Phase plane Physical constant

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

In many cases, the use of ideal equivalent circuits is convenient but not always appropriate. Nonideal behavior might arise from interactions of species, resulting in frequency-dependent capacitances [C((D)]. Under these conditions, the physical process is more accurately described by a range of relaxation time constants instead of a unique value. Such distributed relaxation events are usually manifested as semicircles depressed below the real axis in the complex plane, and the angle of depression is related to the degree of nonideality. Various distribution functions and constant phase elements have been employed to describe such events. These nonidealities are especially evident in biological systems. 

It is often found that the double-layer capacitance or a coating capacitance does not behave like an ideal capacitor, experimentally manifested in the complex plane plot by a depressed semicircle whose center lies below the real axis. This behavior is usually attributed to some distribution (or dispersion) in some physical property of the system (e.g., the porous surface of the metal or the varying thickness or composition of a coating) and is modeled by the use of a constant phase element (CPE) [30]. 

The choice of phase factors in (5.7.1) and (5.7.2) is the one corresponding to the "C-typ " dynamical matrix (eq. (2.1.58) of Ref. 51) the eigenvectors w (5.7.2) correspond then to those of eq. (2.1.60) in Ref. 51. We note that in Ge the amplitudes of both atoms have to be equal by symmetry also the end-points of the dispersion of phases are fully determined by symmetry - but not the variation between them. Closer inspection reveals, however, that the form of the variation is determined (in the non-trivial cases) essentially by the first-neighbor force constants and depends little on interactions with more distant planes this is a physical fact, not predictable from symmetry considerations. 

MD simulations were performed on an antiferromagnetic structure of ice Ih. This structure has a net dipole moment of zero. Haymet and Karim were able to obtain stable ice ih at a reasonable temperature and pressure and hence a stable ice-water interface under physical conditions reasonably close to experiment. (It should be noted that the precise phase coexistence properties of the TIP4P model are not known.) The diffusion constant and density profiles for the oxygen atoms across the interface were shown in Section 2 (Figures 2a and 2b, respectively). The bulk value of the diffusion constant was calculated from a separate simulation with density 1.02 g cm . The letters A-H in Figure 2(a) correspond to individual lattice plane layers through the interface and will be referred to in what follows. 

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