Implications of size dispersion on X-ray scattering of crystalline nanoparticles: CeO2 as a case study

A methodology based on small- and wide-angle X-ray scattering is proposed for simultaneously quantifying size distribution and degree of crystallinity of nanoparticles. Experimental results are shown for cubic-like CeO2 nanoparticles.


ATOMIC DISORDER
The vibrational disorder has been treated within the frozen-in-time approach of the Debye scattering equation (DSE), [1][2][3] as it provides an exact calculation of the X-ray scattering at small and wide angles of randomly oriented nanoparticles (NPs).For materials made of a single element α, such as silicon NPs, the DSE is simply where Q = (4π/λ) sin θ is the reciprocal vector modulus for a scattering angle 2θ and X-rays of wavelength λ.The pair distance distribution function (PDDF) in Eq. (S1) have been calculated by routines asfQ.m and fpfpp.m,both available at Mat-LabCodes(2016). 4 Small disorder in the atomic positions r a = ⟨r a ⟩ + dr were generated by In NPs with sizes above a few nanometers, as the 4 nm diameter NP in Fig. S2, the PDDF obtained for a single δr-disordered NP is very similar to the average PDDF from an ensemble of NPs.The PDDFs are almost identical for the shortest distances, Fig. S2 (left panel).In contrast, the poor statistic from a single particle is more evident for the most extended distances, as seen in Fig. S2 (right panel).However, the NP scattering power P (Q), that is, the X-ray scattering patterns from systems of randomly oriented NPs, exhibit tiny irrelevant differences only, as compared in Fig. S3.Bulk properties dominate for larger particles, and the poor statistic of the most extended distances is even more irrelevant.Therefore, for where of atomic distances r ab = |r b − r a | for NPs with N α atoms; δ() stands for the Dirac delta function.The atomic scattering factor f α the crystal lattice.The random numbers ζ n are in the range [0, 1].The atomic disorder δr produces a broadening of width δr in the histograms of pair distances, as illustrated in Fig. S1 by disordering a single unit cell several times.A Gaussian of standard deviation this broadening, corresponding to an isotropic root-mean-square (RMS) displacement ⟨dr⟩ rms = δr/(4 √ ln 2 ) ≃ 0.3δr around the mean atomic positions.

Figure
Figure S1: (a-c) Broadening in the histogram H α (u) of pair distances caused by atomic vibration with RMS displacement ⟨dr⟩ rms = 0.3δr = 0.014 Å in a silicon unit cell.The histograms (scatter dots) were computed with a bin of 0.001 Å within an ensemble of 1000 unit cells with random disorder of 2%, that is δr = 0.047 Å. Gaussian fits (solid red lines) with standard deviation σ = 0.02 Å, Eq. (S3), are also shown.(d) Silicon unit cell.Atoms indicated by numbers were used to calculate the pair distances r 12 , r 13 , and r 14 along the [111], [110], and [100] directions, respectively.

Figure
Figure S2: PDDF (red dots) of a 2%-disordered 4 nm diameter silicon NP compared to the average PDDF (solid blue line).Left panel: pair distances of the first neighbors.Right panel: longest distances in the NP.PDDF of the whole NP, as well as the NP itself are shown as insets.All distances are seen with a standard deviation σ = 2 pm.Total number of atoms in the NP: H α (0) = 1672 (inset, left panel).Bin width of 0.002 Å.

Figure S3 :
Figure S3: Top panel: Scattering power P (Q), Eq. (S1), of a single 2%-disordered 4 nm diameter silicon NP (blue dots) in comparison to the average scattering from an ensemble of disordered NPs (solid red line).Bottom panel: Difference ∆P of the above curves multiplied by Q 2 to make visible on linear scale tiny differences in the wide-angle region (Q > 1 Å−1 ).

Fig. 7 ,
Fig.S6, respectively.In contrast, Fig.S7shows the X-ray diffraction whole pattern fittings with the GSAS-II Python code.7 The general relationship

Figure S4 :
Figure S4: Sample C1.Line profile fitting (solid red lines) of K α1 and K α2 Bragg peaks (scatter dots) by using the pseudo-Voigt double function V (2θ), Eq. (S5).The FWHM of each V n (2θ) function (dashed lines) are indicated by horizontal solid lines.The residuals of curve fitting are also shown (blue solid lines).

Figure S5 :
Figure S5: Sample B5.Line profile fitting (solid red lines) of K α1 and K α2 Bragg peaks (scatter dots) by using the pseudo-Voigt double function V (2θ), Eq. (S5).The FWHM of each V n (2θ) function (dashed lines) are indicated by horizontal solid lines.The residuals of curve fitting are also shown (blue solid lines).

Figure S6 :
Figure S6: Sample B11.Line profile fitting (solid red lines) of K α1 and K α2 Bragg peaks (scatter dots) by using the pseudo-Voigt double function V (2θ), Eq. (S5).The FWHM of each V n (2θ) function (dashed lines) are indicated by horizontal solid lines.The residuals of curve fitting are also shown (blue solid lines).

Figure S8 :
Figure S8: Examples of line profile fitting (solid red lines) of K α1 and K α2 peaks (scatter dots) by using the pseudo-Voigt double function V (2θ), Eq. (S5).The FWHM of each V n (2θ) function (dashed lines) are indicated by horizontal solid lines.The residuals of curve fitting are also shown (blue solid lines).Both panels display Bragg reflections from the corundum pattern in Fig. S7(a).

Figure S9 :
Figure S9: (a,b) Bragg peak width (scatter points) as a function of Q = (4π/λ 1 ) sin(θ) of the standard sample in Fig. S7(a).Least-square fitting (solid red lines) of the data points leads to the instrumental width W ins (Q) as given in each plot.(a) From individual peak fitting with the pseudo-Voigt double function, Eq. (S5), and (b) with GSAS-II.Peak widths obtained by adjusting instrumental and size-broadening parameters available in the GSAS-II are also displayed (solid blue line).