Mean free path, of electrons

Because a set of binding energies is characteristic for an element, XPS can analyse chemical composition. Almost all photoelectrons used in laboratory XPS have kinetic energies in the range of 0.2 to 1.5 keV, and probe the outer layers of tire sample. The mean free path of electrons in elemental solids depends on the kinetic energy. Optimum surface sensitivity is achieved with electrons at kinetic energies of 50-250 eV, where about 50% of the electrons come from the outennost layer. 

Example 3. The mean free path of electrons scattered by a crystal lattice is known to iavolve temperature 9, energy E, the elastic constant C, the Planck s constant the Boltzmann constant and the electron mass M. (see, for example, (25)). The problem is to derive a general equation among these variables. 

Reynolds, F.W., and G.R. Stilwell Mean free paths of electrons in eva-... 

In recent years, increasing use has been made of in situ methods in EM—as is true of other techniques of catalyst characterization such as IR, Raman, and NMR spectroscopy, or X-ray diffraction. Although the low mean-free path of electrons prevents EM from being used when model catalysts are exposed to pressures comparable to those prevailing in industrial processes, Gai and Boyes (4) reported early investigations of in situ EM with atomic resolution under controlled reaction conditions to probe the dynamics of catalytic reactions. Direct in situ investigation permits extrapolation to conditions under which practical catalysts operate, as described in Section VIII. 

Note that /lo(Es) is the inelastic mean free path of electrons formed in the substrate travelling through the overlayer. In the case that the overlayer is a film of Si02 on a silicon crystal, as in Fig. 3.13, Expression (3-9) reduces to... 

In situ characterization. Catalysts should preferably be investigated under the conditions under which they are active in the reaction. Various reasons exist why this may not be possible, however. For example, lattice vibrations often impede the use of EXAFS, XRD and Mossbauer spectroscopy at reaction temperatures the mean free path of electrons and ions dictates that XPS, SIMS and LEIS are carried out in vacuum, etc. Nevertheless, one should strive to choose the conditions as close as possible to those of the catalytic reaction. This means that the catalyst is kept under reaction gases or inert atmosphere at low temperature to be studied by EXAFS and Mossbauer spectroscopy or that it is transferred to the vacuum spectrometers under conditions preserving the chemical state of the surface. 

A, Mean free path of ions, meters Am Mean free path of electrons, ions, or molecules, meters a Surface tension of liquid or surface energy of solid, N/meter or J/ meter2... 

A Monte Carlo simulation with the parameter A/a, where A is the mean free path of electrons and a is the reaction radius [39,42,114]. 

Mean Free Paths of Electrons as a Function of Kinetic Energy... 

In studying solids the short mean free paths of electrons and their strong dependence on kinetic energy provides a means of differentiating surface from subsurface and bulk phenomena and hence analytical depth profiling by studying core levels... 

In dirty metals at low temperatures, similar dependence is observed everywhere for nanocomposites also it has been reported in a few papers [82-85]. It is specific for nanocomposites that, as it was demonstrated in papers [66,82,85], this dependence could be observed up to very high (room) temperatures. This is due to the small mean free path of electrons in granular metals caused by the strong disorder which are natural for such material. Let us recall that the equation ksT h/x determines the limiting temperature up to which quantum corrections due to an electron interference are actual and the dependence (19) is fair. Here, t is the electron moment relaxation time. However, in some cases at low temperatures the unexpected deviation of experimental data from the dependence (19) was observed [66,82,85]. Discussion of this surprising effect will make a part of the contents of the Section 6. 

Another argument is based on mean free paths of electrons in Auger spectra of He II spectra [47]. In these spectra, the mean free path of electrons is about 3-4 A, while in the case of 4f electrons released due to Mg K< radiations is about 17-18 A. Thus the Auger technique senses the electrons in the outermost two monolayers while XPS is sensitive to about seven monolayers and in such a case, surface stabilization of the divalent state may very well be the case in accordance with the experimental findings [47]. 

Figure 5. The fraction of signal obtained for a given depth below the surface (relative to the signal obtained from a homogeneous infinitely thick solid) calculated from Ii/I" = 1 — exp( / ) and assuming the mean free path of electrons A = 22 A. Note that 63% of the intensity is obtained from the top 22 A (at x = X), 90%) from the top 50 A, and 99% from the top 100 A. Steps are introduced to show scale of atomic dimensions, i.e., jumps every 3 A for a graphite single crystal.

The XPS apparatus is described briefly here. The mean free path of electrons at Imbar pressure is a few millimeters (185). Consequently, if the path length of... 

The samples used for RHEED and FEED are single crystals with carefiilly prepared flat surfaces. For THEED of thin films, the observed areas of the samples must be electron transparent with thickness less than or comparable to the inelastic mean free path of electrons. The inelastic mean free path increases with the electron voltage. The typical sample... 

Photoemission spectroscopy involves measurement of the energy distribution of electrons emitted from a solid under irradiation with mono-energetic photons. In-house experiments are usually performed with He gas discharge lamps which generate vacuum UV photons at 21.2 eV (He la radiation) or 40.8 eV (He Ila radiation ) or with Mg Ka (hv=1284.6 eV) or A1 Ka (hv=1486.6eV) soft X-ray sources. UV photoemission is restricted to the study of valence and conduction band states, but XPS allows in addition the study of core levels. Alternatively photoemission experiments may be performed at national synchrotron radiation facilities. With suitable choice of monochromators it is possible to cover the complete photon energy range from about 5 eV upward to in excess of 1000 eV. The surface sensitivity of photoemission derives from the relatively short inelastic mean free path of electrons in solids, which reaches a minimum of about 5A for electron energies of the order 50-100 eV. 

In an earlier study we had reported the XPS analysis of tungsten oxides formed during anodic polarization experiments. It was determined that even at high applied potentials, the oxide thickness values are less than the mean free path of electrons in the oxides (generally assumed to be between 30 to 50 A ). Clearly the oxide growth in tungsten is a slow process. However, despite the relatively small thickness vsilues, the steady state current density during anodic polarization is restricted to a few tens of microamperes. 

Fig. 6. RPA double-plasmon inverse mean free paths of electrons (solid line) and positrons (dashed line) for = 2.07, versus the velocity of the projectile, as obtained from equations (27) and (28) by either including (electrons) or excluding (positrons) the step function ((Wy q — sp). The dotted line is the high-velocity limit dictated by equation (50).

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