Potential energy of ions

Enhancement of surface diffusion of the growth precursors is considered as one of the beneficial effects of ion bombardment [246,428]. The potential energy of ions, which is released when the ion is neutralized, is typically 10 eV. This energy can be a substantial fraction of the total energy transferred. The release of this ionization energy is sufficient to excite atoms into excited electronic states, thereby weakening their bonds and enhancing their mobilities [429]. 

Now consider ion 2. It wiU be attracted to ions 1 and 3 by the same amount and repelled by 4, although this repulsion will be only half (because the distance is 2r) that of the attraction between 2 and one of its neighbors. Thus, we can conclude that the potential energy of ion 2 must also be negative. 

The process of linearizing the equation for the electric potential it is valid if the potential energy of ions is small compared to their average kinetic energy due to thermal motion. 

The Configuration Coordinate Model. To illustrate how the luminescent center in a phosphor works, a configurational coordinate diagram is used (2) in which the potential energy of the luminescent or activator center is plotted on the vertical axis and the value of a single parameter describing an effective displacement of the ions surrounding the activator, is plotted on the horizontal axis (Fig. 2). At low temperatures, near room... 

Electrostatic Repulsive Forces. As the distance between two approaching particles decreases, their electrical double layers begin to overlap. As a first approximation, the potential energy of the two overlapping double layers is additive, which is a repulsive term since the process increases total energy. Electrostatic repulsion can also be considered as an osmotic force, due to the compression of ions between particles and the tendency of water to flow in to counteract the increased ion concentration. 

To go from experimental observations of solvent effects to an understanding of them requires a conceptual basis that, in one approach, is provided by physical models such as theories of molecular structure or of the liquid state. As a very simple example consider the electrostatic potential energy of a system consisting of two ions of charges Za and Zb in a medium of dielectric constant e. 

Fig. 1.76 Potential energy of an interstitial ion near the metal/oxide interface...

When a molecule is dissociated into a pair of ions, the work done is stored as the mutual potential energy of the ions. In deriving equation (3), we considered a similar process, in which charges of one sign were separated and conveyed across from one condenser plate to the other the work done in this process is stored as the mutual potential energy of... 

In the case of a singly charged atomic ion in aqueous solution we have estimated the mutual potential energy between the ion and an adjacent water molecule when they are of nearly the same size, and have found the value to be about four times as great as the mutual potential energy of two adjacent water molecules. We conclude then that in the vicinity of an atomic ion the water structure will have to build itself round the ion, insofar as this is possible. 

Let us discuss now the conditions required for the electron transfer process. This reaction requires, of course, a suitable electron donor (a species characterized by a low ionization potential) and a proper electron acceptor, e.g., a monomer characterized by a high electron affinity. Furthermore, the nature of the solvent is often critical for such a reaction. The solvation energy of ions contributes substantially to the heat of reaction, hence the reaction might occur in a strong solvating solvent, but its course may be reversed in a poorly solvating medium. A good example of this behavior is provided by the reaction Na -f- naphthalene -> Na+ + naphthalene". This reaction proceeds rapidly in tetrahydrofuran or in dimethoxy... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Our starting point for understanding the interaction between ions in a solid is the expression for the Coulomb potential energy of the interaction of two individual ions (Section A) ... 

FIGURE 2.6 The arrangement used to calculate the potential energy of an ion in a line of alternating cations (red spheres) and anions (green spheres). We concentrate on one ion, the "central ion denoted by the vertical dotted line. 

FIGURE 2.7 The potential energy of an ionic solid, taking into account the coulombic interaction of the ions and the exponential increase in their repulsion when they are in contact. The minimum potential energy is given by the Born-Meyer equation, Eq. 3. 

FIGURE 5.1 The distance dependence of the potential energy of the interaction between ions (red, lowest line), ions and dipoles (brown), stationary dipoles (green), and rotating dipoles (blue, uppermost line). 

The potential energy of the interaction between the full charge of an ion and the two partial charges of a polar molecule is... 

This assumption is no longer valid in its place the wave mechanics provides the simple explanation that the repulsive forces arise from the interpenetration of the atoms. As a simple example, we may consider the hydrogen ion and the chloride ion according to the wave mechanics the potential energy of these two ions at a distance R apart, assuming that no deformation occurs, is6... 

The expression for the potential energy of a potassium ion and a chloride ion, for example, is similar to that of Equation 6, but is still more complicated. 

The potential energy of an ionic crystal (ions of valence z) may be written = —a(e2z2)/i + R), the first term representing the Coulomb energy, and the second the potential of the repulsive forces. Equation 6 suggests a simple form for [Pg.260]

For simplicity we shall assume the fluoride ion to consist of the nucleus, two K electrons very close to it, and eight L21 L22 electrons for as can be seen from the representation of the sodium ion in fig. 3 the Lu electrons show nearly the same distribution along r as the L21 L22 electrons. The potential energy of a hydrogen nucleus at the distance R from the fluorine nucleus is then... 

Expression (8) can be used to calculate the real energy of ions in any solvent, provided the standard potential of the calomel electrode, and the standard potentials of the elements and the A ° P° (Hg, Cl") under study... 

Consider two ions in contact. As they are pulled apart the potential energy of the two ions increases. At some critical point the separation becomes sufficient for a polar solvent molecule to occupy the space between them, which reduces the energy of the system. Further separation increases the energy of the system again. These changes demonstrate that two types of ion-pair exist contact and solvent-separated. 

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