The Harmonic Solid

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.

Protein adlayer thickness on colloidal nanoparticle determined by Rayleigh-Gans-Debye approximation

ABSTRACTReference materials (RM)-assisted Rayleigh-Gans-Debye approximation (rm-RGDA) has been developed and used to in situ determine the size and thickness of the adlayer on the particles in solution. The particle size determined by rm-RGDA is quite close to that measured by electron microscopy but significantly smaller than that measured by DLS. The BSA adlayer absorbed on PS50, PS100 and SiO2 NPs is 3.3, 0.9 and 1.2 nm, respectively, and close to those observed by SEM, which is 4.6, 1.3 and 3.8 nm, respectively. The FTIR analysis results show that the BSA absorbed on larger particles or hydroxyl-abundant surface, e.g. PS100 and SiO2 NPs can lose its secondary structure, e.g. α-helix, to a great extent and that absorbed on a more curve surface, e.g. smaller PS50 particles can largely preserve its secondary structure as its free state. The measurement results show the curvature of the NPs is closely related to the structure change of the adsorbed protein. This method provide a facile and new approach to measure the size and its adlayer change of the hybrid and core-shell structured nanoparticles in a wide range of wavelength.SIGNIFICANCEQuantitative study on the adsorption of the protein on colloidal nanoparticles is an important approach to understand the biophysical effect, compared with other ex situ methods such as TEM and SEM, where the specimen are undergone pre-processing and no longer the original state in measurement. It is, therefore, a big challenge. In order to cope with this challenge, UV-vis based RGDA has been developed and applied to in situ measure the size of the dispersed colloidal nanoparticles and their protein adlayer thickness, where the protein adlayer thickness on the colloidal nanoparticles can be easily determined. We believe this method provide a facile and sensitive way to in situ measure the dimension change of hybrid colloidal nanoparticles.

Comparison of focusing of short pulses in the Debye approximation

We have examined different types of pulses and features of their frequency spectra. Calculations have shown that a significant distinction between the pulses only takes place at a very short pulse duration (shorter than the oscillation period). In this case, the Gaussian pulse becomes nonphysical and one needs to use other types of pulses, for example, the Poisson pulse. We performed comparative modeling of focusing of short pulses by an aplanatic lens for different polarization states and vortex singularity orders in the Debye approximation. We have shown that the polarization state and the presence of vortex phase singularity essentially affect the distribution in the focal area for a subcycle Poisson pulse.