 1. Introduction
 2. Methods
 3. Reciprocalspace functions
 4. Realspace distribution functions
 5. Summary
 A1. Conversions from the radial distribution function, R(r)
 A2. Conversions from the number density, ρ(r)
 A3. Conversions from g(r)
 A4. Conversions from the total radial distribution function, GK(r)
 A5. Conversions from the reduced pair distribution function, G(r)
 A6. Conversions from the differential correlation function, D(r)
 A7. Conversions from the total correlation function, T(r)
 References
 1. Introduction
 2. Methods
 3. Reciprocalspace functions
 4. Realspace distribution functions
 5. Summary
 A1. Conversions from the radial distribution function, R(r)
 A2. Conversions from the number density, ρ(r)
 A3. Conversions from g(r)
 A4. Conversions from the total radial distribution function, GK(r)
 A5. Conversions from the reduced pair distribution function, G(r)
 A6. Conversions from the differential correlation function, D(r)
 A7. Conversions from the total correlation function, T(r)
 References
teaching and education
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Illustrated formalisms for total scattering data: a guide for new practitioners
^{a}Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA, ^{b}National Synchrotron Light SourceII, Brookhaven National Laboratory, Upton, NY, USA, ^{c}Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA, and ^{d}Materials Science and Engineering Department, University of Tennessee, Knoxville, TN, USA
^{*}Correspondence email: petersonpf@ornl.gov, dolds@bnl.gov
The total scattering method is the simultaneous study of both the real and reciprocalspace representations of diffraction data. While conventional Braggscattering analysis (employing methods such as Rietveld refinement) provides insight into the average structure of the material, pair distribution function (PDF) analysis allows for a more focused study of the local atomic arrangement of a material. Generically speaking, a PDF is generated by Fourier transforming the total measured reciprocalspace diffraction data (Bragg and diffuse) into a realspace representation. However, the details of the transformation employed and, by consequence, the resultant appearance and weighting of the realspace representation of the system can vary between different research communities. As the worldwide total scattering community continues to grow, these subtle differences in nomenclature and data representation have led to conflicting and confusing descriptions of how the PDF is defined and calculated. This paper provides a consistent derivation of many of these different forms of the PDF and the transformations required to bridge between them. Some general considerations and advice for total scattering practitioners in selecting and defining the appropriate choice of PDF in their own research are presented. This contribution aims to benefit people starting in the field or trying to compare their results with those of other researchers.
Keywords: total scattering; pair distribution function.
1. Introduction
Pair distribution function (PDF) analysis is a broad term encompassing the use of experimentally generated atomic pair density functions in real space to study shortrange order in materials. The approach has its roots in the work of Warren and coworkers, who developed it to study the short and intermediaterange structure of glasses (Warren, 1934, 1937). Notable extensions have included work on molecular liquids (Narten, 1972), the study of glasses and crystalline materials (Srolovitz et al., 1981), the study of liquids and glasses with neutron diffraction (Wright et al., 1989; Wright, 1990, 1994), the use of highenergy Xrays (Egelstaff, 1967; Root et al., 1986; Neuefeind & Poulsen, 1995; Neuefeind et al., 1996), the adoption of the technique for reverse Monte Carlo (RMC) methods (McGreevy & Pusztai, 1988; McGreevy, 1995, 2001), and the application to nanostructured and disordered crystalline materials (Billinge, 1992; Egami & Billinge, 2012). These advancements and their applications to materials are distinguished in a number of review articles and text books appearing in the past two decades, for liquids and glasses (Neuefeind, 2002; Egelstaff, 2003; Kohara & Suzuya, 2003; Fischer et al., 2005; Kohara et al., 2007; Benmore, 2012), and for disordered crystalline and nanostructured materials (Egami, 2007; Billinge & Kanatzidis, 2004; Billinge, 2004; Billinge & Levin, 2007; Young & Goodwin, 2011; Egami & Billinge, 2012; Playford et al., 2014; Hou et al., 2018).
The PDF is a probability distribution function that measures the probability of finding pairs of atoms separated by a given distance. There are many different forms of the PDF with subtly different functional forms, units, normalizations and use in research communities, but they all contain the same information: the probability of finding atoms separated by a distance (Dinnebier & Billinge, 2008). For example, from a statistical mechanics definition of the radial distribution function, this probability can be given relative to the state, where no correlations exist. Therefore, deviations in such a distribution function will give a factor to multiply the bulk density by to get a local density (McQuarrie, 2000). All the different forms of the PDF give information about the changes in local density with distance and thus insight into the local structure.
The use of PDF analysis has grown exponentially from a specialized technique employed for the study of liquids, glasses and other amorphous materials [where Rietveld (1969) analysis is not possible] to one that encompasses disordered materials more broadly through the study of local atomic structure and disorder in nanocrystalline and crystalline materials. As various material communities have adopted PDF analysis, they have refined the methodology and developed corresponding analysis software to address their specific scientific needs. The weighting and normalization of different features, either from the measured reciprocalspace data or modeled realspace atomic coordinates, has led to no less than eight different published forms of realspace distribution function (Keen, 2001), which can all claim in one way or another to be analogous to the PDF. Despite being functionally similar, the differences between these varied forms has led to some confusion and redundancy across the different communities.
