Box 9.2 (continued)
of which originates from disordered (“incoherent”) scattering of neutrons by the
1
H nuclei of light water. As a result, much better signal-to-noise ratios are
obtained when studying a hydrogenated MP trapped in a contrast-matched
deuterated APol than if the two components are hydrogenated.
– Because X-rays interact with electrons, contrast variation implies chemical or
physical-chemical changes to the sample, such as adding sugar or salts to the
buffers in order to increase their electron density, and it is less widely used.
• Interpretation of scattering curves. In a SAXS or SANS experiment, the intensity
scattered is measured as a function of the angle 2θ between incident and scattered
waves. This intensity depends on the size, shape, composition, etc. of the scattering
particles, on their concentration, but also on the wavelength λ of the radiation. It is useful
to combine the wavelength and the scattering angle into a single parameter, which makes
the analysis independent of λ. Inconveniently, but traditionally, different parameters are
used in SANS and SAXS: SANS data are usually plotted as a function of Q or Q
2
, where
Q ¼ (4π sinθ)/λ, and SAXS data as a function of s or s
2
, with s ¼ (2 sinθ)/λ, Q and
s having dimensions of either nm
À1 or Å
À1 . As to the intensity, I(Q), which, in the case of
independent particles, is proportional to their concentration, C, it is often normalized as
I(Q)/C, where C is expressed in gÁL
À1
, so as to permit easy comparisons between data
sets collected at different concentrations (see e.g. Fig. 9.19A or Fig. 4.14D in Chap. 4).
As in a diffraction pattern, to which scattering curves are closely related, data at high
Q or s contain information about short distances between scattering centers in the
sample, e.g. the distribution of individual atoms within a particle, and data at very
small angles information about long distances, such as the distribution of the particles
in the solution. The presentation of scattering data resorts principally to three types of
graphs, each of which facilitates extracting certain types of information.
– Guinier plots (Guinier and Fournet 1955). A common representation is to plot ln I(Q)
(or ln (I(Q)/C)) vs. Q
2 (or ln I(s) vs. s
2
) (Fig. 9.17). A single look at this plot tells the
experimenter, at a glance, (i) whether the preparation is likely to be homogeneous or not
and, if it is, (ii) what the molecular mass M and radius of gyration R g of the particles are.
M is extracted, knowing the contrast of the particle with the solvent and its concentration
in gÁL
À1
, from the extrapolation of the plot at zero angle (I(0)/C). The slope of the curve
at small angles yields R g , the radius of gyration, which is determined by the spatial
distribution of the scattering centers in the particle (“small angles,” here, means that the
Guinier approximation holds, typically, for QÁR g 1, although for certain shapes it may
extend further in Q).
Deviations from linearity at small angles mean that the preparation is not comprised of
identical, independent particles. Repulsion between particles, for instance, will result in a downward deflection at low angles (Fig. 9.17A; see e.g. Fig. 9.19A and Fig. 11.13 in Chap. 11), whereas
aggregation, or the presence of large contaminants, will create an upward deflection (Fig. 9.17B;
see e.g. Fig. 9.19B and Fig. 4.14D in Chap. 4). These deviations are not too difficult to handle as
long as they do not affect the higher-angle part of the Guinier region. Such is the case, for instance,
for a limited contamination with much larger particles, because they do not scatter significantly
beyond very small angles (for an example of this situation, see Fig. 4.14D in Chap. 4). On the
contrary, contamination with relatively similar particles, such as a proportion of dimers in a
population of monomers, can be difficult to detect, because the deviation from linearity may be
lost in the experimental noise (cf. below; Fig. 9.19B). If this polydispersity is overlooked, it will
bias the analysis toward an overestimation of both the mass and R g of the monomer. It is therefore
essential to characterize carefully the dispersity of SANS and SAXS samples, e.g. using AUC.
