ISO 17867:2020

INTERNATIONAL ISO
STANDARD 17867
Second edition
2020-10
Particle size analysis — Small angle
X-ray scattering (SAXS)
Analyse granulométrique — Diffusion des rayons X aux petits
angles (SAXS)
Reference number
©
ISO 2020
© ISO 2020
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ii © ISO 2020 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 2
5 Principle of the method . 2
6 Apparatus and procedure . 3
7 Preliminary procedures and instrument set-up . 6
8 Sample preparation . 6
9 Measurement procedure . 7
10 Data correction procedures . 8
11 Calculation of the mean particle diameter . 9
11.1 General . 9
11.2 Guinier approximation . 9
11.3 Model fitting .10
12 Size distribution determination .10
12.1 Limitations of size distribution determination from SAXS data .10
12.2 A brief overview of methods .11
12.3 Goodness of fit: evaluating fits .11
12.4 Model-based data fitting .11
12.5 Monte Carlo-based data fitting .12
12.6 Indirect Fourier transform (IFT) method .13
12.7 Expectation maximization (EM) method .13
13 Repeatability .14
14 Documentation and test report.14
14.1 Test report .14
14.2 Technical records .14
Annex A (informative) General principle .16
Annex B (informative) System qualification .24
Bibliography .25
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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iso/ foreword .html.
This document was prepared by Technical Committee ISO/TC 24, Particle characterization including
sieving, Subcommittee SC 4, Particle characterization.
This second edition cancels and replaces the first edition (ISO 17867:2015), which has been technically
revised. The main changes compared to the previous edition are as follows:
— inclusion of various methods for the extraction of particle size distribution by using the SAXS method;
— correction of technical terms.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
iv © ISO 2020 – All rights reserved

