16.1 General Principles 301
1,parity invarianceof the relation (16.1) requires that the parity of the coefficient
tensorChas to be
PC=PaPb.The application of symmetry and parity considerations for the nonlinear electric
susceptibility is treated in Exercise16.1.
Arguments similar to those used for the parity apply to the time reversal behavior.
LetTa,TbandTC, withT..^2 =1, specify the time reversal behavior of the tensors
involved. When time reversal invariance holds true, one has
Tb=TCTa.It is assumed thatTaandTbare given by the physical meaning ofaandb. For time
reversal invariance to be valid, the condition
TC=TaTb, (16.2)has to be fulfilled. When
TC=−TaTb, (16.3)
applies, the relation (16.1) breaks the time reversal invariance, a feature typical for
irreversible processes. In some applications, the tensorCmay not have a unique
time reversal behavior because it contains contributions which are of reversible and
others which are of irreversible character. Of course, this suffices to violate the time
reversible invariance.
Energy considerations may require that certain elements of the tensorChave
to be positive. Thesecond law of thermodynamicsalso imposes conditions on the
sign of coefficients describing irreversible processes. Examples are the Ohm law for
electrical conduction, cf. Sect.14.4.2and the viscosity treated in Sect.16.3.
16.1 Exercise: Nonlinear Electric Susceptibility in a Polar Material
In a medium without hysteresis, the electric polarizationPcan be expanded in powers
of the electric fieldE,cf.(2.59), thus
Pμ=ε 0(
χμν(^1 )Eν+χμνλ(^2 ) EνEλ+...)
.
The second rank tensorχμν(^1 )≡χμνcharacterizes the linear susceptibility. The third
rank tensorχμνλ(^2 ) describes the next higher order contributions toP. Consider a
material whose isotropy is broken by a polar unit vectord. Formulate the expressions
for these tensors which are in accord with the symmetry and with parity conservation.
Consider the casesEparallel and perpendicular tod.