This contribution aims to untangle many of these semantic and terminological confusions through a consistent derivation of the relationships between many different forms of the PDF, demonstrated through simple examples. We reintroduce many fundamental concepts and formalisms, and directly relate them to the physical and distinguishing features they represent.
The equations are presented here with neutron scattering formalism, where there is no Q dependence in the atomic scattering lengths. The added complexity brought by Qdependent Xray form factors (Narten, 1972) has been, broadly, addressed in three approaches: explicit corrections during data normalization (Qiu, Thompson & Billinge, 2004), ad hoc or approximative corrections (Juhás et al., 2013; Billinge & Farrow, 2013) during data normalization, or leaving the data uncorrected and instead forward calculating the effects in a refined model (Gereben et al., 2007; Tucker et al., 2007). The variability of these different approaches and their implications are outside the scope of this paper. Herein, the presented derivations assume that the reduction of Xray scattering data successfully mitigates the effects of these form factors.
For the remainder of this text, the authors make the following assumptions which are being made explicit with the aim of aiding the reader even further in seeing the bridge between various conventions. First, the term `total scattering' was adopted for the PDF method within the past few decades to bring attention to the fact that it provides an examination of both Bragg and diffuse scattering. The modern adoption of this colloquial name should be distinguished from the foundational terminology `total scattering' in use by the general timeofflight neutron scattering community (Squires, 2012). Note also that most neutron and Xray PDF measurements are energyintegrated scattering functions. Throughout this manuscript `total scattering' will be used to refer to the PDF technique in general. Second, as will be further explained in Section 3, the I(Q) presented here is proportional to the differential Note that I(Q) is different from the measured intensity often employed in Rietveld refinements, sometimes utilizing the same nomenclature (Rietveld, 1969).
Previous work went into great detail on the derivation and conversion of different functional forms of the PDF, and is widely cited in the community (Keen, 2001; Fischer et al., 2005). Subtle differences in nomenclature of various approaches have led to some confusion in the PDF communities, particularly when converting between the different formalisms. This manuscript expands on this work, beginning with the conventions of the disordered crystalline material communities (Egami & Billinge, 2012) and bridging to other derivations. Visual comparisons of various functional forms guide the reader and frame discussion of each use case. Effort has been made to reference both initial derivations and examples of utilization in scientific literature.
The derivations and examples in this paper are presented first in A provides a list of simple conversions between the various realspace functions. Appendix B provides details of the simulations used for the liquid example data. Appendix C is an overview of the process of converting from measured intensities to differential cross sections to provide a frame of reference for understanding experimental concerns. Appendix D details calculation of the normalized Laue term employed in some formalisms, while Appendix E details calculation of the Appendix F provides a brief overview of functions.
then in real space. A collection of appendices provide both reference and further detail supporting the derivations. Appendix2. Methods
To illustrate the different functions contained herein, we have simulated neutron nuclear scattering data for two systems of representative material types: bulk binary oxide manganese oxide (MnO) (Sasaki et al., 1979) for crystalline and disordered materials; and liquid argon (Ar) (Yarnell et al., 1973) for liquid, amorphous and glass materials. Mn has a negative neutron scattering length and O has a positive one, which emphasizes certain differentiating characteristics of various forms of the PDF. MnO has a magnetic structure that will be ignored to allow for focusing on the atomic structure. Simulated data from liquid argon (Ar) are also included as a monatomic example. Note that the term `monatomic system' here assumes a single element with a single isotope for neutron scattering and a single element with a single charge state for Xray scattering.
To generate the presented MnO data, realspace patterns were simulated with the PDFgui software (Farrow et al., 2007) using the in Table 1. Patterns were calculated for 0 ≤ r ≤ 160 Å with a bin width of δr = 0.01 Å. This pattern was then inverse transformed to generate the presented reciprocalspace data.

To generate the Ar data, LargeScale Atomistic/Molecular Massively Parallel Simulator (LAMMPS) opensource code (Plimpton, 1995, 2018). Details of the simulations are provided in Appendix B.
simulations were performed using the3. Reciprocalspace functions
We begin the derivation assuming data in the form of the fully corrected and normalized scattering intensity, I(Q), obtained from experimentally measured intensities. Interestingly, Debye himself, along with Menke, performed the first PDF experiments using Xray scattering in 1930 to obtain such data (Debye, 1930). The details for a protocol used to reduce such measured data to fully corrected patterns based on literature and reduction software manuals can be found in Appendix C. I(Q) can be directly related to a set of atomic coordinates through the Debye (1915) scattering equation:
where b_{coh,ν} is the coherent scattering length of atom ν and r_{νμ} = r_{ν} − r_{μ} is the interatomic pairwise vector of atoms ν and μ (Lovesey, 1986; Farrow & Billinge, 2009; Page et al., 2011). Debye's formalism, slightly modified to include the effects of thermal atomic displacements through a Debye–Waller term σ_{νμ} (Debye, 1913; Waller, 1923), is written as
In this formalism I(Q) is the scattering from the sample as a whole. The total scattering community most commonly employs I(Q)/N (the differential cross section), where N is the number of atoms illuminated in the sample (described further in Appendix E).
