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Description	and	Survey	of	Methodologies	for
the	Determination	of	Amorphous	Content	Via
X-Ray	Powder	Diffraction
Article		in		Zeitschrift	für	Kristallographie	·	December	2011
DOI:	10.1524/zkri.2011.1437
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Description and survey of methodologies for the determination
of amorphous content via X-ray powder diffraction
Ian C. MadsenI, Nicola V. Y. ScarlettI and Arnt KernII
I CSIRO Process Science and Engineering, Box 312 Clayton South 3169, Victoria, Australia
II Bruker AXS GmbH, �stliche Rheinbru¨ckenstraße 49, 76187 Karlsruhe, Germany
Received: July 13, 2011, accepted: October 5, 2011
Amorphous material / X-ray diffraction /
Quantitative phase analysis / Calibration
Abstract. The presence of amorphous materials in crys-
talline samples is an increasingly important issue for dif-
fractionists. Traditional phase quantification via the Riet-
veld method fails to take into account the occurrence of
amorphous material in the sample and without careful at-
tention on behalf of the operator its presence would re-
main undetected. Awareness of this issue is increasing in
importance with the advent of nanotechnology and the
blurring of the boundaries between amorphous and crystal-
line species.
The methodology of a number of different approaches
to the determination of amorphous content via X-ray dif-
fraction and an assessment of their performance, is de-
scribed. Laboratory-based, X-ray diffraction data from a
suite of synthetic samples, with amorphous content rang-
ing from 0.0 to 50 wt%, has been analysed using both direct
(in which the contribution of the amorphous component to
the pattern is used to obtain an estimate of concentration)
and indirect (where the absolute abundances of the crystal-
line components are used to estimate the amorphous con-
tent by difference) methodologies. In addition, both single
peak and whole pattern methodologies have been assessed.
All methods produce reasonable results, however the
study highlights some of the strengths, deficiencies and
applicability of each of the approaches.
Introduction
Traditionally, powder X-ray diffraction (XRD) is con-
cerned with the characterisation of poly-crystalline materi-
als which, more often than not, will contain amorphous or
poorly crystalline phases. This is increasingly the case
with the advent of nanotechnology which has blurred the
boundaries between crystalline and amorphous species via
reduced particle size. As a result there is often no clear
dividing line between what is defined as crystalline and
amorphous material.
Crystalline materials are frequently defined as solids
with fixed volume, fixed shape, and long-range order
bringing about structural anisotropy, producing sharp dif-
fraction peaks [1]. Amorphous (or non-crystalline) materi-
als are thus solids with fixed volume, fixed shape, charac-
terized by short-range order which, however, may also
have loose long-range order [1]. The latter embraces disor-
dered materials possessing only one- or two-dimensional,
or lesser, degrees of order [2]. Terms like “short”- and
“long”-range order clearly show the ambiguity of these
definitions, and why there cannot be a sharp dividing line
between crystalline and amorphous materials. To further
increase this ambiguity, even highly crystalline materials
have amorphous surface layers which may be up to 1 to 3
crystallographic units thick. Depending on particle size,
this can represent up to 1 wt% of the total mass of the
sample [3].
A more strict definition for amorphous solids has
been given by Klug & Alexander [2]: “The term, amor-
phous solid, must be reserved for substances that show
no crystalline nature whatsoever by any of the means
available for detecting it”. In other words: the ability to
detect and characterize ordering is dependent upon the
principles of the analytical method and models being used.
Conventional X-ray diffraction loses its power for crystal-
line material structures on the nano-scale; resulting ambi-
guities are paraphrased in literature by the term “X-ray
amorphous” to highlight the limitations of X-ray diffrac-
tion.
The transition from the crystalline to the amorphous
state, and the coexistence of such states, can have signifi-
cant impact on material properties. This is clearly demon-
strated by the development of bulk metallic glasses
(BMG) [4–7]. These are metals cooled in such a way to as
to prevent crystallisation, thereby removing grain bound-
aries and other points of weakness normally associated
with crystalline metals and inducing behaviour more akin
to that of glasses or polymers. Composite materials are
also being developed wherein amorphous BMG acts as a
matrix for a ductile crystalline-phase reinforcement materi-
al [8].
Many areas of industry and materials science require
reliable methodology for the quantification of amorphous
content. In the pharmaceutical context, amorphous content
944 Z. Kristallogr. 226 (2011) 944–955 / DOI 10.1524/zkri.2011.1437
# by Oldenbourg Wissenschaftsverlag, Mu¨nchen
* Correspondence author (e-mail: Ian.Madsen@csiro.au)
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may be induced by processing and thus affect the efficacy
of the compound [9–13]. It can affect a compound’s solu-
bility, tendency to hydrate and chemical stability. It is also
implicated in many physical and chemical transitions [14].
In some instances such content may be desirable for its
influence on dissolution rate and compactability, but it is
often variable and must be determined for process control
and quality assurance purposes. Similarly, a bioceramic
material such as hydroxyapatite must be carefully moni-
tored for amorphous content as this can increase the solu-
bility or biodegradability of the material in vivo [15].
