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E) Calculate the frequencies xs and xa of the symmetric and antisymmetric normal vibration modes. Evaluate the frequencies xs and xa as Taylor series in 2e 2/( f R 3) and truncate the expansions after second order terms. f) The energy of the complete system of two interacting oscillators can be expressed as U = ± 12" h (xs + xa). Derive an expression for the energy of the isolated oscillators and show that this is decreased by an amount c/R 6 when mutual interaction (bonding) occurs. 12 Calculate how the van der Waals bonding of a molecule depends on its distance, d, from a solid surface.

Depending on the temperature and the relative Si-concentration, the alloy exists either as a homogeneous liquid (– for liquidus), as a homogeneous solid (s for solidus) or as a two-phase Fig. 14. Phase diagram for the continuously miscible alloy Ge/Si. In the range bounded by the liquidus and solidus curves a Ge-rich liquid phase coexists with a Si-rich solid phase 38 2 Structure of Solid Matter system with liquid and solid parts (–+s). The realms of existence are marked by the so-called liquidus and solidus lines.

The solid line corresponds to the limiting case in which an intermediate maximum is not yet present. If an intermediate maximum occurs, the system is unstable against phase separation into two phases, one depleted of NB atoms and one enriched with NB atoms. 5) 42 2 Structure of Solid Matter l HB ˆ l HHB ˆ dGH 1 d GH ˆ d NB N d xB HH …2X17† HH dG 1 dG ˆ X d NB N d xB Hence the system decomposes into two phases for which the slope of the free enthalpy (Fig. 17) is equal. Of all (the infinite number of) points on the curve G(xB) that fulfil the condition of an equal slope the total free enthalpy is minimal for the particular pair of concentrations xHB and xHHB through which a common tangent to G(xB) can be drawn.

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Physics- Electricity and Magnetism by Crowell B.

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