Introduction
This document deals with small-angle X-ray scattering (SAXS), which is performed for particle
size analysis in the 1 nm to 100 nm size range. Under certain conditions (narrow size distributions,
appropriate instrumental configuration, and idealised shape) the limit of 100 nm can be significantly
extended. In ideal circumstances, SAXS can determine the mean particle diameter and particle size
distribution, surface area, and sometimes particle shape in a reasonably rapid measurement time. User-
friendly commercial instruments are available worldwide from a number of manufacturers for both
routine and more sophisticated analyses, and state-of-the-art research instruments are available at
synchrotron radiation facilities.
As in all particle size measurement techniques, care is required in all aspects of the use of the instrument,
collection of data, and further interpretation. Therefore, there is a need for an International Standard
that allows users to obtain good interlaboratory agreement on the accuracy and reproducibility of the
technique.
SAXS can be applied to any hetero-phase system, in which the two or more phases have a different
electron density. In most cases, the electron density corresponds reasonably well to the mass density.
SAXS is sensitive to the squared electron density difference. For fixed volume fractions, it does not
matter whether the particles constitute the denser phase and the solvent (or matrix) is the less-dense
phase or vice versa. Thus, pore size distributions can be measured with SAXS in the same way as size
distributions of oil droplets in emulsions or solid particles in suspensions. Core-shell-nanoparticles can
also be investigated, but low density (e.g. organic) shells are not detected if the core has a significantly
higher density. To obtain the outer particle diameter including the shell, other methods should be used.
Although SAXS allows the determination of particle size, size distribution, surface area, and sometimes
particle shape in concentrated solutions, in powders and in bulk materials, this document is limited
to the description of particle sizes in dilute systems. A dilute system in the sense of SAXS means
that particle interactions are absent. In case of long-range interactions (Coulomb forces between the
particles), special care needs to be taken and a reduction of the concentration or the addition of salt can
be necessary.
Since all illuminated particles present in the X-ray beam are measured simultaneously, SAXS results are
ensemble and time averaged across all the particle orientations which are present in the sample.
The shape of the particles can be assigned to a basic geometry: spheroid, disk, or cylinder. This does not
exclude more detailed information about the shape of the particle being obtained. However, the method
of calculation for more detailed shape analysis is very complex to be included in an International
Standard at this time. The sizes of irregularly shaped nanoparticles can be assessed by the radius of
gyration (R ) as obtained by classic Guinier analysis.
g
The size and size distribution of particles with basic shapes (sphere, disk, cylinder, core-shell, etc.) can
be determined from curve fitting for relatively narrow size distributions. The reliability of the method
of calculation for broader distributions depends on prior knowledge of the distribution.
This document assumes isotropically oriented nanoparticles of any shape in a test procedure. No
dimension of the nanoparticle shall be larger than defined by the scattering accessible to the specific
SAXS instrument. This generally limits the largest measurable particle size of the conventional
technique to 100 nm, although this limit can be significantly extended in samples with a very narrow
size distribution.
Small-angle neutron scattering is not described in this document but can be used without restriction
because the theory and application are similar.
A list of suitable references for further reading is given in the Bibliography.
INTERNATIONAL STANDARD ISO 17867:2020(E)
Particle size analysis — Small angle X-ray scattering (SAXS)
1 Scope
This document specifies a method for the application of small-angle X-ray scattering (SAXS) to the
estimation of mean particle sizes in the 1 nm to 100 nm size range. It is applicable in dilute dispersions
where the interaction and scattering effects between the particles are negligible. This document
describes several data evaluation methods: the Guinier approximation, model-based data fitting, Monte-
Carlo–based data fitting, the indirect Fourier transform method and the expectation maximization
method. The most appropriate evaluation method is intended to be selected by the analyst and stated
clearly in the report. While the Guinier approximation only provides an estimate for the mean particle
diameter, the other methods also give insight in the particle size distribution.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 26824, Particle characterization of particulate systems — Vocabulary
ISO/TS 80004-2, Nanotechnologies — Vocabulary — Part 2: Nano-objects
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 26824, ISO/TS 80004-2 and
the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
3.1
particle
minute piece of matter with defined physical boundaries
Note 1 to entry: A physical boundary can also be described as an interface.
Note 2 to entry: A particle can move as a unit.
Note 3 to entry: This general particle definition applies to nano-objects.
[SOURCE: ISO 26824:2013, 1.1]
3.2
particle size
x
d
linear dimension of a particle determined by a specified measurement method and under specified
measurement conditions
Note 1 to entry: Different methods of analysis are based on the measurement of different physical properties.
Independent of the particle property actually measured, the particle size is reported as a linear dimension, e.g. as
the equivalent spherical diameter.
Note 2 to entry: Examples of size descriptors are those based at the opening of a sieve or a statistical diameter,
e.g. the Feret diameter, measured by image analysis.
Note 3 to entry: In ISO 9276-1:1998, the symbol x is used to denote the particle size. However, it is recognized that
the symbol d is also widely used to designate these values. Therefore the symbol x may be replaced by d.
3.3
radius of gyration
R
g
square root of the ratio of the moment of inertia to the particle mass
Note 1 to entry: Guinier radius (i.e. radius of gyration) is expressed in nanometres. Typical average radii are in
the range of 1 nm to 50 nm.
[SOURCE: ISO 26824:2013, 10.2]
4 Symbols
Symbol Description Unit
d Volume-squared-weighted mean particle diameter nm
vs
d Number-weighted mean particle diameter nm
num
g (r) Intensity-weighted particle size distribution
int
g (r) Number-weighted particle size distribution
num
g (r) Volume-weighted particle size distribution
vol
I Primary beam intensity with sample
out
I Primary beam intensity without sample
in
I(q) Scattered intensity (or scattering intensity)
M Number of degrees of freedom in fitting
N Number of particles
P(q, r) Particle form factor as functions of q-value and particle radius, r
Momentum transfer or q-value, magnitude of the scattering vector given by
−1
q nm
q = 4π/λ sin(θ)
−1
q Small angle resolution, minimum accessible q-value nm
min
−1
q Maximum accessible q-value nm
max
r Particle radius nm
R Radius of gyration (Guinier radius, see A.4) nm
g
t Optimum sample thickness mm
o
T Transmission
V Volume of particle nm
2θ Scattering angle deg or rad
λ Wavelength of the incident X-rays in vacuum nm
−1
μ Linear absorption coefficient mm
ρ(r) Electron density distribution nm
σ Standard deviation of size distribution nm
5 Principle of the method
When electromagnetic radiation passes through matter, a small fraction of the radiation may be
scattered due to electron density differences in the matter. The scattered radiation intensity profile (as
a function of the scattering angle or momentum transfer, q), contains information that can be used to
deduce morphological characteristics of the material. When X-rays are used to probe a geometrically
2 © ISO 2020 – All rights reserved

