Direct determination of amorphous number density from the reduced pair distribution function

Graphical abstract


Method details
In the case of perfectly periodic materials, it is possible to utilize pair correlation statistics in order to estimate the material's bulk number density, r 0 . For these materials, the reduced PDF, symbolized as G(r), is the sum of peak information and of a straight sloping baseline expressed as À4pr 0 r and intrinsically feasible to work out from the PDF shape. However, upon increasing deviation from bulk crystallinity, this baseline becomes affected by the dominating presence of an attenuating particle shape factor, g(r), and is, then, expressed as À4pr 0 g(r)r [1]. The extreme case of linearity deviation towards high values of the real space coordinate, r, is that of completely amorphous materials [2][3][4][5][6][7], in which case r 0 may not be estimated from the tangent of the baseline as the latter is no longer a straight line [8]. In some cases, numerical estimation (i.e. the identification of PDF peaks and, as a result, of what constitutes baseline) has been cited as possible via modelling [1], but in practice this is not readily applicable to the amorphous state. Additionally, although in principle the PDF baseline may be estimated from a Fourier transform of Small Angle Scattering (SAS) data [8], the pair density function, r(r), which feeds into the calculation requires use of ab initio structural models, and we are not aware of such paradigms applied to amorphous materials in the literature.
Aiming to provide a reliable alternative for the determination of amorphous density from diffraction intensity datasets, here we propose a simple but functional methodology which relies on a numerical manipulation of the reduced PDF. Following we outline the proposed methodology in two steps.
1 Collection of diffraction intensity and relation to G(r). To derive an approximation of amorphous number density, we employ the fundamental relation between diffraction and the PDF, by first performing a diffraction experiment, correcting the collected spectra for incoherent, multiple and background scattering and then Fourier transforming (FT) the coherent/elastic part of the collected intensity, I(Q), to obtain the reduced PDF, G(r), as where F(Q) is the reduced structure factor, for which F(Q) 2 =I(Q) and F(Q)=Q[S(Q)-1] where S(Q) is the material's structure factor [3]. Eq. (1) is independent of the radiation source and gives best results primarily for electron diffraction (driven by recent breakthroughs in Transmission Electron Microscopy ePDF analysis of nanoparticles and disordered materials [9]) but also for monochromatic powder X ray diffraction (XRD). I(Q) is experimentally collected over a range of reciprocal space vectors, Q, the lower and upper limits of which -Q min and Q max respectivelyare functions of the limitations set on the range of scattering angles, 2u, and on the radiation wavelength, l, by the experimental apparatus, via Q = 4psin (u)/l. To derive physically meaningful PDF curves, the FT in Eq. (1) mandates that Q max is chosen such that F(Q max ) = 0 (or S(Q max ) = 1), in order to avoid finite-size effects/artifacts in the low r range. The best practice would then be to compare G(r) shapes calculated across a range of Q max cut-off values, sampling the reduced PDF at real space intervals no larger than the Nyquist rate, equal to p/Q max (oversampling will not cause harm, however), while always seeking to disregard peaks/shoulders which do not consistently appear on all PDF curves under consideration. Additionally, Q max cut-off's causing noticeable G(r) peak position shifting should be disregarded. The part of G(r) common to all Q max cut-off's can then be safely considered for use in the next step.
2 Real space G(r) sampling and numerical manipulation. PDF importance lies in the pair correlation statistics which connect G(r) to the material's (bulk) number density, r 0 , via the atomic where r 0 =N/V, N is the number of atoms in the structure contained in volume V [10]. However, by definition, the atomic PDF is also the ratio of the non-negative pair density function, r(r), divided by Hence, r(r) may possess a local minimum for which Eq.
(3) will be close to zero, and from Eq. (2) it follows that for this local minimum, G(r) % -4prr 0 ; on the additional provision that G(r) is sampled for r values at least equal to r min = 2p/Q max , we may write The current methodology is applicable to any amorphous system, as all of its individual steps -i.e. a) collecting diffraction intensity, I(Q), b) correcting the collected intensity dataset depending on the diffraction method (Q max ), c) Fourier-transforming I(Q) to G(r): via Eq. (1), d) calculating the ratio G(r)/ 4pr following Eqs. (2)-(4) and e) calculating amorphous density from the value of the minimum of G (r)/4pr -are system independent and, as a result, generally applicable. In essence, the current contribution is aimed at making explicit two points of increased merit regarding amorphous G(r); these are: 1 That it is feasible to infer amorphous density via the ratio G(r)/(4pr) and we explain this in the main body of the paper (Eqs. (2)-(4)). These equations are system independent, i.e. they are valid for any purely amorphous system, or for amorphous parts of, say, quasicrystalline materials which may be detectedfor exampleon a TEM. 2 That an estimate of amorphous density may be inferred from the value of the G(r)/(4pr) minimum of the G(r) usable range. However, for our methodology to provide accurate density values, the I(Q) corrections need to be optimal, as G(r) shape is entirely dependent on I(Q) corrections and on the shape of S(Q) dataset towards the high Q's; the latter will affect the local minimum yielding the number density of the low r's of the G(r).
The ability of the current methodology to provide an accurate density estimate is intrinsically dependent on the quality of the diffraction dataset. This quality is partially dependent on the attainable Q max which, by extension, affects the low r region through Eq. (1) and the 'usable' G(r) range -i.e. the part which is physically meaningful and which is most likely to contain the most negative value for the ratio G(r)/(4pr). We note here that as r min = 2p/Q max it follows that G(r) does not have any physical significance below r min , because Q max is the resolution limit of the diffraction method; hence the lower limit of the usable G(r) range is inherently set by r min . As the G(r) curves towards increasingly positive values with increasing r, after the first coordination peak (also see reference [8] and references therein), it is justified to consider the position of the first coordination peak as a natural upper limit of the usable G(r) range; as a reminder, the usable G(r) range is comprised of the most negative G(r)/(4pr) values, which are systematically located between r min and the first coordination peak for all amorphous spectra, with no exceptions.

