Mean inner potential

Table 2. Values of the Electrostatic potentials (V) at the nuclear positions in crystals and free atoms and corresponding mean inner potentials ((po) ...

In studies on electron diffraction, however,-yet another potential is considered, called the mean inner potential . The metal may be considered as a potential box, the distance of the uppermost level below the potential just outside the metal, V, the electrostatic potential already referred to, being equal to x> the thermionic work function. The lowest level (at low temperatures) is, according to the new statistics, a distance below the upper equal to [/1], where... 

After passing a sample with a thickness t, the phase of the transmitted beam Acp is shifted with respect to the vacuum. If a charged dislocation is present, the mean inner potential Vo of the material is locally modified by... 

If k is the wavevector of the incident wave in vacuum, then K is the magnitude of k after correction for the mean inner potential Uq of the crystal. The fact that K is different from k means that the crystal has a refractive index, and it is clear that the mean refractive index n must be related to Uq. Direct measurements show that n is of the order of 1 x 10 In the dynamical theory of electron diffraction, which we develop in this chapter, we formally make all the Fourier components Kg complex quantities, that is, we replace Kg by Vg + iV. The full physical significance of the procedure will become clear in due course, but for the moment it will be helpful to consider the consequences of making Kq complex. If Kq is complex, then the mean refractive index n must also be complex, and so we write n = n +in". 

The first and fourth terms in the dispersion equations represent the difference between the square of the wavevectors Kq and Kg inside the crystal and the square K, which is the vacuum value k corrected for the mean inner potential. If there is no difference, there is no unique solution. Thus, the refractive index for Kq and Kg waves must be different from the average refractive index. This is the crux of the dynamical theory. 

If the construction is now performed taking into account the mean inner potential of the crystal, then the center Q of the sphere is determined such that the magnitude of the wavevectors to 0(000) and to G khl) is K. It can be seen from Eq. (4.15) that K>k, so the point Q is somewhere to the left of the Laue point L in Figure 4.1. The Ewald sphere construction with Q as center and radius K is shown in Figure 4.2. The diagram also includes the point L, but the distance between Q and L has been greatly... 

K" represents the attenuation due to the imaginary part of the mean inner potential. As there is always attenuation and not amplification, K"> o. The attenuation of the wave with wavevector Kq is given by Kg. 

The selective attenuation of waves originating on different branches of the dispersion surface can thus be incorporated into the expressions for /q and /g, Eqs. (4.55) and (4.58), by making 1/tg and a complex. However, a uniform absorption associated with the mean inner potential C/q (discussed in Section 4.2) must also be included. This absorption is described by K" which, from Eq. (4.70), is given by... 

Inclusions of the order of 2 nm diameter or larger can also be made visible by a phase contrast mechanism. When a crystal is oriented so that no strong, low-order diffracted beeuns are operating, all the Fourier coefficients Kg of the potential are negligibly small except the mean inner potential Kq, which is effectively the refractive index of the crystal for electrons (as explained in Sections 4.1 and 4.2). If the mean inner potential 0i of an inclusion is different from the mean inner potential Kq of the matrix, then the inclusion can be considered as a phase object. In the light microscope, a phase object is usually barely visible at exact focus but if the objective lens is slightly defocused, it will be seen with high contrast. 

The weak phase object approximation (eq. (1.6)) is satisfied when Fp(= FoAz) <1.2 X 10 VA for E = 200 kV. This means Az < 120 40 A since Fq (mean inner potential) = 10 30 V for most inorganic crystals. In reality, however, structure images are obtained mostly for thickness between 15 and 50 A, depending on the material, crystal structure and orientation, and accelerating voltage. That is to say, the condition for the weak phase object is not satisfied in most actual cases. 

Figure 14.5 The mean inner potential of Au NPs as a function of the height, o, spherical Au NPs on activated carbon , the line profile of the phase shift by Au NPs was linear fitted from 3D model , top half of the truncated octahedron of Au on Ti02 , spherical Au NPs on Ti02 [29]. Shaded area with dotted lines shows calculated values for bulk Au. Shaded area without lines shows experimental values for bulk Au.