ordered group of particles or molecules (“crystals”), the well-known X-ray diffraction pattern is obtained
at wide scattering angles, which can be used to characterize the unit cell and lattice constants of such
crystalline material. In the small-angle regime (typically 2θ < 5 °; wavelength dependent), information
on the particle or pore dimensions within the material is available from the elastic scattering arising
from the electron density contrast between the particles and the medium in which they reside. This
is analogous to static light scattering and small-angle neutron scattering. A diagrammatic form of the
angular dependence of the X-ray scattered intensity of a titanium dioxide mixture (rutile and anatase)
is shown in Figure 1.
Key
X scattering angle 2θ (in degrees)
Y intensity
1 SAXS range
2 XRD range
Figure 1 — X-ray scattering diagram illustrating the small-angle SAXS region (left hand side) and
the wide-angle XRD (X-ray diffraction) region (right hand side) of a titanium dioxide powder
At low concentration, the small-angle scattering region contains information about the particle
morphology, which may be evaluated to extract either the particle size (distribution), or particle shape.
Only in very few cases is it possible to obtain both size and shape information. Different regions in
q are dominated by signals from a particular length range, and so can contain distinct (exploitable)
information aspects. In the low-q range, the Guinier approximation can be applied as an indicative
method to get an intensity weighted mean size, provided the particles are smaller than 2π/q . Model
min
fitting can be applied in the full range of q to compute a traceable particle size and size distribution
with associated uncertainties. Both methods can fail depending on data quality and particle properties.
At increased concentrations, i.e. those higher than typically one volume %, particle-particle interactions
and inter-particle interference can be relevant. Such interactions require sophisticated data modelling
and expert knowledge for data interpretation, which is beyond the scope of the present document.
In practice, a concentration ladder may be explored to determine the dependence of reported size on
concentration. If available, each sample shall be measured twice: in its original concentration, and
diluted 1:1 to allow identification of concentration artefacts. The result of both measurements shall be
arithmetically averaged and the uncertainty enhanced by the variation. If dilution is not possible for
technical reasons, this shall be stated in the report. In particular, the Guinier approximation is highly
sensitive to concentration-induced scattering effects.
6 Apparatus and procedure
A diagrammatic form of a SAXS instrument is shown in Figure 2.
Key
X 2θ or q
Y scattered intensity
1 X-ray source
2 optics
3 collimation system
4 sample
a
2θ.
Figure 2 — Diagrammatic form of a SAXS instrument, consisting of X-ray source, optics,
collimation system, sample holder, beam stop, and X-ray detector
The SAXS set-up consists of X-ray source, optics, collimation system, sample holder, beam stop,
and detector. In order to extract meaningful information from the measurement, the following key
parameters define the capability of the system:
— q-range: q and q ; number of sampled points in the Guinier region for Guinier approximation;
min max
— detector sensitivity and system background noise.
Most available X-ray sources produce divergent beams which shall be collimated for SAXS measurements.
With laboratory X-ray sources, multilayer optics are commonly used but basic SAXS measurements can
also be achieved with slit collimation. The X-ray flux on the sample is generally higher when optics is
used. Furthermore, multilayer coated optics can be used to generate a monochromatic X-ray beam.
The greatest challenge in SAXS is to separate the unscattered, transmitted beam (“direct beam”) from
the scattered radiation at small angles (around 0,1°). The direct beam is normally blocked by a beam
stop and parasitic scattering should be eliminated. The need for separation of primary and scattered
beam makes collimation of the primary beam mandatory.
There are two main options to collimate an X-ray beam (see Figure 3).
— Point collimation systems have multiple pinholes or crossed slits that limit the shape of the X-ray
beam to a low divergence and a small dimension (typically, the beam spot on the sample is less
than 0.8 mm in diameter). The scattering is normally centro-symmetrically distributed around
the primary X-ray beam. For isotropic samples, the scattering pattern in the detection plane
perpendicular to the X-ray beam exhibits circular contour lines around the point of incidence of the
primary beam. The illuminated sample volume is smaller than in line-collimation. Point collimation
allows the study of isotropic and anisotropic systems.
— Line-collimation instruments confine the beam in one dimension so that the beam profile is a long
and narrow line. The beam dimensions can be adapted to accommodate a given sample geometry.
Typical dimensions are 20 mm × 0,3 mm. The illuminated sample volume is larger compared to
point-collimation and the scattered intensity at the same flux density is proportionally larger. If
the system is isotropic, the resulting smearing can be reverted using a deconvolution procedure,
albeit at the cost of magnified uncertainties of the observed intensities. The investigation of
4 © ISO 2020 – All rights reserved