Amorphous content present in thin films of superconduc-
tors has also been observed to affect the transition tem-
perature of these materials [16] and must therefore be de-
termined. The degree of crystallinity of silica determines
its potential to be respirable and, therefore, deleterious to
health. Methods for the determination of respirable silica
are the subject of standard operating procedures and are
frequently being reviewed [17–19].
There are numerous methods for characterising amor-
phous content within materials including differential scan-
ning calorimetry[6, 10, 20], near infra-red spectroscopy
[13, 21], thermally stimulated current spectroscopy [11],
water vapour sorption analysis [9] and XRD methods in-
cluding conventional diffraction [9, 13–16, 18, 22–26]
and pair distribution function analysis [12, 27]. Combina-
tions of techniques may also be employed [9, 22] for
amorphous content determination.
In the diffraction context it is worth highlighlighting
the recent work of Cline et al. [3] who have described the
certification of a NIST standard (SRM 676a) developed
specifically for use as an internal standard for quantitative
phase analysis (QPA). One of the key outcomes of that
study was the measurement and certification of the amor-
phous content of the standard.
This paper presents a survey of conventional labora-
tory-based XRD techniques on the basis of accessibility
and ease of implementation. It considers a suite of syn-
thetic mixtures to which various methods of QPA have
been applied, where applicable, to the same data. The sam-
ple suite was selected to pursue what results could be ob-
tained under “ideal” conditions through the elimination of
as many potential sources of error as possible. Since mi-
croabsorption is one of the greatest impediments to accu-
racy in QPA [28, 29] a system with constant chemistry
was chosen so that the overall mass absorption contrast
did not vary as a function of phase composition, hence
minimising the chances of inducing a microabsorption pro-
blem. Such a system would almost certainly not represent
all suites of materials, for example, most mineralogical
systems where there is a wide range in the phase chemis-
try of individual phases. However, the findings are still
relevant to these systems albeit with a reduced level of
accuracy. The techniques are directly applicable to a wide
range of other sample types where there may be a mixture
of crystalline polymorphs and amorphous material with
the same or similar chemical composition. Examples in-
clude nucleation and growth in glassy metals, polymorph-
ism in pharmaceuticals and so on. The relative accuracy
and precision of the methods are compared and recom-
mendations made accordingly.
Experimental
Samples
The samples used in this study were simple three-phase
mixtures consisting of approximately 50 wt% corundum
(a-Al2O3) and 50 wt% silica (SiOx) composed of varying
ratios of crystalline quartz and amorphous silica flour. The
corundum was the same as that used in the IUCr Commis-
sion on Powder Diffraction QPA round robin [28] and was
estimated to be �99% crystalline. The quartz was a high
purity, acid washed sand which was micronised for 5 min
per gram in �3 g batches and then mixed prior to weigh-
ing and sample preparation. No estimate of its crystallinity
has been made and so is assumed, for this exercise, to be
100% crystalline. This is a reasonable assumption since
this is a comparative study rather than one involving abso-
lute determination of amorphous content. The silica flour
was sourced from BDH (product number 30057, batch
52490) and used as supplied. These three phases were cho-
sen since they have similar mass absorption coefficients
(31.4 and 35.8 cm2 gm�1 for Al2O3 and SiO2 respectively)
for CuKa and hence should not induce significant micro-
absorption effects during measurement. In addition, since
the overall mass absorption coefficient is the almost the
same for each sample, methods which employ simple line-
ar calibration models were included in the study.
The samples were prepared by weighing appropriate
amounts (summing to approximately 10 g) and mixing the
components in ethanol for 20 minutes in a McCrone mi-
cronising mill. The resultant slurry was centrifuged for
15 minutes, the excess alcohol decanted and the sample
dried in an oven at 80 �C. The dried material was mixed
in a hand mortar and pestle in case there had been any
phase separation during centrifuging. The as-weighed con-
centrations of the 3 phases in the 9 mixtures prepared are
given in Table 1.
Data collection
XRD data were collected using a Philips X’Pert diffract-
ometer (173 mm radius) fitted with a Cu LFF tube oper-
ated at 40 kV and 40 mA, a curved graphite post-diffrac-
tion monochromator and a sealed proportional counter.
The beam path was defined using a fixed 1� divergence
Determination of amorphous content via X-ray powder diffraction 945
Table 1. The as-weighed concentrations (as wt%) of the three phases
in the nine mixtures used in this study.
Sample Corundum Quartz Silica Flour
A 50.01 0.00 49.99
B 49.98 15.03 34.99
C 49.99 30.00 20.00
D 50.00 40.00 10.00
E 50.00 45.00 5.00
F 50.01 48.00 1.99
G 50.00 49.00 1.00
H 50.02 49.48 0.50
I 49.87 50.13 0.00
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slit, a 0.3 mm receiving slit and a 1� antiscatter slit. Soller
slits were fitted to the incident beam, but not in the dif-
fracted beam in order to maximise the observed intensi-
ties. Data were collected in Bragg-Brentano geometry
from 10 to 140� 2q in increments of 0.02� at 3 seconds
per step from three separate sub-samples of each mixture.
A typical diffraction pattern (Sample B1, �35 wt% silica
flour) can be seen in Fig. 1 while a pattern of the raw
silica flour can be seen in Fig. 2.
Data analysis
There are a number of different approaches to the determi-
nation of amorphous content using XRD based methods.