Peak profile e.g. Rietveld analysis) most commonly define the scattering per sample, rather than scattering per atom. This convention can be traced back to Rietveld's initial aims of fitting models against the peak profiles of the relative intensities as directly generated from instrument measurements. Use of an arbitrary scale factor during modeling was convenient and sufficient for this purpose. The total scattering formalism, on the other hand, allows for a fully atomistic model comparison with data. Accurate corrections to remove experiment artifacts are required to compare the data with atomistic models, the importance of which has been shown elsewhere (Egelstaff, 1992; Wright, 1994; Fischer et al., 2005). However, in practice, many practitioners studying crystalline materials also apply an arbitrary scale factor to data during modeling (Farrow et al., 2016). In fact, the departure of a scale factor from unity for standard (known) samples is sometimes applied as a quality criterion for assessing the success of data reduction procedures (Peterson et al., 2003).
methods (When defined only through isotropic atomic displacement parameters (commonly referred to as U_{iso}), the Debye–Waller term σ_{νμ} can be written as (Jeong et al., 1999, 2003; Proffen & Billinge, 1999)
where σ_{ν} and σ_{μ} are the amplitudes of the uncorrelated thermal motion of atoms ν and μ. This relationship is more complicated in the case of anisotropic atomic displacement (Dunitz et al., 1988; Jeong et al., 2003), but its effect on the normalized intensity is similar: exponential dampening of the Bragg intensities at high Q.
Another form of the normalized and corrected scattering data is the `structure function', S(Q). This form of the scattering data is employed in the generation of many atomic pair–pair representations of data, which accounts for its widespread description in past work (Yarnell et al., 1973; Billinge & Egami, 1993; Keen, 2001; Peterson et al., 2003; Farrow & Billinge, 2009; Page et al., 2011; Olds et al., 2015) where details of the derivation can be found. The structure function is related to the normalized I(Q) function through the relationship
where is the average total scattering power of the system, σ_{tot} is the total and 〈b_{coh}〉^{2} is the average coherent scattering power of all atoms in the sample. Appendix D provides a more complete discussion of the total scattering length term, . The second term in equation (4) contributes a constant factor called the normalized Laue monatomic diffuse scattering term. The normalized Laue term, often written simply as L, is
L is zero in the case of singleelement scattering (though not strictly so for neutron scattering, since naturally abundant elemental samples often comprise a mix of isotopes). L is calculated for our simulated examples in Appendix D.
The limits of S(Q) are
and
where η is unitless and proportional to the isothermal compressibility of the sample (Lovesey, 1986; Egelstaff, 1992; Wang et al., 2014). Thermodynamically, η is defined as
where κ is the isothermal compressibility (equal to the inverse of the bulk modulus, K_{0}), ρ_{0} is the k_{B} is Boltzman's constant, T is temperature and P is pressure. This is only strictly correct for monatomic, homogeneous, isotropic systems and is incorrect for a fluid close to its Additional details and references for more complex cases such as mixtures of molecular liquids and ions in aqueous solution are given by Fischer et al. (2005). The isothermal compressibility can be calculated as
where V is the volume and is the variance of the volume. η is often negligible (Bhatia & Thornton, 1970; Wagner, 1985; Egelstaff, 1992; Keen, 2001), as shown in Table 2 for our simulated examples. Note that in cases where nanostructured features exist (such as materials where smallangle scattering is present) the measured lowQ behavior will deviate (Mildner & Carpenter, 1984; Farrow & Billinge, 2009; Olds et al., 2015).

The `reduced total scattering structure function' is defined as
This representation of reciprocalspace data has a limit of 0 at high Q and is linearly weighted by Q, such that noise and resolution are highlighted features that have dramatic effects on the resultant realspace PDF). Another advantage to this formalism is that associated uncertainties increase linearly with Q (Egami & Billinge, 2012; Olds et al., 2018).
In the derivations of Keen (2001), a similar function also referred to as F(Q) is presented. This alternative function, here referred to as F_{K}(Q), is scaled by 〈b_{coh}〉^{2} and not by Q. Thus, these three reciprocalspace function are related as
The normalized and corrected intensity, I(Q), is related to F(Q) and F_{K}(Q) as
and
A visual comparison of S(Q), F(Q) and F_{K}(Q) is shown for the case of MnO in Fig. 1 and for the case of Ar in Fig. 2. A summary of the limiting behaviors of these functions and I(Q) can be found in Table 3.

4. Realspace distribution functions
The pair distribution function is a general concept describing the distribution of distances between pairs of objects contained in a volume. Zernike & Prins (1927) were the first to report the theoretical expression for the atomic density at a given separation in real space via their Fourier transform relationship, leading to the origin of the PDF formalisms. Yet, throughout the years of literature on the PDF, many different functional forms have spawned from this origin.
When defined independently of the atomic origin, μ, this is termed a radial distribution function (RDF), an entity that finds prevalent use as a descriptor for the atomic structure of amorphous, liquid, disordered and nanocrystalline materials. The same name can be associated with different functional forms, ever increasing confusion. For example, the name `radial distribution function' is associated with both equation (14) and equation (22) in previous literature (Thorpe et al., 1998; McQuarrie, 2000). In this section, we derive and relate a number of functions used in various research communities for representing realspace PDFs (see Figs. 3 and 4 below). These are generally related by multiplicative or additive constants and thus contain the same underlying information. We will explain some of the relative merits and related preferences for these formalisms at the close of the section.