Figure 14.7 Reprinted from Carter, C.B., Fdll, H., Ast, D.G., and Sass, S.L. (1981) Electron diffraction and microscopy studies of the structure of grain boundaries in silicon, Phil. Mag. A 43, 441. With permission from Taylor and Francis, http //www.tandf.co.uk/journals Figure 14.8 Reprinted from Sass, S.L. and Riihle, M. (1984) The detection of the change in mean inner potential at dislocations in grain-boundaries in NiO, Phil. Mag. 49, 759, with permission from Taylor and Francis. http //www.tandf.co.uk/journals...

The phase shift due to the magnetic induction is about 10 times less than the one corresponding to the mean inner potential and its value across the wire (0.03 rad) eorresponds to a nanowire fiilly magnetized along its axis with B = 1.7 tesla. The magnetization Ues along the wire axis which is the Co-hcp c axis (Fig. 17.11(b)) while it presents a peeuUar vortex structure around the junction of the Co wires in Fig. 17.11(d). 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

Strictly speaking the exponents should not contain the inner potential 4> but the so-called potential of mean force, but this subtlety is only important at high electrolyte concentrations and high potentials, where other weaknesses of this theory also become important. Substituting Eqs. (3.3) and (3.2) into Eq. (3.1) gives ... 

All quantities in Eq. (12.6) are measurable The concentrations can be determined by titration, and the combination of chemical potentials in the exponent is the standard Gibbs energy of transfer of the salt, which is measurable, just like the mean ionic activity coefficients, because they refer to an uncharged species. In contrast, the difference in the inner potential is not measurable, and neither are the individual ionic chemical potentials and activity coefficients that appear on the right-hand side of Eq. (12.3). 

As explained in Section 6.3.11, the inner potential difference—A( )—seems to encompass all the sources of potential differences across an electrified interface—Ax and A jf—and therefore it can be considered as a total (or absolute ) potential across the electrode/electrolyte interface. However, is the inner potential apractical potential First, the inner potential cannot be experimentally measured (Section 6.3.11). Second, its zero point or reference state is an electron at rest at infinite separation from all charges (Sections 6.3.6 and 6.3.8), a reference state impossible to reach experimentally. Third, it involves the electrostatic potential within the interior of the phase relative to the uncharged infinity, but it does not include any term describing the interactions of the electron when it is inside the conducting electrode. Thus, going back to the question posed before, the inner potential can be considered as a kind of absolute potential, but it is not useful in practical experiments. Separation of its components, A% and A f, helped in understanding the nature of the potential drop across the metal/solution interface, but it failed when we tried to measure it and use it to predict, for example, the direction of reactions. Does this mean then that the electrochemist is defeated and unable to obtain absolute potentials of electrodes ... 

The inner layer is a concept within the framework of the classical Gouy-Chap-man-Stern model of the double layer [57]. Recent statistical-mechanical treatments of electrical double layers taking account of solvent dipoles has revealed a microscopic structure of inner layer" and other intriguing features, including pronounced oscillation of the mean electrostatic potential in the vicinity of the interface and its insensitivity at the interface to changes in the salt concentration [65-69]. 

Sometimes it is useful to break the inner potential into two components called the outer (or Volta) potential, if/, and the surface potential, x- Thus, (f) = if/ + x- There is a large, detailed literature on the establishment, the meaning, and the measurement of interfacial potential differences and their components. See references 23-26. Although silver chloride is a separate phase, it does not contribute to the cell potential, because it does not physically separate silver from the electrolyte. In fact, it need not even be present one merely requires a solution saturated in silver chloride to measure the same cell potential. 

Hence, the Fermi level in the solution is a quantity that is not directly applicable to the semiconductor-solution interface at this time, though it may be possible in future research to evaluate it by means of measurements of individual inner potential. 

In the previous section, we used electrode potential without any strict definition. However, it should be defined in advance, in order to better understand it. Originally, the electrode potential was closely related to Gibbs free energy change of a reaction at the interface (electrode). It is also related to chemical affinity. In this electrochemical case, we might say that the electrode reaction should be related to electrochemical affinity. So what is electrochemical affinity To answer this question and others would lead to a complete understanding of what electrode potential is and what it really means. We would like to introduce the concept of inner potential for the explanation. 

A chemical equilibrium implies that the chemical potential of a species is the same in all phases. As regards electrons in a system, this also means that their chemical potentials (or electrochemical potentials if the inner potential can not be neglected) must be equal, although they may have different energies. Thus the chemical potential of the electrons in general, pe. must be equal to the ehemieal potential of valence eleetrons, eonduetion electrons, etc.. 

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