– Shape information. Beyond the Guinier region – at wider angles, that is – the scattering
curve contains information about the shape of the particles. This is illustrated in
Fig. 9.18A, which compares the scattering curves of three particles of the same volume,
9.3 A Walk Through Selected Solution Studies
429
of which originates from disordered (“incoherent”) scattering of neutrons by the
1
H nuclei of light water. As a result, much better signal-to-noise ratios are
obtained when studying a hydrogenated MP trapped in a contrast-matched
deuterated APol than if the two components are hydrogenated.
– Because X-rays interact with electrons, contrast variation implies chemical or
physical-chemical changes to the sample, such as adding sugar or salts to the
buffers in order to increase their electron density, and it is less widely used.
• Interpretation of scattering curves. In a SAXS or SANS experiment, the intensity
scattered is measured as a function of the angle 2θ between incident and scattered
waves. This intensity depends on the size, shape, composition, etc. of the scattering
particles, on their concentration, but also on the wavelength λ of the radiation. It is useful
to combine the wavelength and the scattering angle into a single parameter, which makes
the analysis independent of λ. Inconveniently, but traditionally, different parameters are
used in SANS and SAXS: SANS data are usually plotted as a function of Q or Q
2
, where
Q ¼ (4π sinθ)/λ, and SAXS data as a function of s or s
2
, with s ¼ (2 sinθ)/λ, Q and
s having dimensions of either nm
À1 or Å
À1 . As to the intensity, I(Q), which, in the case of
independent particles, is proportional to their concentration, C, it is often normalized as
I(Q)/C, where C is expressed in gÁL
À1
, so as to permit easy comparisons between data
sets collected at different concentrations (see e.g. Fig. 9.19A or Fig. 4.14D in Chap. 4).
As in a diffraction pattern, to which scattering curves are closely related, data at high
Q or s contain information about short distances between scattering centers in the
sample, e.g. the distribution of individual atoms within a particle, and data at very
small angles information about long distances, such as the distribution of the particles
in the solution. The presentation of scattering data resorts principally to three types of
graphs, each of which facilitates extracting certain types of information.
– Guinier plots (Guinier and Fournet 1955). A common representation is to plot ln I(Q)
(or ln (I(Q)/C)) vs. Q
2 (or ln I(s) vs. s
2
) (Fig. 9.17). A single look at this plot tells the
experimenter, at a glance, (i) whether the preparation is likely to be homogeneous or not
and, if it is, (ii) what the molecular mass M and radius of gyration R g of the particles are.
M is extracted, knowing the contrast of the particle with the solvent and its concentration
in gÁL
À1
, from the extrapolation of the plot at zero angle (I(0)/C). The slope of the curve
at small angles yields R g , the radius of gyration, which is determined by the spatial
distribution of the scattering centers in the particle (“small angles,” here, means that the
Guinier approximation holds, typically, for QÁR g 1, although for certain shapes it may
extend further in Q).
Deviations from linearity at small angles mean that the preparation is not comprised of
identical, independent particles. Repulsion between particles, for instance, will result in a downward deflection at low angles (Fig. 9.17A; see e.g. Fig. 9.19A and Fig. 11.13 in Chap. 11), whereas
aggregation, or the presence of large contaminants, will create an upward deflection (Fig. 9.17B;
see e.g. Fig. 9.19B and Fig. 4.14D in Chap. 4). These deviations are not too difficult to handle as
long as they do not affect the higher-angle part of the Guinier region. Such is the case, for instance,
for a limited contamination with much larger particles, because they do not scatter significantly
beyond very small angles (for an example of this situation, see Fig. 4.14D in Chap. 4). On the
contrary, contamination with relatively similar particles, such as a proportion of dimers in a
population of monomers, can be difficult to detect, because the deviation from linearity may be
lost in the experimental noise (cf. below; Fig. 9.19B). If this polydispersity is overlooked, it will
bias the analysis toward an overestimation of both the mass and R g of the monomer. It is therefore
essential to characterize carefully the dispersity of SANS and SAXS samples, e.g. using AUC.
– Shape information. Beyond the Guinier region – at wider angles, that is – the scattering
curve contains information about the shape of the particles. This is illustrated in
Fig. 9.18A, which compares the scattering curves of three particles of the same volume,
9.3 A Walk Through Selected Solution Studies
429