The first decision to be made is whether to use a single
peak approach, where an estimate of the amorphous con-
tribution to the observed data is obtained from a limited
2q range of data, or to use a “whole pattern” approach. In
the latter case, a choice between crystal model (Rietveld)
based and pattern summation methods may also need to
be made. In addition, some methods use a direct ap-
proach, where the amorphous contribution to the pattern is
used to directly estimate its concentration, while others
use an indirect approach, where the absolute concentration
of the crystalline phases is determined and the amorphous
content estimated by difference.
While this paper describes and assesses some of these
analytical approaches, it does not cover all methodology
available. Instead, the work seeks to address methodolo-
gies that are accessible to as wide a range of diffraction-
ists as possible, namely, those with access to laboratory-
based XRD equipment.
Single peak method
This approach uses a single diffraction peak, rather than
the whole pattern, to obtain an estimate of the amorphous
content. It relies on the availability of a suite of samples
from which calibration constants may be determined, and
on the direct estimation of the nett contribution of the sili-
ca flour to the diffraction pattern using the following
steps:
1. Estimation of the gross intensity of the amorphous
peak
P
(Ipk) near its maximum (Fig. 3). In this case
the peak maximum at about 22� 2q is free from
interference by crystalline phases. The counting sta-
tistics were improved by averaging the intensity
from all data points between 21.5 and 23.5� 2q.
2. Estimation of the pattern background
P
(Ibgd) at a
point on the pattern free from interference from either
the amorphous content or the crystalline phases.
Again, the counting statistics were improved by aver-
aging all intensities from the data points between 12
and 14� 2q.
3. In order to determine the background under the
amorphous peak position,the background slope was
estimated using the ratio of the intensities from
steps 1 and 2 above for Sample I which has no added
silica flour:
Slope ¼
P
IpkIP
IbgdI
: ð1Þ
4. For each sample a, the nett peak height can be cal-
culated from
Netta ¼
P
Ipka � Slope �
P
Ibgda : ð2Þ
5. A calibration curve may then be prepared relating
nett peak height to amorphous content. In this case
Samples A and I have been used for the calibration
and the others treated as unknowns.
In principle, this method is the easiest to implement
since the diffractometer could be programmed to measure
946 I. C. Madsen, N. V. Y. Scarlett and A. Kern
Fig. 1. Typical XRD pattern for Sample B1. Note the presence of the
broad peak from the amorphous component at �22� 2q.
Fig. 2. XRD pattern for pure silica flour.
Fig. 3. Showing the method for the determination of nett counts for
the silica flour peak. The black line is the diffraction pattern from
Sample B containing �35 wt% silica flour and the grey line is that
from Sample I containing no silica flour.
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the intensities at two points in the pattern provided that
these regions are free from excessive overlap from crystal-
line diffraction peaks.
This method provides a direct measure of the silica flour
content. However, it does rely on the presence of suffi-
cient amorphous material for its contribution to be ob-
served above the pattern background. In addition, if it is
to be applied to a sample suite with variable chemical
composition, then a term must be included which corrects
for the change in overall sample mass absorption coeffi-
cient (MAC).
Whole pattern methods
The methods described in this section utilised a whole pat-
tern mode of data analysis. The data from the three repli-
cates of each of the nine mixtures were modelled using
Rietveld methodology [30–31] via TOPAS software [32]
which incorporates a fundamental parameters [33] ap-
proach to the modelling of peak width and shape. Using
this method, the instrument contribution to the pattern is
determined from a standard material free of physical ef-
fects such as small crystallite size or microstrain which
would contribute to the observed peak width. The instru-
ment parameters may then be fixed allowing the sample
contribution to the pattern to be determined.
The crystal structures for corundum and quartz were
obtained from the ICSD entry numbers 31548 and 31228
respectively [34]. Some phase specific parameters (unit cell
dimensions, crystallite size and strain) were refined using
data collected from the pure starting materials. These para-
meters were then tightly constrained during subsequent
analysis. The crystal structure parameters (atom coordi-
nates, site occupation factors and atomic displacement
parameters) were fixed at the values reported in the ICSD.
In order to ensure best fits between observed and calcu-
lated patterns, the contribution of the amorphous silica
also needed to be modelled in a manner which would be
stable in subsequent analysis of the samples. To achieve
this, a model was developed using the diffraction pattern
of pure silica flour (Fig. 4). This model consisted of:
1. A refinable four-parameter Chebychev polynomial
background function with an additional parameter in
the 1=2q function to account for the increase in
background at low 2q values,
2. Thirteen pseudo-Voigt peaks selected to represent
the major observable features in the diffraction pat-
tern. The peaks were constrained to have the same
full width at half maximum (FWHM) and a single
mixing parameter between Gaussian and Lorentzian
forms. The refined FWHM was 7.13� and the peaks
exhibited about 55% Lorentzian character. The over-
all scale factor for the peak group was fixed at unity
and the positions and intensities of individual peaks
allowed to refine during this stage of the analysis.