A conceptually different quantity is the RDF which, containing no relationship to scattering weights and thus not directly measurable, is presented here for comparison. We begin by defining a configuration of N atoms arranged such that each atom has a position defined through the vector r_{ν}. The interatomic distance between any pair of atoms, μ and ν, is thus r_{μν} = r_{ν} − r_{μ}. An unweighted radial distribution function, labeled here RDF(r), can be constructed through the sum of Dirac delta functions, δ, which describe the full set of these pair–pair distances [of which there will be N(N − 1)/2 total pairs]. RDF(r) can be written as
A radial PDF can be generated from the measured scattering intensities of various physical measurements, including r) is straightforward to calculate but is only straightforward to measure with monatomic systems. More easily measured, radiationspecific PDFs can be calculated from atomic models by accounting for the scattering power of each atom. This results in the weighted radial distribution function, R(r), defined as
electron diffraction, Xray diffraction and neutron diffraction, with the last three all used to produce atom–atom PDFs. RDF(For the case of monatomic systems, the weighting prefactor becomes unity and R(r) simplifies to the equation for RDF(r).
A similar formalism often encountered is the density function, ρ(r), which is the radial distribution function normalized by the surface area of a sphere of radius r, such that (Warren, 1990)
For isotropic and threedimensional systems, the density function can be directly related to S(Q) through the following pair of transforms (Warren, 1990; Billinge, 1992):
and
The heavily used form of the PDF encountered in disordered crystalline material literature is the reduced pair distribution function, G(r) (Egami & Billinge, 2012), which is defined in relation to the density function as
Here, ρ_{0} is the average of N atoms in the volume V such that ρ_{0} = N/V, and γ_{0}(r) is the characteristic shape function or nanoparticle form factor (Guinier & Fournet, 1955; Azaroff, 1968; Farrow & Billinge, 2009; Olds et al., 2015). In the case of bulk materials, γ(r) = 1.0, and thus the term is often neglected in the literature.
This reduced pair distribution function can be generated from reciprocalspace data via the sine transform of F(Q), such that
Therefore, the Fourier inversion theorem holds that
An alternative formalism of the PDF often encountered in studies of amorphous and liquid materials is g(r). It is frequently called the pair distribution function by the liquids/amorphous community and the pair density function by the disordered crystalline materials community (Benmore, 2012). g(r) is functionally identical to the density function ρ(r); however, it has been scaled by the average resulting in the relationship
Note that the g(r) function is related to isothermal compressibility, defined in equation (9), via
Up to this point, all described atomic PDFs have assumed a sum over all atom–atom pairs in a defined volume. However, one can define a `partial PDF', g_{μν}(r), which includes contributions from only those atoms in a given pair type. By definition, the sum of all possible partial PDFs will reconstruct the corresponding allatom PDF. The most common convention requires that `Faber–Ziman factors' be calculated for each atom pair. Yet other formalisms exist, each with their respective advantages. The Bhatia–Thornton formalism is an alternative representation of the system as the mean square fluctuations in the particle number, fluctuations in concentration and the correlation between these two correlations (Bhatia & Thornton, 1970). These can be, for a twocomponent system, directly mapped to Faber–Ziman using the equations in Bhatia and Thornton's seminal 1970 work. The Ashcroft–Langreth formalism is another that is commonly used in theoretical and computational work owing to its connection to direct correlation functions (Ashcroft & Langreth, 1967). The mapping of the Ashcroft–Langreth to the Faber–Ziman equations is most readily accessible in equation 2.35 of the review paper by Fischer et al. (2005) The connection to the Faber–Ziman factors and both the total scattering and partial pair distribution functions is presented in Appendix F. The weighted sum of the partial PDFs will result in a g(r) such that
where W_{μν} is the associated weighting factor for the pair of atoms μ and ν. Note that g_{μν}(r) is a `true' distribution function as it does not include weighting by scattering lengths. However, this is not a distribution function in the statistical sense as the normalization is
rather than one (McQuarrie, 2000). Different communities employ different normalization schemes for these weighting factors. Some communities will add an `x' or `n' superscript to g(r) to denote the weighting. Herein, we normalize them such that unless explicitly noted otherwise. For monatomic systems, the weighting factor is always one.
A form of confusion within the greater PDF community is the differences between the reduced pair distribution function, G(r), and the total radial distribution function, which is often also labeled G(r) (Keen, 2001). For clarity, we here refer to the total radial distribution function as G_{K}(r). This form of the PDF is constructed from the sum of all partials, g_{μν}(r), weighted according to concentration of atomic species, c, and associated coherent scattering power, b_{coh}, such that
The relationship between G_{K}(r) and G(r) is therefore
An important note, and an example of where confusion can occur for new practitioners, is that G_{K}(r) in this work is equivalent to G(R) in the review paper of Fischer et al. (2005) This can be seen by comparing equation (27) in this work with equation 2.40 of Fisher et al.
A third variation commonly found in the crystalline PDF community, referred to as the differential correlation function, D(r) (Tucker et al., 2007, 2017), is identical to G(r) apart from a constant scaling factor such that
Related to the previous note about G_{K}(R), D(r) here is different from the D(r) found in the review paper of Fischer et al. (2005) The D(r) in this work is equivalent to G(R) in the review paper of Fischer et al. This can be seen by comparing G(r) in equation (34) in this work with equation 2.26 of Fisher et al.
