Spinh Manifolds

Spinh Manifolds††This paper is a contribution to the Special Issue on
Differential Geometry Inspired by Mathematical Physics in honor of Jean–Pierre
Bourguignon for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html

H. Blaine LAWSON Jr.

H.B. Lawson Jr.

Stony Brook University, Stony Brook NY, USA<EMAIL_ADDRESS>https://www.math.stonybrook.edu/~blaine/

Received January 25, 2023, in final form March 06, 2023; Published online
March 19, 2023

The concept of a ${\rm Spin}^{h}$-manifold, which is a cousin of Spin- and
${\rm Spin}^{c}$-manifolds, has been at the center of much research in recent
years. This article discusses some of the highlights of this story.

Spin-manifold; ${\rm Spin}^{c}$-manifold; obstructions; embedding theorems;
bundle invariants; ABS-isomophism

53C27; 55P99

It is a great pleasure for me to contribute a paper to this volume dedicated
to Jean-Pierre Bourguignon on his seventy-fifth birthday. Jean-Pierre has been
a mathematical colleague and a dear friend for most of my mathematical life,
and throughout that time I have been in awe of his abilities to support
mathematics, and science in general, in so many ways.

Our various interests in mathematics have been close over the years, from
manifolds of negative curvature, minimal varieties, Yang–Mills fields, spin
manifolds and Dirac operators. I want to say that the best result in our
collaboration on Yang–Mills [10, 11] was entirely Jean-Pierre’s.

In light of his great interest in spin geometry, I thought it would be
appropriate to recount some recent work in that area by various young people
at Stony Brook.

Let’s begin by recalling that for $n\geq 3$ the group ${\rm Spin}(n)$ is the
universal covering group

$\pi\colon\ {\rm Spin}(n)\to{\rm SO}(n)$

of the special orthogonal group. An oriented Riemannian manifold $X$ of
dimension $n$ is a Spin-manifold if there exists a principal ${\rm
Spin}(n)$-bundle $P_{\rm Spin}\to X$ on $X$, and a ${\rm Spin}(n)$-equivariant
bundle map

$P_{\rm Spin}\to P_{\rm SO}(X),$

where $P_{\rm SO}(X)$ is the bundle of positively oriented, orthonormal
tangent frames on $X$. Each such “square root” of the principal bundle $P_{\rm
SO}(X)$, up to equivalence, is called a Spin-structure on $X$. The existence
of such structures depends only on the topology of $X$. That is, Spin-
structures exist for any Riemannian metric on $X$ if and only if the first two
Stiefel–Whitney classes satisfy $w_{1}(X)=w_{2}(X)=0$.

Now the notion of a Spin-manifold played an important role in the
establishment of the Atiyah–Singer index theorem. It was a result of Atiyah
and Hirzebruch [5] that every compact Spin-manifold has the property that its
$\hat{\mathbb{A}}$-genus is an integer. This characteristic invariant is
generally a rational number. (For example, for the complex projective plane
${\mathbb{P}}^{2}({\mathbb{C}})$, it is $-1/8$.) Atiyah thought that this
might be a consequence of the fact that $\hat{\mathbb{A}}(X)$ was the index of
an elliptic operator on $X$ that could not be defined on non-Spin manifolds.
Indeed Singer found such an operator, and this lit the path to the theorem and
its proof.

These operators were obtained by defining vector bundles on $X$ as associated
bundles

${\mathbf{V}}_{\lambda}\equiv P_{\rm Spin}\times_{\lambda}V\to X,$

where $\lambda\colon{\rm Spin}(n)\to{\rm SO}(V)$ is a finite-dimensional
representation of ${\rm Spin}(n)$ which is not pulled back from a
representation of SO$(n)$. Such representations can be obtained from
representations of the Clifford algebra ${\rm
C}\ell({\mathbb{R}}^{n},\langle\cdot,\cdot\rangle)$.

We recall [17] that a Riemannian manifold $(X,\langle\cdot,\cdot\rangle)$ has
a natural bundle of Clifford algebras ${\rm C}\ell(X)$ whose fibre at $x\in X$
is the Clifford algebra ${\rm C}\ell(T_{x}X,\langle\cdot,\cdot\rangle)$. For
Clifford algebra representations $\lambda$, the associated bundle
${\mathbf{V}}_{\lambda}$ is a bundle of modules for the bundle of Clifford
algebras ${\rm C}\ell(X)$. This fact allows one to define differential
operators via the Dirac construction.

It turns out that the notion of a Spin-manifold has analogues over the complex
numbers and the quaternions. In the complex case they are called ${\rm
Spin}^{c}$-manifolds, and they have been of central importance for quite some
time. Those in the quaternion case I will call ${\rm Spin}^{h}$-manifolds, and
they will be the center of discussion for the rest of the paper.

To begin we shall consider just the groups themselves (or, equivalently,
suppose the manifold to be just a point). Let’s start by writing ${\rm
Spin}(n)$ trivially as ${\rm
Spin}(n)\times_{{\mathbb{Z}}_{2}}{\mathbb{Z}}_{2}={\rm
Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm O}(1)$ where ${\mathbb{Z}}_{2}$ acts
diagonally. Then we can define (with ${\mathbb{Z}}_{2}$ always acting
diagonally)

$\displaystyle{\rm Spin}^{r}(n)={\rm Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm
O}(1),$ $\displaystyle{\rm Spin}^{c}(n)={\rm
Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm U}(1),$ $\displaystyle{\rm
Spin}^{h}(n)={\rm Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm Sp}(1),$

the real, complex and quaternionic Spin groups. Dividing further by the group
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ (where the original
${\mathbb{Z}}_{2}$ is the diagonal) gives us surjective homomorphisms

$\displaystyle{\rm Spin}^{r}(n)\longrightarrow{\rm SO}(n),$ $\displaystyle{\rm
Spin}^{c}(n)\longrightarrow{\rm SO}(n)\times{\rm U}(1),$ $\displaystyle{\rm
Spin}^{h}(n)\longrightarrow{\rm SO}(n)\times{\rm SO}(3).$

Now a ${\rm Spin}^{c}$-manifold is an oriented Riemannian $n$-manifold,
equipped with a principal ${\rm Spin}^{c}(n)$-bundle $P_{{\rm Spin}^{c}}$, and
a ${\rm Spin}^{c}(n)$-equivariant bundle map

$P_{{\rm Spin}^{c}}\to P_{{\rm SO}}(X)\times P_{{\rm U}(1)}(L),\qquad\text{
(fibre product)}$

where $L$ is a complex line bundle on $X$, called the canonical bundle of the
structure, and $P_{{\rm SO}}(X)$ is as above. Such a structure exists with
canonical bundle $L$ if and only if $w_{2}(X)=\rho(c_{1}(L))=0$ where $\rho$
is mod-2 reduction (cf. [17, p. 391]). Now complex line bundles on $X$ are in
1-to-1 correspondence with elements of $H^{2}(X,{\mathbb{Z}})$. Thus a
manifold carries a ${\rm Spin}^{c}$-structure if and only if $w_{2}(X)$ is the
mod-2 reduction of an integer class, which is equivalent to the third integral
Stiefel–Whitney class satisfying $W_{3}(X)=0$. Note that given one ${\rm
Spin}^{c}$-structure on $X$, we get all the rest by tensoring $L$ with complex
line bundles having $w_{2}=0$. We point out that every almost complex manifold
$X$ has a canonical ${\rm Spin}^{c}$-structure where $L$ is the anticanonical
bundle of $X$.

From this point of view ${\rm Spin}^{h}$-manifolds are defined analogously. A
${\rm Spin}^{h}$-manifold is an oriented Riemannian $n$-manifold, equipped
with a principal ${\rm Spin}^{h}(n)$ bundle $P_{{\rm Spin}^{h}}$, and a ${\rm
Spin}^{h}(n)$-equivariant bundle map

$P_{{\rm Spin}^{h}}\to P_{{\rm SO}}(X)\times P_{{\rm SO}(3)}(E),$

where $E$ is an oriented Riemannian 3-plane bundle on $X$, called the
canonical bundle of the structure. Such a structure exists if and only if one
can find such an $E$ with $w_{2}(X)+w_{2}(E)=0$.

Now for ${\rm Spin}^{h}$-manifolds there is an analogy with the fact that an
almost complex manifold is ${\rm Spin}^{c}$. A manifold of dimension $4n$ is
almost quaternionic if its structure group can be reduced to ${\rm
Sp}(n)\cdot{\rm Sp}(1)={\rm Sp}(n)\times_{{\mathbb{Z}}_{2}}{\rm Sp}(1)$. Every
such manifold has a canonical ${\rm Spin}^{h}$-structure. In dimensions 8$n$,
the manifold is in fact spin. In dimensions 4 (mod 8) the canonical bundle $E$
is a 3-dimensional subbundle of ${\rm End}(TX)$ which locally has an
orthonormal basis $I$, $J$, $K$ with $I^{2}=J^{2}=K^{2}=IJK=-{\rm Id}$. See
[1, Remark 2.2] for a detailed discussion.

Note there are natural homomorphisms ${\rm Spin}(n)\to{\rm Spin}^{c}(n)\to{\rm
Spin}^{h}(n)$. This shows that any Spin-manifold has a ${\rm
Spin}^{c}$-structure by choosing the canonical bundle to be trivial, and any
${\rm Spin}^{c}$-manifold has a ${\rm Spin}^{h}$-structure with canonical
bundle $L\oplus{\mathbb{R}}$. Furthermore, the notion of a ${\rm
Spin}^{c}$-manifold is definitely more general than Spin, since
${\mathbb{P}}^{2}({\mathbb{C}})$ is complex and therefore ${\rm Spin}^{c}$,
but not Spin, as we have discussed.

In a similar vein there are manifolds which are not ${\rm Spin}^{c}$ but are
${\rm Spin}^{h}$. A basic example is the Wu manifold $X={\rm SU}(3)/{\rm
SO}(3)$ which is not ${\rm Spin}^{c}$ ($W_{3}\neq 0$), but is ${\rm
Spin}^{h}$. This was first noticed by Xuan Chen in his thesis [12, p. 26ff].
The general question of understanding manifolds with this property is treated
in great generality in the work of Albanese and Milivojević below.

Now just like Spin- and ${\rm Spin}^{c}$-manifolds, ${\rm Spin}^{h}$-manifolds
also carry Dirac operators. Let ${\rm C}\ell_{n}$ denote the Clifford algebra
of ${\mathbb{R}}^{n}$ with the standard inner product, and set

${\rm C}\ell_{n,{\mathbb{H}}}\equiv{\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}}.$

Then there is an embedding

${\rm Spin}^{h}(n)\subset{\rm C}\ell_{n,{\mathbb{H}}}$

given as the projection of ${\rm Spin}(n)\times{\rm Sp}(1)\subset{\rm
C}\ell_{n}\times{\mathbb{H}}$. Suppose now that $X$ is a ${\rm
Spin}^{h}$-manifold. Then given any representation $\rho\colon{\rm
C}\ell_{n,{\mathbb{H}}}\to{\rm End}(V)$ on a finite-dimensional vector space
$V$, we restrict to ${\rm Spin}^{h}$ and form the associated bundle

${\bf V}_{\rho}=P_{{\rm Spin}^{h}}\times_{\rho}V.$

If the representation $\rho$ is irreducible, the bundle ${\bf V}_{\rho}$ is
called fundamental. There is always a Dirac operator defined on sections of
${\bf V}_{\rho}$. In certain dimensions the restriction of an irreducible
$\rho$ to the even part of ${\rm C}\ell_{n,{\mathbb{H}}}$ splits into two
irreducible representations, and the Dirac operator interchanges the sections
of the corresponding bundles. For example, this happens when $n\equiv 0$
$({\rm mod}\ 8)$. For this question, and other as well, there is a periodicity
of order 8 in dimensions, as in the real case. However one must be careful. An
irreducible real representation of ${\rm C}\ell_{n}$, when tensored with
${\mathbb{H}}$, may not be an irreducible real representation of ${\rm
C}\ell_{n,{\mathbb{H}}}$.

There are ${\rm C}\ell_{n,{\mathbb{H}}}$-linear Dirac operators with indices
in ${\rm KSp}_{*}$ which has been worked out by Jiahao Hu [16] and is
discussed below.

There is an interesting story surrounding Spin-manifolds and their cousins. It
was a “folk” theorem for some years that an oriented Riemannian manifold $X$
has a Spin-structure if and only if there exists a vector bundle of real ${\rm
C}\ell(X)$-modules $\cancel{S}\to X$ which are irreducible at each point. That
such a bundle $\cancel{S}$ exists on a Spin-manifold was clear, and I asked my
graduate student Xuan Chen, some years ago, to find a good geometric proof of
the converse.

Xuan first showed that if one replaces Spin-structures with ${\rm
Spin}^{c}$-structures, and the existence of a real locally irreducible ${\rm
C}\ell(X)$-module with the existence of a complex one, then the converse is
always true. Furthermore, the proof of this result has a very geometric
flavor. Xuan also found a proof in the real case whenever the dimension of $X$
is congruent to 0, 6 or 7 (mod 8). However, to my great surprise he showed
that in all other dimensions $>8$ the converse is false!

