X-ray Scattering at Grazing Incidence

Detlef Smilgies


Introduction

The interaction of x-rays with matter is weak. Absorption lengths are typically on the order of 1-10 mm. Typical length scales for surface and interface studies are 2-20 Å and 20-2000 Å for thin films. So why would anybody in his/her right mind use x-rays to study interfaces or thin films?  Well, there are some important advantages:


Scattering from surfaces and interfaces

The refractive index for x-rays is given by

n = 1 - δ + iβ
The real part 1-d is given by the high-frequency limit of scattering an electromagnetic wave from matter
n = 1 - ωP2 / ω2
where w and wP are frequency of the x-ray photon and the plasma frequency of the material, respectively. The latter is a function of the average electron density of the material, which in turn depends on the atomic number Z, the atomic weight A, and the mass density r:
ωP2  = const.  Z/A  ρ
For metals, typical plasmon energies are around 10 eV, i.e. due to their finite mass, electrons are way to slow to follow the electric field of an x-ray photon. A resonator far above resonance will actually respond out-of-phase with the excitation, and hence n gets reduced below 1.  From the typical plasmon energy we estimate d values should be of to order of 1e-6 for 10 keV photons.

Some representative actual values are shown in the following table for 10 keV photons (source: CXRO).
 
Table 1. Refractive properties of selected materials for 10 keV photons
element Z d b ac (deg) kzc (Å-1)
C (diamond) 6 4.6e-6 4.5e-9 0.173 0.0153
Si 14 4.9e-6 7.4e-8 0.180 0.0159
Cu 29 1.6e-5 1.9e-6 0.326 0.0288
Au 79 3.0e-5 2.2e-6 0.443 0.0383


Note that the real part of n is ever so slightly less than one. This means that at grazing incidence angle there will be total external reflection of the x-ray wave. The critical angle can be calculated from Snell's law. The result is

αc = sqrt(2δ)
Representative values for ac are shown above. From the above expression for n in the high-frequency limit we see that d scales with the photon energy E as
δ ~ E-2
This means that ac also depends on energy E or wave length l as
αc ~ E-1 ~ λ
and ac values for other photon energies can be easily derived from Table 1. If we consider the normal component
of the wave vector k at the critical angle
kzc = (2p/λ) sin(αc) » (2p/λ) αc
we obtain a quantity, that  is independent on energy/wave length of the radiation to first order, and only depends on the material.

The imaginary part of n is related to x-ray absorption and will be further analyzed in the next section.


What is the nature of the wave scattered at or below the critical angle? For once, the refracted wave cannot propagate into the material; instead there will be an evanescent wave travelling parallel to the surface, with an exponentially decaying amplitude away toward the interior of the material. Hence the penetration depths of the incident wave is limited to about 50-100 Å. This is important, if the diffuse background from defect scattering in the bulk is an issue, in particular for scattering from a liquid surface or a glass substrate. For scattering from single crystalline substrates often an incident angle of 2-3 ac is chosen, which is more convenient to work with, but still reduces the scattering from bulk defects.

Important applications of scattering at grazing incidence are