3. During subsequent data analysis, the relative intensi-
ties, FWHM and mixing parameter of the silica
flour peaks were fixed to the values determined dur-
ing step 2 above. For each sample, an overall scale
factor for the peak group was refined in order to
estimate the total contribution of the silica flour to
the observed pattern. Using this approach a single
parameter can be extracted to reflect the total contri-
bution of the silica flour to each diffraction pattern.
In this way a basic model was developed and then ap-
plied consistently to the 27 data sets in order to minimise
the effect of operator bias during analysis.
For most of the whole pattern methods described in de-
tail in following sections, the amorphous content was de-
termined post-analysis using the refined scale factors for
each phase, including that for the amorphous peak group
described above.
Internal standard method
A previous study [29] revealed that by far the most com-
mon approach to the QPA of amorphous material via
XRD is the internal standard method. This is also the one
most commonly enabled in the interfaces of many Riet-
veld analysis packages. In this method, the sample is spiked
with a known mass of standard material and the QPA nor-
malised accordingly. This approach is reliant upon using a
standard having a similar mass absorption coefficient to
the sample in order to minimise the effects of microab-
sorption.
For the internal standard method, the weight fraction
Wa of the crystalline phases present in each sample is first
estimated using the algorithm of Hill and Howard [35]:
Wa ¼ SaðZMVÞaPn
j¼1
SjðZMVÞj
: ð3Þ
Where Sa ¼ the Rietveld scale factor for phase a,
ZM ¼ the mass of the unit cell contents,
V ¼ the volume of the unit cell,
n ¼ the number of phases in the analysis.
This is the most commonly used approach for QPA via the
Rietveld method and it relies on the assumption that all
phases in the sample are crystalline and have been in-
cluded in the analysis. Therefore, Eq. (3) sums the ana-
lysed concentrations to unity. The analysed concentration
of the standard can be:
� Overestimated relative to the weighed amount, indi-
cating that amorphous/unidentified material is pre-
sent,
Determination of amorphous content via X-ray powder diffraction 947
Fig. 4. Observed XRD pattern (grey dots) for silica flour (CuKa)
showing the thirteen pseudo-Voigt peaks used to model its contribu-
tion to the calculated diffraction pattern (solid line overlaying the ob-
served data).
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� Equal to the weighed amount, indicating that there is
probably no amorphous/unidentified material present,
� Underestimated relative to the weighed amount, indi-
cating there is probably an error in the analysis pro-
cedures.
The presence of a known weight fraction of a crystal-
line internal standard material in the sample allows these
reported concentrations to be corrected proportionately ac-
cording to:
Corr ðWaÞ ¼ Wa STDknown
STDmeasured: ð4Þ
Where Corr (Wa) ¼ the corrected weight percent,
STDknown ¼ the weighed concentration of the
standard in the sample, and
STDmeasured ¼ the analysed concentration
derived from Eq. (3).
Once the corrected concentrations have been calculated,
the weight fraction of amorphous material Wamorphous can
then be derived from:
Wamorphous ¼ 1�
Pn
j¼1
Corr ðWjÞ : ð5Þ
A significant difference between this approach and the sin-
gle peak approach, is that the absolute amounts of the
crystalline phases are first estimated and then the amount
of silica flour determined by difference. There is no direct
measurement of the silica flour; in this case the refined
peaks in its pattern serve only to improve background fit-
ting. For the examples used here, the known corundum
content served as the internal standard. The model for the
silica flour was included only for the purpose of obtaining
the best fit between the observed and calculated patterns
and its refined scale factor discarded.
External standard method
This method closely follows the approach in the internal
standard method in that it attempts to put the determined
crystalline components on an absolute scale and derives
the amorphous content by difference. Like the internal
standard method, it uses only the refined Rietveld scale
factors of corundum and quartz in subsequent analysis and
discards the refined scale factor for the silica flour.
However, unlike the internal standard method, this
method uses an external standard in the manner described
by O’Connor and Raven [36]. In this method an external
standard, either a pure material or mixture in which the
chosen standard is present in known quantity, is used to
determine a normalisation constant (K) which allows the
calculated weight fractions to be placed on an absolute
scale. This is derived via:
Wa ¼ SaðZMVÞa mm*
K
: ð6Þ
Where mm* ¼ the mass absorption coefficient of the
entire sample, and
K ¼ the normalisation constant used to put
Wa on an absolute basis.
K is dependant only on the instrumental and data collec-
tion conditions and is independent of individual phase and
overall sample-related parameters. Therefore, a single meas-
urement should be sufficient to determine K for a given
instrumental configuration. For the samples examined here
K has been determined from a sample of pure corundum.
Linear Calibration Model (LCM) method
Unlike the internal and external standard methods, this
method provides a direct measure of the silica flour con-
tent but is still based on the analysis of wide range diffrac-
tion data. It is similar to those methods in that it also uses
only some of the refined parameters. However, here it is
the information pertaining to the crystalline phases which
is discarded and only the refined scale factor for the silica
flour is used in subsequent analysis. A simple linear cali-
bration model which relates the amount of the silica flour
Wamorphous to the scale factor (equivalent to a measure of
the total area contributed to the pattern by the silica flour)
can be derived from the relationship between peak inten-
sity and concentration [2]:
Iamorph ¼ Camorph Wamorph
ramorphmm*
: ð7Þ
Where Camorph ¼ constant for the group of
reflections comprising the silica flour,
Wamorph ¼ the weight fraction of silica flour,
ramorph ¼ the density of silica flour,
mm* ¼ the mass absorption coefficient of the
entire sample.