Xuan then came up with the notion of a ${\rm Spin}^{h}$-structure on a
manifold, as explained above. He showed that the canonical bundles $E$ are
characterized by the condition on the Stiefel–Whitney classes that:
$w_{2}(TX)=w_{2}(E)$. He then showed that ${\rm Spin}^{h}$-structures can
exist on manifolds which have no ${\rm Spin}^{c}$-structure, and in dimensions
2, 3, or 4 (mod 8), a ${\rm Spin}^{h}$-structure implies the existence of a
real irreducible ${\rm C}\ell(X)$-module.

I found this idea of a quaternionic analogue of a ${\rm Spin}^{c}$-manifold
enticing. Michael Albanese and Aleksandar Milivojević, students at Stony Brook
at the time, showed that the primary obstruction to the existence of a ${\rm
Spin}^{h}$-structure is the fifth integral Stiefel–Whitney class $W_{5}$. They
then showed that:

  1. Every compact oriented manifold of dimension $\leq 7$ admits a ${\rm Spin}^{h}$-structure.

  2. However, in every dimension $\geq 8$, there are infinitely many homotopy types of compact simply connected manifolds which do not admit ${\rm Spin}^{h}$-structures.

They have similar results for non-compact manifolds:

  1. Every oriented $5$-manifold is ${\rm Spin}^{h}$.

_Note:_ Every oriented 4-manifold is ${\rm Spin}^{c}$ [15] (see also [19, 20],
for compact manifolds, and the case of non-compact manifolds was done in
[25]), and every oriented 3-manifold is Spin:

  1. An oriented manifold of dimension 6 or 7 is ${\rm Spin}^{h}$ if and only if the primary obstruction $W_{5}$ vanishes.

See [1, 2] for these results, and [2] for further discussion of the non-
compact case.

It turned out that the notion of ${\rm Spin}^{h}$-manifolds had been found
previously (but the results mentioned here are new). See, for example, [9,
21]. Recent math papers include [1, 2, 12, 13, 18].

Now in fact, it turned out that physicists had also found ${\rm
Spin}^{h}$-manifolds, and were interested in them for physical reasons. See,
for example, [13, 23], and in dimension 4 [7, 8, 14, 26]. In [22], a ${\rm
Spin}^{h}$-structure on a 4-manifold was used in constructing a quaternionic
version of Seiberg–Witten theory.

In fact the work in [1] goes much further than my ${\rm Spin}^{h}$ discussion.
They define a notion of a ${\rm Spin}^{k}$-manifold for all integers $k>0$
where $k=1,2,3$ correspond to the three cases above. For $k\geq 3$ the group
is defined by

${\rm Spin}^{k}(n)\equiv{\rm Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm Spin}(k)$

with ${\mathbb{Z}}_{2}$ acting diagonally. They prove that:

  1. For every $k$ there exists a compact simply connected smooth manifold which does not admit a ${\rm Spin}^{k}$-structure.

Their results on ${\rm Spin}^{k}$ manifolds are non-trivial and basic for
their results in the ${\rm Spin}^{h}$ case. They also have the corollary that:

  1. There is no integer $k$ such that every manifold embeds into a Spin-manifold with codimension $k$.

Note that:

  1. Every manifold $X$ embeds into a orientable manifold with codimension $1$

by embedding $X$ as the zero-section of its orientation line bundle. Asking
for the spin analogue of this was part of the motivation for examining ${\rm
Spin}^{k}$-manifolds, and the answer they found was striking.

Recently ${\rm Spin}^{h}$-manifolds played a role in the work of Freed and
Hopkins [13] which discusses quantum field theories, Thom spectra, and much
else. In their paper, ${\rm Spin}^{h}$-manifolds, Thom classes of ${\rm
Spin}^{h}$-bundles, ${\rm Spin}^{h}$-cobordism, its natural transformation to
symplectic ${\rm K}$-theory, etc. are all defined, but as special cases of a
more general discussion. The ${\rm Spin}^{h}$-case appears, for example, in
case 4 of Table 9.35. Certainly this shows that ${\rm Spin}^{h}$-theory is
relevant in modern quantum field theories.

Very recently, with the work of Jiahao Hu in his Stony Brook thesis [16],
${\rm Spin}^{h}$-manifolds have become a basic tool in topology.

This resulted from his addressing the following problem: Find enough
invariants to know that two real vector bundles over a space $X$ are stably
isomorphic. Of course the bundles must have the same characteristic classes,
but that is not enough. (For example, there exists a non-stably trivial real
vector bundle over $S^{9}$ with vanishing Stiefel–Whitney and Pontryagin
classes.) Jiahao Hu has solved this problem and done it in a very geometric
way, using “integration over cycles”.

Now stable equivalence classes of real bundles on a reasonable space $X$ form
the real ${\rm K}$-ring ${\rm KO}(X)$ which expands to a generalized
cohomology theory ${\rm KO}^{*}(X)$. One of Hu’s important contributions was
showing that:

  1. Classes in a generalized cohomology theory are determined by their periods over cycles for the “Anderson dual theory”.

Now the Anderson dual of ${\rm KO}$-theory turns out to be symplectic ${\rm
K}$-theory ${\rm KSp}^{*}$, which is defined like ${\rm KO}^{*}$ but by using
quaternionic vector bundles in place of real ones [3].

This led Hu to a study of symplectic ${\rm K}$-theory and ${\rm
Spin}^{h}$-cobordism with many new results. One of his basic theorems is that:

  1. The Atiyah–Bott–Shapiro isomorphism [4] holds for quaternionic Clifford modules and symplectic ${\rm K}$-theory.

Jiahao then defined a Thom class for ${\rm Spin}^{h}$-vector bundles in
symplectic K-theory, using
$(\Delta^{+}-\Delta^{-})\otimes_{\mathbb{R}}{\mathbb{H}}$ where $\Delta^{\pm}$
are the real irreducible spinor bundles for ${\rm Spin}(8n)$ and
${\mathbb{H}}$ is the canonical representation of ${\rm Sp}(1)$. He
constructed the Thom spectrum ${\rm MSpin}^{h}$ for ${\rm
Spin}^{h}$-cobordism, and defined a map

${\rm ind}^{h}\colon\ {\rm MSpin}^{h}\longrightarrow{\rm KSp},$

which allowed him to define the cycles needed for real ${\rm K}$-theory. (As
he found out later, and as mentioned above, these things had already been done
in [13].)

His next important step was to show that:

  1. There exist Dirac operators on ${\rm Spin}^{h}$-manifolds with indices in ${\rm KSp}_{*}({\rm pt})$.

To do this he followed Atiyah–Singer’s index theorem for elliptic operators on
${\rm C}\ell_{k}$-bundles, where the algebra ${\rm C}\ell_{k}$ acts
universally on the bundle [6]. (One could also see [17, p. 139ff].) In this
context, for the Dirac operator on a Spin $n$-manifold $X$, one defines a
${\rm C}\ell_{n}$-bundle

${\mathbb{S}}(X)=P_{{\rm Spin}}(X)\times_{\ell}{\rm C}\ell_{n},$

where $\ell\colon{\rm Spin}(n)\to{\rm End}({\rm C}\ell_{n})$ is given by left
multiplication, and ${\rm C}\ell_{n}$ acts on the bundle by right
multiplication. The kernel is a real ${\mathbb{Z}}_{2}$-graded ${\rm
C}\ell_{n}$-module which can be identified with ${\rm KO}_{n}$(pt) by the ABS
isomorphism.

On a ${\rm Spin}^{h}$-manifold $X$ one defines

${\mathbb{S}}_{{\mathbb{H}}}(X)=P_{{\rm Spin}^{h}}(X)\times_{\ell}({\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}}),$

where $\ell\colon{\rm Spin}^{h}\equiv{\rm
Spin}(n)\times_{{\mathbb{Z}}_{2}}{\rm Sp}(1)\longrightarrow{\rm End}({\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}})$ is given by left Clifford
multiplication of Spin($n$) on ${\rm C}\ell_{n}$ and of Sp(1) on
${\mathbb{H}}$. This is a bundle of ${\mathbb{Z}}_{2}$-graded ${\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}}$-modules. The standard Dirac
operator commutes with this action of ${\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}}$, and so its analytic index lies in
the Grothendieck group of ${\mathbb{Z}}_{2}$-graded $({\rm
C}\ell_{n}\otimes_{\mathbb{R}}{\mathbb{H}})$-modules, which is isomorphic to
${\rm KSp}_{n}$(pt) by the quaternionic version of the ABS isomorphism. One
now needs the symplectic version of the Atiyah–Singer index theorem, which
Jiahao provides in [16].

From here Jiahao was able to prove the following:

  1. ${\rm Spin}^{h}$-manifolds provide enough cycles for symplectic ${\rm K}$-theory to distinguish real vector bundles up to stable equivalence.

By a Spin-cycle on a space $Z$ (a finite CW-complex) we mean a map $f\colon
X\to Z$ where $X$ is a compact Spin-manifold. For a real vector bundle $E\to
Z$ we define its period over the Spin-cycle $f$ to be $\langle
f,E\rangle\equiv$ the ${\rm KO}$-index of the Dirac operator on $X$ twisted by
$f^{*}E$, which takes values in ${\mathbb{Z}}$ or ${\mathbb{Z}}_{2}$ depending
on dimension. A ${\rm Spin}^{h}$-cycle and the period of a real vector bundle
over a ${\rm Spin}^{h}$-cycle are defined analogously using ${\rm
Spin}^{h}$-manifolds and the Dirac operator on ${\rm Spin}^{h}$-manifolds.

We think of these as integer invariants. There are also torsion invariants
defined in much the same way. The assertion above says that the periods of a
real vector bundle $E$ over ${\rm Spin}^{h}$-cycles and torsion ${\rm
Spin}^{h}$-cycles determine the stable isomorphism class of $E$.

The proof uses the fact that Jiahao’s quaternionic version of the
Atiyah–Bott–Shapiro isomorphism is equivariant with respect to the real ABS
isomorphism. This implies that ${\rm ind}^{h}$ above is equivariant with
respect to the Atiyah–Bott–Shapiro orientation ${\rm ind}\colon{\rm
MSpin}\to{\rm KO}$. Since
ind${}^{h}\big{(}{\mathbb{P}}^{1}({\mathbb{H}})\big{)}$ generates ${\rm
KSp}_{4}({\rm pt})\cong{\mathbb{Z}}$, one can transfer periods of real bundles
defined over Spin-cycles $f\colon X\to Z$ to periods over ${\rm
Spin}^{h}$-cycles $f\times f_{0}\colon
X\times{\mathbb{P}}^{1}({\mathbb{H}})\to Z\times{\rm pt}$.

This work gives an answer to the question posed above. The history of this
question goes back to Dennis Sullivan’s study of Hauptvermutung for manifolds
[24]. For that purpose, he analyzed ordinary cohomology, complex ${\rm
K}$-theory and real ${\rm K}$-theory at odd primes. His arguments relied on
knowing that these theories are Anderson self-dual and that, for every $p$,
the associated theories with ${\mathbb{Z}}_{p}$-coefficients have a cup
product. Both of these fail for real ${\rm K}$-theory at prime 2. Jiahao
overcame this difficulty. His result at the prime 2 is really surprising.

It is common in topology that prime 2 is more difficult than odd primes.
Jiahao formulated and proved a general statement about determining cohomology
classes in a generalized cohomology theory by their periods over cycles for
its Anderson dual theory. In the real ${\rm K}$-theory case, he needed cycles
for symplectic ${\rm K}$-theory to define invariants taking values in
${\mathbb{Z}}$ and ${\mathbb{Z}}_{2}$, which forced him to consider ${\rm
Spin}^{h}$-manifolds and to define the KSp-index of Dirac operators on ${\rm
Spin}^{h}$-manifolds.

In another direction, Jiahao has also computed the
${\mathbb{Z}}_{2}$-cohomology of the classifying space ${\rm BSpin}^{h}$. It
turns out to be very similar to the ${\mathbb{Z}}_{2}$-cohomology of BSpin and
${\rm BSpin}^{c}$. He found, interestingly, that

$\displaystyle H^{*}({\rm BSpin};{\mathbb{Z}}_{2})=H^{*}({\rm
BSO};{\mathbb{Z}}_{2})/\big{(}\nu_{2},Sq^{1}\nu_{2^{r}},r\geq 1\big{)}$
$\displaystyle H^{*}\big{(}{\rm BSpin^{c}};{\mathbb{Z}}_{2}\big{)}=H^{*}({\rm
BSO};{\mathbb{Z}}_{2})/\big{(}Sq^{1}\nu_{2^{r}},r\geq 1\big{)}$ $\displaystyle
H^{*}\big{(}{\rm BSpin^{h}};{\mathbb{Z}}_{2}\big{)}=H^{*}({\rm
BSO};{\mathbb{Z}}_{2})/\big{(}Sq^{1}\nu_{2^{r}},r\geq 2\big{)},$

where $\nu_{k}$ is the $k^{\rm th}$ Wu class. He has also computed the first
several ${\rm Spin}^{h}$ cobordism groups.