Another version of the PDF, primarily used in the liquids and glass community and referred to as the total correlation function, is T(r) (Soper, 1989; Hannon et al., 1990). T(r) is related to G(r) as
Additional minor variations of pair distribution function relationships can be found, but while some do surface occasionally in the modern literature, many are no longer actively utilized.
Figs. 3 and 4 graphically display examples of R(r), ρ(r), g(r), G_{K}(r), G(r) and T(r) for the cases of crystalline MnO and liquid Ar, respectively. The inherent information content of all forms is the same. All PDFs show atomic pair–pair correlations as peaks centered at average pairwise distances in real space, with the height of these peaks informing on the frequency of these pairwise distances (often with the scattering power of atoms involved) and the widths related to the distribution of the pairwise distances. The functions feature different limiting behaviors at low and high r, summarized in Table 4. These limits, and the accompanying scaling of peak intensities as r increases, emphasize different features of interatomic order. Preferred usage has developed in various research communities according to some of these distinguishing behaviors.

The weighted radial distribution function, R(r), shown at the top of Figs. 3 and 4, and the radial distribution function, RDF(r), find limited use because they rapidly increase towards infinity with increasing r. Thus it is a challenge to visually inspect the local correlations on the same scale as the midtolongrange correlations.
In materials which lack longrange order, few important structural details exist at high r, and several functions are commonly used. We introduced definitions of pair distribution functions based on the density function, ρ(r), which is a straightforward quantity to calculate from atomistic simulations and models. It is shown in the panels second from the top in Figs. 3 and 4. g(r) is simply the divided by the average density, and it has found wide adoption in the amorphous and liquids community (Benmore, 2012). The limits of g(r) are absolutely defined to be zero prior to the first pair correlation and 1 at high r, which in practice can aid with data reduction and normalization procedures. An example of g(r) can be seen in the third panel from the top of Figs. 3 and 4. G_{K}(r), shown fourth from the top in Figs. 3 and 4, shares many qualitative features with g(r), and while they appear nearly identical at first glance, they feature different units, limits and scaling behavior (refer to Appendix A, Figs. 3 and 4, and Table 4 for details).
To resolve certain details of local structure, researchers occasionally find it useful to preferentially weight a structural r, ignoring or downplaying longerrange features. While there is nothing implicitly wrong with this approach, this is a decision best applied at the time of modeling and stated clearly in analysis discussions. The issue with functions that inherently carry their own rdependent weighting [such as G_{K}(r) damping as 1/r] is that they require additional rdependent normalization of residuals to uniformly treat misfit at different length scales. It can be argued that representations of the measured data should not themselves contain such an rdependent feature bias.
towards features at lowThe reduced pair distribution function, G(r) (second from the bottom in Figs. 3 and 4), is the most prevalent formalism used in the study of disordered crystalline materials and nanocrystalline materials and is the version of data compatible with the popular realspace PDF modeling program PDFgui (Farrow et al., 2016). G(r) is also sometimes used in glass or molecular liquid studies, particularly when longerrange ordering is present. Arguably, the most advantageous feature of the G(r) formalism is that the amplitude of the oscillations is independent of R value. This means that the nature of a material's structural coherence can be readily interpreted via visual inspection of G(r). It also means that residual differences between models and data are equally weighted at all R values.
It is sometimes asserted that G(r) is the most directly calculable function from experimental data (Egami & Billinge, 2012), as it is the direct Fourier transform of S(Q) and does not require any assumptions of or average scattering power. In practice, the data reduction procedures employed to generate G(r) typically involve a number of optimization steps, which effectively estimate various sampledependent corrections either analytically (Peterson et al., 2000; Jeong et al., 2001; Qiu, Božin et al., 2004) or via ad hoc methods (Neuefeind et al., 2012; Juhás et al., 2013).
D(r) shares many of the same features as G(r), as it is equivalent to 〈b_{coh}〉^{2}G(r). Because of this subtle difference, there has been some confusion in the community about when to use D(r) compared with G(r) in different analysis methods. In practice, employing either D(r) or G(r) when using smallbox modeling approaches (where a scale parameter for the data set can be freely refined) will produce identical model results. However, they cannot be used interchangeably in those methods that rely on absolutely normalized data, such as RMCbased modeling. This can be particularly tricky in those cases where 〈b_{coh}〉^{2} is near to one, as the results of a largebox modeling approach may appear to be converging, but the results will be incorrect. Researchers are advised to carefully verify what form of the PDF they are employing, especially when using data from a new beamline (where data reduction protocols may differ) or employing a new form of analysis.
The neutron glass community tends to favor the T(r) formalism shown at the bottom of Figs. 3 and 4 (Hannon et al., 1990; Ellison et al., 1993). T(r) [and G(r)] scale relative to the as a function of r, as opposed to functions such as g(r) or R(r) (see Appendix A4). In the harmonic approximation of atomic motion, peaks are broadened symmetrically in T(r) [and G(r)] by thermal motions (Wright et al., 1989), which is cited as a considerable advantage in differentiating between static and thermal disorder (Hannon et al., 1990). This symmetry is noted by the glass community to lead to more accurate fitting for extracting coordination numbers (Benmore, 2012). However, T(r)'s overall rdependent scaling means it is not practical for viewing wide ranges in real space, and thus it is not commonly used in studies of nanocrystalline or disordered crystalline materials.
