Mathematical Methods for Physics and Engineering - Matematica.NET

4.9 HINTS AND ANSWERSfind a closed-form expression for α, the Madelung constant for this (unrealistic)lattice.4.35 One of the factors contributing to the high relative permittivity of water to staticelectric fields is the permanent electric dipole moment, p, of the water molecule.In an external field E the dipoles tend to line up with the field, but they do notdo so completely because of thermal agitation corresponding to the temperature,T , of the water. A classical (non-quantum) calculation using the Boltzmanndistribution shows that the average polarisability per molecule, α, isgivenbyα = p E (coth x − x−1 ),where x = pE/(kT) andk is the Boltzmann constant.At ordinary temperatures, even with high field strengths (10 4 Vm −1 or more),x ≪ 1. By making suitable series expansions of the hyperbolic functions involved,show that α = p 2 /(3kT) to an accuracy of about one part in 15x −2 .4.36 In quantum theory, a certain method (the Born approximation) gives the (socalled)amplitude f(θ) for the scattering of a particle of mass m through an angleθ by a uniform potential well of depth V 0 and radius b (i.e. the potential energyof the particle is −V 0 within a sphere of radius b and zero elsewhere) asf(θ) = 2mV 0(sin Kb − Kbcos Kb). 2 K3 Here is the Planck constant divided by 2π, the energy of the particle is 2 k 2 /(2m)and K is 2k sin(θ/2).Use l’Hôpital’s rule to evaluate the amplitude at low energies, i.e. when k andhence K tend to zero, and so determine the low-energy total cross-section.[ Note: the differential cross-section is given by |f(θ)| 2 and the total crosssectionby the integral of this over all solid angles, i.e. 2π ∫ π0 |f(θ)|2 sin θdθ.]4.9 Hints and answers4.1 Write as 2( ∑ 1000n=1 n − ∑ 499n=1n) = 751500.4.3 Divergent for r ≤ 1; convergent for r ≥ 2.4.5 (a) ln(N + 1), divergent; (b) 1 [1 − 3 (−2)n ], oscillates infinitely; (c) Add 1 S 3 N to theS N series; 3 [1 − 16 (−3)−N ]+ 3 4 N(−3)−N−1 ,convergentto 3 . 164.7 Write the nth term as the difference between two consecutive values of a partialfractionfunction of n. Thesumequals 1 (1 − 2 N−2 ).4.9 Sum the geometric series with rth term exp[i(θ + rα)]. Its real part is{cos θ − cos [(n +1)α + θ] − cos(θ − α)+cos(θ + nα)} /4sin 2 (α/2),which can be reduced to the given answer.4.11 (a) −1 ≤ x