Damping length

In slab calculations, a finite number of layers mimicks the semi-infinite system, with a two-dimensional (2D) translational periodicity. A minimal thickness dmin is required, so that the layers in the slab centre display bulk characteristics. Practically speaking, dm-,n should be at least equal to twice the damping length of surface relaxation effects, which depend upon the surface orientation. In plane wave codes, the slab is periodically repeated... 

Ferry, 1980] where co is the angular frequency in rad/s, r = X/ lnXo) where X is the shear wave length, Xg is the exponential damping length and v is the shear wave phase velocity. For known a> and p, G and G" may be obtained from equation 2.34 by measiuement of v and Xg. 

In order to calculate the damping length, consider the stack of circular cells and plates of radius L. For the rest of this section, the subscript (3 in the symbol Vp is omitted V is thus the local stack potential. 

Equations (5.172) mean symmetry at f = 0 and the absence of normal current through the side surface of the stack, respectively. Equations (5.173) describe the disturbance of potential at 2 =0 and the decay of this disturbance at large distance. We assume that the damping length is much smaller than the stack length. 

At fixed 5, the terms in (5.180) represent the modes of the f-shape of the potential. The Bessel function oscillates and thus a s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance 5 the characteristic damping length is / = l/a , or, in the dimensional form... 

The values a rapidly increase with n thus the high-frequency modes (the modes with n > 1) die out along i faster. The maximal damping length... 

We see that the modes with large a rapidly decay with the distance z. Therefore, the damping length is determined by the low-frequency mode n= 1. Taking into account (5.202), for this length we find... 

In Eq. (5.216), 0 L and w L. Therefore, if the term with a is small, we have L L, i.e. the damping length is proportional to the transversal size of the stack. In this situation the stacks with large cell area are more transparent for temperature disturbances. 

When J —> Jiim the damping length tends to infinity, that is to say the temperature disturbance is constant along z. Fiu-thermore, for J > Jum the power of the exponent of the first (long-wave) mode in Eq. (5.212) becomes complex, which means that this mode oscillates along 2. 

Length of the Nusselt number relaxation in the chaimel (m) Length of the domain exposed to degradation, Section 4.4 (m) Damping length of the disturbance in the stack Reaction penetration depth (2.14) and (4.158) (m)... 

As the mechanical properties of the torsional wire (retarding moment, internal damping, length) depend on temperature, calibration of the pendulum is necessary for all temperatures at which measurements should... 

If a nozzle is attached at the opening, it also offers reinforcement area available for replacing that area removed from the vessel at the opening. Although the vertical limits are different in various codes, all are based on the wave damping length of a beam on an elastic foundation. For a cylindrical shell, this length is a function of 1/, where j8 for a poisson s ratio of 0.3 is equal to 1.285/Vrt. 

Fig. 5.10. MIGS density of states at the interface no and MIGS damping length Ip, for energies at the gap centre, as a function of the oxide ionicity (ec — f a)/P, for two crystal structures (NaCl full lines ZnS dashed lines). The anion level position 6a is taken at 6 eV above the bottom of the metal conduction band and the oxide lattice parameter is a = 5 A (after Bordier and Noguera, 1991).

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