anisotropic nanostructure with such line-collimated instruments is not as straightforward as for
point collimation.
In addition, both the point and line collimation systems can use either a parallel or focused beam (see
Figure 4).
The scattered radiation (containing the morphological information, as described in Clause 5) forms a
pattern that contains the information on the size and structure of the sample. This pattern is detected
typically by a 1-dimensional or 2-dimensional flat X-ray detector situated behind the sample and
perpendicular to the direction of the primary beam. Some multipurpose diffractometers that combine
SAXS and diffraction use a scanning point (0-dimensional) detector. Detector classes commonly used
include photon-counting and integration type detectors, although in practice, the least problematic/
distorted measurements come from high-dynamic range, direct-detection, photon-counting detectors.
Key
1 X-ray source
2 collimation system
3 sample
Figure 3 — Point and line collimation types used in SAXS

a) Parallel beam
b) Point (2D) or line (1D) focused beam
Key
1 X-ray source
2 mirror
3 sample
Figure 4 — Focused and parallel beam set-up
7 Preliminary procedures and instrument set-up
Wavelength calibration (see Annex B) can be performed before conducting an experiment and thus
would be classified as a preliminary procedure, but this is only required for polychromatic sources.
If characteristic X-ray emission lines (e.g. copper Kα or molybdenum Kα lines) are used, a suitable
absorber can be used to check that the right emission line has been selected correctly (Nickel for Cu Kα,
Zirconium for Mo Kα). Utilization of a calibration material, such as silver behenate, should form part of
a full system qualification and fit-for-purpose specification as noted in Annex B.
Only if, in addition to the mean particle diameter and the particle size distribution, the (absolute)
concentration or volume fraction of scatterers is to be determined, the intensity shall be scaled to
absolute units. For this purpose, a variety of auxiliary calibration materials are available, including
water and glassy carbon. Alternatively, some instruments can determine this directly by means
of calibrated detectors, or measurement of the unattenuated primary beam intensity on the SAXS
detector. The use of semi-transparent beamstops to measure the beam intensity is not recommended
due to the radiation hardening effects of such, which can lead to inaccurate values.
All calibrations should be described in the analysis report.
8 Sample preparation
Sample preparation is simple and fast for SAXS measurements. The required sample volumes are small,
typically in a range of 5 μl to 50 μl for liquids and pastes, if copper radiation is used. Solid samples
2 2
require an area of (1 × 1) mm to (1 × 20) mm . The sample thickness is typically smaller than 1 mm and
can be tuned to optimize the scattering and limit the X-ray absorption, depending on the composition of
the sample.
Liquid samples are usually measured inside a thin-walled capillary, the diameter of which is typically
between 0,5 mm to 2 mm when the liquid primarily contains water or hydrocarbons. Solvents that
6 © ISO 2020 – All rights reserved