I hope I have been able to attract your interest to ${\rm
Spin}^{h}$-manifolds. I think they play an important role in understanding
certain aspects of geometry.

## Acknowledgements

I want to thank Michael Albanese, Aleksandar Milovojević and Jiahao Hu for
their careful reading of this manuscript and for the many serious improvements
they have suggested. I would like to thank the Simons Foundation for support
during the writing of this paper.

## References

  * [1] Albanese M., Milivojević A., ${\rm Spin}^{h}$ and further generalisations of spin, J. Geom. Phys. 164 (2021), 104174, 13 pages, arXiv:2008.04934.
  * [2] Albanese M., Milivojević A., Corrigendum to “${\rm Spin}^{h}$ and further generalisations of spin”, J. Geom. Phys. 184 (2023), 104709, 4 pages.
  * [3] Anderson D., Universal coefficient theorems for K-theory, Mimeographed Notes, University of California, Berkeley, CA, 1969, aviable at https://faculty.tcu.edu/gfriedman/notes/Anderson-UCT.pdf.
  * [4] Atiyah M.F., Bott R., Shapiro A., Clifford modules, Topology 3 (1964), suppl. 1, 3–38.
  * [5] Atiyah M.F., Hirzebruch F., Riemann–Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281.
  * [6] Atiyah M.F., Singer I.M., Index theory for skew-adjoint Fredholm operators, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 5–26.
  * [7] Avis S.J., Isham C.J., Generalized Spin structures on four-dimensional space-times,
hrefhttps://doi.org/10.1007/BF01197630textitComm. Math. Phys. 72 (1980),
103–118.

  * [8] Back A., Freund P.G., Forger M., New gravitational instantons and universal spin structures, Phys. Lett. B 77 (1978), 181–184.
  * [9] Bär C., Elliptic symbols, Math. Nachr. 201 (1999), 7–35.
  * [10] Bourguignon J.-P., Lawson Jr. H.B., Yang–Mills theory: its physical origins and differential geometric aspects, in Seminar on Differential Geometry, Ann. of Math. Stud., Vol. 102, Princeton University Press, Princeton, N.J., 1982, 395–421.
  * [11] Bourguignon J.-P., Lawson Jr. H.B., Simons J., Stability and gap phenomena for Yang–Mills fields, Proc. Nat. Acad. Sci. USA 76 (1979), 1550–1553.
  * [12] Chen X., Bundles of irreducible Clifford modules and the existence of spin structures, Ph.D. Thesis, sStony Brook University, 2017.
  * [13] Freed D.S., Hopkins M.J., Reflection positivity and invertible topological phases, Geom. Topol. 25 (2021), 1165–1330, arXiv:1604.06527.
  * [14] Hawking S.W., Pope C.N., Generalized spin structures in quantum gravity, Phys. Lett. B 73 (1978), 42–43.
  * [15] Hirzebruch F., Hopf H., Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958), 156–172.
  * [16] Hu J., Invariants of real vector bundles, Ph.D. Thesis, Stony Brook University, 2023\.
  * [17] Lawson Jr. H.B., Michelsohn M.-L., Spin geometry, Princeton Math. Ser., Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  * [18] Lazaroiu C.I., Shahbazi C.S., Real pinor bundles and real Lipschitz structures, Asian J. Math. 23 (2019), 749–836, arXiv:1606.07894.
  * [19] LeBrun C., Twistors, self-duality, and spinc structures, SIGMA 17 (2021), 102, 11 pages, arXiv:2108.01739.
  * [20] Morgan J.W., The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, Math. Notes, Vol. 44, Princeton University Press, Princeton, NJ, 1996.
  * [21] Nagase M., ${\rm Spin}^{q}$ structures, J. Math. Soc. Japan 47 (1995), 93–119.
  * [22] Okonek C., Teleman A., Quaternionic monopoles, Comm. Math. Phys. 180 (1996), 363–388, arXiv:alg-geom/9505029.
  * [23] Shiozaki K., Shapourian H., Gomi K., Ryu S., Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries, Phys. Rev. B 98 (2018), 035151, 55 pages, arXiv:1710.01886.
  * [24] Sullivan D., On the Hauptvermutung for manifolds, Bull. Amer. Math. Soc. 73 (1967), 598–600.
  * [25] Teichner P., Vogt E., All 4-manifolds have spinc structures, available at https://math.berkeley.edu/~teichner/Papers/spin.pdf.
  * [26] Wang J., Wen X.-G., Witten E., A new ${\rm SU}(2)$ anomaly, J. Math. Phys. 60 (2019), 052301, 23 pages, arXiv:1810.00844.


F

<EMAIL_ADDRESS>Schmuckler Chen [c]Geng
<EMAIL_ADDRESS>Queen [b]Simon J<EMAIL_ADDRESS>[a]Institute of Chemical Sciences and Engineering (ISIC), École Polytechnique
Fédérale de Lausanne (EPFL), Sion, CH-1950, Switzerland [b]Department of
Applied Physics and Applied Mathematics, Columbia University, New York, NY
10025, USA [c]X-ray Laboratory, 3M Corporate Research Analytical Laboratory,
3M Center, St. Paul, MN 55144, USA

# Protocols for Obtaining Reliable PDFs from Laboratory x-ray Sources Using
PDFgetX3

Daniel  Yucong  Wendy L

###### Abstract

In this work, we explored data acquisition protocols and improved data
reduction protocols using PDFgetX3 to obtain reliable data for atomic pair
distribution function (PDF) analysis from a laboratory-based Mo $K_{\alpha}$
x-ray source. A variable counting scheme is described that preferentially
counts in the high-angle region of the diffraction pattern. The effects on the
resulting PDF are studied by varying the overall count time, the use of Soller
slits, and limiting the out-of-plane divergence of the incident beam. The
protocols are tested using an amorphous silica and a quartz sample. We also
present a modification to the current PDFgetX3 data corrections to take care
of sample absorption, which was previously neglected in the use of that
program for high-energy synchrotron x-ray data. We show that, despite
limitations in the $Q$-range and flux of laboratory instruments, reasonable
data for PDF model fits may be obtained using the best protocols in a few
hours of counting.

###### keywords:

Data Reduction, PDF, Pair distribution function

Various protocols are assessed for obtaining optimal data from a laboratory
based Mo $K_{\alpha}$ instrument for subsequent atomic pair distribution
function (PDF) analysis. Modifications to the PDFgetX3 data reduction
protocols are also explored.

##  1 Introduction

The atomic pair distribution function (PDF) analysis of powder diffraction
data is a widely used technique for studying the structure of nanomaterials
[keenCrystallographyCorrelatedDisorder2015,
a.youngApplicationsPairDistribution2011,
billingeNanoscaleStructuralOrder2008d, lindahlchristiansenThereNoPlace2020a],
amorphous materials [wagnerDirectMethodsDetermination1978,
benmoreReviewHighEnergyXRay2012] and liquids
[furukawaRadialDistributionCurves1962, fischerNeutronXrayDiffraction2005], and
increasingly to study nanostructures in crystals [keenTotalScatteringPair2020,
lin;prb05, beecherDirectObservationDynamic2016b]. It is currently most widely
applied to diffraction experiments carried out at x-ray synchrotron sources
and neutron sources which are large-scale national and international shared
facilities. However, data useful for PDF analysis may also be obtained from
laboratory-based instruments that have relatively short wavelength sources
such as Mo or Ag $K_{\alpha}$ x-rays [egami;b;utbp12, dykhn;phmr11,
thomaePushingDataQuality2019, tsymbarenkoQuickRobustPDF2022a,
prinzHardXraybasedTechniques2020, irvingAdvantagesCurvedImage2021a,
confalonieriComparisonTotalScattering2015]. These types of PDFs are set to
grow in popularity because of access challenges at the national facilities,
and the emergence of specialized instruments for PDF analysis on the
laboratory x-ray diffractometer market [D8ADVANCE, D8DISCOVER,
EmpyreanXRDInstrument, Stadi, Rigaku]. For instance,
[irvingAdvantagesCurvedImage2021a] demonstrated that PDF from nanoparticles
and liquids could be obtained with an acquisition time of 2-3 hours using an
in-house single-crystal diffractometer equipped with an Ag $K_{\alpha}$
source. Despite offering a lower accessible $Q$-range than Ag $K_{\alpha}$, Mo
sources offer a number of advantages over Ag sources, such as higher flux and
longer source life, and can also yield PDFs with sufficient real-space
resolution for many scientific problems [dykhn;phmr11,
tsymbarenkoQuickRobustPDF2022a]. Though a number of groups have explored the
use of specialized and standard laboratory diffractometers for obtaining
quantitative PDFs [thomaePushingDataQuality2019,
tsymbarenkoQuickRobustPDF2022a, prinzHardXraybasedTechniques2020,
irvingAdvantagesCurvedImage2021a] less attention has been given to optimal
data acquisition and analysis protocols. This paper deals with protocols for
acquiring and analyzing data from such lab instruments to obtain accurate PDFs
using the latest-generation data reduction software packages.

A challenge in PDF analysis is to obtain a high-quality PDF from raw
diffraction data since the quality of the resulting PDF depends not only on
the choices made during data acquisition but also on how the data reduction is
carried out from the raw data to the PDF. Two contrasting approaches to data
reduction are often followed. One makes corrections for experimental effects
by explicitly calculating them from the understood physics and geometry and
making algebraic corrections to the data to account for those effects
[egami;b;utbp12]. The problem with this approach is that many of the
experimental factors and their contributions to the scattering signal are only
approximately known. It has been shown that making certain ad hoc corrections
that are justified by the physics of the situation and the fact that
reasonable assumptions about the experiment hold can result in PDFs that are
equivalent to those obtained using the physics-based data reduction approach
[juhas;jac13]. In fact, the ad hoc correction approach can be superior in some
cases, for example, when subtracting large backgrounds to extract weak signals
of interest, when it is not guaranteed that the foreground and background
measurement geometries are exactly reproduced. This allowed, for example,
signals to be extracted from nanosized organic drug particles in aqueous
solution at concentrations of 0.2 wt% [terba;nan15] and even the observation
of different hydrogen bonding networks in differently amorphized polymeric
samples [terbanLocalStructuralEffects2020].

Here, we explore how to use the ad hoc program, PDFgetX3, to obtain reliable
PDFs from a Bruker D8 Discover laboratory instrument. We also compare and
contrast some different data acquisition protocols to understand best
practices for laboratory instrument data acquisition and analysis. Some of the
approximations inherent in the use of PDFgetX3, which was developed for high-
energy synchrotron experiments, are not necessarily valid for the lab data and
additional corrections for these effects are discussed.

##  2 _Ad hoc_ analysis procedure in PDFgetX3

The ad hoc approach used in PDFgetX3 for obtaining the PDF from powder
diffraction data is described in detail in [juhas;jac13]. It was shown
[juhas;jac13], in the case of high energy synchrotron data, that this approach
resulted in PDFs that were identical to those obtained using a conventional,
physics-based, analysis using the PDFgetX2 program [qiu;jac04i].

We summarize the pertinent aspects of the PDFgetX3 method here. In general,
multiplicative and additive/subtractive corrections need to be applied to a
measured diffraction pattern to turn it into the relevant scattering function,
$S(Q)$. Multiplicative corrections deal with things such as sample absorption
and geometric effects that affect the illuminated sample volume
[egami;b;utbp12]. Additive corrections deal with effects of multiple
scattering, fluorescence, Compton scattering, and so on [egami;b;utbp12]. In
many cases we don’t care about the absolute value of the overall scale factor
since that parameter can be refined to fit structural models and doesn’t
affect the refined values of structural parameters [juhas;jac13]. If we don’t
care about an overall scale-factor, it is only the $Q$-dependence of the
multiplicative and additive aberrations, $Q$ being the magnitude of the
scattering vector, $Q=\frac{4\pi\sin\theta}{\lambda}$, where $\theta$ is half
the scattering angle and $\lambda$ is the x-ray wavelength, that is important.
For the case of high-energy synchrotron radiation experiments, in most cases,
the absorption and sample volume effects give very weak $Q$ dependencies and
can be neglected. On the other hand, especially for RA-PDF (rapid acquisition
PDF) experiments [chupasRapidacquisitionPairDistribution2003f] carried out
with large area 2D detectors, the additive effects are significant. The
PDFgetX3 approach uses a polynomial function to fit the $Q$-dependence of
these signals and removes it by subtraction. The flexibility of the polynomial
is very important. It needs to be flexible enough to remove slowly varying
incoherent backgrounds that do not contain structural information, but not to
remove any meaningful structural signals. In PDFgetX3 the flexibility is
determined by the order of the polynomial function that is fitted, where a
higher order results in a more flexible fitting curve. There is no structural
signal in $S(Q)$ that has a frequency lower than $2\pi/r_{nn}$, where $r_{nn}$
is the inter-atomic distance of the nearest neighbor peak in the PDF, and so
limiting the order of the polynomial to some value such that no frequencies
higher than this can be removed ensures no loss of structural information. The
polynomial order, and thus the fitting flexibility, is controlled by the
$rpoly$ parameter in the program. After removal of the additive aberrations,
the signal is normalized to obtain $S(Q)$ by satisfying its known asymptotic
behavior, so that its low- and high-$Q$ behavior is correct.

