Biomedical Engineering Reference
In-Depth Information
Table 2.1 Characteristics of scattered wavelengths.
q range (nm
−
1
)
Radiation
Incident wavelength (nm)
Spatial resolution (2
π
/q) (nm)
5 x 10
-
4
-
3x10
-
2
Visible light
400
-
600
200
-
10 000
5 x 10
-
2
X-rays
0.1
-
0.4
0.1
-
150
-
50
5 x 10
-
2
Neutrons
0.1
-
3
0.01
-
150
-
500
different shapes, which is the usual way of identifying these, in particular when scattering
vectors such as q > R
g
−
1
can be used.
In (
2.9
), the scattered intensity decays exponentially versus the square of the scattering
vector. The characteristic length R
g
such that q
1/R
g
determines the limit of validity of
the Guinier approximation. This means that the range of scattering vectors to be used for
characterizing the size of the particles has to be selected according to the expected
average size.
As an example, if R
g
= 10 nm, the range of q vectors to be chosen is around q
≈
0.1 nm
−
1
.
Considering
Figure 2.2
, when using classical light scattering equipment the usual range
of
≈
= 633 nm, this gives a q range of
7x10
-
3
< q <2.5x10
−
2
nm
−
1
, which is obviously too small. In this case light scattering is
not the best technique. Instead, using small-angle X-ray scattering (SAXS) with a CuK
α
X-ray source, the incident wavelength is
θ
is between 30° and 150°, so with a He-Ne laser,
λ
= 0.154 nm; with a specially designed small-
angle collimation camera, the scattering angle 0.65 < 2
λ
<5x10
-
3
rad, so the q range in this
case is 0.02 < q <0.2nm
−
1
, which is a much better choice.
Table 2.1
summarizes the ranges of scattering vectors for various techniques. It also
indicates the spatial resolution, 2
θ
π
/q, meaning that any detail on the structure within a
distance smaller than 2
π
/q cannot be observed with this technique.
2.2.3
Effect of particle concentration
Increasing the concentration of particles in solution increases the scattered intensity. The
structure factor re
ects the interactions between particles or the correlation between the
relative positions of a pair of particles within a coordination shell. When dealing with
spherical particles (such as colloidal particles), the function S(q) exhibits a peak whose
position is directly related to the interaction potential between the particles. An example
is shown in
Figure 2.3
. The intensity scattered by a single spherical particle of radius R
and the overall measured intensity are both shown, and the contribution of the factor S(q)
is well identi
ed. These curves correspond to scattering spheres interacting with a hard
core potential with a distance 2R. The maximum in S(q) should correspond approx-
imately to 2
/q=2R when the concentration is rather high. In this example, the distance
between centres of a pair of spheres is larger than 2R, corresponding to a volume fraction
of spheres of
π
= 12.5%.
The effect of concentration on the intensity scattered per particle in a suspension of
spherical particles interacting with a hard core potential is shown in
Figure 2.4
. As can be
ϕ
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