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Next: 7. Description of the Physical Properties Up: An Introduction to Crystal Physics (Description Previous: 5. The Value of a Physical

6. Higher Order Effects

The relationship between two physical properties is not necessarily linear. The relation between the dependent and independent physical variables can be often expressed with Taylor expansion. Thus for instance the electric field dependence of the electric polarization in a strong field is described with expansion in power series

$\begin{displaymath} P_i = \chi^0_{ij}E_j + \chi_{ijk}E_jE_k + \chi_{ijkl}E_jE_kE_l + \cdots \end{displaymath}$ (6.1)

where the tensor [ $\chi^0_{ij}$ ] describes the linear or first-order effect, the tensor [ $\chi_{ijk}$ ] stands for the second-order effect and so on. (The second-order effect explains the generation of double frequency light waves whenever light passes through crystals without symmetry centres).

There is some freedom in deciding the order of an effect, which depends upon the aspect the effect is studied. Thus in the above example if instead of the tensor [ $\chi^0_{ij}$ ] the differential quotient with respect to the electric field of the electric polarization vector (i.e. the [ $\chi_{ij}$ ] tensor) is considered as dielectric susceptibility, the previously second-order effect may be regarded as a first-order effect, which describes the electric field dependence of the dielectric susceptibility. This becomes obvious from the equation

$\begin{displaymath} \frac{{\partial}P_i}{{\partial}E_j} = \chi_{ij} = \chi^0_{ij} + \chi_{ijk}E_k + \chi_{ijkl}E_kE_l. \end{displaymath}$ (6.2)

The dependence of the electrical resistivity on the magnetic field is similarly

$\begin{displaymath} \rho_{ik}(H) = \rho^0_{ik} + \rho_{ikl}H_l + \rho_{iklm}H_lH_m + \rho_{iklmn}H_lH_mH_n \end{displaymath}$ (6.3)

where the [ $\rho^0_{ik}$ ] tensor represents the electrical resistivity in the absence of a magnetic field; the tensor [ $\rho_{iklm}$ ] describes the change of electrical resistivity due to a magnetic field, and the tensors [ $\rho_{ikl}$ ]and [ $\rho_{iklmn}$ ] refer to the first- and second-order Hall effects.

Finally it should be noted that in the theory of elasticity the coefficients of the second-order effect are called third-order elastic stiffnesses, because it is more suitable to start the discussion of the nonlinear stress-deformation relationship with the energy function whose third-order derivatives supply the coefficients of the principally second-order effects.

Next: 7. Description of the Physical Properties Up: An Introduction to Crystal Physics (Description Previous: 5. The Value of a Physical

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