In principle, we could use the same approach for data from laboratory
instruments, but the assumption of negligible x-ray absorption by the sample
is no longer valid in general due to the longer wavelength x-rays. This both
increases the importance of the absorption effects overall but additionally
introduces significant $Q$ dependencies to them because data have to be
measured over wide angular ranges to obtain reasonable values of
$Q_{\mathrm{max}}$, and the effective illuminated volume of the sample becomes
significantly angle-dependent [egami;b;utbp12]. For quantitatively accurate
PDFs, new steps need to be added to the PDFgetX3 algorithm to account for this
issue.

To test these issues we carried out a set of Mo $K_{\alpha}$ experiments from
samples of quartz and silica glass. These were chosen as a first choice
because of their relatively moderate, though not negligible, absorbing power,
for Mo $K_{\alpha}$ radiation. The results can be extended to more absorbing
samples later. Quartz was chosen because it is crystalline and it is therefore
possible to model it quantitatively using PDFgui [farro;jpcm07]. Silica glass
was chosen because, on the other hand, it is amorphous and contains only
diffuse scattering. Such data with weak signals can be more challenging for
carrying out data corrections to obtain PDFs and so the combination of a
crystalline and an amorphous test sample covers both cases.

To benchmark the quality of the results, we compare the resulting PDFs to
expected results from quartz structural models and against PDFs obtained from
the same samples measured using high-energy synchrotron data using a standard
RA-PDF approach.

##  3 Laboratory Experiments

Mo $K_{\alpha}$ data was obtained with a Bruker D8 Discover diffractometer
equipped with a line focus Mo $K_{\alpha}$ radiation source ($\lambda=0.7107$
Å), and an EIGER2 R 500K detector operated in 1D mode. The transmission PDF
configuration includes a focusing Goebel mirror, a 1.0 mm divergence slit, a
2.5∘ axial Soller slit, a scattering guard after the source, a capillary
spinner on a centric Eulerian cradle, and an optional 2.5∘ axial Soller slit
before the detector. The source-to-sample distance is 200 mm, and the
detector-to-sample distance is 198.5 mm. X-ray generator settings of 40 kV and
40 mA were employed.

Whilst flat plate geometries such as Bragg-Brentano can give increases in
detected scattered flux, they can introduce data correction challenges in PDF
because of the difficulty in estimating the illuminated sample volume
[egami;b;utbp12]. To avoid this, and at the expense of a loss in measured
intensity, we chose to use the capillary geometry, which is the least
susceptible to these uncertainties, with the powdered samples contained in
polyimide capillaries with an internal diameter (ID), $d_{c}=1.5$ mm, and of
approximate length 50 mm. The choice of $d_{c}$ is important and depends on
the absorbing properties of the samples. Guidelines for choosing $d_{c}$ are
described below in Section 7.1.

Scans were conducted using an EIGER2 detector operated in Continuous 1D mode
with 69 mm (equatorial) $\times\,24$ mm (axial) opening, which is a fast scan
mode that does not stop between steps and with a standard over-travel by half
of the detector opening. A special low-angle scan was conducted using the same
detector but operated in no-over-or-under-travel standard continuous 1D mode.
This allowed us to collect the lowest angle data whilst avoiding a collision
between the detector arm and the goniometer floor/platform which would have
happened if we had extended the standard continuous scans all the way to the
lowest angles. This was not required for the optimal protocol but was a detail
of specifics of the diffractometer platform itself.

Based on our estimate that the beam-height on the detector was around 0.5 mm,
which corresponds to an angular height of 0.15∘, we chose an angular step size
of 0.025∘ as a convenient round number less than 0.15/5 for the step size to
ensure sufficient sampling of each peak. A larger step size up to 0.07∘ may be
used without danger of under-sampling the pattern.

The scans were conducted using a staircase-counting-time (SCT) measurement
strategy. The goal is to increase counting statistics in the high-$Q$ region.
Because of the way that the signal/noise ratio decreases with increasing $Q$,
we desire the detected flux to increase in an exponential way with $Q$ to
maintain a roughly constant $Q$-dependence of the signal/noise ratio in
$F(Q)$. Various approaches may be taken to accomplish this, for example,
increasing slit size in front of the detector with angle, and increasing the
time collecting data per point, $t_{\mathrm{point}}$, or both. Such optimal
approaches are not currently designed into the instrument data collection and
subsequent analysis protocols. The SCT strategy is a more convenient strategy
for getting an approximately exponential increase in detected flux with
increasing angle using the current instrument control software. It involves
making $i$ measurements each in the range from
$2\theta_{\mathrm{min}}^{i}<2\theta<2\theta_{\mathrm{max}}^{i}$ with the same
slit settings, step-size, and a collection time per point
$t_{\mathrm{step}}^{i}$. We chose a strategy where
$2\theta_{\mathrm{max}}^{i}=2\theta_{\mathrm{max}}$ is a constant. In our case
$2\theta_{\mathrm{max}}=140^{\circ}$. $2\theta_{\mathrm{min}}^{i}$ varies in
such a way as to make a staircase-shaped total count time vs $2\theta$ plot
with more steps in the staircase at higher-$2\theta$, as shown in Fig. 1

Figure 1: The staircase-count-time, SCT, protocol used for our experiments.
The red lines indicate the range of $2\theta$ for each step, and the blue
lines indicate the range of $Q$ for each step, with total time per point
$t_{\mathrm{point}}$ which are plotted in units of the fixed count-time
interval, $\Delta t$. $\Delta t$ may be varied from experiment to experiment
to change the overall measurement time but this is done without changing other
aspects of the protocol.

The values of $2\theta_{\mathrm{min}}^{i}$ we used in our protocol are
reproduced in Table 1. To enhance the exponential increase in count-time the
$t_{\mathrm{step}}^{i}$ is increased with increasing $i$. To make calculations
of the estimated total measurement time easier, we used a protocol where we
define a fixed count-time interval, $\Delta t$, and this was scaled by a
factor to get $t_{\mathrm{step}}^{i}$ for a particular step, i.e.,
$t_{\mathrm{step}}^{i}=c_{i}\Delta t$. All the scale factor values, $c_{i}$,
we used for our SCT protocol are reproduced in Table 1 for convenience.

Table 1: The SCT protocol used for our experiments. $t_{\mathrm{step}}^{i}=c_{i}\Delta t$ and $2\theta_{\mathrm{min}}^{i}$ are the count time per point and the minimum $2\theta$ angle for the scan of the $i$th step. In each case, the maximum $2\theta$ value was 140 ∘, and the step size was $\Delta 2\theta=0.025^{\circ}$. In addition to the staircase indicated in the Fig. 1 a low-angle scan in a different mode was collected from -4∘ to 15∘ to get all the low-$Q$ region to the edge of the beamstop whilst avoiding a diffractometer collision. $N_{p}$ is the number of points in the step.  Step |  $t_{\mathrm{step}}$ ($\Delta t$) |  $2\theta_{\mathrm{min}}$ (∘) |  $2\theta_{\mathrm{max}}$ (∘) | $N_{p}$
---|---|---|---|---
0 | 2 | -4 | 15111Special low-angle scan to avoid a collision of the diffractomter, please see text for details. | 761
1 | 1 | 10 | 140 | 5201
2 | 2 | 75 | 140 | 2601
3 | 4 | 107 | 140 | 1321
4 | 8 | 124 | 140 | 641
5 | 16 | 132 | 140 | 321

This SCT strategy allows us to easily vary the total collection time from 1.5
hours per measurement for $\Delta t=0.1$ s to a 25.5 hour measurement for
$\Delta t=1.8$ s by just varying $\Delta t$ without changing other parameters
in the SCT protocol. In general, we may want to design a scan for a fixed
amount of available measurement time, $T_{m}$. With a step-size of $\Delta
2\theta$ our protocol resulted in the number of $2\theta$ steps per stair-step
given in Table 1 as $t_{\mathrm{step}}$. The entire protocol then takes a
measurement time of

$\displaystyle T_{m}$
$\displaystyle=\sum_{i}N_{p}^{i}(t_{\mathrm{step}}^{i}\Delta t+t_{o})$ (1)
$\displaystyle=N_{\Delta t}\Delta t+N_{p}t_{o}$ (2)

seconds, where $t_{o}$ is the fixed time overhead per point, $N_{p}^{i}$ is
the number of points in the $i$th step, $N_{p}$ is the total number of points,
and the total number of $\Delta t$’s, $N_{\Delta
t}=\sum_{i}t_{\mathrm{step}}^{i}N_{p}^{i}$. Inverting this equation, and
knowing $t_{o}$, we can predict a $\Delta t$ to use for the protocol given a
fixed amount of measurement time, $T_{m}$. In our case, we found that the
diffractometer setup had an overhead time per point that scaled not only with
the number of points but also with $\Delta t$. This could be true if there is
a fixed overhead time to read out the detector and the readout happens on a
fixed frequency, for example, every 0.1 s. In this case we need to modify Eq.
2 to take this into account, getting

$\displaystyle T_{m}$
$\displaystyle=\sum_{i}N_{p}^{i}(t_{\mathrm{step}}^{i}+t_{o})\Delta t$ (3)
$\displaystyle=(N_{\Delta t}+N_{p}t_{o})\Delta t,$ (4)

and

$\Delta t=\frac{T_{m}}{(N_{\Delta t}+N_{p}t_{o})}.$ (5)

In our case, $N_{\Delta t}=27,473$ and $N_{p}=10,846$ and based on actual
observed measurement times we get $t_{o}$, in units of $\Delta t$, to be 2.3
on average. Thus for our case, we can estimate a $\Delta t$ given a target
$T_{m}$ from

$\displaystyle\Delta t$ $\displaystyle=\frac{3,600T_{m}}{52,419}$ (6)
$\displaystyle=\frac{T_{m}}{14.56}.$ (7)

where the 3,600 factor is if we input $T_{m}$ in units of hours instead of
minutes. This just gives a useful estimate of the $\Delta t$ to use for a
target total measurement time. It would have to be determined for each
different diffractometer setup using either Eq. 2 or Eq. 4 depending on
whether the overhead time for the setup in question was constant per
measurement point or scaled also with the count-time per point. XRD raw data
from the SCT scan were integrated from the 2D detector, normalized in
counts/second (cps), and merged into one data set, and then exported as a .xy
file by use of the Bruker DIFFRAC.EVA software.

To assess different collection strategies and instrument configurations we
carried out an extensive set of different measurement protocols. These are
summarized in Table 2. They all used the SCT protocol laid out above with a
value of $\Delta t$ shown in Table 2.

Table 2: Data collection protocols for the Mo $K_{\alpha}$ experiments. A staircase-counting-time (SCT) protocol was used, described in Section 3. $2\theta_{max}$ is always set to 140∘ and five steps in the stair were measured following the SCT protocol described in the text. $\Delta t$ is the count-time interval described above. The beam-height on the detector was 0.5 mm corresponding to a solid angle of 0.15∘, and we used a step size of 0.025∘ resultin in a total elapsed time for each experiment reproduced here. The secondary beam Soller slit was a 2.5∘ slit where Y and N indicate whether the slit was installed or not, respectively. The illumination length refers to the length of the capillary sample that was exposed to the beam. Protocol |  $\Delta t$ (s) | Elapsed time (hours) | Soller | Illumination length (mm)
---|---|---|---|---
P1 | 0.1 | 1.5 | N | 25
| 0.3 | 4.5 | N | 25
| 0.9 | 12.75 | N | 25
P2 | 0.3 | 4.5 | Y | 25
| 0.9 | 12.75 | Y | 25
| 1.8 | 25.5 | Y | 25
P3 | 0.1 | 1.5 | N | 15
| 0.3 | 4.5 | N | 15

We carried out data collections with and without a 2.5∘ Soller slit in the
incident path and by varying the length of the sample capillary that was
illuminated by the beam. The latter was accomplished by cutting the edges of
the incident beam using lead tape upstream of the sample to limit the length
of the incident beam on the sample. This led to a set of four acquisition
protocols, each with various $\Delta t$ times, that are summarized in detail
in Table 2.

A background measurement was performed with an empty polyimide capillary,
using the same experimental configuration and measuring conditions, to account
for the signal produced by the capillary material, air-scattering, or by any
other source different from the sample itself.