Arguably, the density function is one of the most straightforward functions to calculate from an atomic model. We here present the conversions from this common quantity, ρ(r), and the other six common allatom PDFs we have discussed herein. Table 5 provides conversions between some of the real space functions. The full list of conversions originating from each function in turn can be found in Appendix A.
Finally, PDF data are often interpreted using the N(r). This is the number of atoms between r_{min} and r_{max}. The is described mathematically in terms of the partial g_{μν}(r) functions (Soper, 2010):
which explicitly does not include the scattering lengths. A closely related function is the accumulation of this summed over all atoms:
This quantity is shown in Fig. 5. Note that is not weighted by scattering length, such that it cannot be easily transformed to other realspace functions without employing partial PDF functions.
5. Summary
This paper has provided a resource to understand the relationships of and convert between eight realspace pair distribution functions commonly found in the scientific literature. This purely instructional work offers direct mathematical conversions, graphical representations and a practical discussion of function characteristics, meant as an updated stepbystep reference for new practitioners and those seeking to compare their results with those of other researchers. While the appearance and weighting of these representations can vary, often to emphasize certain features of interest, the inherent structural information must be the same among these different forms. Although we respect the decisions of individual researchers to use whichever PDF formalism they find most appropriate, we encourage convergence and standardization where possible. On the basis of the comparison and discussion presented herein, we endorse the use of two primary formalisms: G(r) and g(r). The reduced pair distribution function, G(r), is recommended because of its uniform weighting at all R values. It has been broadly employed by the disordered crystalline and nanocrystalline communities. In contrast, the pair distribution function, g(r), is recommended as it is conveniently bounded at 0 and 1 (simplifying normalization procedures), features symmetric peaks and emphasizes lowr features. Thus it has been widely adopted by the liquids, amorphous and glass communities.
We also encourage that, when reporting a PDF, authors overtly define which realspace distribution functions they present. Reciprocalspace functions should be described in terms of S(Q) (which has an agreed upon and consistent definition across fields). For example, `we show the structure function, F(Q) = Q[S(Q) − 1]'. Similarly, realspace functions should be described in terms of ρ(r). For example, `we fit the pair distribution function, G(r) = 4π[ρ(r) − ρ_{0}]'. It is our hope that the derivations, tables and figures presented in this work help serve as a reference tool for researchers to easily navigate the PDF landscape and guide towards a convergence of total scattering data formalisms.

APPENDIX A
Conversions between realspace functions
This appendix will provide transformations between the various realspace functions. Several notes are needed before the listing of equations. First, we will assume a shape function of γ_{0}(r) = 1. This is true for sufficiently bulk material measurements (i.e. material structures exhibit translational periodicity and thus do not have finite size, shape or morphology effects contributing to the experimental PDF). Second, the correlation number, N(r), will not be mentioned since it can only be correctly calculated from the partial distributions. Note that the rise of N(r) at the location of an isolated peak corresponds to the coordinate number of that pairwise peak. Third, the normalization of g(r) provided in the main text is applicable here.
The original purpose of this appendix was to provide simple recipes for fellow researchers to transform their data or model calculations between the realspace forms. An additional benefit of generating this appendix, as well as the limits in Tables 3 and 4, was further validation that the conversions between conventions and the presentation of the conventions themselves are correct.
APPENDIX B
Details of liquid argon
simulationsFor all Ar simulations, the system consisted of a cubic simulation cell with 50 000 atoms and a velocityVerletlike (Verlet, 1967) time integrator was used with a 1 fs timestep. A Lennard–Jones pair potential with a 15 Å cutoff was used with a tail correction applied, with ε = 0.238067 kcal mol^{−1} and σ = 3.405 Å (Yarnell et al., 1973). Initially, the atoms were randomly placed in the box and then force minimized with a force tolerance of 10^{−8} kcal (mol Å)^{−1}. To produce the real and reciprocalspace patterns, canonical (NVT) ensemble simulations were carried out with T = 86.56 K and ρ_{0} = 0.02138 atoms Å^{−3} to reproduce previous results (Yarnell et al., 1973). A Nosé–Hooverstyle thermostat (Nosé, 1984; Hoover, 1985) was used to keep constant temperature with a of 0.1 ps. For the isothermal compressibility calculations, isobaric isothermal (NPT) simulations were carried out at the same temperature as the NVT simulation but with different pressures. A Nosé–Hooverstyle barostat (Nosé, 1984; Hoover, 1985) was used to keep constant pressure with a of 1 ps. The PDF was then calculated over a realspace range 0.075 ≤ r ≤ 100 Å with a bin width of δr = 0.05 Å. The reciprocalspace data were generated through inverse Fourier transform. The reciprocalspace data are scaled such that forward Fourier transforms recreate the original PDFs precisely.
