the atomic layers without any microscopic measurements. This process is done for different points on a
single wafer and because this process is fast when
using synchrotron sources that provide a large flux of
X-rays, materials scientists can rapidly characterize
hundreds or thousands of materials a day using X-ray
diffraction. An example of the apparatus for taking
these XRD measurements can be seen in figure 2.
A key challenge in high-throughput materials discovery is to solve the phase-mapping problem, which
identifies the characteristic XRD patterns of the
materials (or basis patterns or crystal structures of the
materials) that demix the XRD signals from the high-throughput experiments, some signals of which may
be pure in that they represent one phase, while others are a linear combination of the pure phases. A
visual description of the phase-mapping problem can
be seen in figure 3.
Mathematically, the measured XRD pattern in the
j-th sample point can be characterized by a one
dimensional signal Aj(q). The scattering vector mag-
nitude (q) is a monotonic transformation of the dif-
fraction angle, and is directly related to the spacing
of atoms in a crystal. The phase-mapping problem is
to find a small number of phases W1(q),…, WK(q),
and their corresponding activation coefficients hij,
such that the XRD patterns at each sample point can
be explained by a linear combination of phases:
Aj q ( ); hijWi
; ;ij,q ( )
The physical process of alloying complicates the linear combination by introducing additional scaling
factors λij. Alloying typically can be approximated by
a multiplicative scaling of the XRD pattern of a
specific phase in the q domain; we also refer to this
process as peak shifting. We use the term Wi(λij,q) to
allow for the phases to scale slightly according to
parameter λij at each sample point. In addition to
peak shifting, there are a number of other constraints
on the solution of the phase-mapping problem, arising from the fact that the solution must describe a
system constrained by the laws of physics. One
important precept is the Gibbs phase rule, which limits the number of phases present to at most k phases
per sample point, in a system involving k elements.
Therefore, in a k-element system, no more than k
coefficients among hij for fixed j may be nonzero.
Additionally, feasible spatial variation of hij by composition, as well as the shapes that each Wi may take,
are constrained by the relevant physics.
Fundamentally novel techniques are required to
solve the phase-mapping problem quickly and accurately. Historically, the phase-mapping problem has
been solved by hand, which can take days or months
for a single system, and has become the bottleneck of
the entire materials discovery workflow. A number of
automatic techniques have been developed in recent
years, which can be broadly grouped into clustering,
constraint reasoning, and factor decomposition
approaches. Proposed clustering methods such as
hierarchical clustering (Long et al. 2007), dynamic
time warping kernel clustering (Le Bras et al. 2011),
and mean shift theory (Kusne et al. 2014) produce
maps of phase regions, but fail to resolve mixtures or
Figure 2. Apparatus for Taking XRD Measurements.
To the left is an image of the X-ray diffraction apparatus being used to characterize a composition library in the high-throughput experiment. On the right is an illustration of X-ray diffraction. Incoming X-rays hit the composition sample. They then diffract and are detected by the XRD detector.