The merged XRD data were converted to $G(r)$ using PDFgetX3 [juhas;jac13]
within xPDFsuite [yang;arxiv14]. A $Q_{\mathrm{max}}=16.6$ Å-1 was used in the
Fourier transform from $F(Q)$ to $G(r)$. The Fourier transform was carried out
on a grid of $\Delta r=0.01$ Å and also on to the Nyquist-Shannon (NS) grid
($\Delta r=\frac{\pi}{Q_{\mathrm{max}}}=0.189$ Å) [farro;prb11]. The former
grid gives a smoother PDF which is easier to interpret visually whilst the
latter results in data points in the PDF that are as statistically
uncorrelated as possible without loss of information, and therefore results in
better estimates of uncertainties during model fitting.

##  4 Synchrotron Experiments

Total scattering experiments were carried out on at the 28-ID-1 beamline at
the National Synchrotron Light Source II (NSLS-II) at Brookhaven National
Laboratory. A 2D Perkin-Elmer amorphous silicon detector ($2048\times 2048$
pixels, $200\times 200$ micron pixel size) was placed perpendicular to the
incident beam path 205 mm downstream of the sample. Samples were fine powders
loaded in Kapton tubes with an inner diameter of 1 mm. A bent Laue
monochromator was used to select the photon energy. The incident wavelength
was $\lambda=0.1665$ Å. The instrument geometry was calibrated using a
polycrystalline nickel standard using the pyFAI program [ashio;jac15]. Data
were collected for 15 minutes by summing 4,500 0.2 second frame exposures.
Dark frames were collected before the actual exposures by exposing the
detector in the same way but with the shutter closed. These were subtracted
from the bright images to remove detector dark current effects. 2D diffraction
images were masked, and the raw data were integrated along arcs of constant
diffraction angle using pyFAI with full pixel splitting, solid angle
correction off, and 15000 radial bins to produce 1D powder diffractograms.
Total scattering data from an empty 1 mm Kapton (polyimide) tube was collected
to determine the background intensity due to the sample container and any air
scattering. The PDFs, $G(r)$, were obtained following standard data reduction
protocols described in detail elsewhere [juhas;jac13] using PDFgetX3 within
xPDFsuite [yang;arxiv14]. The background signal was scaled and subtracted as
part of this process in PDFgetX3.

$Q_{\mathrm{min}}$ = 0.8 and $Q_{\mathrm{max}}$ = 26.0 Å-1 were used in the
Fourier transform that takes $F(Q)$ to $G(r)$. The Fourier transform was done
onto a grid of $\Delta r=0.01$ Å and also on to the NS grid ($\Delta
r=\frac{\pi}{Q_{\mathrm{max}}}=0.12$ Å)[farro;prb11].

##  5 PDF Modelling

Structure modeling was done using Diffpy-CMI [juhas;aca15]. The python script
for carrying out fitting is available in the supplementary information. The
structure model of $\alpha$-quartz was taken from the crystallographic open
database (ID 1011097). The parameters $Q_{\mathrm{damp}}$,
$Q_{\mathrm{broad}}$, the correlated motion parameter $\delta_{2}$, the
lattice parameters $a$ and $b$, the atomic displacement parameters (ADPs), the
atomic fractional coordinates, $X$, $Y$ and $Z$, and a scaling variable
$s_{1}$, were refined. The lattice parameters, ADPs, and atomic positions were
constrained to satisfy the symmetries of the space group (P3121). The diagonal
ADPs U11, U22, and U33 were initialized with a value of 0.005. The off-
diagonal ADPs U12, U13, and U23 were initialized with a value of 0.00. Other
parameters were initialized with their values in the CIF file for the first
refinement, but to increase computational efficiency, subsequent refinements
used the outputs of previous refinements to initialize the variables.
Refinements were carried out on the Nyquist-Shannon (NS) grid, where the data
were sampled onto the NS grid using the Whittaker-Shannon interpolation that
uses sinc functions.

The refinements were done with a full error propagation, with initial standard
errors assigned to the points in the PDF data by calculating the root-mean-
square ($rms$) of the fluctuations in the measured PDFs between $300<r<400$ Å
(see section 6.2.1 for justification for this). This results in uncertainties
on the refined parameters that are shown in parentheses in Table 3. This
process is expected to result in statistically reasonable uncertainty
estimates when the data is in the count-rate limited regime that applies to
the laboratory data. This does not apply to the synchrotron data where the
high counting statistics mean that the fluctuations in the PDF at high-$r$ are
dominated by systematic errors. In this case, the estimated uncertainties
shown in the parentheses result in a significant underestimation of actual
uncertainties, though they are reproduced in the table for completeness.

##  6 Results

###  6.1 Comparison of Lab and synchrotron data

We start by making a direct comparison of the 1D diffraction patterns from
quartz obtained from the lab instrument and the synchrotron. For the lab data,
we consider the data from the measurement protocol P2 with $\Delta t$ = 1.8 s,
the 25.5 h measurement with Soller slits in place. The results are plotted in
Fig. 2.

Figure 2: Comparison of the measured intensities from a quartz measurement
using RA-PDF synchrotron (red) and a Mo $K_{\alpha}$ lab (blue) data. The lab
data were collected with protocol P2 and $\Delta t=1.8$ s (a) The data are
plotted over the entire $Q$-range measured in each case. The inset shows the
intermediate range of $Q$ on an expanded scale. (b) The low-$Q$ range for the
same data is shown on an expanded scale. In both plots the synchrotron and Mo
data have been scaled to 1 by dividing by the maximum peak such that the
amplitude of the low-$r$ peaks in $G(r)$ are approximately equal in the two
curves.

As expected, we see that the accessible $Q$-range from the Mo $K_{\alpha}$
measurement is much smaller than for the synchrotron data. The Mo x-rays
($\lambda=0.7107$ Å) give us an achievable $Q_{\mathrm{max}}=16.615$ Å-1 after
measuring up to the maximum angle of $2\theta=140^{\circ}$. This will limit
the real-space resolution of the PDF from the Mo $K_{\alpha}$ measurement.

Close examination of Fig. 2(a) suggests that overall, the signal seems quite
well reproduced between the two measurements, though we observe a stronger
upturn in the Mo $K_{\alpha}$ measurement with increasing=$Q$ that is not
present in the synchrotron data. This is shown on an expanded scale in the
inset. We hypothesize that this upturn is, in part, a result of sample
absorption which will be less important in the backscattering region
($2\theta>90^{\circ}$) than the forward scattering and therefore lead to a
relatively stronger signal with increasing-$Q$. We explore this effect more
quantitatively in Section 7.1.

The low-$Q$ region of the diffraction pattern is shown on an expanded scale in
Fig. 2(b). We see that the $Q$-space resolution of the Mo $K_{\alpha}$ data is
higher than that of the RA-PDF synchrotron data. This will result in a
structural signal in $G(r)$ that extends to higher-$r$ in the lab data than
the synchrotron data.

The raw data were reduced to $F(Q)$ and $G(r)$ using PDFgetX3. The resulting
functions are compared in Fig. 3 where, as before, the synchrotron data are
shown in red and the Mo $K_{\alpha}$ in blue.

Figure 3:  Comparison of (a) $F(Q)$ and (b) $G(r)$ functions of crystalline
quartz collected via a synchrotron measurement (red) using the RA-PDF method
(total measurement time 15 m) and the Mo $K_{\alpha}$ lab measurement (blue),
collected with protocol P2 and $t_{\mathrm{step}}=1.8$ s (total time 25.5
hours). The insets show regions of the curves on an expanded scale.

As before we see that the synchrotron data extends to higher $Q$ than the Mo
$K_{\alpha}$ data. There is now no upturn of the data in the high-$Q$ region
since this slowly varying dependence was removed by PDFgetX3. However, as we
argue in Section 7.1, the PDFgetX3 algorithm is leaving an uncorrected
multiplicative correction due to absorption. For high incident x-ray energy
synchrotron measurements, neglecting this is a reasonable approximation.
However, for Mo $K_{\alpha}$ this is no longer true in general. Such an
uncorrected absorption effect would lead to diffraction intensities that are
overestimated in the low-$Q$ region and underestimated in the high-$Q$ region
of $F(Q)$ in the Mo $K_{\alpha}$ data compared to the RA-PDF data. In Fig.
3(a) we have re-scaled the red $F(Q)$ curve by $1.5\times$ to compare the two
curves. This factor was chosen by making the $G(r)$ functions scale onto each
other in the region $0<r<15$ Å. This results in an $F(Q)$ curve that is a bit
higher in amplitude at low-$Q$ (Fig. 3 main panel), and lower in amplitude at
high-$Q$ (Fig. 3 inset), compared to the red curve, consistent with the
expectation of an incorrectly handled absorption correction.

The impact of this will be to produce an extra broadening of the peaks in
$G(r)$ in the Mo $K_{\alpha}$ data. In the left inset to Fig. 3(b), we show a
low-$r$ region of the PDFs on an expanded scale and, indeed, the PDF peaks
from the RA-PDF data are narrower. However, this is also because of the larger
$Q_{\mathrm{max}}$ of the synchrotron data, and so the effect of the
uncorrected absorption correction is not directly visible. When structural
models are fit to the data, it is expected to result in overestimated ADPs in
resulting model fits. This is discussed in Section 7.

On the other hand, as expected due to the higher $Q$-space resolution, the
structural signal in the Mo $K_{\alpha}$ $G(r)$ extends further in $r$ than
does the synchrotron PDF in red. The signal in the synchrotron data dies out
by about $r=80$ Å and is replaced by noise, compared to the Mo $K_{\alpha}$
$G(r)$ that clearly extends to above $r=200$ Å. High $Q$-resolution
measurements may be made at synchrotrons using different setups
[fitchHighResolutionPowder2004], but these experiments are slow, and fitting
of models to PDFs are rarely done beyond an $r_{\mathrm{max}}=40$ or 50 Å and
the RA-PDF setup [chupa;iucrcpd03] is predominantly preferred.

The signal/noise level in the synchrotron data is much better than from the
lab data despite the collection time being almost $50\times$ less. This is
clearly evident in the right-hand inset in Fig. 3(b) that shows both PDF
curves in the region from $250<r<300$ Å. There is no structural signal left in
this region in either the synchrotron or the Mo $K_{\alpha}$ data and so the
ripples in the PDF are completely coming from noise in the measurement. The
two PDFs have been scaled onto each other such that the structural signal is
weighted the same at low-$r$, and so the much smaller amplitude noise signal
of the red curve in this inset clearly shows that even comparing a 15 minute
synchrotron scan with a 12.75 hour Mo $K_{\alpha}$ scan, the noise levels are
still much lower in the synchrotron data. The high throughput of synchrotron
measurements is a clear advantage for rapid experiments. The effect may also
be seen directly in the $F(Q)$ data shown in Fig. 3(a), especially in the
inset, where the weak diffraction signal is clearly better resolved in the red
curve than the blue curve in this high-$Q$ region.

Despite this comparison, which appears to reflect negatively on the Mo
$K_{\alpha}$ data, below we explore optimizing counting statistics and argue
that in many cases, especially for crystalline samples, sufficiently good PDFs
may be obtained from the lab setup even in 1.5 hours rather than the 12.75 h
that are compared to the synchrotron data here. This is because we are often
interested in the low-$r$ signal in the PDF which has a very good signal/noise
ratio in both measurements. The highly penetrating nature, high flux, and
high-$Q_{\mathrm{max}}$ allowed by the synchrotron radiation make it highly
desirable for high throughput and in situ experiments, and when higher real-
space resolution is needed. However, Mo $K_{\alpha}$ and Ag $K_{\alpha}$ lab-
based setups can give high-quality data.

We now assess the quality of model fits to the synchrotron and Mo $K_{\alpha}$
data. In Fig. 4 we show visually the quality of the fits. Fig. 4(a) and (b)
show fits to the Mo $K_{\alpha}$ data without and with Soller slits,
respectively, and the synchrotron data are shown in Fig. 4(c). The best
agreement between model and data is evidently the Mo $K_{\alpha}$ data
measured with the Soller slit in place (Fig. 4(b)) and the synchrotron data
Fig. 4(c)).

Figure 4: Comparison of best-fit calculated and measured PDFs for quartz. The
measured PDF is shown as blue circles and the best-fit PDF as a solid red
line. The difference between the two is shown offset below. The data are (a)
with no Soller slit (protocol P1) and 4.5 h measurement time, (b) with Soller
slit (protocol P2) and 25.5 h measurement time, and (c) the synchrotron data.

The refined parameters are reproduced in Table 3 where we compare the P2(1.8)
and the synchrotron columns.

Table 3: Results of the quartz model refinements to the data measured with
selected protocols. The protocols are written as P$N$($\Delta t$), where $N$
is the number of the protocol from Table 2. Abs indicates the dataset that has
had an absorption correction applied. In the model Silicon. The uncertainties
are one standard deviation. The estimated errors on the input data are
computed by evaluation the root-mean-square fluctuations of the PDF in the
region from $300\leq r\leq 400$ Å on the Nyquist-Shannon grid. The
justification for this for the Mo $K_{\alpha}$ data is in the text.

