Chemistry Reference
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φ(
)
a simple box model of
x
can be used for the central interval of the object plane
[
x
min
,
x
max
]
φ(
)
is non-zero. To simplify matters the phase
contrast is a constant value given by the the electron density contrast of the oil and
the surrounding water. This simplification holds for planar membranes which have
not thinned yet and contain an interlayer of solvent/oil.
To obtain the complete field
E
, where the phase shift
x
(
x
)
=
E
c
(
x
)
+
E
l
(
x
)
+
E
r
(
x
)
, the fields
E
l
(
x
)
and
E
r
(
due to the source points in the left and right half-planes (bordering the
object interval
x
)
in the sample plane) have to be added. For these regions
constant (in practice often zero) relative phase shifts
[
x
min
,
x
max
]
φ
r
can be assumed.
From the above equations we see that the intensity will increase as the wavelength
φ
l
and
λ
this improvement will be
compensated by less interaction between photons and the sample with higher photon
energies. Regarding
z
, the fringe distance will shrink on reduction of the propagation
distance and finally lead to a loss in resolution. Since the energy is mostly fixed at
synchrotron beams, we can overcome the limits in resolution for small
z
by using a
divergent beam geometry.
and the propagation distance
z
decrease. In terms of
λ
4.3.2 Divergent Beam Imaging
In this work we apply divergent beam propagation imaging to the presented
membrane system. By using the geometry of a divergent beam, yielding a magnified
phase contrast image of the object [
26
,
27
], one can compensate for wavelength and
propagation distance dependent effects. This leads to an increasing phase contrast at
the expense of fringe distance and, consequently, less resolution, as discussed before.
Thus, it allows one to circumvent the limited resolution, which is mainly defined by
the detector and is, in the best case, in the range of 1
m [diameter of the point
spread function (PSF)] [
16
]. Especially in the case of Fresnel fringes originating
from the bilayer, the resolution of high spatial frequencies at high diffraction angles
is increased. Thus, the propagation distance can be reduced without further loss in
resolution. It was shown [
28
] that a divergent beam experiment is equivalent to a
parallel beam geometry when introducing an effective pixel size
p
eff
, which is the
physical pixel size
p
s
divided by the magnification
M
. Furthermore, the propagation
distance
z
is replaced by an effective propagation distance
z
eff
.
M
and
z
eff
are defined
as follows [
28
,
29
]:
µ
z
1
+
z
2
z
1
z
2
z
1
+
M
=
,
z
eff
=
(4.6)
z
1
z
2
where
z
1
is the distance from the focal point of the beam to the sample and
z
2
is
the distance from the sample to the detector as shown in Fig.
4.1
. Experimentally, a
divergent beam geometry can be achieved in different ways. It has been described
for hard X-ray synchrotron radiation using wave guides (WG) [
28
-
30
], compound
refractive lenses (CRL) [
31
-
33
] and Kirkpatrick-Baez-mirrors (KB) [
26
,
27
]. For