By equating Iamorph with the silica flour scale factor SF and
including the density and sample mass absorption coeffi-
cient into the calibration constant, the following can be
derived:
Wamorphous ¼ A � SF � B : ð8Þ
Where A and B ¼ the slope and any residual offset
of the calibration, respectively
SF ¼ the refined scale factor (or other
measure of intensity).
In this study, the three replicates of samples A and I (50.0
and 0.0 wt% silica flour, respectively) were used to cali-
brate this method.
It should be noted that the linear calibration model
works effectively in this context as there is minimal varia-
tion of absorption contrast between the phases. However,
for systems where there is significant variation in mass
absorption coefficient between phases, then an overall
sample absorption correction must be included. It should
also be noted that this method has the potential to mini-
mise the impact of any residual sample related effects
such as microabsorption on the analysis, as these aberra-
tions will be included in the calibration constants.
PONKCS (Partial or No Known Crystal Structure)
method
This method follows the same general form as that used in
the internal standard method but now includes the silica
948 I. C. Madsen, N. V. Y. Scarlett and A. Kern
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flour in Eq. (3). However, since there is no defined crystal
structure for the silica flour, the ZMV for the phase must
be derived empirically. This can be achieved by using a
known mixture of the amorphous phase and an internal
standard s for which ZMVs is known:
ðZMVÞamorph ¼
Wamorph
Ws
Ss
Samorph
ðZMVÞs : ð9Þ
Where Wamorph and Ws ¼ the known weight fractions
of silica flour and standard respectively
Samorph and Ss ¼ the refined scale factors for
silica flour and standard respectively.
The ZMV value for silica flour was determined using an
average of the three replicates of Sample A (�50 wt%
silica flour) and defining corundum as the standard. This
method provides a direct measure of the silica flour con-
tent, but combines it with the analysis of the crystalline
phases present in the sample.
The background to, and application of, this method has
been described in detail in [37].
Degree of Crystallinity (DOC) method
This method is Rietveld-based but does not require the
application of any calibration constants. Instead, it relies
on the estimation of the total intensity or area contributed
to the overall pattern by each component in the analysis
(Fig. 5). The DOC is then calculated from the total areas
under the defined crystalline and amorphous components
from:
DOC ¼ Crystalline Area
Crystalline Areaþ Amorphous Area : ð10Þ
The crystalline area comprises the sum of all area for phases
that are not flagged as amorphous while the amorphous
area is the converse. The weight fraction of amorphous
material can be calculated from:
Wamorphous ¼ 1� DOC : ð11Þ
This methodology is described in Riello [38] and refer-
ences therein.
Full structure method
This method relies on finding a crystal structure which
adequately models the positions and relative intensities of
the observable peaks in a diffraction pattern of the amor-
phous component. Allowing the crystallite size and strain
to refine to small and large values respectively provides
peak widths and shapes which represent the broad features
in the observed data.
In this work, a number of SiOx structures were trialled
but the one which provided the best fit was the cristobalite
structure of Kim Yong-Il et al. (2005) ICSD, entry number
153886 [34]. The crystallite size (lorentzian ¼ 1.87 nm,
gaussian ¼ 2.04 nm) and microstrain (gaussian only ¼ 5.0)
were refined using a data set collected from pure silica
flour. These values were then fixed for use in the subse-
quent analysis of the remaining samples. The fit for the
XRD data collected from Sample A using this method is
shown in Fig. 6.
Since this method treats all componentsas crystalline
and included in the analysis, the derivation of phase abun-
dance can be obtained using Eq. (3) detailed above in the
description of the internal standard method.
Determination of calibration constants
For some of the methods described above, there are resi-
dual experimental difficulties in the determination of the
calibration constants with a resultant decrease in accuracy.
For the external standard method, the determination of K
relies on (i) data from material which is 100% crystalline
(or at least with known crystallinity) or (ii) having a sam-
ple with the phase of interest present in known concentra-
tion.
Rather than estimating K from individual phases in a
sample where there may be uncertainty about their con-
centration or sample related effects such as microabsorp-
tion are present, it is possible to use a whole-sample ap-
proach to its determination. By summing Eq. (6) for all n
phases in the sample, the following relationship is ob-
tained:
Pn
i¼1
Wi ¼ mm*
K
Pn
i¼1
SiðZMVÞi : ð12Þ
Determination of amorphous content via X-ray powder diffraction 949
Fig. 5. A portion of the diffraction pattern for Sample I showing the
individual contributions of corundum (thin solid line) and quartz
(thick solid line) to the observed data (dots). The total area under
each of the components is used to derive the degree of crystallinity
(DOC).
Fig. 6. A section of the XRD pattern for Sample A showing the re-
fined fit using the cristobalite structure for the amorphous component.
The upper and lower peak markers show the positions of the corun-
dum and cristobalite reflections, respectively.
