Biomedical Engineering Reference
In-Depth Information
order of the refl ection. The scattered intensity is usually measured as a func-
tion of the scattering vector (or momentum transfer)
Q
, which is related 2
θ
by the relation:
Q
= (4π sin
θ
)/
λ
. Combining this defi nition of
Q
with Bragg's
law we have:
Q
= 2π/
d
. Although Bragg's law does not apply to amorphous
materials that do not have crystallographic planes, the inverse relationship
between the real space (
d
) and the reciprocal space (
Q
) is applicable to
all types of scattering, whether they are Bragg refl ection, X-ray diffrac-
tion (XRD) or diffuse scattering (X-ray scattering). Thus, data at lower
Q
(smaller scattering angles) probe longer length scales in the material. X-ray
scattering can reveal the structure of materials at three different length
scales: at the level of atoms and molecules (
Q
> 0.5 Å
−1
,
d
= 0.1-10 Å), at the
level of aggregates of atoms and molecules (0.005 <
Q
< 0.5 Å
−1
,
d
~ 10-100
nm), and large colloidal length scales (
Q
< 0.005 Å
−1
,
d
~ 0.5 µm) approach-
ing electron microscopy. The three techniques that are used to determine
the structure at these length scales are:
wide-angle X-ray scattering (WAXS);
small-angle X-ray scattering (SAXS);
ultra-small-angle X-ray scattering (USAXS).
Numerous types of X-ray diffractometer are in use, each of them designed
specifi cally for a particular type of measurement. The most common, how-
ever, is the powder diffractometer used with specimens with random ori-
entations, or in instances in which the orientation effects are not pertinent.
The data are collected either in refl ection (parafocus; Bragg-Brentano) or in
transmission geometry. The intensity of the diffracted X-rays is measured as a
function of the scattering angle from about 5° to 135° either by step scanning
with a proportional scintillation detector or scanning with a solid state detec-
tor. One-dimensional (1D) or two-dimensional (2D) detectors are also being
used to increase throughput. A typical WAXS set-up is shown in Fig. 2.4a.
X-ray sources are typically sealed tubes or rotating anodes. The wavelength
of the radiation is ~1 Å. Synchrotron radiation is now widely used for special-
ized studies, as well as for studying small sample, weakly scattering systems
and for time- or temperature-resolved experiments.
A few basic examples of the use of X-ray scattering will be described here.
The availability of synchrotron sources has provided an opportunity to explore
other aspects of phase behavior not discussed here. These include the use of
microbeam X-ray diffraction in which a beam less that 5 µm can be used to
map the phase separation on micrometer length scales. Glancing angle tech-
niques are used to study the changes in the phase behavior as a function of
depth near surfaces. Combining techniques, such as SAXS/WAXS with thermal
and mechanical tests, provides further opportunities to study phase behavior
with heat, and the effect of phase morphology on mechanical properties.
