Mean-free-path, dependence temperature

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],... 

As the temperature is increased, electron-phonon scattering becomes dominant. The mean free path for such scattering varies as 71"" with n larger than unity. The mean free path of electrons at room temperature is typically on the order of 100 A. The mean free path depends on the material but is independent of the sample, since electron-phonon scattering is an intrinsic process. As a result of electron-phonon scattering, thermal conductivity of metals decreases at higher temperatures. 

The slip flow near the boundary surface can be analyzed based on the type of fluids, i.e., gas and Newtonian and non-Newtonian liquids. The sUp flow in gases has been derived based on Maxwell s kinetic theory. In gases, the concept of mean free path is well defined. Slip flow is observed when characteristic flow length scale is of the order of the mean free path of the gas molecules. An estimate of the mean free path of ideal gas is /m 1/(Vlna p) where p is the gas density (here taken as the number of molecules per unit volume) and a is the molecular diameter. The mean free path / , depends strongly on pressure and temperature due to density variation. Knudsen number is defined as the ratio of the mean free path to the characteristic length scale... 

The continuum model assumes continuous and indefinitely divisible matter. For gases the continuum model is valid when the mean free pathlength of the molecules K is much smaller than the characteristic length of the flow I. The mean free path, depending on pressure and temperature of a gas molecule modeled as a rigid sphere. 

Randrianalisoa and Baillis (2008) used Monte Carlo simulation to model heat conduction in porous Si. In their method, an original 3D pore network that reproduces the morphology of mesoporous Si was developed. The nonlinear phonon dispersion curves of Si and the phonon mean-free path dependent on temperature, frequency, and polarization were also considered. The model of steady-state phonon transport through the pore network was simulated. Their results were compared with experimental results of porous Si thin films on a p" -type Si substrate. 

Note that in Eq. (3.34), Aioverain kc> and ki are constants, k denotes the reaction rate that occurs in unit time, which depends upon the pressure and temperature at which we operate the semibatch reactor. Since, for most reactions, we operate semibatch reactors at constant pressure and temperature, Atrxd does not change, ka and Atl depend upon the mean free path of a molecule through the gas phase and through the liquid phase, respectively. Both mean free paths depend upon the density of their respective phases. If the phase densities are constant, then kc and ki will be constant. Therefore, any fluctuation in /coveraii must arise from fluctuations in ac and... 

The rate of dissociation is a function of temperature, increasing rapidly above 2000°C. It also increases with decreasing pressure.f °l The rate of recombination (i.e., the formation of the molecule) is rapid since the mean-free-path dependent half-life of atomic hydrogen is only 0.3 s. 

Unlike rj or x, D is density dependent since diffusion, as discussed in the introduction to this chapter, is essentially a direct measure of the mean free path. The temperature dependence is a consequence of the fact that faster molecules diffuse more rapidly. From Table 2.1 we see that the ratio mnDlf] is predicted to be constant in dilute gases this is completely different from Walden s rule in liquids, Drj const. 

The temperature dependence of the upper critical field in the presence of RE-impurities is determined by the following pair-breaking mechanisms a) the effect of the magnetic field acting on the electron orbits. This leads to Abrikosov flux lines in type II superconductors. This effect is mean free path dependent ... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The deposition rate increases upon increasing the pressure. This is explained by noting that the impingement rate per unit area, r,, of molecules on the filament is linearly dependent on the pressure as r, = pj 2nksT, with the gas temperature. However, as the pressure becomes higher, the collisional mean free path of the silane becomes smaller, and the silane supply to the filaments becomes restricted. Moreover, the transport of deposition precursors to the substrate is restricted as well. The mean free path of silane was estimated to be 2.5 cm at a pressure of 0.02 mbar [531]. i.e.. the mean free path about equals the distance between filament and substrate. Indeed, a maximum in deposition rate is observed at this pressure. This corresponds to a value of pdk of 0.06 (cf. [530]). The microstructure parameter plotted as a function of pd has a minimum around Ms = 0.06 0.02 [530]. 

Since the number of phonons increases with temperature, the electron-phonon and phonon-phonon scattering are temperature dependent. The number of defects is temperature independent and correspondingly, the mean free path for phonon defect and electron defect scattering does not depend on temperature. 

Growth of particles by accumulation on existing particles can be classed as two broad processes. If the precursor is supersaturated, growth will occur at a rate limited by vapor diffusion, which depends on the supersaturation, the temperature, the particle size, and the accommodation coefficient at the surface. The proportionality of particle size changes with the ratio of particle diameter to mean free path of the suspending... 

Informations on the vibrational and electron mean free path properties. Such analysis is possible only if the interface phase is very well defined, and if temperature dependent measurements are done and compared. Debye Waller effects can be tangled with ordering transformation of the interface phase as a function of temperature and so on. If a single phase interface with order at least to the second nearest neighbour is recognised, then a temperature dependent Debye Waller, and mean free path analysis can be attempted. 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

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