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By rearrangement, K can be calculated from:
K ¼ mm*
Pn
i¼1
SiðZMVÞi
Pn
i¼1
Wi
: ð13Þ
For samples which are assumed to be fully crystalline, or
those samples in which an amorphous or PONKCS phase
makes up the balance of the sample, Eq. (13) can be sim-
plified by setting
P
Wi to unity. In this study, the expres-
sion can be further simplified by ignoring mm* since it is
effectively constant for all samples in this series. Then, the
value of K is simply the
P
(SiZMVi) for all analysed
phases. Note, however, that this is not a generalisation and
mm* needs to be retained if the sample suite being exam-
ined has variable absorption contrast.
In this example, only the corundum and quartz have
well defined crystal structures and hence known values for
ZMV. A modified value of K, denoted K0 ¼P(SiZMVi) for
corundum and quartz only (i.e. the known and analysed
crystalline components in the sample) can be calculated. A
plot of K0 (Fig. 7) shows that, as the amorphous content
increases, the amount of scattering accounted for in
Eq. (13) decreases and hence there is a decrease in the
calculated value for K0.
In a suite of samples which have been prepared for the
purpose of calibration such as those discussed here, it is
possible to determine the true value of K by calculating a
line of best fit to some or all of the calculated K0 values
as a function of the known amorphous content. The inter-
cept then represents K for a sample with no amorphous
content.
The difference between K0 for each sample and the true
value of K can be used to estimate the amorphous/uniden-
tified content using the relationship:
Wamorphous ¼ 1� K
0
K
: ð14Þ
While this approach is useful when a set of appropriate
standards with known amorphous contents can be pre-
pared, it may be cumbersome when a limited number of
samples need to be analysed.
For the PONKCS method, it is normally necessary to
obtain an estimation of an empirical ZMV using a known
addition of a standard to a “pure” sample of the phase of
interest. In many cases, it is not possible to obtain a pure
sample, or even one with known concentration. However,
if the sample suite being analysed has amorphous material
or a PONKCS phase with (i) the same composition and
(ii) a wide range of concentration, an empirical ZMV can
be derived without the need to prepare additional stan-
dards simply by using the natural variability of phases in
the samples. This requires the following steps during ana-
lysis:
1. Collect the diffraction data from the sample suite
under the same set of instrumental conditions. This
is critically important since changes to the instru-
ment intensity will impose their own variation in the
derived value of K.
2. Conduct the Rietveld analysis and extract the Riet-
veld scale factor for each phase (including the amor-
phous or PONKCS phase).
3. Define an arbitrary value for the empirical ZMV for
the amorphous/PONKCS phase.
4. For each sample, calculate an initial value of K using
all phases in the sample (including the amorphous/
PONKCS phase) using Eq. (13) with
P
Wi set to
unity (since all material is now included in the ana-
lysis). If the chemical composition of the samples is
variable, a value for mm* for each sample must be
included in these calculations.
5. Obtain a mean and standard deviation (SD) for K
over all samples. Since K is an experiment constant,
it should be the same for all samples. However, at
this stage K will vary widely and the SD should be
relatively large due to the inaccuracy in the initial
arbitrary value of ZMV chosen for the amorphous
phase.
6. Now adjust the empirical ZMV for the amorphous
phase in order to minimise the SD of the calculated
K values. One easily accessible way of achieving
this is to use the Solver Tool in MicroSoft-Excel.
7. This empirical ZMV value can then be used in
Eq. (3) while the optimised K value can be used in
the external standard method embodied in Eq. (6)
provided that data from subsequent samples has
been collected under the same instrumental condi-
tions as those used in its determination.
This whole sample approach to the determination of
these constants eliminates the need to prepare standard
samples since it only relies on knowledge of (i) refined
Rietveld scales and ZMV values for the crystalline phases
and (ii) an overall scale factor (or other measure of inten-
sity) for the amorphous/PONKCS phase. Prior knowledge
of the concentration of any phase is not required but a
wide variability in phase concentration will serve to im-
prove stability in the determination of K and the empirical
ZMV.
Assessment of methods
Figure 8 shows the measured values compared to the
weighed values for all samples included in the study.
Superficial inspection shows that all methods provide rea-
950 I. C. Madsen, N. V. Y. Scarlett and A. Kern
Fig. 7. Plot of
P
(SiZMVi) for corundum and quartz with respect to
silica flour concentration.
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sonable approximations of the amorphous content at most
concentration levels. However, Fig. 8 is not adequate for
examination of the detailed performance of each of the
methods.
In order to better assess the individualperformance of
each method, the bias in the amorphous content was cal-
culated from:
Bias ¼ Analysed �Weighed : ð15Þ
Figure 9 shows that most of the analyses fall within
�2 wt% of the weighed amount. However, this is an abso-
lute measure and does not take into account the level at
which the amorphous content is present. An absolute error
of 2 wt% is appreciably less significant in a phase present
at the 50 wt% level where it represents a relative bias of
4% than it is in a phase present at the 1 wt% level where
the relative bias is now 200%.
The reported concentration in those samples where no
amorphous material is present is useful in assessing the
lower limit of detection. Sample I, which contained no
amorphous material, was used to assess the level of bias
for each of the methods. Figure 10 shows the bias re-
turned for each of the replicates and each of the methods
considered here. The most poorly performing method for
this sample was the external standard method, with consis-
tent reporting in excess of 2 wt% amorphous in a sample
containing none. The single peak and LCM methods were
the best performers, but this is to be expected given that
the sample with no added silica flour was used as part of
their calibration routine. The other methods returned val-
ues of around þ0.5 wt% for amorphous content when none
is present.
