Mayer

Chu W-K, Mayer J W and Nicolet M-A 1978 Backscattering Spectrometry (New York Academic)... 

The nth virial coefficient can be written as sums of products of Mayer/-fiinctions integrated over the coordinates and orientations of n particles. The third virial coefficient for spherically syimnetric potentials is... 

In the graphical representation of the integral shown above, a line represents the Mayer function f r.p between two particles and j. The coordinates are represented by open circles that are labelled, unless it is integrated over the volume of the system, when the circle representing it is blackened and the label erased. The black circle in the above graph represents an integration over the coordinates of particle 3, and is not labelled. The coefficient of is the sum of tln-ee tenns represented graphically as... 

For hard spheres, the coefficients are independent of temperature because the Mayer/-fiinctions, in tenns of which they can be expressed, are temperature independent. The calculation of the leading temiy fy) is simple, but the detennination of the remaining tenns increases in complexify for larger n. Recalling that the Mayer /-fiinction for hard spheres of diameter a is -1 when r < a, and zero otherwise, it follows thaty/r, 7) is zero for r > 2a. For r < 2a, it is just the overlap volume of two spheres of radii 2a and a sunple calculation shows tliat... 

The principle ideas and main results of tlie theory at the level of the second virial coefficient are presented below. The Mayer/-function for the solute pair potential can be written as the sum of temis ... 

Coulomb potential multiplied by -p. The graphical representation of the virial coefficients in temis of Mayer/ -bonds can now be replaced by an expansion in temis ofy bonds and Coulomb bonds ). 

Going beyond die limiting law it is found that the modified (or renonnalized) virial coefficients in Mayer s theory of electrolytes are fiinctions of the concentration through their dependence on k. The ionic second virial coefficient is given by [62]... 

Weak electrolytes in which dimerization (as opposed to ion pairing) is the result of chemical bonding between oppositely charged ions have been studied using a sticky electrolyte model (SEM). In this model, a delta fiinction interaction is introduced in the Mayer/-fiinction for the oppositely charged ions at a distance L = a, where a is the hard sphere diameter. The delta fiinction mimics bonding and tire Mayer /-function... 

By integrating over the hard cores in the SL expansion and collecting tenns it is easily shown this expansion may be viewed as a correction to the MS approximation which still lacks the complete second virial coefficient. Since the MS approximation has a simple analytic fomi within an accuracy comparable to the Pade (SL6(P)) approximation it may be more convenient to consider the union of the MS approximation with Mayer theory. Systematic improvements to the MS approxunation for the free energy were used to detemiine... 

Mayer J G and Mayer M G 1940 Statistical Mechanics (New York Wiley)... 

Mayer J 1950 Theory of ionic solutions J. Chem. Phys. 18 1426... 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

The McMillan-Mayer theory allows us to develop a fomialism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

The McMillan-Mayer theory offers the most usefiil starting point for an elementary theory of ionic interactions, since at high dilution we can incorporate all ion-solvent interactions into a limitmg chemical potential, and deviations from solution ideality can then be explicitly coimected with ion-ion interactions only. Furthemiore, we may assume that, at high dilution, the interaction energy between two ions (assuming only two are present in the solution) will be of the fomi... 

From these results, the thennodynamic properties of the solutions may be obtamed within the McMillan-Mayer approximation i.e. treating the dilute solution as a quasi-ideal gas, and looking at deviations from this model solely in temis of ion-ion interactions, we have... 

All the previous discussion in this chapter has been concerned with absorption or emission of a single photon. However, it is possible for an atom or molecule to absorb two or more photons simultaneously from a light beam to produce an excited state whose energy is the sum of the energies of the photons absorbed. This can happen even when there is no intemrediate stationary state of the system at the energy of one of the photons. The possibility was first demonstrated theoretically by Maria Goppert-Mayer in 1931 [29], but experimental observations had to await the development of the laser. Multiphoton spectroscopy is now a iisefiil technique [30, 31]. 

Gdppert-Mayer M 1931 Concerning elementary processes with two quanta Ann. Phys. 9 273-94... 

Hoffmann H, Mayer U and Krisohanitz A 1995 Struoture of aikyisiioxane monoiayers on siiioon surfaoes investigated by externai refieotion infrared speotrosoopy Langmuir 1304-12... 

The 4>(r) can be expressed in various fomis [30], e.g. the Bolir, Bom-Mayer, Thomas-Femii-Firsov and Moliere models, as well as the universal potential of Ziegler, Biersack and Littmark known as the ZBL potential [31]. The ZBL potential fiinction is expressed as... 

Feldman L C and Mayer J W 1986 Fundamentals of Surface and Thin Film Analysis (Amsterdam Elsevier)... 

Csepregi L, Kennedy E F, Gallagher T J, Mayer J W and Sigmon T W 1978 Substrate orientation dependence of the epitaxial regrowth rate from Si-implanted amorphous Si J. Appi. Phys. 49 3906... 

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