Mean free time

Figure 3 Escape probability as a function of the initial electron-cation distance. The lower broken curve is calculated from the Onsager equation [Eq. (15)]. The numerical results for different mean free times x were taken from Ref. [22]. The unit of x is rJ(ksTlmf, where m is the electron mass. The upper broken curve was calculated using the energy diffusion model. (From Ref. 23.)...

Computer simulation has also been used to calculate the external electric field effect on the geminate recombination in high-mobility systems [22]. For the mean free time x exceeding -0.05, the field dependence of the escape probability was found to significantly deviate from that obtained from the diffusion theory. Furthermore, the slope-to-intercept ratio of the field dependence of the escape probability was found to decrease with increasing x. Unlike in the diffusion-controlled geminate recombination, this ratio is no longer independent of the initial electron-ion separation [cf. Eq. (24)]. 

Figure 6 Dependence of the electron-ion recombination rate constant on the external electric field F, calculated for different values of the electron mean free time r. The unit of Cis, and that of r is r l k-g,Tlnif. The simulation results of Morgan [45] for liquid methane at 120 K are shown by crosses. (From Ref. 48.)...

For the purpose of this work, we will call a complex of two or more interacting atoms/molecules a supermolecule. Supermolecules may exist for a short time only, e.g., the duration of a fly-by encounter ( 10-13 s). Alternatively, supermolecules may be bound by the weak van der Waals forces and thus exist for times of the order of the mean free time between collisions ( 10-10 s), or longer. In any case, it is clear that, in general, supermolecules possess a spectrum of their own, in excess of the sum of the spectra of the individual (non-interacting) molecules that make up the supermolecule. These spectra are the collision-induced spectra, the subject matter of this monograph. 

Summarizing, it may be said that virial expansions of spectral line shapes of induced spectra exist for frequencies much greater than the reciprocal mean free time between collisions. The coefficients of the density squared and density cubed terms represent the effects of purely binary and ternary collisions, respectively. At the present time, computations of the spectral component do not exist except in the form of the spectral moments see the previous Section for details. 

This time also exceeds by many times the mean free time of molecules in the gas in accordance with the fact that any reaction, whether because of a significant heat of activation or because of a complicated mechanism,... 

Rankine assumes that there is a linear relation of pressure and volume in the change of state in a shock wave. The deviation from the adiabatic law, pvK = const, is due to the effect of heat conduction at the shock wave front. In fact, in a shock wave both heat conduction and viscosity act simultaneously so that the law of state change differs both from Poisson s adiabate and from a linear relation. In addition, the very question of a relation between pressure and volume for compression in a shock wave makes sense only for small-amplitude waves for a large amplitude the change of state takes place over a time of the order of the molecular mean-free time. For further details see the author s monograph [9]. 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

The spin relaxation time for paramagnetic Mn2+ in aqueous solution is about 3 x 10 9 s the rotational correlation time is about 10 11 s the mean free time for a proton to reside in a hydration sphere is about 2 x 10 8s. 

As described above, the electrons in a semiconductor can be described classically with an effective mass, which is usually less than the free electron mass. When no gradients in temperature, potential, concentration, and so on are present, the conduction electrons will move in random directions in the crystal. The average time that an electron travels between scattering events is the mean free time, Tm. Carrier scattering can arise from the collisions with the crystal lattice, impurities, or other electrons. However, during this random walk, the thermal motion is completely random, and these scattering processes will therefore produce no net motion of charge carriers on a macroscopic scale. 

In principle, the Boltzmann transport equation (BTE) can cover the regime where die lengdi and time scales are larger than carrier mean free time rand mean free length A. However, tremendous computational efforts are required in practice when the system length scale L and the process time scale t are getting larger. The BTE is, thus, usually... 

Nc. Carrier number density, m T, Mean free time, s... 

Note that even though in (7.5) we use a continuous representation of position and time, the nature of our physical problem implies that Ax and AZ are finite, of the order of the mean free path and the mean free time, respectively. 

The motion of the exciton wavepacket causes the transport of energy. In order to find the appropriate energy diffusion coefficient we must estimate the mean free path and the mean free time of the wavepackets. This situation is quite similar to that of phonon heat conductivity (see, for example, (12)). 

The states have an uncertainty in energy 6e v fi/x, where x is the mean free time for scattering by impurities, phonons or particle-particle interactions. A FrShlich-Peierls type transition, opening up a gap 2A with a distortion corresponding to 2k, is possible only if the states are sufficiently well defined so that 6e < m A. The same applies to the gap from hybridization of inverted bands on donor and acceptor chains. A 3D type band structure is appropriate if 6e < m tj. To the extent that 6e depends on thermal fluctuations a ID band structure may be appropriate at high temperatures and a 3D at low. 

Drift-diffusion for ions The drift-diffusion approximation (Eq. 25) is made for ions, e.g., ion inertia is neglected. This appears to be a good approximation for Kn < 0.2. Eq. (37) provides the field that drives ions in the plasma. Also, a constant mobility is assumed for ions. Actually, when collisions are infrequent (high Kn number), it is better to use a variable mobility model, for which the ion mobility is inversely proportional to the magnitude of the directed velocity of the ion [6]. This model results by assuming a constant mean-free-path of ion-neutral collisions. The constant mobility model results by assuming a constant mean-free-time for ion-neutral collisions. 

Mean free time x Wave Microscopic particle ... 

Note that the diffusion timescale xd contains a characteristic size of an object L. This ties in the length scale of the problem. Similarly, the mean free path can be associated with mean free time between collisions x by the relation l = vx. Note that this is a statistical quantity since collision distances are not fixed. However, the probability p that a particle emerging from a collision travels a distance x without a collision is related to the mean free path as p = exp(-x/ ). Associated with the relaxation time x, is a length scale which is the characteristic size of a volume over which local thermodynamic equilibrium can be defined. Generally, the hierarchy of the length scales is X < < ,. 

The velocity relevant for transport is the Fermi velocity of electrons. This is typically on the order of 106 m/s for most metals and is independent of temperature [2], The mean free path can be calculated from i = iyx where x is the mean free time between collisions. At low temperature, the electron mean free path is determined mainly by scattering due to crystal imperfections such as defects, dislocations, grain boundaries, and surfaces. Electron-phonon scattering is frozen out at low temperatures. Since the defect concentration is largely temperature independent, the mean free path is a constant in this range. Therefore, the only temperature dependence in the thermal conductivity at low temperature arises from the heat capacity which varies as C T. Under these conditions, the thermal conductivity varies linearly with temperature as shown in Fig. 8.2. The value of k, though, is sample-specific since the mean free path depends on the defect density. Figure 8.2 plots the thermal conductivities of two metals. The data are the best recommended values based on a combination of experimental and theoretical studies [3],... 

Cahill and Pohl [8,9] recently developed a hybrid model which has the essence of both the localized oscillators of the Einstein model and coherence of the Debye model. In the Cahill-Pohl model, it was assumed that a solid can be divided into localized regions of size A./2. These localized regions were assumed to vibrate at frequencies equal to to = 2kvsIX where v, is the speed of sound. Such an assumption is characteristic of the Debye model. The mean free time of each oscillator was assumed to be one-half the period of vibration or x = it/to. This implies that the mean free path is equal to the size of the region or XI2. Using these assumptions, they derived the thermal conductivity to be... 

Effective density ofstates in the valence band p = Hole concentration in a semiconductor pi = Hole concentration in an intrinsic semiconductor at equilibrium q = Charge on a carrier a = Electrical conductivity of a semiconductor T = Transmittance Tm = Mean free time for scattering Ud = Drift... 

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