While Figs. 9 and 10 provide the detailed outcomes of
all analyses included in this study, it is difficult to esti-
mate which method provided the “best” overall outcome.
To assist in this, the square of the bias was averaged over
all 27 determinations to provide a single figure by which
different methods could be compared (Fig. 11). Compari-
son of the results indicates that the external standard,
DOC and full-structure methods are the least accurate,
with PONKCS, single peak, LCM and internal standard
methods all performing at a similar, more accurate, level.
While Fig. 11 provides a measure of the overall level of
accuracy, it does not give any indication regarding the pro-
pensity of a method to under- or over-estimate the amor-
phous content. Figure 12 on the other hand, shows the
average bias for each method and reveals that the DOC
method returns a small negative value while all other meth-
ods consistently overestimate the amount of amorphous
material.
Factors affecting accuracy
During data analysis, it became apparent that a number of
factors had the potential to reduce the overall accuracy of
Determination of amorphous content via X-ray powder diffraction 951
Fig. 8. Plot of the measured vs weighed values (wt%) of silica flour
for all samples.
Fig. 9. Plot of the bias vs. the weighed values of silica flour for all
samples in the analysis.
Fig. 10. Assessment of accuracy of individual methods for the deter-
mination of amorphous content in Sample I (which contained no
amorphous material).
Fig. 11. Plot of the average of the square of the bias for the 27 deter-
minations (9 samples � 3 replicates) for each method. The thin verti-
cal lines represent the standard deviation of the mean.
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some methods. In particular, the atomic displacement para-
meters (ADPs) in crystal structure models used to generate
the calculated patterns (in this case for corundum and
quartz) correlate strongly with the Rietveld scale factor.
Hence, the ADP values used during analysis will impact
on the final phase abundances derived from these scale
factors.
XRD data from pure sub-samples of quartz and corun-
dum were collected under the same conditions as for the
test samples. During analysis of these data, full crystal
structure refinement was undertaken including atom coor-
dinates and individual ADPs. For corundum, all refined
values were within 1–2 standard deviations of the reported
values in the structure database [34]. Hence, the analysis
proceeded with reported database values.
For quartz, the database reported ADPs of about 0.4 A˚2
for both Si and O. However, the refined ADPs were sig-
nificantly higher with values of 0.83 and 1.24 A˚2 for Si
and O respectively. The refined ADPs brought improve-
ments in Rwp from 9.54 to 7.79% and in RBragg from 6.32
to 3.83%. These improvements were sufficient to convince
the authors that the refined ADPs, in spite of being higher
than expected, more accurately reflected the details of the
structure of the material used in this study than those re-
ported in the database. Scrutiny of the refined quartz pat-
tern parameters failed to identify any other potential
sources of error in the refinement that might be causing
the high ADPs.
All data were reanalysed using the refined ADPs for
quartz while the corundum ADPs were held at the data-
base values. Figure 13 and Table 2 show the outcome of
the analysis. For those methods that derived the amor-
phous content by difference (internal and external standard
methods) there is a significant decrease in accuracy, espe-
cially at the lower levels of concentration.
This accrues from (i) an increase in the quartz scale
factor to compensate for the “loss” of calculated intensity
due to the higher ADPs and (ii) a subsequent increase in
the analysed quartz content using Eq. (3). Hence, there is
a subsequent decrease in the corundum content due to nor-
malisation of the total weight fraction of unity. This effect
is compounded when the corrected values are calculated
according to Eq. (4).
For the analyses undertaken using the refined ADPs for
quartz, the analysed corundum content derived was less
than the weighed amount for most of the samples with
low concentrations of silica flour. The normalisation pro-
cess then results in the large negative estimates of amor-
phous content for these samples (Fig. 13). These negative
values were well beyond the expectations of random error.
The use of the internal standard in this manner shows
that it is accurate at high concentrations of amorphous ma-
terial. In the limiting case of Sample A, which is reduced
to a binary mixture of two crystalline phases, the analysis
will be exact. However, it is highly sensitive at low con-
centrations to aberrations in the Rietveld analysis condi-
tions, especially those which impact on the calculated in-
tensity.
It is worth highlighting the cause of these errors. Fig-
ure 14 shows the raw XRD data for pure corundum and
quartz. For corundum, the intensity decrease as a function
of 2q is relatively small (a factor of �7x for peaks above
and below 70� 2q). Hence, there is strong observed inten-
sity at high angles to stabilise the refinement of parameters
which have a strong angular dependence, such as the
ADPs. However, for quartz, the observed data is dominated
by a few strong low angle peaks with a large decrease in
intensity at higher angles (�40x). Any factors, such as
extinction, preferred orientation or graininess, which affect
the intensity of the low angle peaks intensities will conse-
quently have a strong impact on the refined ADPs.
Therefore, care must be taken when refining parameters
which correlate strongly with the Rietveld scale factor. It
952 I. C. Madsen, N. V. Y. Scarlett and A. Kern
Fig. 12. Plot of the sum of average bias for the 27 determinations
(9 samples � 3 replicates) for each method. The vertical lines represent
the standard deviation of the mean.