contain heavy atoms, for example, chlorine in chloroform, should be measured in smaller diameter
capillaries as the atoms strongly absorb the incident radiation, or higher energy radiation should be
used. Viscous samples can be measured better in a paste cell. It is strongly recommended to measure
liquid samples in a re-fillable or flow-through container, as this greatly reduces the risks of errors
encountered in the background subtraction procedure (incorrigible effects may lead to inaccurate
subtraction if non-identical sample containers are used for the two measurements).
Pastes, powders, and vacuum sensitive materials can be mounted into a sample holder with windows,
which shall be transparent to X-rays and exhibit little scattering themselves. Frequently used window
materials include polyimide films, mica or silicon nitride (see Reference [18]). Care should be taken
that the scattering from the window material does not affect the result of the measurement and can
be appropriately subtracted. For example, polyimide films exhibit a broad small-angle diffraction peak
−1
in the vicinity of q approximately 0,7 nm , which shall be correctly subtracted in the background
subtraction procedure.
Solids can be clamped onto frames with or without additional window foils for protection against the
vacuum. The sample thickness shall be chosen in line with the respective absorption of the material
(see Reference [17]). The optimum thickness, t , is given by
o
t = (1)
μ
where μ is the linear absorption coefficient of the material. The optimum specimen thickness
corresponds to a ratio of the primary beam intensity with and without sample, I and I , of:
out in
−−μ t 1
II/%==ee ~37 (2)
outin
Thus, the ideal specimen will transmit about 37 % of the incident radiation, and the specimen thickness
can be adjusted accordingly to optimize transmission. Any sample pre-treatment (for example,
dilution, sonication, annealing or centrifugation) may affect the particle size distribution and should be
described in the analysis report.
9 Measurement procedure
Every SAXS particle-sizing experiment consists of at least two measurements using the same sample
holder and preferably the same acquisition time:
a) a sample measurement (containing signals from the particles, the solvent or matrix, the sample cell
windows, the parasitic instrument radiation, background radiation and detector noise);
b) a background measurement (containing signals from the solvent or matrix, the (same) sample cell
windows, the parasitic instrument radiation, background radiation and detector noise).
This is the minimum requirement for determining the scattering of the particles, which is the difference
between the two scattering measurements. A typical example for this procedure is given in Figure 5.
Care shall be taken that the scattering of the window material of the sample cell, the parasitic scattering
of the SAXS instrument, and the dark count rate of the detector are removed. The transmission from
the sample and background/matrix material and efficiency variation over the detector shall be taken
into account.
Key
−1
X q (in nm )
Y scattered intensity (in a. u.)
1 solvent
2 particle scattering
3 particle dispersion
Figure 5 — Typical SAXS profiles of a particle dispersion, the solvent and the difference (the
corrected signal only due to particle scattering)
The statistical quality of the scattering pattern improves with increasing intensity and complies with
standard statistics for signals obtained by the subtraction of two independent measurements.
10 Data correction procedures
As the data analysis methods are sensitive to the quality of the data, the scattering signal of the sample
shall be carefully extracted from the total measured signal. These correction steps are performed
by the software provided with commercial instruments, or through custom software. At best, they
propagate the data uncertainties through the correction steps, resulting in a final data set complete
with uncertainty estimates on the data points.
The minimum amount of data corrections to be considered or applied by this software to the data
before background subtraction are corrections for: invalid pixels, dark counts, time, X-ray flux and X-ray
transmission. This is followed by the background subtraction (complying with standard statistics
for signals obtained by the subtraction of two independent measurements). After the background
subtraction, a normalization for sample thickness can be performed, as well as an optional scaling to
absolute intensity units. These corrections may be combined in their implementation, but a discussion
of the separate implementation is described in Reference [24]. An overabundance of data points may be
reduced using a binning procedure. For size-disperse samples, 100 bins per decade of q are typically
sufficient. These bins may be linearly or logarithmically spaced.
8 © ISO 2020 – All rights reserved

For the analysis report, it is sufficient to mention the applied corrections, and the software name and
version they have been applied with.
11 Calculation of the mean particle diameter
11.1 General
After background subtraction and desmearing (if required, see Annex A), the mean particle diameter
can be estimated according to two different approaches.
11.2 Guinier approximation
For the Guinier approximation (explained in detail in A.2), ln(I) is plotted as a function of q (Guinier
plot). As the scattered intensity at very small angles is approximated by a Gaussian function
 
Iq =−IRexp q (3)
()
0 g
 
 3 
which can be transformed to
ln Iq() =lnIR− q (4)
[]
 
