The screened interaction

Having reduced all quantities to their vert.ex irreducible parts we can introduce the screened interat. tion. In eadi order of the loop expaii.sion we have to resum a series of diagrams. For instance, the series of one loop graphs 

In the last line we used Eep (3.22) to introduce the Debye function. We will find in Sect. 5.4 that wo(q) shows all the aspects of screening. Here we want to show that uo(q) cmi be introduced to replace consistently all vertices 

The loop expansion of a vertex irreducible quantity containing external vertices is represented by all vertex irreducible diagrams coiitahiing only screened interactions. Diagrams, in which acliain-like structure not contain- 

Grand Canonical Description of SoJutions at Finite Concentration 

Again thcj extcirnal vertices are lieeded to make the procedure unique. Thus tin result does not apply to the osmotic pressure, or equiva- 

Grand Canonical Description of Solutions at Finite Concentration 

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The interaction between two ions in a metal is screened by the gas of conduction electrons. Although corrections for exchange and correlation are required, the features of the screened interaction are what one would expect from the preceding calculation of the... 

Fig. 5. 13. The screened interaction u0 q)fu0 as function of q2/q 2. uqc 0.001 Ar — 100,1000, oo (from above). The brvken line gives uo(q) = (dilute limit)...

It should be commented that two distinct readjustments of the reference 0 state are implicit in Eqn. (2.2.4), both relating to the change of the lower limit of integration from k to pN. When applied to the integral of the screened interactions [last integral in Eqn. (2.2.4)], this change reflects the adoption of the unperturbed real chain instead of the phantom chain. When applied to the three-body integral [see Eqn. (2.2.7)], it implies a small shift of the 0 temperature from the phantom chain value... 

U J > N, we are in the precollapse region here the effect of the screened interactions and the three-body repulsions vanishes. This may be understood if we remember that the integrals respectively multiplying K and in Eqs. (2.2.4) and (2.2.7) reduce to zero because their lower limit (= J) exceeds the overall chain length. In the strong-compression limit J < N, substitution of Eqn. (2.2.12) into (2.2.5) gives... 

A question may be whether the three-body or the two-body screened interactions are more effective in resisting chain collapse. An analysis based on Eqn. (2.2.22) shows that the screened interactions predominate for lower molecular weights whereas the three-body repulsions prevail for longer chains [52]. The number of chain atoms for which the two components have equal weight in the collapsed state just after transition is... 

Fig. 10. From top to bottom, magnitudes in atomic units imaginary part of the screened interaction Im[ — VF(x, V,

For well-behaved free electron metals these quantities can be calculated using the SRM which, as explained above, has shown to constitute a valid approximation to obtain the energy loss of fast protons. Expressions for the screened interaction within the SRM can be found in Ref. [21]. In the subsequent construction, the electronic surface is located using the standard prescription that consists of placing it at a distance rZ/2 in front... 

In a semi-dilute solution, the chains screen one another. Then, in order to determine the end-to-end distance of a polymer, we can perform an approximate perturbation calculation, replacing the real interaction by the screened interaction. In the following, we reproduce a calculation first made by Edwards.48 It consists in determining the mean square end-to-end distance by perturbation to first order with respect to the screened interaction b(q) or more precisely its approximate value (13.2.143)... 

Fig. 13.24. Size of a polymer in solution. First-order diagram contributing to (fc, — k 0,S S). The interaction line is made of small crosses in order to symbolize the screened interaction b q).

Screening of electronic interactions can be qualitatively understood, but hardly subject to numerical estimates. The difficulty arises from the inhomogeneous nature of the medium, since the reduced interaction must be described at distances comparable to chemical bonds. Neither is it sufficient to consider only local effects of screening, nor to screen independently the various wavelengths in the Fourier transform of the coulomb interaction. Hubbard formulated an integral equation for the screened interaction, but only very approximate solutions seems to be feasible. We will demonstrate that some numerical evidence supports the determination of interaction integrals based on screening theory. 

We now have equations of motion for the one- and two-particle Green s functions. They depend on the Hartree-exchange-correlation self-energy. Its Hartree part is trivial, but a practical way of calculating its exchange-correlation part is needed. Hedin [36] proposed a scheme that yields to a set of coupled equations and allows in principle for the calculation of the exact self-energy. This scheme can be seen as a perturbation theory in terms of the screened interaction W instead of the bare Coulomb interaction v. We show a generalization of this derivation for the case of a nonlocal potential. 

Wpr,qr It has been used that the (Coulomb interaction v and the screened interaction W are spin independent = pw,rui =... 

The Flory approach to this problem is simply to consider that all monomers of the long chain interact between themselves through the screened interaction for the excluded volume parameter v of Chapter I we must substitute the virial coefficient A2s- More precisely we have... 

Fig.11, containing the largest matrix element of the screened interaction as function of ti), tells us that the assumption in COHSEX of an independent real part of W (and ImW =0) is very reasonable to about 8 (eV) from there on we have dynamic screening effects to include, which becomes obvious near the plasmon pole. 

We would like to stress here that most of the above tren s have already been established in our previous diamond calculation. However, there exist also some important differences one example is furnished by the RPA calculation for S, leaving out the e-h attraction in the screened interaction W. 

The calculation of the screened interaction tensors T-" and the dispersion energy AE12 between two macroscopic particles 1 and 2 is greatly simplified if these particles are homogeneous and isotropic. Isotropy entails that the susceptibilities Xj(co) corresponding to the different positions j are scalars which can be separated from the interaction tensors... 

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