APPENDIX C
From experimentally observed scattering intensities to differential cross sections
The authors mostly ignored the steps involved in reducing data from measured scattering intensities (Section 3) to reciprocalspace functions in the main text. This appendix will shed light on some of the corrections necessary to transform measured intensities to a single scattering differential (DCS), , which can then be used in equation (4). For Rietveld analysis, this level of rigor for corrections is not usually needed for most samples as these effects are frequently addressed as part of the measurement's `background intensity', which is normally fitted with a polynomial to reduce its influence on the profile However, for realspace total scattering measurements, these `minor' effects can produce a signal comparable to the reciprocalspace features of local disorder and therefore must be accounted for.
where I(Q) is the scattering from the sample (using the same convention as Rietveld). Its relation to the differential (DCS) is simply (Egami & Billinge, 2012)
where N = ρ_{0,eff}V is the number of atoms illuminated, V is the illuminated volume and ρ_{0,eff} is the effective as described in Appendix E. Because corrections are easier to explain in terms of the DCS, we rewrite equation (4) as
The DCS can only be found by fully correcting the measured intensity. Proper treatment of total scattering data requires correcting for effects such as attenuation through the sample material and its environment, subtraction of multiple scattering events, and recoil or inelastic effects. That being said, the notations for the various corrections do not have a standard set of definitions. The associated measurements for these corrections (e.g. empty container, empty sample environment, normalization) are no longer as unusual to perform as they used to be. Table 6 describes the standard set of measurements required and associates subscripts to refer to them. Additionally, Paalman & Pings (1962) developed a detailed notation for the attenuation correction terms which will be used here. Table 7 describes the notation for the correction terms applied to the experimental measurements.

For the measurement of the sample and container combined there are four terms of scattering and attenuation (also called absorption) in the experiment. For example, the scattering that occurs in only the sample but is attenuated by both the sample and container component is denoted as A_{s,sc}. Similarly, the multiple scattering is denoted as M_{μ} for the μth measurement, and thus the sample and container are denoted M_{sc}. Finally, the inelastic recoil correction is denoted as for the νth species; thus the container inelastic correction would be P_{c}^{ie}.
Having the various functions and subscript definitions in hand, the description of what is actually experimentally measured, I^{E}_{i}, can be introduced. The experimentally measured intensities are summarized in Table 8.

These experimentally measured intensities are represented by the functional forms below with the necessary correction terms included from Table 7:
Note that the doubledifferential ω being the energy loss. For the ultimate goal of producing a suitable reciprocalspace function, one must integrate over ω to obtain the energyintegrated DCS. If all scattering were purely elastic from the sample, we would simply have
term appears in these equations, with ℏYet, there are undoubtedly ω is given as
events that occur and necessary terms must be included in the above equation for the specific experimental technique. The equation for an experimental integration overwhere β(k_{i}, k_{f}) accounts generally for terms such as detector efficiency, incident spectrum and other terms needed based on the nature of the experimental method (i.e. constant wavelength or time of flight). P^{ie}_{ i} is the inelastic correction term that occurs due to both the recoil of the atomic nuclei from the neutron collision and, for time of flight, the fact that the measurement is carried out at a fixed angle and not fixed momentum transfer, Q (Soper, 2009). For the functional forms of β(k_{i}, k_{f}) for constant wavelength or time of flight, we refer the reader to reports in the series by Powles (1973, 1978a,b) or the GUDRUN manual (Soper, 2010).
Realspace techniques are often colloquially referred to as `total scattering' because they use all of the reciprocalspace information rather than just the Bragg peaks. As mentioned in the main text, this term has a second meaning when it is applied to describe integration over
This is not the same as the which can only be properly measured when the incident and final energy of the probe are known. Integrating the energy yields the following set of equations:Solving each of the equations above for the respective DCS [and simplifying to just ] provides
Reducing the equations into terms that only involve the corrections and the measured intensities, the sample DCS is
the container DCS is
and the sample environment apparatus DCS is
Note that here the normalization term, Φ, has not been made explicit. Thus, an additional experimental measurement is needed to characterize the normalization term. Using our definitions in equations (45), we can define this normalization measurement as
The normalization measurement performed will differ according to the probe used, Xrays or neutrons. Yet, both serve a similar purpose in that they factor out the dependence of detector efficiencies, solid angle coverage etc. As a specific example, vanadium is often utilized for neutron scattering, mainly because it has a small coherent scattering length, implying that the distinct scattering signal is small compared with the selfscattering and thus it exhibits a relatively smooth diffraction pattern. Since selfscattering does not vary much with Q or 2θ, the differential is essentially , where b_{tot,v} is the total scattering length of vanadium from equation (53). For more detail on reasons to use vanadium, we refer the reader to Section 3.8.1 of the GUDRUN manual (Soper, 2010) and for more information regarding the details of the normalization, we refer the reader to both Section A5 of the text by Egami & Billinge (2012) and Section 3.8 of the GUDRUN manual (Soper, 2010). By substituting 〈b_{tot,v}^{2}〉 into equation (50) for dσ^{n}/dΩ and the I^{E}_{e} term from equation (45), we can derive the following:
Solving for Φ gives
which defines the final Φ term needed in equations (47), (48) and (49).
Solving equation (47) from the experimentally measured intensities then allows one to arrive at the differential also commonly defined as I(Q)/N in the total scattering community. This is the same as that defined in equation (1) and throughout the rest of this work. For further details on the data reduction described briefly herein and all necessary corrections, the reader is referred to more complete discussions in the literature (Howe et al., 1989; Hannon et al., 1990; Soper, 2010; Egami & Billinge, 2012). There are also other excellent references that deviate in detail from the reduction described herein, specifically in the steps where corrections are applied. (Windsor, 1981) Naturally, of the methodology has occurred with time.
