Chemistry Reference
In-Depth Information
2.1 INTERFERENCEOFX-RAYS
The phenomena of interference result from the superposition of two beams
(double-beam interference) or even more than two beams (multiple-beam
interference). They are generally explained in the wave picture. In the region of
superposition, the resulting wave field can show a pattern with maxima and
minima. These fluctuations will be highly distinct if two superimposing waves
are monochromatic and coherent, that is, if they have the same wavelength and
a fixed phase difference. If the phase difference is an odd multiple of
π
, the
amplitudes are subtracted to a minimum. This kind of interference is called
destructive, and points with minima are called nodes. On the other hand,
interference is called constructive if the phase difference is an even multiple of
π
and the amplitudes sum up to a maximum. The corresponding points are
called antinodes. Nodes and antinodes can be extended to nodal and antinodal
lines or planes, respectively.
A most simple way to produce interference is the superposition of two
beams propagating on the same straight line. This two-beam interference can
be affected by reflection of X-rays at the upper and lower boundaries of a thin
layer deposited on a thick substrate. It can become multiple-beam interference
if a multilayer is chosen with a reflection at various boundaries. Furthermore,
the scattering of X-rays at different atoms of a crystal can lead to multiple-beam
interference. The specific behavior can be explained by reflection at the various
lattice planes, which results in the well-known Bragg's law. This interference of
coherently scattered X-rays is usually called diffraction.
2.1.1Double-BeamInterference
A thin homogeneous layer may be deposited on a thick and flat substrate. If
a monochromatic X-ray beam hits this layer at grazing incidence, double-
beam interference can be observed. Such experiments require a glancing
angle of about 0.1
°
and a layer thickness of 1 nm to 1
μ
m. Furthermore, a
beam width of
∼
10
μ
m and a layer size of
∼
10 mm are required for a wide
superposition.
Figure 2.1 depicts the paths of X-rays within the three media denoted by
subscripts 0, 1, and s: the first medium, from which the X-rays are coming, is
assumed to be vacuum or air, the second medium is a thin plane-parallel layer
of thickness
d
, and the third medium is a thick substrate. Two cases A and B can
be distinguished. In case A, the layer is optically denser than the substrate
(
n
´
´
´
is the real part of the refractive index
n
) (see Section 1.5.1). In
case B, the layer is optically thinner than the substrate (
n
1
>
n
s
where
n
´
´
s
).
In case A, the glancing angle
α
1
is smaller than
α
0
and
α
s
is even smaller than
α
1
, so total reflection is possible at
both
interfaces: air-layer and layer-
substrate. The first happens at or below a small angle determined by
α
01
=
1
<
n
p
2
δ
1
when
α
1
becomes zero or imaginary (see Section 1.5.3). The last happens below
the greater angle
α
0s
=
p
p
2
δ
s
when
α
1
becomes
2(
δ
s
δ
1
) and
α
s
becomes zero
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