Parameter P1(0.1) P1(0.3) P2(1.8) P2(1.8) Abs P2(0.3) P3(0.3) synchrotron
scale 0.190(20) 0.17547(33) 0.215(5) 0.229(6) 0.2129(9) 0.183(12) 0.180083(22)
$Q_{\mathrm{damp}}$ -0.0(2.1)e+11 -0(5)e+10 -0(7)e+12 -0(5)e+13 -0(7)e+12
-0(7)e+12 -0(4)e+09 $\delta_{2}$ -0.1(4) 7.64(4) 7.56(10) 7.6(7) 7.65(24)
7.77(29) 7.659(4) a 4.9099(13) 4.91225(10) 4.90377(17) 4.909(7) 4.9038(6)
4.9032(8) 4.913671(4) c 5.4113(6) 5.41105(7) 5.40251(15) 5.4075(11) 5.4023(12)
5.411(4) 5.414290(10) Si(U11) 0.0162(14) 0.01209(15) 0.00907(17) 0.009(12)
0.0084(9) 0.0099(9) 0.006129(6) Si(U22) 0.0041(32) 0.00745(16) 0.0071(5)
0.00797(31) 0.0076(18) 0.0069(17) 0.005675(10) Si(U33) 0.000569(27)
0.000555(17) 0.00743(14) 0.006(12) 0.0073(17) 0.0049(4) 0.004727(5) Si(U23)
0.0017(17) 0.00213(4) -0.0009(10) -0.001(6) -0.0012(7) 0.0001(14) -0.000737(8)
O(U11) 0.0331(6) 0.0234(4) 0.0286(19) 0.029(20) 0.0279(16) 0.038(7) 0.03953(4)
O(U22) 0.0271(5) 0.01947(31) 0.0221(7) 0.020(15) 0.0209(13) 0.0100(18)
0.021365(18) O(U33) 0.0280(5) 0.02364(35) 0.0192(5) 0.023(16) 0.0200(10)
0.0235(16) 0.020988(21) O(U12) 0.0300(6) 0.02141(34) 0.0126(8) 0.012(18)
0.0104(13) 0.0190(16) 0.028642(26) O(U13) 0.0304(5) 0.01757(31) 0.0051(17)
0.005(10) 0.0055(10) 0.0043(25) 0.014979(23) O(U23) 0.0273(5) 0.01640(27)
0.0091(8) 0.010(9) 0.0087(10) 0.0067(11) 0.014778(16) Si($X$) 0.47780(23)
0.47532(12) 0.4718(10) 0.4719(22) 0.4718(4) 0.4726(5) 0.472189(6) O($X$)
0.41218(35) 0.42301(21) 0.4164(14) 0.4167(17) 0.4151(13) 0.4143(12)
0.418908(16) O($Y$) 0.27242(31) 0.27390(18) 0.2664(4) 0.2671(15) 0.2661(4)
0.2716(8) 0.271559(11) O($Z$) 0.21621(23) 0.21646(15) 0.2140(12) 0.214(13)
0.2141(5) 0.2136(6) 0.215568(9) $Q_{\mathrm{max}}$ 16.599 16.599 16.599
16.610061 16.599 16.599 26.0 Grid 0.189264 0.189264 0.189264 0.189138 0.189264
0.189264 0.12083 $R_{w}$ 0.425604 0.475678 0.355678 0.352295 0.360612 0.451993
0.411143 $\chi^{2}_{\mathrm{red}}$ 249.433819 900.845769 637.199518 416.620333
117.009988 60.120852 103180.097323

These fits were done on PDFs computed on the Nyquist-Shannon sampling grid,
the case for which points in the PDF are as statistically uncorrelated as
possible [toby;aca04], which will result in the most reliable estimates of
uncertainties on refined parameters. The refined values are all very close
between the two measurements. The most statistically significant variation is
with the lattice parameters. This may indicate an issue with the calibration
of one or both of the setups, though the effects are small, with the
synchrotron lattice parameters being larger by $\approx 0.01$ Å.

###  6.2 Comparison of different protocols for Mo $K_{\alpha}$ data

We now turn to a comparison of the different Mo $K_{\alpha}$ data acquisition
protocols to obtain the best Mo $K_{\alpha}$ data for PDF analysis.

####  6.2.1 Effect of counting statistics

We would like to understand the effects of changing the count-time, and
therefore the counting statistics, on the resulting PDF and its information
content. We do this by collecting Mo $K_{\alpha}$ data with identical
experiment configuration but different count times and assess the effect on
the PDF, and on structural parameters extracted from the PDF. Data was
collected using protocol P1 with dwell-times per point of $\Delta t=0.1$, 0.3,
and 0.9 s, corresponding to total elapsed experiment times of 1.5, 4.5, and
12.75 hours, respectively.

The longer counting time does result in a reduction of noise in the PDF. This
is most easily seen in the $G(r)$ of the amorphous silica data in the high-$r$
region where the structural signal has gone to zero, and only noise remains in
the measured PDF. We can make a visual comparison of the effects of the
different count times by looking at the region of the PDF above 17 Å in Fig.
5(d). The region $9<r<35$ Å is shown on an expanded scale in the inset. Below
$r\sim 20$ Å, there is a structural signal, but it is gone in the region above
this. The fluctuations in this $r$ region above $r=10$ Å are all due to noise
in the measurement.

Figure 5: Comparison of the effects of count time on the Mo $K_{\alpha}$ data
where all experiments used protocol P1 but with dwell times per step of
$t_{\mathrm{step}}=0.1$ s (red), 0.3 s (dark blue) and 0.9 s (pale blue)
corresponding to total experiment durations of 1.5, 4.5, and 12.75 hours,
respectively. (a) and (b) show data from crystalline quartz, and (c) and (d)
show data from amorphous silica. The $F(Q)$ functions are shown in (a) and (c)
and the PDFs, $G(r)$, are shown in (b) and (d). Specific regions of the curves
are shown on expanded scales in the insets. Horizontal dashed lines in the
insets of (b) and (d) show the root-mean-squared at high-r of
$t_{\mathrm{step}}=0.1$ s (dots), $0.3$ s (dashes), and $0.9$ s (long dashes).

In this region, we do see a reduction in the amplitude of the rapid noise
fluctuations with increasing count times, where they are largest in red
($\Delta t=0.1$ s), less so in dark blue ($\Delta t=0.3$ s) and smallest in
pale blue ($\Delta t=0.9$ s), consistent with the noise signal being dominated
by the counting statistics and therefore reducing in amplitude when the
counting time increases.

If the noise in this region is coming from counting statistics alone, we
expect that the rms amplitude of the ripples will decrease as the square-root
of the count time. This is indeed approximately the case. We computed the
root-mean-square fluctuations in the $\Delta t=0.1$, 0.3 and 0.9 s data in the
region from $300<r<400$ Å for the silica glass data. We get
$\Delta_{\mathrm{rms}}(0.1)=0.0079$, $\Delta_{\mathrm{rms}}(0.3)=0.0048$, and
$\Delta_{\mathrm{rms}}(0.9)=0.0027$. We would expect ratios of $\sqrt{\Delta
t(1)/\Delta t(2)}=\Delta_{\mathrm{rms}}(2)/\Delta_{\mathrm{rms}}(1)$. The
results are presented in Table 4.

Table 4: Expected and actual root-mean-square (rms) fluctuations in the PDF for different ranges of the measurements of the silica glass sample. The ”Expected” column contains values that would be expected if the signal is dominated by random counting statistics, $\sqrt{\Delta t(1)/\Delta t(2)}$, and the ”Actual” column is the measured ratio, $\Delta_{\mathrm{rms}}(2)/\Delta_{\mathrm{rms}}(1)$, of the rms fluctuations in the signal over the range indicated. $\Delta t$ ratio | Range (Å-1) | Expected | Actual
---|---|---|---
0.3/0.1 | 300-400 | 1.73 | 1.65
0.9/0.1 | 300-400 | 3.00 | 2.93
0.9/0.3 | 300-400 | 1.73 | 1.78
0.3/0.1 | 2-10 | 1.73 | 1.00
0.9/0.1 | 2-10 | 3.00 | 1.00
0.9/0.3 | 2-10 | 1.73 | 1.00

The actual rms ratios are close to what is expected if they were coming from
random noise. The actual ratios tend to be slightly smaller than those
expected from random counting statistics. This may be due to the presence of
some small amount of systematic error in the measured signal, but for the case
of the Mo $K_{\alpha}$ data, it is rather small. The noise in the Mo
$K_{\alpha}$ data is counting-statistics limited, which is often not the case
in the synchrotron data.

In the low-$r$ region where the structural signal is strong, the rms ratios
are 1.00, which is what would be expected in the opposite limit, where the
signal/noise ratio is much greater than one. In this case, the fluctuations in
the data are coming from a repeatable structural signal, which doesn’t change
with count time.

Theoretically, noise in $G(r)$ is expected to be roughly constant in $r$
[toby;aca04], which is one of the attractive aspects of this function
[egami;b;utbp12]. It means that features, for example, in the difference
curves of a fit, can be compared at different values of $r$, and their
significance is the same. This is not true for other related pair correlation
functions [keenComparisonVariousCommonly2001b] such as the radial distribution
function. We do, indeed, see this in our current data where we see that, after
the structural signal disappears ($r=20$ Å for the silica glass and $r\sim
250$ Å for the quartz) the peak-to-peak amplitude of the noise fluctuations
are constant in $r$. To help illustrate this, in the inset to Fig. 5(b) and
(d), we plot dashed lines to indicate the computed rms values from the noise-
dominated regions of the PDF data. In the regions without structural signals,
the amplitude of the noise fluctuations is constant in $r$.

To make better choices about how long to count for in an experiment, we would
like to have a clearer measure of when the noise fluctuations are sufficiently
small for our needs, at which point we can stop counting. Rather than the
absolute value of the noise amplitude itself, the key issue is actually the
signal/noise ratio on some signal of interest to us scientifically such as the
presence or absence of small features in the PDF structural signal. To
investigate this, we need to look at a region of $G(r)$ where there is a
structural signal.

For a well-ordered crystal such as crystalline quartz, the structural signal
dies out slowly with $r$, as evident in Fig. 5(b) main panel, and the worst
signal/noise ratio will be at higher-$r$ values. In the inset, we show the
region $220<r<270$ Å, where the structural signal begins to disappear into the
noise for the red curve but not for the blue curve. It is not always possible
to determine by eye the difference between a weak structural signal and noise,
though in this comparison of two measurements of the same sample, features
that repeat in both curves are likely of structural origin, allowing a visual
assessment of the signal/noise ratio. If we were interested in the structural
signal in this region or in small modulations to the structural signal at this
level elsewhere in the PDF, this would be a reason to count for a longer time.
This might be important, for example, when estimating the size of
nanoparticles in samples. Similarly in the silica glass sample, the small
valley in the signal at around $r\sim 16$ Å (Fig. 5(d) inset) would be very
difficult to detect from the red data ($\Delta t=0.1$ s) but is very clear in
the pale blue ($\Delta t=0.9$ s). In the case of nanomaterials and amorphous
materials, or cases where we do not have a good model, the ability to assess
signal-noise directly means that better counting statistics are likely needed.

We can also assess the effect of the counting statistics on the value and the
precision of refined structural parameters. This is most relevant when the
scientific objective is extracting some parameter values from modeling. The
refined structural parameters are reproduced in Table 3. For this comparison
we can compare the P1(0.1) with the P1(0.3) columns, or the P2(0.3) and the
P2(1.8) columns. For the P2 comparison (with Soller slits), the refined
results are all within statistical uncertainties, as expected. There is less
good agreement for the P1 protocol (no Soller slits) comparison. This will be
discussed further in the next Section. This analysis establishes that, for the
case of the well-ordered quartz sample and the case where we are interested in
refined parameters, the P2(0.3) 4.5 h protocol gives comparable results to the
much longer P2(1.8) 25.5 h measurement.

A challenge when making measurements for PDF analysis is that we measure in
$Q$ but the signals we are interested in are in $G(r)$ and so we would like to
build up an intuition of what signals look like in $F(Q)$ when they give
sufficiently good statistics in $G(r)$ for our particular experimental setup.
The best reciprocal space function for doing this is $F(Q)$ because the data
have been fully normalized in the way they will be before the Fourier
transform. We show these functions in Fig. 5(a) and (b). The way that these
PDFs (b) appear depends on the step-size of the counting scan and can be
misleading in a visual comparison. For example, because we chose a step-size
that oversampled the signal in $G(r)$ (see Section 3) we could resample the
data onto a grid that was $2\times$ courser without loss of information. This
would result in a plot where the apparent noise was reduced by $\sqrt{2}$, but
actually, we had not changed the information in the data. It is therefore
important to develop your intuition about what are acceptable noise levels
through visual assessment of $F(Q)$, or the raw intensity, for your preferred
setup of scan parameters. To do this, make the assessment on PDFs plotted on
the Nyquist grid and then plot $F(Q)$ or the raw intensities and use this as a
guide, much as we have done here. Since we are doing PDF analysis, it is
ultimately the noise level in the PDF that should govern our decisions.

