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In other words, zeta potential is the potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle. The zeta potential is caused by the net electrical charge contained within the region bounded by the slipping plane, and also depends on the location of that plane .
Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency ...
The density of states related to volume V and N countable energy levels is defined as: = = (()). Because the smallest allowed change of momentum for a particle in a box of dimension and length is () = (/), the volume-related density of states for continuous energy levels is obtained in the limit as ():= (()), Here, is the spatial dimension of the considered system and the wave vector.
In 1923, Peter Debye and Erich Hückel reported the first successful theory for the distribution of charges in ionic solutions. [7] The framework of linearized Debye–Hückel theory subsequently was applied to colloidal dispersions by S. Levine and G. P. Dube [8] [9] who found that charged colloidal particles should experience a strong medium-range repulsion and a weaker long-range attraction.
Substituting this length scale into the Debye–Hückel equation and neglecting the second and third terms on the right side yields the much simplified form () = ().As the only characteristic length scale in the Debye–Hückel equation, sets the scale for variations in the potential and in the concentrations of charged species.
The general form of a one-dimensional periodic potential equation is Hill's equation: [19] + =, where f(t) is a periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation .
In general the dispersion relation cannot be approximated as parabolic, and in such cases the effective mass should be precisely defined if it is to be used at all. Here a commonly stated definition of effective mass is the inertial effective mass tensor defined below; however, in general it is a matrix-valued function of the wavevector, and ...
The Lennard-Jones potential is a simple model that still manages to describe the essential features of interactions between simple atoms and molecules: Two interacting particles repel each other at very close distance, attract each other at moderate distance, and eventually stop interacting at infinite distance, as shown in the Figure.