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Determination of Nanomaterial Electronic Structure via Variable-Temperature Variable-Field Magnetic Circular Photoluminescence (VTVH-MCPL) Spectroscopy
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2020
Patrick J. Herbert, Kenneth L. Knappenberger
An A-term denotes differential absorption or emission into unoccupied Zeeman split states. Figure 20.6a shows the differential absorption of left- and right-circularly polarized light into unoccupied states and the resulting derivative line shape in the differential spectrum. For a Gaussian transition profile, absorption or emission peak energies are shifted depending on the Zeeman splitting energy. Assuming a similar transition dipole value, the magnitude of left- and right-circularly polarized absorption or emission is equal. For MCD and MCPL, a rigid shift (RS) approximation is used to predict differential signal line shape. The RS approximation states that while the center peak energy will shift in energy as a function of applied field strength, the spectral line profile will be unchanged. Subtraction of equal Gaussian profile’s offset in energy results in a symmetric derivative line shape. This symmetric derivative line shape is a specific experimental signature of A-terms. Because the differential response arises from Zeeman splitting of unoccupied states, A-terms are not subject to Boltzmann thermal distributions and are accordingly temperature independent.
Uniqueness for Spherically Convergent Multiple Trigonometric Series
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
Here are the steps of Cantor’s proof. Form the Riemann function F and use the Cantor-Lebesgue Theorem to see that it is a continuous function.Observe that the second symmetric derivative of F is identically 0.By a theorem of Schwarz, F must be a linear function.Apply the L2 theory of uniqueness to F to see that all the coefficients of F are 0. Therefore all the coefficients of S are 0 also.
Buckling of Beams and Plates
Published in L.M.B.C. Campos, Higher-Order Differential Equations and Elasticity, 2019
involving: (i) the Poisson ratioσ as moduli of elasticity as well as the Young modulus E; (ii) the strain tensor (II.4.12a–d) ≡(6.304c) that equals half the symmetric derivative of the displacement vector; (iii) the repeated index, which implies a summation (6.304d) in the strain tensor (6.304c) and corresponds to the trace of the matrix (6.304e) and specifies the volume change(6.304f): 2Sij=∂iuj+∂jui:Skk≡∑k=13Skk=Sxx+Syy+Szz≡D3=∇.u⇀,
Biaxial ordering in the supercooled nematic phase of bent-core mesogens: effects of molecular symmetry and outer wing lateral groups
Published in Liquid Crystals, 2020
Robin Harkins, Tatum Tauscher, Jason Nguyen, Sunny Lewis, Fabrizio C. Adamo, Michela Pisani, Daniel Hermida-Merino, Edward T. Samulski, Francesco Vita, Oriano Francescangeli, Eric Scharrer
In order to probe the effects of the different substitution patterns on the local biaxial order in this class of compounds, XRD measurements were performed on the N phase of the three non-symmetric derivatives and one symmetric derivative, Ph(3MeODBP). Representative examples of XRD patterns taken on cooling from the isotropic melt upon application of an aligning magnetic field (horizontal in all figures) are shown in Figures 9(a–c) and Figures 10(a–c) for compounds OC4 2MePh(mono(3,5diMe)ODBP) and OC4 Ph(3MeODBP), respectively. All of the investigated compounds showed the four-spot small-angle pattern typical of tilted, i.e. Smectic C-like, cybotactic order 22–26]. The four-spot pattern was present throughout the N phase, becoming more evident as the temperature decreased.