####  6.2.2 Effect of Soller slits

We now turn our attention to the effect of Soller slits. Soller slits reduce
the divergence of the incident and scattered beams. A 2.5∘ incident beam
Soller slit is a standard part of the experimental setup for the instrument
used in this work. However, an additional 2.5∘ Soller slit may be inserted or
left out in the secondary flight path after the sample. The effects of this
secondary Soller slit on the Bragg peaks are readily seen in Fig. 6.

Figure 6:  Comparison of (a) $F(Q)$ and (b) $G(r)$ of quartz with (red and
light blue) and without (dark blue) a 2.5∘ soller slit. The data were
collected with protocol P1 with $\Delta t=0.3$ s (dark blue), and with
protocol P2 with $\Delta t=0.3$ s (red) and $\Delta t=1.8$ s (light blue).
Expanded views of regions of the curves are shown in the insets.

For the Bragg peaks in the low-$Q$ region, comparing the red and pale blue
curves (with Soller slits) against the dark blue curve (without) we see there
is a significant broadening of the Bragg peaks with a pronounced low-$Q$ tail
in the absence of Soller slits. For this reason, Soller slits are often
preferred for conventional powder diffraction measurements, especially when
low-$Q$ is the scientific focus. In this region, the signal/noise ratio is
high, and there are large gains in line-shape improvements at essentially no
cost in doing this. However, we see (Fig. 6(a) main panel and middle inset)
that the peak-shape effect diminishes with increasing $Q$ and, above around
$Q=5$ Å-1, the Soller slits do little to improve the Bragg peak line-shape. On
the other hand, the Soller slits result in considerably worse counting
statistics. A comparison of the data collected for the same $\Delta t=0.3$ s
with and without Soller slits is given by comparing the red and dark blue
curves, respectively (Fig. 6(a) especially in the mid-high-$Q$ region, second
and third insets). The noise fluctuations are significantly greater in the red
curve compared to the dark blue curve for the same count times with and
without the Soller slits. In fact, we see that with the Soller slits, we have
to collect data for around six times longer (25.5 h, pale blue curve) to get
approximately the same counting statistics as without the Soller slits (4.5 h
dark blue curve).

The question then arises whether the effect on the Bragg peaks at low-$Q$, or
the reduced counting statistics, has a larger effect on the PDF. The sharper
low-$Q$ Bragg peaks obtained with the Soller slits do have a significant
effect on the range (the extent in $r$) of the structural signal in the
resulting PDF, as evident in Fig. 6(b), where the features in the pale-blue
PDF extend further in $r$ than the dark blue. This comparison is for datasets
with (pale-blue) and without (dark blue) but with approximately the same
counting statistics. Even though for much of the $Q$-range, the Soller slits
do not actually sharpen the Bragg peaks significantly, the sharpening in the
low-$Q$ region still results in PDF extending to significantly higher-$r$.

On the other hand, in the low-r̊egion (Fig. 6(b) upper inset) there is
essentially no effect on the amplitude or the width of the PDF peaks due to
the addition of the Soller slits (compare dark blue and light blue curves).

Another comparison is to consider collecting with and without Soller slits for
a fixed experiment time. Inserting the slits for the same count-time results
in much noisier data (comparing the red curve (with slits) to the dark blue
curve (without slits) both collected for 4.5 h). Again, we see that the signal
extends to higher-$r$ with the slits, but the noise in the PDF is higher, and
the argument about how good the counting statistics need to be takes over, as
discussed in Section 6.2.1.

We first note that, for the Mo $K_{\alpha}$ data with no Soller slit, there is
an evident oscillatory background in the PDF as can be seen in Fig. 4(a). This
likely comes from the shifting of the intensity away from the correct position
of the Bragg peaks in the low-$Q$ region of the diffraction pattern. Indeed,
the strongest signal in $G(r)$ seems to be a sinusoidal variation of the
difference curve with a wavelength of around 5 Å Fig. 4(a), which would come
from an issue in $F(Q)$ at around $Q=2\pi/5\approx 1.2$ Å-1. This is the
position of the strongest low-$Q$ Bragg peak that is significantly affected by
the absence of the Soller slits, as shown in Fig. 6(a). This would be less of
a concern if it did not result in significant differences in refined parameter
values. We can assess this by comparing columns P1(0.3) (no Soller) and
P2(1.8) (with Soller) in Table 3. The refined parameters are quite close but
do differ beyond the estimated uncertainties. The results with the Soller slit
are closer to those of the synchrotron measurement than are the ones taken
without the Soller slit. For example, both the synchrotron and the P2 results
indicate that the Si ADPs are approximately isotropic, whereas without the
Soller slits the fit returns very anisotropic Si ADPS. The symmetric
coordination of Si suggests that it should have rather isotropic ADPs, which
further supports the idea that the synchrotron and with-Soller fits are
getting it right but not the refinements without the Soller slits.

In summary, for the case we studied, the Soller slits did extend the PDF to
significantly higher-$r$ but at the cost of increasing the count time by a
factor of about six to obtain the same counting statistics. Furthermore, the
low-$Q$ diffraction line-shape aberrations resulted in a distortion to the
PDF, which affected the refined parameters during modeling. It seems that a
setup with a more parallel beam is to be preferred even though it is only
low-$Q$ peaks that are affected.

For diffuse signals, this effect will produce a smaller aberration on the
signal, and the importance of better statistics may argue in favor of making
measurements with a looser angular divergence (no Soller slits), but for
crystalline materials, measurements with Soller slits are clearly preferred.

####  6.2.3 Effect of beam length on the sample

We can also look at the effect of using a mask to limit the length of the
incident beam along the capillary. In the current case, a mask was crafted
using lead tape to restrict the beam length on the sample from 25 mm to 15 mm.
This masks out some of the out-of-plane beam divergence. The results are shown
in Fig. 7.

Figure 7:  Comparison of (a) $F(Q)$ and (b) $G(r)$ of quartz with a 15 mm
illumination length on the sample (red and light blue) and with 25 mm
illumination length (dark blue). The data were collected with protocol P3 with
$\Delta t=0.1$ s (red) and $\Delta t=0.3$ s (light blue) and P1 with $\Delta
t=0.1$ s (dark blue), and with protocol. Expanded views of regions of the
curves are shown in the insets.

The results of this test are very similar to the case of the axial Soller
slits, which limit the out-of-plane divergence of the beam. Inserting the Pb-
mask results in slightly sharper Bragg peaks in the low-$Q$ region of the
diffraction pattern but not in the mid-high $Q$ regions and results in noisier
data for the same count time. In the low-$r$ region of the PDF, the peak
shapes and amplitudes are essentially unaffected by the removal of the Pb-mask
but are slightly more noisy. The sharpening of the Bragg peaks is less acute
than in the case of the axial Soller slits, and so the extension of the signal
in the high-$r$ region is less pronounced.

The effect on refined parameters can be assessed by comparing columns P3(0.3)
with P1(0.1). The two result in quite comparable refinements, though the
P3(0.3) protocol gave results slightly closer to the synchrotron and Soller
slit cases.

Overall, the trade-off between count time and resolution seems to favor the
use of the full out-of-plane divergence (no axial Soller slits or Pb-mask) for
PDF measurements. However, for the most accurate measurements on highly
crystalline materials, they may benefit from limiting both out-of-plane and
in-plane (along $2\theta$) beam divergence at the cost of longer count-times.

##  7 Multiplicative correction

###  7.1 Absorption effects

Here, we explore the modification to the intensity due to sample absorption
for our capillary setup. In general, the measured intensity is proportional to
the effective sample volume probed by the x-ray beam, assuming a uniform
density of the sample. In the absence of significant x-ray absorption by the
sample this effective sample volume is just the volume of the physical region
of the sample that is illuminated by the x-rays. For common geometries used in
RA-PDF measurements at synchrotrons, cylindrical samples, or flat pill-shaped
samples perpendicular to the beam with the detector behind the sample, the
illuminated volume is not sattering-angle dependent and therefore not $Q$
dependent. No multiplicative correction is therefore needed to correct for
this issue in the PDFgetX3 algorithm.

When absorption by the sample is significant, the effective volume deviates
from the physical volume probed because there are different path-lengths that
the x-rays have to take to reach different sub-regions, or voxels, of that
physical volume. The measured scattered intensities from the different voxels
then become weighted by how many x-rays are absorbed by the sample on the path
the x-rays take to reach that voxel, and then the path to exit the sample
again to reach the detector. This results in an angle dependence on the
effective volume of the sample, which requires an angle-dependent
multiplicative correction to the measured intensities. In the forward-
scattering directions, this angle dependence is weak, which further helps the
approximation of no multiplicative correction for the RA-PDF geometry.
However, when intensities are collected over wide angular ranges, the effects
can be large. Part of the upturn in the Mo $K_{\alpha}$ data evident in Fig. 2
will originate from this effect because we expect the effective sample volume
to be larger in backscattering because more of the sample voxels can be
accessed by short path-lengths through the sample.

###  7.2 Effective volume correction

To determine the effective sample volume, $V_{e}$, for the capillary geometry
we have here we first determine the x-ray path length into and out of each
voxel at each scattering angle, $2\theta$, $\ell_{v}(2\theta)$. We can then
determine the absorption for the $v^{\mathrm{th}}$ voxel, $A_{v}(2\theta)$,
from

$A_{v}(2\theta)=e^{-\mu_{s}\ell_{v}(2\theta)}$ (8)

where $\mu_{s}$ is the sample linear x-ray attenuation coefficient, where we
assume a macroscopically uniform sample so it does not depend on the location
of the voxel. The normalized scattering is then obtained by dividing the
measured intensity by the effective sample volume given by

$V_{e}=\sum_{v}A_{v}\Delta V_{v},$ (9)

where $\Delta V_{v}$ is the volume of the $v^{\mathrm{th}}$ voxel and the sum
is over all the voxels in the illuminated region of the sample. The volume-
corrected intensity, $i_{c}$ is then given by

$i_{c}(2\theta)=\frac{I_{c}(2\theta)}{V}=\frac{I_{m}(2\theta)}{V_{e}(2\theta)},$
(10)

for a measured intensity $I_{m}$.

Figure 8: Quartz Mo $K_{\alpha}$ diffraction pattern with and without
correction for absorption. The raw, uncorrected, signal is dark blue and the
absorption corrected signals is red. A version of the absorption corrected
signal that has been scaled down to lie on top of the raw signal is shown in
light blue. The high-$Q$ behavior of the raw and scaled-corrected curves are
shown on an expanded scale in the inset. In the second inset we show the
effective volume, $v_{e}$, from the absorption correction for a 1.5 mm
diameter tube of quartz with and Mo $K_{\alpha}$ radiation.

In Fig. 8 we show the angle-dependence of $V_{e}(2\theta)$, $v_{e}(2\theta)$,
for the case of our Mo $K_{\alpha}$ experiments on the silica samples. This
was computed for the sample composition SiO2 and an assumed packing fraction
for the powder of 1.0 resulting in $\mu_{s}=7.39$ cm-1, and a 1.5 mm ID
capillary. To get $v_{e}(2\theta)$ we divide $V_{e}(2\theta)$ by the physical
illuminated volume, $V=\sum_{v}\Delta V_{v}$,

$\displaystyle v_{e}$ $\displaystyle=V_{e}/V$ (11)
$\displaystyle=\frac{\sum_{v}e^{-\mu_{s}\ell_{v}(2\theta)}\Delta
V_{v}}{\sum_{v}\Delta V_{v}}.$ (12)

If all the voxels are the same size, $\sum_{v}\Delta V_{v}=N_{v}\Delta V$ and
we get

$v_{e}=\frac{1}{N_{v}}\sum_{v}e^{-\mu_{s}\ell_{v}(2\theta)},$ (13)

and our corrected intensity is

$\displaystyle I_{c}$ $\displaystyle=\frac{I_{m}}{v_{e}}$ (14)
$\displaystyle=\frac{N_{v}I_{m}(2\theta)}{\sum_{v}e^{-\mu_{s}\ell_{v}(2\theta)}}$
(15)

We plot $v_{e}$ for our sample in the inset to Fig. 8. The absolute value of
$v_{e}$ for our case Mo $K_{\alpha}$ radiation and a 1.5 mm tube of finely
powdered quartz ($\mu d\sim 1.1$ assuming packing fraction 1, where $d$ is the
diameter of the tube) is a little less than 0.5, with a $Q$-dependence from
low-$Q$ to high-$Q$ of a little less than 0.05. Thus, for $\mu d$ of the order
of unity, the effects on the resulting PDF are small.