The derivations herein integrated the scattering intensities over all measurement energies, where inelastic and multiple scattering effects are handled through data corrections. Several authors have noted the implications of this simplification (Page et al., 2011). Dynamic PDF has emerged as an extension of total scattering, making explicit use of the energy dependence (Egami & Billinge, 2012). This falls outside the scope of the present work.
APPENDIX D
Calculating the normalized Laue term
There is often confusion when determining the normalized Laue term of equation (5). Lovesey (1986) introduced quantities related to the total of the material,
and the coherent
Here, the subscript coh stands for coherent and tot stands for total, where must be inferred from the total ). In the Faber–Ziman scheme, 〈b_{coh}〉^{2} comes from the sum of the partials and the pupose of the term is to remove the selfscattering from the differential (Faber & Ziman, 1965). These are the terms that appear in the normalized Laue term with the factor of 4π canceled out. Note that 〈σ_{coh}〉 ≠ 4π〈b_{coh}〉^{2}, but Lovesey's notation obscures this fact. Being more explicit, it is straightforward to calculate the terms as originally intended. First we introduce a normalized concentration
(Sears, 1992where N_{α} is the number of atoms of type α and . This provides for the normalization that . The two quantities needed in the normalized Laue term are simply
and
This formulation of 〈b_{coh}〉^{2} makes more explicit that the complex scattering lengths, with sign, are averaged. Sears (1992) offers a useful discussion of calculating average cross sections. Unique to neutron measurements, this can lead to materials where the normalized Laue term becomes infinite because of atoms with a negative scattering length. Ti_{2.08}Zr is a classic example of 〈b_{coh}〉^{2} = 0 and .
An additional complication is found when is calculated. Since 1 barn = 100 fm^{2}, people often calculate by simply dividing by 4π and ignoring the units. While the factor of 100 will cancel out in the normalized Laue term, the values of the individual terms will be listed in units of 10 fm (or dekafemtometres) rather than fm. Tables 9 and 10 show calculations for the example materials.


APPENDIX E
Determining number density
Often when processing or analyzing data, the effective C. Since it is difficult to directly measure the packing fraction precisely, one can see why it is often tuned during data reduction.
or packing fraction, is treated as an adjustable parameter. Here we demonstrate how to calculate it and give guidance on reasonable limits. The effective affects absorption and multiple scattering corrections as well as the total number of illuminated atoms as described in AppendixFirst we introduce the concept of effective
which is based on the crystallographic ρ_{0}, using a packing fraction f ∈ (0, 1]. For measurements of crystalline powders, the packing fraction rarely exceeds f = 0.5. The for a crystalline sample is often phrased in terms of the number of atoms per called the Z parameter, and the unitcell volume. Using standard crystallographic conventions (Giacovazzo, 1992)
For liquids, ρ_{0} is not normally known. The technique for establishing it is to calculate ρ_{0,eff} from the ρ_{m}. This is done by weighing the empty sample container, adding the sample, and then weighing the filled container and measuring how full the container is. This measurement will give a with the packing fraction correctly accounted for. Then the effective is
where N_{A} is Avogadro's number, m_{α} is the and c_{α} is the normalized concentration of element α or the atomic fraction as defined in Appendix D. This procedure is often used for crystalline samples as well. An alternative to this approximation is to measure the volume by displacement in a fluid that the material does not react with and is insoluble within. However, it is likely to be difficult to recover a sample from this type of measurement so it is recommended that it is done after the scattering measurement is completed.
APPENDIX F
functions
To use the prevalent Faber–Ziman ; Suck et al., 1993; Egami & Billinge, 2012)
functions, one must first define weights (Faber & Ziman, 1965which are normalized such that . Note that the same symbol is used with an alternative normalization in some communities with . The Faber–Ziman
functions are related to the total scatteringwith A_{νμ}(Q) = S_{νμ}(Q) in the nomenclature that Egami uses. The benefit of this normalization of S_{νμ}(Q) is that it has the same asymptotes as S(Q) with
and
In equation (61), 〈b_{coh}〉^{2} is the average scattering length for the material and the numerator is the average scattering length for the atoms contributing to the partial.
Then it is straightforward to define a partial reduced pair distribution function as
which observes a summation rule of
The other realspace correlation functions follow analogous forms.
To clarify a related point, difference correlation functions are a distinct concept. Instead of being the correlation of two atomic species, it is all correlations with an atom of type ν at the origin. In other words
which follows the summation rule
A more detailed description is given by Egami & Billinge (2012, Sections 3.1 and 3.2).
For a twoatom system with atoms labeled α and β, the summation rule for the weights in equation (61) is
For a monatomic system, there are no partial functions. See Table 11 for explicit calculation of the values. Fig. 6 shows the partial G(r).

Acknowledgements
Much of the diffraction group of ORNL's Neutron Scattering Division gave advice during the production of this document. The authors acknowledge and express their thanks for the consultation of Matt G. Tucker, Joerg C. Neuefeind and Andrei T. Savici.
Funding information
This material is based upon work supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under contract No. DEAC0500OR22725.
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