# Optical response functions¶

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The optical response functions couple an external electric field, $$E_\mathrm{ext}$$, with the internal electric field arising from the response of the crystal. It is convenient to introduce the displacement field, $$\bf{D}$$, which determines the electric field from external charges, $$\rho_\mathrm{ext}$$. The displacement field is related to the internal electric field, $$\bf{E}$$, through the polarization, $$\bf{P}$$:

$\bf {D} = \epsilon_0 \bf {E} + {\bf P},$

where $${\bf -P}$$ is the electric field from the internal charges, $$\rho_{int}$$.

The polarization is related to the dipole moment of the material, $${\bf p}$$,

${\bf P } \equiv {\bf p }/V,$

where $$V$$ is the volume of the material.

## Linear response coefficients¶

The linear response coefficients $$\chi$$ (susceptibility), $$\epsilon_r$$ (dielectric constant), and $$\alpha$$ (polarizability) relate the electrodynamic quantities outlined above to each other:

${\bf P } = \epsilon_0 \chi {\bf E },$${\bf D } = \epsilon_r \epsilon_0 {\bf E },$${\bf p } = \alpha {\bf E }.$

### Optical conductivity¶

For a perturbation $${\bf E}({\bf r}) = {\bf E}_0 \exp({\rm i}{\bf q} \cdot {\bf r} )$$, the linear response current in the long wave-length limit ($$q \ll 1/a$$ , where $$a$$ is the lattice constant) is given by [Wis63]:

${\bf j}({\bf r}, \omega) = \sigma(q=0, \omega) {\bf E}({\bf r}, \omega),$

where $$\sigma$$ is the optical conductivity.

### Units¶

The unit of the linear response coefficients are:

Coefficient Unit
$$\alpha$$ C2/Nm5
$$\epsilon_r$$ 1
$$\chi$$ 1
$$\sigma$$ C2/Nsm2

### Relation between the linear response coefficients¶

All the response coefficients follow from the susceptibility, $$\chi$$:

$\epsilon_r(\omega) = (1 + \chi(\omega) ),$$\alpha(\omega) = V \epsilon_0 \chi(\omega),$$\sigma(\omega) = - i \omega \epsilon_0 \chi(\omega).$

The derivation of the last relation can be found in [Mar04].

 [Mar04] Richard M. Martin. Electronic structure: Basic theory and practical methods. Cambridge University Press, New York, 2004.
 [Wis63] N. Wiser. Dielectric constant with local field effects included. Phys. Rev., 129:62–69, Jan 1963. doi:10.1103/PhysRev.129.62.