#### Optics with X-rays

The energy of x-rays is much larger than the energy needed for ionisation of light elements.
Thus one can describe the interaction of x-ray radiation with matter with the Drude theory of the
free electron gas and the reflexion of x-rays by the classical Fresnel reflectiviy of interfaces between
media with different index of refraction. Since the frequency of x-ray is much larger than the plasma frequency
one gets a simple equation describing the index of refraction:

An index of refraction smaller than one leads to total external reflexion at angles lower than
the critical angle a_{c}. The Fresnel reflectivity R_{F} of a simple
interface decreases strongly at angles a larger than the critcal angle
R_{F} ~ (a_{c}/2a)^{4}.
Therefore it is usual to normalize the reflectivity R a complex layer by the Fresnel reflectivity of an idealized smooth interface.

#### Measuring Setup

x-ray reflexion of thin films on solid support as well as of Langmuir monolayers at the air water
interface is measured at incident angles between 0.1° and 5°. This leads to a variation of the horizontal wave vector transfer of
0.01 up to 0.65Å^{-1}. Liquid interfaces cannot be tilted and should not be moved (excitation of waves), so in our setup
the sample (i.e. the langmuir trough) is fixed. Angle of incidence and of evanescence are determined by tilting and lifting
of two slit systems S1 & S2 and S3 & S4. In front of the detector
Cu K_{a} radiation is selected by a graphite crystal and a monochromator
slit S5.

#### Interpretation

The little difference d ~ 10^{-5} to 1 in the refractive index is proportional
to the electron density r. The smallness of both, d and the used
range of angles allow some approximations, which lead to a simple relation between normalized electron density and normalized
reflectivity.

The normalized reflexion is some kind of fourier transform of the electron density ( the absolute square of its first
deriative, to be exact). Yet
taking the absolute square, all phase information ist lost and a retransformation
is impossible- *problem of inversion*. So one has to guess a model of the system and to adjust
its parameters, until the calcultated reflexion matches the measured.