Method validation
In order to exemplify the use of Eq. (4), here we determine the number density of a mixed Fe-Si-Al-Ca-Mg-Cr-Cu-Ni oxide system in the vitreous state, which we originally discussed elsewhere 7 . For completeness, we note that the sample was aerodynamically levitated by a compressed air flow and melted to 1500 C by a 100 W CO 2 laser source. While in flight, the laser beam was switched off, leading to rapid quenching of the sample by the air stream, producing a glassy near-spherical particle with a density of up to 0.088 atoms/Å 3 (3.54 g/cm 3 ) based on a CCD measurement of the particle's diameter. During levitation, high-energy XRD intensities were collected at a photon wavelength of 0.20194 Å (61.39 keV) yielding a Q range of up to 21 Å À1 at low scattering angles (up to 40 2u).
The G(r) was extracted from I(Q) based on Eq. (1) (the diffraction dataset was Fourier-transformed via IGOR PRO TM running a custom subroutine [7], in the absence of window functions in order to avoid spurious effects). The calculation of amorphous number density based on G(r) is implemented in the spreadsheet included as supplementary material. The procedure first involves inputting the G(r) vs. r dataset (spreadsheet cells downwards from B9 and A9 respectively) and the determination of r min (see spreadsheet cell H3 and purple dashed line in Fig. 1a and b) via Q max (spreadsheet cell H1), followed by identification of the usable G(r) range, i.e. the part of the spectrum lying from r min up to some interatomic distance for which G(r) intercepts the x-axis. Here, we set this distance equal to the position of the first coordination peak (see spreadsheet cell H2 and yellow dashed line in Fig. 1a and b). The usable part of G(r) is designated by a red line superimposed on the PDF profile in Fig. 1a (spreadsheet column C). We then apply Eq. (4) to the usable G(r) range by first dividing G(r) by 4pr (spreadsheet column D) and considering only the negative part of the G(r)/(4pr) range (spreadsheet column F and spectrum designated by the red line in Fig. 1b) for which we proceed to identify all of its local minima. This is done by first calculating the tangent for all pairs of consecutive points on the G(r)/ (4pr) plot (spreadsheet column G) and then identifying a tangent sign reversal for each of these pairs, the sign reversal flagging the local minimum (spreadsheet column H). We then store the position, r, as well as the G(r)/(4pr) value for each of the local minima identified. On the basis of Eq. (4), the local minimum with the most negative value, r d , is selected as the closest approximation to the material's number density, r 0 , (point designated by the green circle in Fig. 1b and its absolute value in spreadsheet cell H6). In order to visualize the PDF baseline of the glass sample, we may then draw the -4pr 0 r term between r = 0 and r d (see green solid line in Fig. 1a) and extend it towards high r values (dashed green line in Fig. 1a), the baseline touching the local minimum prior to the first coordination shell, which appears to be a plausible fit to the G(r) profile. In this example, the calculated r 0 compared favorably against the experimentally established number density [6] to within 15.3%.

Appendix A. Supplementary data
Supplementary data associated with this article can be found, in the online version, at https://doi. org/10.1016/j.mex.2019.03.005.