###  7.3 Effect of the $v_{e}$ correction on the measured signal

Fig. 8 shows the raw and corrected diffraction curves for our Mo $K_{\alpha}$
quartz. Dividing by $v_{e}$ increases the overall scale factor of the
corrected signal. This is seen in the main panel where the raw signal is shown
in dark blue and the corrected signal in red. Since the PDFgetX3 algorithm is
scale agnostic, this overall change in scale will not result in a change in
the resulting PDF. However, as we have discussed, there is also
$I(Q)$-dependence to the absorption correction where the effective volume is
larger in the back-scattering region. When we divide by $v_{e}$ we will
therefore preferentially scale down the high-$I(Q)$ signal compared to the
low-$Q$ signal. This is shown in the inset where we have taken the corrected
curve in red and scaled it down so that it lies on top of the raw signal in
the low-r region. We see that in the high-$Q$ region the absorption corrected
signal increases by less. We see from the plot of $v_{e}$ that the
$Q$-dependence of the curve is an order of magnitude less than the absolute
value. It increases from 0.4 to $\sim 0.45$.

The comparison of refined parameters with and without the absorption
correction can be assessed by comparing the columns P2(1.8) and P2(1.8) ”Abs”
in Table 3. All of the refined parameters are the same within estimated
uncertainties. For the case of our experimental setup the effects of
neglecting the absorption correction were negligible.

##  8 Summary and Conclusions

We have carried out a systematic assessment of different protocols for
measuring data on a laboratory Mo $K_{\alpha}$ source. We present a particular
variable counting scheme, the staircase count time scheme, that results in
preferential counting in the high-$Q$ region where the signal/noise ratio is
worst for PDF data. We show that for crystalline materials and PDF modeling,
the Mo $K_{\alpha}$ data gave comparable results to the synchrotron
measurement. For PDF model refinements on crystalline materials, limiting the
angular divergence, in our case with the use of Soller slits in the secondary
flight-path, gave more accurate results. We report additional corrections that
may be introduced in the data reduction process to account for sample
absorption. For our case, with a $\mu d\sim 1.1$ computed with a packing
fraction of 1, the effects were small. It is recommended to assess sample
capillary thickness to obtain a $\mu d$ close to unity or below at the energy
of the laboratory x-ray source. However, for more absorbing samples, we
present software that may be inserted into the data reduction steps to correct
for these effects. The uncertainties in the Mo $K_{\alpha}$ diffraction
patterns were shown to be dominated by counting statistics, whereas the
synchrotron data fits are model limited and uncertainty estimates do not
reflect the true accuracy.

Funding information

Work in the Billinge group was supported by a grant from 3M Corporate Research
Analytical Laboratory. Synchrotron x-ray PDF measurements were conducted on
beamline 28-ID-1 of the National Synchrotron Light Source II, a U.S.
Department of Energy (DOE) Office of Science User Facility operated for the
DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-
SC0012704. Till Schertenleib acknowledges support from the Swiss National
Science Foundation under grant number 200021_188536.

Data availability

Data used for all the plots in the manuscript are available at Zenodo at
https://doi.org/10.5281/zenodo.11060384

## References

  * [1] A. Young L. Goodwin2011a.youngApplicationsPairDistribution2011 A. Young, C. L. Goodwin, A. 2011.  _Journal of Materials Chemistry_ , 21(18), 6464–6476.
  * [2] [Ashiotis _et al._]Ashiotis, Deschildre, Nawaz, Wright, Karkoulis, Picca Kieffer2015ashio;jac15 Ashiotis, G., Deschildre, A., Nawaz, Z., Wright, J. P., Karkoulis, D., Picca, F. E. Kieffer, J. 2015.  _J. Appl. Cryst._ 48(2), 510–519.
  * [3] [Beecher _et al._]Beecher, Semonin, Skelton, Frost, Terban, Zhai, Alatas, Owen, Walsh Billinge2016beecherDirectObservationDynamic2016b Beecher, A. N., Semonin, O. E., Skelton, J. M., Frost, J. M., Terban, M. W., Zhai, H., Alatas, A., Owen, J. S., Walsh, A. Billinge, S. J. L. 2016.  _ACS Energy Lett._ 1(4), 880–887.
  * [4] Benmore2012benmoreReviewHighEnergyXRay2012 Benmore, C. J., 2012.  A Review of High-Energy X-Ray Diffraction from Glasses and Liquids.  https://www.hindawi.com/journals/isrn/2012/852905/.
  * [5] Billinge2008billingeNanoscaleStructuralOrder2008d Billinge, S. J. L. 2008.  _Journal of Solid State Chemistry_ , 181(7), 1695–1700.
  * [6] Bruker D8 Advance2023D8ADVANCE Bruker D8 Advance, 2023.  https://www.bruker.com/en/products-and-solutions/diffractometers-and-x-ray-microscopes/x-ray-diffractometers/d8-advance-family/d8-advance.html.  Online; accessed 15 November 2023.
  * [7] Bruker D8 Discover2023D8DISCOVER Bruker D8 Discover, 2023.  https://www.bruker.com/en/products-and-solutions/diffractometers-and-x-ray-microscopes/x-ray-diffractometers/d8-discover-family/d8-discover-plus.html.  Online; accessed 15 November 2023.
  * [8] [Chupas _et al._]Chupas, Grey, Hanson, Kim, Rodriguez, Qiu, Billinge Lee2003 _a_ chupa;iucrcpd03 Chupas, P. J., Grey, C. P., Hanson, J. C., Kim, J.-Y., Rodriguez, J., Qiu, X., Billinge, S. J. L. Lee, P. L. 2003 _a_.  _Commission on Powder Diffraction Newsletter, International Union of Crystallography_ , (29), 24–25.
  * [9] [Chupas _et al._]Chupas, Qiu, Hanson, Lee, Grey Billinge2003 _b_ chupasRapidacquisitionPairDistribution2003f Chupas, P. J., Qiu, X., Hanson, J. C., Lee, P. L., Grey, C. P. Billinge, S. J. L. 2003 _b_.  _J. Appl. Cryst._ 36(6), 1342–1347.
  * [10] [Confalonieri _et al._]Confalonieri, Dapiaggi, Sommariva, Gateshki, Fitch Bernasconi2015confalonieriComparisonTotalScattering2015 Confalonieri, G., Dapiaggi, M., Sommariva, M., Gateshki, M., Fitch, A. N. Bernasconi, A. 2015.  _Powder Diffraction_ , 30(S1), S65–S69.
  * [11] [Dykhne _et al._]Dykhne, Taylor, Florence Billinge2011dykhn;phmr11 Dykhne, T., Taylor, R., Florence, A. Billinge, S. J. L. 2011.  _Pharmaceut. Res._ 28, 1041–1048.
  * [12] Egami Billinge2012egami;b;utbp12 Egami, T. Billinge, S. J. L. 2012.  _Underneath the Bragg Peaks: Structural Analysis of Complex Materials_.  No. 16 in Pergamon Materials Series. Amsterdam: Elsevier, 2nd ed.
  * [13] [Farrow _et al._]Farrow, Juhas, Liu, Bryndin, Božin, Bloch, Proffen Billinge2007farro;jpcm07 Farrow, C. L., Juhas, P., Liu, J. W., Bryndin, D., Božin, E. S., Bloch, J., Proffen, T. Billinge, S. J. L. 2007.  _J. Phys.: Condens. Matter_ , 19(33), 335219.
  * [14] [Farrow _et al._]Farrow, Shaw, Kim, Juhás Billinge2011farro;prb11 Farrow, C. L., Shaw, M., Kim, H.-J., Juhás, P. Billinge, S. J. L. 2011.  _Phys. Rev. B_ , 84, 134105.
  * [15] [Fischer _et al._]Fischer, Barnes Salmon2005fischerNeutronXrayDiffraction2005 Fischer, H. E., Barnes, A. C. Salmon, P. S. 2005.  _Reports on Progress in Physics_ , 69(1), 233–299.
  * [16] Fitch2004fitchHighResolutionPowder2004 Fitch, A. N. 2004.  _J Res Natl Inst Stand Technol_ , 109(1), 133–142.
  * [17] Furukawa1962furukawaRadialDistributionCurves1962 Furukawa, K. 1962.  _Reports on Progress in Physics_ , 25(1), 395.
  * [18] [Irving _et al._]Irving, Keen Light2021irvingAdvantagesCurvedImage2021a Irving, D. J. M., Keen, D. A. Light, M. E. 2021.  _Review of Scientific Instruments_ , 92(4), 043107.
  * [19] [Juhás _et al._]Juhás, Davis, Farrow Billinge2013juhas;jac13 Juhás, P., Davis, T., Farrow, C. L. Billinge, S. J. L. 2013.  _J. Appl. Cryst._ 46, 560–566.
  * [20] [Juhás _et al._]Juhás, Farrow, Yang, Knox Billinge2015juhas;aca15 Juhás, P., Farrow, C., Yang, X., Knox, K. Billinge, S. 2015.  _Acta Cryst. A_ , 71(6), 562–568.
  * [21] Keen2001keenComparisonVariousCommonly2001b Keen, D. A. 2001.  _J. Appl. Cryst._ 34(2), 172–177.
  * [22] Keen2020keenTotalScatteringPair2020 Keen, D. A. 2020.  _Crystallography Reviews_ , 26(3), 143–201.
  * [23] Keen Goodwin2015keenCrystallographyCorrelatedDisorder2015 Keen, D. A. Goodwin, A. L. 2015.  _Nature_ , 521(7552), 303–309.
  * [24] [Lin _et al._]Lin, Bǒzin, Billinge, Quarez Kanatzidis2005lin;prb05 Lin, H., Bǒzin, E. S., Billinge, S. J. L., Quarez, E. Kanatzidis, M. G. 2005.  _Phys. Rev. B_ , 72, 174113.
  * [25] [Lindahl Christiansen _et al._]Lindahl Christiansen, R. Cooper Ø. Jensen2020lindahlchristiansenThereNoPlace2020a Lindahl Christiansen, T., R. Cooper, S. Ø. Jensen, K. M. 2020.  _Nanoscale Advances_ , 2(6), 2234–2254.
  * [26] PANalytical Empyrean2023EmpyreanXRDInstrument PANalytical Empyrean, 2023.  https://www.malvernpanalytical.com/en/products/product-range/empyrean-range/empyrean.  Online; accessed 15 November 2023.
  * [27] [Prinz _et al._]Prinz, Schwensow, Wendholt, Jentys, Bauer, Kleist Zobel2020prinzHardXraybasedTechniques2020 Prinz, N., Schwensow, L., Wendholt, S., Jentys, A., Bauer, M., Kleist, W. Zobel, M. 2020.  _Nanoscale_ , 12(29), 15800–15813.
  * [28] [Qiu _et al._]Qiu, Thompson Billinge2004qiu;jac04i Qiu, X., Thompson, J. W. Billinge, S. J. L. 2004.  _J. Appl. Cryst._ 37, 678.
  * [29] Rigaku XtaLAB Synergy-S2023Rigaku Rigaku XtaLAB Synergy-S, 2023.  https://www.rigaku.com/products/crystallography/synergys.  Online; accessed 15 November 2023.
  * [30] STOE STADI P2023Stadi STOE STADI P, 2023.  https://www.stoe.com/products/stadi-mp/.  Online; accessed 15 November 2023.
  * [31] [Terban _et al._]Terban, Johnson, DiMichiel Billinge2015terba;nan15 Terban, M. W., Johnson, M., DiMichiel, M. Billinge, S. J. L. 2015.  _Nanoscale_ , 7, 5480–5487.
  * [32] [Terban _et al._]Terban, Russo, Pham, Barich, Sun, Burke, Brum Billinge2020terbanLocalStructuralEffects2020 Terban, M. W., Russo, L., Pham, T. N., Barich, D. H., Sun, Y. T., Burke, M. D., Brum, J. Billinge, S. J. L. 2020.  _Mol. Pharmaceutics_ , 17(7), 2370–2389.
  * [33] [Thomae _et al._]Thomae, Prinz, Hartmann, Teck, Correll Zobel2019thomaePushingDataQuality2019 Thomae, S. L. J., Prinz, N., Hartmann, T., Teck, M., Correll, S. Zobel, M. 2019.  _Review of Scientific Instruments_ , 90(4), 043905.
  * [34] Toby Billinge2004toby;aca04 Toby, B. H. Billinge, S. J. L. 2004.  _Acta Cryst. A_ , 60, 315–317.
  * [35] [Tsymbarenko _et al._]Tsymbarenko, Grebenyuk, Burlakova Zobel2022tsymbarenkoQuickRobustPDF2022a Tsymbarenko, D., Grebenyuk, D., Burlakova, M. Zobel, M. 2022.  _J. Appl. Cryst._ 55(4), 890–900.
  * [36] Wagner1978wagnerDirectMethodsDetermination1978 Wagner, C. N. J. 1978.  _Journal of Non-Crystalline Solids_ , 31(1), 1–40.
  * [37] [Yang _et al._]Yang, Juhas, Farrow Billinge2014yang;arxiv14 Yang, X., Juhas, P., Farrow, C. L. Billinge, S. J. L. 2014.  _arXiv:1402.3163 [cond-mat]_.
  * [38]