Chemistry Reference
In-Depth Information
instantaneous picture of the interference pattern with a graded scale for the
strength
E
int
of the electromagnetic field. This pattern moves from the left to
the right parallel to the surface. In all directions normal to the surface, however,
standing waves can be observed with nodes (zero field strength) and antinodes
(maximum field strength). Parallel to the surface, there are stationary nodal
lines with zero amplitude (gray) that follow one another at a constant distance.
The stationary antinodal lines with extreme amplitudes (crests are white and
troughs are black) are exactly between the nodal lines.
The pattern of course depends on the angle at which the primary beam is
incident. Four examples are demonstrated in Figure 2.8. For
α
=
45
°
, the nodal
lines are the same distance apart as crests and troughs are, so that a checkered
pattern arises (Figure 2.8a). For glancing angles
α
<
45
°
, the nodal lines are
pulled apart while the crests and troughs of the antinodal lines move closer
together. This pattern is formed in X-ray optics because of the small angles at
total reflection (Figure 2.8b). For angles
α
>
45
°
, the nodal lines are compressed
while the crests and troughs move further apart (Figure 2.8c). At normal
incidence, that is,
α
=
90
°
, the crests and troughs are horizontally fixed and the
nodal lines follow one another at a distance of half a wavelength. This pattern
represents the most simple and familiar picture of a standing wave as is known
from light optics (Figure 2.8d).
For a mathematical description of the pattern some assumptions have to
be made. The substrate is first regarded as a totally reflecting mirror with a
reflectivity
R
=
100%. The
xy
plane is taken to be the surface plane; the
z
-axis, a normal of that plane; and
α
, the glancing angle of an X-ray beam
incident from the vacuum and reflected at the substrate. The incident and
reflected beam represent a monochromatic plane wave with wavelength
λ
,
velocity
c
0
, and maximum amplitude
E
0
of the electric field strength. In
front of the substrate, both beams cross at an angle 2
α
. Based on these
conditions, the interference can be described by the electric field strength
E
int
according to
E
int
2
E
0
sin
k
0
z
sin
α
φ
cos
k
0
ct
k
0
x
cos
α
(2.11)
where
k
0
=
2
π
/
λ
is the wave number;
t
, the time;
φ
, a fixed phase difference
between incoming and reflected waves; and
z
, the height above the
xy
plane [2].
The formula, which is independent of
y
, represents a standing wave for any
fixed
x
value but a propagating wave for any fixed
z
value. Its amplitude is 2
E
0
sin (
k
0
z
sin
α
+
φ
), which periodically varies with height
z
due to the sine factor.
There are nodal and antinodal planes parallel to the surface where the sine
factor becomes zero or unity, respectively. These planes follow one another at
a distance or period
a
vertical
, normal to the surface plane:
λ
2 sin
α
hc
0
E
1
2 sin
α
a
vertical
(2.12a)
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