respect to normal coordinates, absolute values of a. and only can be evaluated from the experiment. Extended basis set ab initio MO calculations of polarizability derivatives are usually employed in solving the sign ambiguity problem. 

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