A permanent dipole of moment in a local field at temperature has orientational energy
PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electric fields in matter · Electric fields in matter
The Boundary-Value Problem, Ch Molecular polarisability, permanent dipoles and the Clausius-Mossotti relation
Problem
A permanent dipole of moment p in a local field Eloc at temperature T has orientational energy U(θ)=−pEloccosθ, with θ the angle between p and Eloc. Let x≡pEloc/kBT.
((a)) Starting from the Boltzmann weight excosθ averaged over solid angle, show that the mean dipole moment along the field is ⟨pz⟩=pL(x) with L(x)=cothx−1/x (the Langevin function), and that for x≪1 this reduces to ⟨pz⟩=p2Eloc/3kBT.
((b)) A liquid of number density Nv contains molecules with electronic polarisability α and permanent dipole moment p. Using pavg=(α+p2/3kBT)Eloc, P=Nvpavg, and the Lorentz local field Eloc=E+P/3ε0, derive the polar form of the Clausius-Mossotti relation, εr+2εr−1=3ε0Nv(α+3kBTp2).
((c)) Show that this relation gives an infinite εr (a spontaneous, self-sustaining polarisation with no applied field) once the right-hand side reaches 1, and solve for the temperature Tc at which this happens, Tc=9ε0kB(1−3ε0Nvα)p2Nv. Evaluate Tc for liquid water at its actual density, taking Nv=3.34×1028m−3, α/4πε0=1.5×10−30m3 and p=6.2×10−30Cm, and comment on what this predicted value means physically, given that water at room temperature is not spontaneously polarised.