Distance of closest approach

Examining (3.30) we see that e is a dimensionless energy unit. Physically, s gives a measure of how energetic the collision is and how close the ion gets to the nucleus of the target atom. For example, the value of the Thomas-Fermi screening distance, aTF, for He on Si is 

This large value of e is consistent with the very small value of the collision diameter. For calculation purposes, (3.30) can be simplified and rewritten as 

Nastasi, M., Mayer, J.W., Hirvonen, J.K. Ion-Solid Interactions Fundamentals and Applications, chap. 3. Cambridge University Press, Cambridge (1996) 

Sigmund, P. Collision theory of displacement damage, ion ranges and sputtering. Rev. Roum. Phys. 17, pp. 823, 969 and 1079 (1972) 

As an example, for 1 MeV He (Zi = 2) ions incident on Si (Z2 = 14), the CM energy, E =M2EJ(Mi +M2), equals 875 keV and the distance of closest approach (the collision diameter) equals Z Z2 IE = 4.6 x 10 nm, which has a value much smaller than otf- K is informative to write d relative to the screening distance, otf, where the parameter a Yld is referred to as e, the reduced energy, given by 

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A = /W//Wp, P is impact parameter and Tq is the distance of closest approach (apsis) of the collision pair. The transformations from the CM coordinates (scattering angle y) to the laboratory coordinates with the scattering angle 0 for the primary particle and (]) for the recoiled surface atoms Is given by... 

Figure 8.10 Excluded volume for two spheres (dotted surface) as determined by the distance of closest approach.

If this repulsion exceeds the polari2abihty attraction at the distance of closest approach, determined by vicinal fluid, the particles can be prevented from agglomerating. For example, water with =81, containing NaCl at 7 = mol/m, creates a counterion layer with = 0.15 nm. The maximum... 

Eigure 3 schematically depicts the stmcture of the electrode—solution interface. The inner Helmholtz plane (IHP) refers to the distance of closest approach of specifically adsorbed ions, generally anions to the electrode surface. In aqueous systems, water molecules adsorb onto the electrode surface. 

The outer Helmholtz plane (OHP) refers to the distance of closest approach of non specifically adsorbed ions, generally cations. The interactions of the ions of the OHP with the surface are not specific and have the character of longer range coulombic interactions. Cations that populate the outer Helmholtz plane are usually solvated and are generally larger in size than the anions. 

The scattering cross-section is considerably different from the Rutherford cross-section, because the distance of closest approach, Ri i , is rather large at low energies. Thus, electronic screening of the interaction between the nuclei is important. The screened scattering potential V(r) reads ... 

The Orientation of Water Molecules Adjacent to an Ion. Order and Disorder in the Vicinity of Solute Particles. Coulomb Attraction and Repulsion between Ions. Activity Coefficients. The Distance of Closest Approach. Activity Coefficients of Various Solutes. Forces Superimposed on the Coulomb Forces. 

Values of the distance of closest approach derived from experimental values of the activity coefficients are given in column 2 of Table 40. It will be seen that for the lithium and sodium salts the value is greater than the crystal-lattice spacing (given in column 4) by rather more than 1 angstrom, as is expected. For the salts of cesium, rubidium, and potassium, on the other hand, the distance of closest approach... 

In Table 40 we notice that likewise for rubidium iodide the value adopted for the distance of closest approach is smaller than the lattice spacing in the crystal. 

Strictly speaking, the size of an atom is a rather nebulous concept The electron cloud surrounding the nucleus does not have a sharp boundary. However, a quantity called the atomic radius can be defined and measured, assuming a spherical atom. Ordinarily, the atomic radius is taken to be one half the distance of closest approach between atoms in an elemental substance (Figure 6.12). 

The impact parameter, b, is defined to be the perpendicular distance between the initial relative path (along g) and the line parallel to g through the force center (b would be the distance of closest approach of the particles, if there were no interaction) the initial angular momentum is just pbg. Conservation of angular momentum 6 is thus ... 

This determines the distance of closest approach in terms of the initial relative velocity, the impact parameter, and the dynamical quantities (masses and force law constants). The equation for the orbit of the relative motion is found from the first of Eqs. (1-8) and (1-9), using the identity (fj6) = (drjdd), as follows ... 

The lower limit of the integration assumes that the central ion has a finite size with a, the radius of the ion, representing a distance of closest approach of other ions to the central ion. 

The electrical double layer at Hg, Tl(Ga), In(Ga), and Ga/aliphatic alcohol (MeOH, EtOH) interfaces has been studied by impedance and streaming electrode methods.360,361 In both solvents the value ofis, was independent of cei (0.01 < cucio4 <0.25 M)and v. The Parsons-Zobel plots were linear, with /pz very close to unity. The differential capacity at metal nature, but at a = 0,C,-rises in the order Tl(Ga) < In(Ga) < Ga. Thus, as for other solvents,120 343 the interaction energy of MeOH and EtOH molecules with the surface increases in the given order of metals. The distance of closest approach of solvent molecules and other fundamental characteristics of Ga, In(Ga), Tl(Ga)/MeOH interfaces have been obtained by Emets etal.m... 

C08-0073. Repeat the calculation of Problem 8.37 for K and I, using 500 kJ/mol as the estimated second electron affinity of iodine and assuming no change in distance of closest approach. 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

In relation with these avoided crossings, the radial coupling matrix elements present sharp peaks at respectively 5.4, 6.6, 7.55 and 9.5 a.u. (Fig. 5). We may notice that these radial couplings are almost insensitive to the choice of the origin of electronic coordinates. The most sensitive one is the g23 function at short internuclear distance range, but we may expect weak translational effects for such electron capture processes dominated by collisions at large distance of closest approach. 

Ion pairs can form only when the distance of closest approach, a, of the two ions is less than r . For 1 1 electrolytes for which = 0.357 nm, this condition is not always fulfilled, but for others it is. The fractions of paired ions increase with increasing concentration of solutions. In nonaqueous solutions which have lower values of permittivity e than water, the values of and the fractions of paired ions are higher. In some cases the values of coincide with the statistical mean distance between the ions (i.e., the association of the ions is complete). 

Grahame introdnced the idea that electrostatic and chemical adsorption of ions are different in character. In the former, the adsorption forces are weak, and the ions are not deformed dnring adsorption and continne to participate in thermal motion. Their distance of closest approach to the electrode surface is called the outer Helmholtz plane (coordinate x, potential /2, charge of the diffuse EDL part When the more intense (and localized) chemical forces are operative, the ions are deformed, undergo partial dehydration, and lose mobility. The centers of the specifically adsorbed ions constituting the charge are at the inner Helmholtz plane with the potential /i and coordinate JCj < Xj. 

The successive equilibria are characterized by K12 and K23, respectively, and when Kl2 (often denoted K0) cannot be directly determined, it may be estimated from the Fuoss equation (3), where R is the distance of closest approach of M2+ and 1/ (considered as spherical species) in M OH2 Um x) +, e is the solvent dielectric constant, and zM and zL are the charges of Mm+ and Lx, respectively (20). Frequently, it is only possible to characterize kinetically the second equilibrium of Eq. (2), and the overall equilibrium is then expressed as in Eq. (4) (which is a general expression irrespective of mechanism). Here, the pseudo first-order rate constant for the approach to equilibrium, koba, is given by Eq. (5), in which the first and second terms equate to k( and kh, respectively, when [Lx ] is in great excess over [Mm+]. When K0[LX ] <11, koba - k,K0[Lx ] + k.it and when K0[LX ] > 1, fc0bs + k l. Analogous expressions apply when [Mm+] is in excess. 

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