0 g
A straight line can fit the data in the Guinier region which is typically up to qR around 1. The slope is
g
then equal to − R
g
For monodisperse homogeneous spherical particles, the volume-squared-weighted mean particle
diameter can be calculated from R according to:
g
dR=2 (5)
vs g
According to ISO 9276-2 and Reference [3], d corresponds to D and x .
vs 8,6 26,
There are several caveats which concern the validity of this approximation. Firstly, if a straight line
is not found in the Guinier region, the approximation does not apply, and no values can be estimated
using the Guinier approximation. Secondly, while linearization may be appropriate for on-site review
of data, values quoted in an analysis report should be the result of a fit of the Guinier approximation
[Formula (3)] to the unlinearized data, at best weighted by the datapoint uncertainty. Thirdly, a linear
relationship in the Guinier region is no guarantee for applicability, Data analysis according to 11.3 is
required beyond this point, using the Guinier approximation values as an optional guide.
11.3 Model fitting
For model fitting, the full range of q can be fitted by a model function for a polydisperse ensemble of
particles according to:
∞
Iq =NPΔρ qr, gr dr+c (6)
() () ()
num
∫
For homogenous spheres, the form factor is given by:
4π 
Pq,sr =−inqr qrcosqr (7)
() ()
 
q
 
The most common distributions are lognormal and Gaussian. A Gaussian size distribution is described by:
2 2
   
rd− /2 ρ−d /2
∞
() ()
 numn  um 
gr =exp /exp dρ (8)
()
num   ∫  
2 2
2σ 2σ
   
   
and a lognormal distribution can be written as:
 
 
1 
 
lnrd−ln
 ln 
 
 2 
1  
 
gr()=−exp (9)
 
num
2πrσ
2σ
ln
 ln 
 
 
where the mean diameter d can be transformed to the number-weighted mean particle diameter of a
ln
Gaussian distribution d according to:
num
dd= exp/σ 2 (10)
()
num ln ln
N, c, the standard deviation of the size distribution (σ or σ ) and the mean particle diameter ( d or
ln num
d ) are the fit parameters. From the determined size distribution, the volume-weighted and intensity-
ln
weighted mean particle diameters can also be calculated.
According to ISO 9276-2, d corresponds to D and x .
num 1,0 1,0
Information on the particle size can also be obtained from other evaluation methods in real space or
Fourier space as explained in A.5.
12 Size distribution determination
12.1 Limitations of size distribution determination from SAXS data
When considering the analysis of scattering patterns with the intent to extract size distributions, there
are several limitations to consider. The main limitations include:
a) the upper and lower size bounds, as defined by the data limits;
b) the practical necessity for uncertainty estimates on the data;
c) the required assumptions on the scatterer shape and (in some cases) the mathematical form of the
size distribution; and
d) the inherent size-weighting of information in the SAXS data. These limitations will be briefly
addressed.
10 © ISO 2020 – All rights reserved

The range of dimensions that can be probed with SAXS is in general limited by the measurement range
(a second, less common limitation to the upper limit in size is given by the transversal coherence
length of the radiation). The maximum and minimum probed dimensions are defined in Formula (A.3).
Information obtained beyond these limits may not be grounded in actual data, but rather will be present
due to the assumptions made.
In the data fitting procedures described herein, the agreement between the model and the measured
data are normalized by the uncertainty estimates of the measured data values. While it is technically
possible to bypass the normalization, such a circumvention is strongly discouraged as it can easily
lead to overfitting and misinterpretation (skewed weighting) of the information. The normalization by
the uncertainty ensures that the data are weighted by the accuracy of each individual data point, and
furthermore helps prevent overfitting due to its provision of a clear cut-off criterion.
Due to the information loss inherent in the process of small-angle scattering, additional external
information shall be provided in order to arrive at a unique solution. The information required to
obtain a size distribution includes a shape assumption (globular by default), an assumption on the inter-
scatterer interaction (dilute, i.e. no interaction by default), and, for one of the methods, an assumption
on the mathematical form of the size distribution (typically log-normal, Gaussian or Schultz-Zimm).
This information can be either assumed (on the basis of reasonable expectations) or provided by
external methods (e.g. microscopic methods). For mixtures of scatterers differing in shape, very strict
assumptions (or rather provision of external information) may still allow an arrival at the correct
solution.
The exact contribution is dependent on the shape of each scatterer, the measurement range, and the
angle-dependent collection efficiency of the instrument. The overall effect, however, is that the detection
limits of small scatterers deteriorates with increasing presence of large scatterers. For spherical
particles with homogeneous electron density and narrow size distribution, the volume-weighted and
the number-weighted size distribution can be obtained. For other particles, the number-weighted size
distribution is not easily accessible without additional assumptions (see Reference [25]).
12.2 A brief overview of methods
The methods are separated into model-based and non model-based methods. The model-based data
fitting is the easiest to implement but requires the assumption both the scatterer shape as well as the
mathematical form of the distribution. The non model-based methods can derive the distribution form-
free, requiring only the assumption on the general shape of the scatterer.
12.3 Goodness of fit: evaluating fits
For all fitting methods, the agreement between the data [I (q)] and the model [I (q)] can be
meas model
evaluated using the reduced chi-squared value (χ ). This value is large for a large discrepancy between
r
the two, but approaches unity when the model fits the data on average to within the uncertainty [σ q
()
i
] of the data. For most data fitting methods, the reduced chi-squared value is used in the optimization
procedure directly. The calculation of χ follows:
r
N
d
I qI− q
 () ()
meas iimodel
χ = (11)
 
