Electrons quasi-free

Electron Beam Resist Reactions of CMS. The lifetime of the excimer fluorescence of CMS observed in pulse radiolysis of CMS solutions in cyclohexane and tetrahydrofuran (THF) is almost independent of chloromethylation ratio from 0% to 24%. The intensity of the excimer fluorescence decreases with increasing degree chloromethylation indicating that the precursor of the excimer is scavenged by the chloromethylated part of CMS. In this case, an electron (quasi-free electron in cyclohexane and solvated electron in tetrahydrofuran, which are the precursors of the excimer), is scavenged by the chloromethyl group. The excited singlet state... 

Where b is Planck s constant and m and are the effective masses of the electron and hole which may be larger or smaller than the rest mass of the electron. The effective mass reflects the strength of the interaction between the electron or hole and the periodic lattice and potentials within the crystal stmcture. In an ideal covalent semiconductor, electrons in the conduction band and holes in the valence band may be considered as quasi-free particles. The carriers have high drift mobilities in the range of 10 to 10 cm /(V-s) at room temperature. As shown in Table 4, this is the case for both metallic oxides and covalent semiconductors at room temperature. 

The interpretation of these remarkable properties has excited considerable interest whilst there is still some uncertainty as to detail, it is now generally agreed that in dilute solution the alkali metals ionize to give a cation M+ and a quasi-free electron which is distributed over a cavity in the solvent of radius 300-340 pm formed by displacement of 2-3 NH3 molecules. This species has a broad absorption band extending into the infrared with a maximum at 1500nm and it is the short wavelength tail of this band which gives rise to the deep-blue colour of the solutions. The cavity model also interprets the fact that dissolution occurs with considerable expansion of volume so that the solutions have densities that are appreciably lower than that of liquid ammonia itself. The variation of properties with concentration can best be explained in terms of three equilibria between five solute species M, M2, M+, M and e ... 

A three-dimensional (3D) piece of metal can be considered as a crystal of infinite extension in the directions x, y and z with standing waves with the wave numbers k, ky and k, each being occupied with two electrons as a maximum. In a piece of bulk metal the energy differences Sk y are so small that A->0, identical with quasi-free continuously distributed electrons. Since the energy of free electrons varies with the square of the wave numbers, its dependence on k describes a parabola. Figure 4a shows these relations. 

An electron in a condensed medium is considered localized if the lowest energy in that state is less than V, the ground state energy of the quasi-free electron. According to Springett et al. (1968), the condition for localization is expressed as... 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

The authors assume different and statistically independent mechanisms of electron-ion recombination in the quasi-free and trapped states. Thus P = w, where and wt are respectively the probabilities of escaping recombination in the quasi-free and trapped states. Based on some heuristic and not entirely plausible arguments, wq( is approximately equated to 1/2. The probability of finding a trapped electron at a distance between r and r + dr from the positive ion is given by (crtP dr/v) exp(-crPr/v), where P, is again the probability of finding an... 

In an alternative model, quantum-mechanical tunneling of the electron is invoked from trap to trap without reference to the quasi-free state. The electron, held in the trap by a potential barrier, may leak through it if a state of matching... 

The electron in a condensed medium is never entirely free, being in constant interaction with the molecules. It is designated quasi-jree when its wave function is delocalized and extends over the medium geometry. Such quasi-free electrons do... 

The more incisive calculation of Springett, et al., (1968) allows the trapped electron wave function to penetrate into the liquid a little, which results in a somewhat modified criterion often quoted as 47r/)y/V02< 0.047 for the stability of the trapped electron. It should be noted that this criterion is also approximate. It predicts correctly the stability of quasi-free electrons in LRGs and the stability of trapped electrons in liquid 3He, 4He, H2, and D2, but not so correctly the stability of delocalized electrons in liquid hydrocarbons (Jortner, 1970). The computed cavity radii are 1.7 nm in 4He at 3 K, 1.1 nm in H2 at 19 K, and 0.75 nm in Ne at 25 K (Davis and Brown, 1975). The calculated cavity radius in liquid He agrees well with the experimental value obtained from mobility measurements using the Stokes equation p = eMriRr], with perfect slip condition, where TJ is liquid viscosity (see Jortner, 1970). Stokes equation is based on fluid dynamics. It predicts the constancy of the product Jit rj, which apparently holds for liquid He but is not expected to be true in general. 

In high-mobility liquids, the quasi-free electron is often visualized as having an effective mass m different fron the usual electron mass m. It arises due to multiple scattering of the electron while the mean free path remains long. The ratio of mean acceleration to an external force can be defined as the inverse effective mass. Often, the effective mass is equated to the electron mass m when its value is unknown and difficult to determine. In LRGs values of mVm 0.3 to 0.5 have been estimated (Asaf and Steinberger,1974). Ascarelli (1986) uses mVm = 0.27 in LXe and a density-dependent value in LAr. 

Schmidt (1976) has given a classical model for the field dependence of quasi-free electron mobility that predicts p(E) in the high-field limit. At any... 

It is clear that in low- and intermediate-mobility liquids Xt Xf and P x/xt. If the trapped electron energy is lower than VQ, the smallest energy of quasi-free electrons, by an amount eQ, the binding energy in the trap, then one gets approximately Tt = ktf-1 = v 1exp(e T). In a classical activation process, 0 is an activation energy and V would correspond to vibrational frequency in the trap. However, these associations are not precise, because of the stated... 

Equation (10.6) for the mobility in the two-state model implicitly assumes that the electron lifetime in the quasi-free state is much greater than the velocity relaxation (or autocorrelation) time, so that a stationary drift velocity can occur in the quasi-free state in the presence of an external field. This point was first raised by Schmidt (1977), but no modification of the two-state model was proposed until recently. Mozumder (1993) introduced the quasi-ballistic model to correct for the competition between trapping and velocity randomization in the quasi-free state. 

The new model is called quasi-ballistic because the electron motion in the quasi-free state is partly ballistic—that is, not fully diffusive, due to fast trapping. It is intended to be applied to low- and intermediate-mobility liquids, where the mobility in the trapped state is negligible. According to this, the mean... 

As for the trap density, a lower limit of 1018 cm-3 has been taken, based on the fall of trapped electron yield in hydrocarbon glasses at a dose 1020 eV/gm (Willard, 1975). An upper limit of trap density 1020 may be argued on the basis of Berlin and Schiller s (1987) finding that a quasi-free electron interacts with... 

In applying the quasi-ballistic model to electron scavenging, Mozumder (1995b) makes the plausible assumption that the electron reacts with the scavenger only in the quasi-free state with a specific rate fesf. Denoting the existence... 

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