A three-dimensional nanoparticle tracking technique using ratiometric total internal reflection fluorescence microscopy (R-TIRFM) is presented to experimentally examine the classic theory on the near-wall hindered Brownian diffusive motion. An evanescent wave field from the total internal reflection of a 488-nm bandwidth argon-ion laser is used to provide a thin illumination field on the order of a few hundred nanometers from the wall. Fluorescence-coated polystyrene spheres of 200±20 nm diameter (specific gravity=1.05) are used as tracers and a novel ratiometric analysis of their images allows the determination of fully three-dimensional particle locations and velocities. The experimental results show good agreement with the lateral hindrance theory, but show discrepancies from the normal hindrance theory. It is conjectured that the discrepancies can be attributed to the additional hindering effects, including electrostatic and electro-osmotic interactions between the negatively charged tracer particles and the glass surface.

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Near-wall hindered Brownian diffusion of nanoparticles examined by three-dimensional ratiometric total internal reflection fluorescence microscopy (3-D R-TIRFM)

This chapter discusses how to extinguish spin–orbit interactions and/or scalar relativistic effects from four-component relativistic molecular calculations in order to assess their importance on molecular properties. It is pointed out that standard non-relativistic calculations use the non-relativistic free-particle Hamiltonian H ˆ p , but the relativistic Hamiltonian H ˆ int which describes the interaction between particles and fields. In the strict non-relativistic limit, electrodynamics reduce to electrostatics, that is there are no effects of retardation and no magnetic interactions. It is, however, perfectly reasonable from a pragmatic point of view to introduce both scalar and vector potentials in a non-relativistic framework. Non-relativistic theory can perfectly well accommodate magnetic sources, including spin, but does not provide a mechanism for generating them. We demonstrate that the pragmatic approach leads to some inconsistencies in that non-relativistic theory cannot describe spin–same orbit interactions, but spin–other orbit interactions. We also emphasize that the distinction between spin–orbit interactions and other spin interactions is somewhat artificial and highly dependent on the chosen reference frame. In a previous paper [L. Visscher and T. Saue, J. Chem. Phys. , 2000, 113 , 3996] we demonstrated how to eliminate spin–orbit interaction from four-component relativistic calculations of spectroscopic constants by deleting the quaternion imaginary parts of matrix representations of the modified Dirac equation. In this chapter, we discuss the extension of this approach to second-order electric and magnetic properties. We will demonstrate the elimination of poles corresponding to spin-forbidden transitions from the dispersion of the dipole polarizability of the mercury atom. More care is needed when considering second-order magnetic properties in that the elimination of quaternion imaginary parts will extinguish all spin interactions. A procedure is developed which allows us to demonstrate important spin–orbit effects on the NMR shielding polarizabilities of the xenon atom. It is also possible to extinguish all spin interactions in relativistic calculations, but only within the framework of the Sternheim approximation, that is when calculating the diamagnetic contribution as an expectation value.

Spin-Interactions and the Non-relativistic Limit of Electrodynamics

We have calculated the double Gamow–Teller (DGT) matrix element M2νGT and half-life T2ν1/2 for two neutrino double beta decay (β+β+/β+EC/ECEC) of 124,126Xe and 130,132Ba in the Hartree–Fock–Bogoliubov framework. The reliability of the intrinsic wavefunctions of 124,126,130,132Xe, 124,126Te and 130,132Ba nuclei has been tested by comparing the theoretically calculated spectroscopic properties with the available experimental data. The calculated half-lives for the 0+→ 0+ transition range from 1022 years to 1029 years, varying from mode to mode indicating that β+ EC may be the process with the most potential to be detected using present experimental setups. Moreover, it can be inferred from the present study that deformation plays an important role in the quenching of M2νGT and hence is very crucial to be considered in the framework used to study double beta decay.

Study of two neutrino β+β+/β+ EC/ECEC decay of 124, 126Xe and 130, 132Ba for 0+→ 0+ transition in PHFB model

In this paper, we show that Lorentz boosts are direct isometries according to the recent mathematical definitions of direct and indirect isometries and of chirality, working for any metric space. Here, these definitions are extended to the Minkowski spacetime. We also show that the composition of parity inversion and time reversal is an indirect isometry, which is the opposite of what could be expected in Euclidean spaces. It is expected that the extended mathematical definition of chirality presented here can contribute to the unification of several definitions of chirality in space and in spacetime, and that it helps clarify the ubiquitous concept of chirality.

https://www.mdpi.com/2073-8994/11/10/1320/pdf

About Chirality in Minkowski Spacetime

As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation — the so-called Thomas (or Wigner) rotation — and a singl...

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

The composition of two pure Lorentz transformations (boosts) parametrized by non-parallel velocities is equivalent to a boost combined with a pure spatial rotation - the Thomas rotation. Thirty years elapsed from Thomas's 1926 calculation for the precessional application until an explicit result for the Thomas rotation angle appropriate to two finite boost velocities appeared in the literature. Over the years there have been a number of papers that have produced results by various methods for the Thomas rotation angle but none by direct Lorentz matrix methods. This paper repairs that deficiency by presentation of an explicit Lorentz matrix proof of sufficient conciseness to show that the much-vaunted complexity of the direct approach has been considerably overstated. The demonstration fills a gap at a fundamental level in the development of the basics of special relativity.

W L Kennedy

Papers

Thomas rotation: a Lorentz matrix approach