r ∑
NM− σ q
()
 
d i
i=1
where N denotes the number of data points, M denotes the number of degrees of freedom in the fitting
d
procedure, and i denotes the index of a given data point.
12.4 Model-based data fitting
In the model-based data fitting procedure, the goodness-of-fit value is minimised (and thereby the
agreement between model and data maximized) by adjusting a limited number of parameters of a model
function. One common model function describing the scattering of a sample containing size-disperse
scatterers is given in Formula (6), combining the form factor of a particularly shaped scatterer with a
monomodal, single-parameter size distribution. This equation is evaluated numerically to compute the
model scattering function.
There are several minimization methods available to find the model parameters that minimize the
goodness-of-fit. The most common of these is the Levenberg-Marquardt nonlinear least-squares solver,
for which libraries are readily available. This minimization method furthermore provides uncertainties
on the final values of the (minimised) fit parameters. However, it may become unstable when models
are fitted with many parameters in complex models. For this reason, modern, more stable adaptations
of the Levenberg-Marquardt methods should be implemented where possible.
The model-based data fitting procedure is the most commonly implemented data analysis method in
SAXS and can be found in a wide variety of free and open-source data fitting software programs. These
programs differentiate mostly in the datatypes they can read, and the models that are implemented
therein. However, each of these programs requires the selection of several assumptions.
Firstly, the scatterer shape model needs to be selected. The choice of model is ideally driven by
information from complementary methods. Secondly, the variable dimension needs to be assigned to a
particular form of the distribution (often, the form can be assumed based on the formation process of
the scatterers). A fitting range and variable constraints can optionally be added.
Starting parameters can lead the minimization method to a local minimum rather than a global
minimum, and a variety of starting parameters needs to be chosen to ensure a global minimum is
reached. As for uniqueness, any number of combinations of shape and distribution may fit a measured
scattering pattern, and it is tempting to assume that the combination leading to the best fit is correct.
It is therefore important to realize that it is theoretically impossible to retrieve information on both
shape as well as the mathematical form of the distribution. One of these two parameters shall therefore
be given and supporting information shall be provided on why this choice was made.
12.5 Monte Carlo-based data fitting
A Monte Carlo based data fitting procedure can be used to remove the requirement to select a
mathematical form of the size distribution. Using solely information on the elementary scatterer
shape, it can retrieve any size distribution within the size limits dictated by the measurement. These
procedures are therefore less reliant on assumptions and are more data-driven.
This procedure relies heavily on the goodness of fit measure and is therefore reliant on the provision
of representative uncertainty estimates. Furthermore, since no assumption is made on the asymptotic
behaviour of the size distribution, extrema of the size distribution carry a higher uncertainty
than the central component. Lastly, since the representation of scatterers in scattering data are not
proportional to their amount, but weighted proportional to their size, Monte Carlo methods are often
unable to positively confirm the existence of small numbers of small scatterers in the presence of larger
scatterers. They are best used to determine volume-weighted distributions, which closely approaches
the informat
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