Fig. 13. Plot of the bias for 27 determinations (9 samples, 3 repli-cates) of amorphous content. The difference between this plot and
Fig. 9 relates to the use of published ADPs for Si and O for quartz in
Fig. 9 and refined ADPs in this plot.
Table 2. Sum of the square of the bias for all methods except the Single Peak method.
Quartz ADP’s Internal Std External Std LCM PONKCS DOC Full Structure
Database 7.2 45.7 5.6 10.2 26.7 19.7
Refined 121.4 100.9 5.1 10.4 28.3 11.1
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is further worth highlighting that some entries for ADPs in
databases are set to arbitrary values such as 0.5 or 1.0 A˚2.
Verification of the database or refined values is a critically
important step at this stage of the analysis if the highest
levels of accuracy are to be obtained.
Conclusions
For the determination of amorphous material, in general,
the problem will dictate the method(s) used. All methods
discussed in this study are, in principle, capable of deter-
mining the concentration of what has been defined as
amorphous material in mixtures with the similar levels of
accuracy and precision as for crystalline phases in general;
in ideal cases even down to 1% absolute or better. The
limitations are the same as for QPA of crystalline phases
and are dictated by sample properties and the analytical
techniques used. Table 3 provides a summary of the re-
quirements and applicability of the methods described
here.
Intensity contributions of amorphous phases to the dif-
fraction pattern are not always evident, especially at low
concentrations. Even when their presence is apparent, it
can be difficult to resolve their contribution other compo-
nents of the diffraction pattern such as pattern back-
ground. However, Williams et al. [39] have recently de-
monstrated that is is possible to distinguish and quantify
two amorphous phases in a series of geopolymers. In addi-
tion, it is reasonable to assume that all materials possess a
non-diffracting surface layer with some degree of disorder
or contain some surface reaction products and adsorbed
species. Such a layer can easily account for a mass frac-
tion of �1 wt% percent in a finely divided solid as the
amount of this surface layer relative to the bulk sample
will increase as the particle size decreases [3].
Some recommendations resulting from this study in-
clude:
1. Where the intensity contribution of the amorphous
content to the diffraction pattern is not evident, it is
better to use one of the indirect methods (internal or
external standard method).
2. For indirect methods, any errors in the analysis of
the crystalline phases will decrease the overall accu-
racy attainable since the amorphous phase is deter-
mined by difference.
3. Where intensity contributions of amorphous phases
are evident, any method based on the direct model-
ling of the amorphous component provides im-
proved accuracy relative to the indirect methods.
4. Calibration based methods usually have the potential
to achieve the highest accuracy, as many residual
aberrations in the data, for example, microabsorp-
tion, are now included in the calibration function.
Caution is advised here as the magnitude of these
residual errors may change with different sample
suites so a calibration function derived for one sam-
ple suite may not be generally applicable.
5. The analysis of single samples do not usually afford
the luxury of making an extensive calibration suite
Determination of amorphous content via X-ray powder diffraction 953
Fig. 14. Raw XRD data (CuKa) for corundum (upper) and quartz
(lower).
Table 3. Summary of the applicability of various methods for the determination of amorphous content.
Method Approach Standardisation Residual errors
absorbed in calibration?
Multiple amorphous
phases?
Single Peak Direct Calibration suite Yes Yes
Internal Standard Indirect Addition of standard of known crystallinity
to each sample
No No
External Standard Indirect Need data from standard of known crystallinity* No No
LCM Direct Calibration suite Yes Yes
PONKCS Direct Addition of standard of known crystallinity
to single sample *
Yes Yes
DOC Direct Case dependant No Yes
Full Structure Direct Structures from literature No Yes
* Note potential for whole sample calibration approach discussed in Section “Determination of calibration constants”.
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so the external standard, DOC or full structure
methods may be the most relevant.
6. Usually a sample of pure amorphous material, or a
sample where the amorphous content is high, is re-
quired to establish an accurate model for direct
methods dependent upon the measurement of a
scale factor for the amorphous component.
The traditional Rietveld method using Eq. (3) delivers
relative phase amounts by default and in the presence of
amorphous/unidentified crystalline phases, the analysed
crystalline weight fractions may be significantly overesti-
mated. Most phase abundances reported in literature,
obtained via Rietveld analysis, are provided in a manner
suggesting absolute values. However, where no specific
allowance for amorphous/unidentified phases has been
made and reported, it is better to assume that the reported
values are correct relative to one another, but may be over-
estimated. Therefore, standard practise in QPA should be
to use methodology which produces absolute rather than
relative phase abundances. Any positive difference be-
tween unity and the sum of the absolute weight fractions
will alert the analyst to the presence of non-analysed ma-
terial in the sample.
References
[1] Pecharsky, V. K.; Zavalij, P. Y.: Fundamentals of Powder Dif-
fraction and Structural Characterisation of Materials. 1 ed. 2003,
Massachusetts: Kluwer Academic Publishers. 713.
[2] Klug, H. P.; Alexander, L. E.: X-ray diffraction procedures: for
polycrystalline and amorphous materials. 1974, New York: Wi-
ley. 966.
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