Electron work function, equation

Equation (17) shows the relationship between electrode potentials and electronic energy. The electrode potential is measured by the electron work function of the metal, modified by the contact with the solution (solvent). This establishes a straightforward link, not only conceptually but also experimentally, between electrochemical and UHV situations.6,32 In many cases, electrochemical interfaces are synthesized in UHV conditions55-58 by adding the various components separately, with the aim possibly of disentangling the different contributions. While the situation can be qualitatively reproduced, it has been shown above that there may be quantitative differences that are due to the actual stmctural details. 

Equation (9.2) can be used to calculate the metal s surface potential. The value of the electron work function X can be determined experimentally. The chemical potential of the electrons in the metal can be calculated approximately from equations based on the models in modem theories of metals. The accuracy of such calculations is not very high. The surface potential of mercury determined in this way is roughly -F2.2V. 

This equation states that the ratio of the rates of ion evaporation to atom evaporation is determined by the temperature and by the difference between the electron work function and the ionization potential it depends only indirectly on ipj, J, Np, and N. This conclusion agrees with thermodynamical reasoning. 

The reasoning which led to these conclusions was as follows. From the photographs we deduce whiesh fraction of the total current comes from particular areas, such as the 111 plane, the 100 plane, the 100 region, and the 110 region. We then compute current densities for these areas for each photograph. These current densities are converted to electron work functions ip by means of the well-established Fowler-Nordheim equation. ... 

In a Thought Experiment, the junction is disassembled (Fig. 6.32) by division through the insulator and the two halves are first treated as electrically isolated objects. In the ensuing equations, we use the common symbol for the work function of a material. There are three electron work functions to be considered that of palladium 0pd, that of an arbitrary metal which does not interact with hydrogen 0m, and that of silicon 0su The insulator is considered to be ideal which means that it does not contain mobile charges. Therefore, it does not have a defined Fermi level. Because the two halves are not connected, their energy levels are in an arbitrary undefined position with respect to each other. On the other hand, metal M and palladium (as well as the M and silicon) form ohmic junctions, meaning that the... 

When the electrochemical approach to describing the work function (equation (8.5.1)) is compared to the physical approach (equation (8.5.3)), the following relationships are obtained. The chemical potential of the electron is related to bulk properties so that... 

Brodsky and Frumkin estimated the possible thermoemission rates (from electrodes into aqueous solutions), using the Richardson-Sommerfeld equation and the electronic work function for the metal-water system, determined from photoemission measurements. They have shown that in the range of potentials typical of cathodic reactions in aqueous solutions the emission rates should be very small and need not ensure cathodic evolution of hydrogen via the thermoemission stage, i.e., via the intermediate formation of hydrated electrons. 

The electron work function is related to oxygen pressure A = ylogC0t Substituting this value into the equation for the rate of ethylene oxide formation, we obtain... 

Suppose there is, on the surface, an electronegative species X (or electron-acceptor) like oxygen and whose electron affinity A is greater than the electronic work function O of a semiconductor it will tend to trap an electron coming from the solid, according to the following equation ... 

The origin of the chemical signal can be expressed in thermodynamic terms. At equilibrium the number of moles n of all species and their chemical potentials // in a phase (e.g. in a chemically selective layer) are related through the Gibbs-Duhem equation which says that if a new species enters the organic layer the chemical potentials of all species in that layer must change. These include the change of the electrochemical potential of the electron - the Fermi level and therefore the electron work function. 

In electrode kinetics, as empirically represented by Tafel s equation, a basic feature is the potential-dependence of the reaction rate (current-density). This effect arises in Gurney s representation in a fundamental and general way as the electric potential V, of the electrode metal is changed by AV relative to that of the solution (in practice, measured relative to the potential of a reference electrode at open-circuit), the effective value of the electron work function 4> of the metal is changed according to... 

The first two terms can be interpreted as an electronic work function. Using Equation 13.16, we can write... 

The final technique addressed in this chapter is the measurement of the surface work function, the energy required to remove an electron from a solid. This is one of the oldest surface characterization methods, and certainly the oldest carried out in vacuo since it was first measured by Millikan using the photoelectric effect [4]. The observation of this effect led to the proposal of the Einstein equation ... 

A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

Although we do not wish to imply that equation (6.20) is a general fundamental equation, we are also not aware of any published exceptions to the physical meaning it conveys, i.e. that the enthalpy of adsorption and thus, according to any isotherm, the coverage of an electron acceptor/donor adsorbate decreases/ increases with increasing work function O and thus decreasing Fermi level EF. 

Equation 6.34 expresses the fact that the electrochemical potential of an electron donor (Xj>0) is lowered with anodic (UWr>0) potential or with increasing work function (A<1 0). This favours adsorption. Similarly for an electron acceptor adsorbate (A.j<0) anodic potential (UWR>0) or increasing work function (Aelectrochemical potential of the adsorbate. This hinders adsorption. 

Work functions refer traditionally not to one mole of electrons (with the charge -F) but to one electron with the charge -Q°, and usually are stated in electrical units of electron volts (1 eV = 1.62 X 10 J). In equations of the type of (2.32), therefore, the value of also refers to one electron. 

The kinetic energy kin A (see Fig. 1), however, will be measured by use of an analyzer and may differ at the analyzer from the kinetic energy Ekin the electron had at the sample. Therefore the work function A of the analyzer, which can be considered as constant for a given measuring period, has to be used in equation (2). The work function of the sample has no influence in this simple picture on the kinetic energy measured for an electron excited from the bulk of the sample, because fikin is measured with respect to r/>A of the analyzer. 

Werner Heisenberg stated that the exact location of an electron could not be determined. All measuring technigues would necessarily remove the electron from its normal environment. This uncertainty principle meant that only a population probability could be determined. Otherwise coincidence was the determining factor. Einstein did not want to accept this consequence ("God does not play dice"). Finally, Erwin Schrodinger formulated the electron wave function to describe this population space or probability density. This equation, particularly through the work of Max Born, led to the so-called "orbitals". These have a completely different appearance to the clear orbits of Bohr. 

Most of the quantities appearing in this equation are measurable or calculable. Thus, the potential of zero charge for the cell is measured as 2.51 V, and the work function of mercury as 4.51 V, while the chemical potential of the electron in sodium can be reliably... 

Equation (32) is extremely important to the calculation of work functions. The work required to remove an electron from the interior of the metal to vacuum outside is... 

This helps explain58,73 why simple variational calculations can give good work functions only the electronic tail, spilling over from the positive background, is involved in the surface contribution. Equation (32) also holds72 for the charged interface, in the form... 

Equation (1) suggests that tunnel junctions should be ohmic. This is true only for very small bias. A much better description of the tunneling current results when the effects of barrier shape, changes in barrier with applied potential, and effective mass of the electron are all included. An example of such an improved relationship is given by (2), where / is the current density, a is a unitless parameter used to account empirically for non-rectangular barrier shape and deviations in the effective electron mass, and barrier height given by B = (L + work function of the left-hand metal ... 

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