A. Case of opaque specimen holder

It is assumed that the width of the incident X-ray beam along the orthogonal direction to the rotation axis is given by B = RΦ _DS for the formulation of sample transparency aberration, where R is the goniometer radius and Φ _DS is the divergence slit open angle. The effect of sample transparency aberration depends on the apparent diffraction angle 2Θ, the width W, thickness t, and linear attenuation coefficient μ of the sample.

There are five cases (a), (b), (c), (c’), and (d) (see Figure 2) for the formulation of the sample transparency aberration, when an opaque sample holder is used. The five cases are classified by the relation between (i) the specimen width W, (ii) the irradiated width at the sample face Ω = B/sin Θ, and (iii) the width at the sample face viewed from the backside face at the viewing angle of π– 2Θ, τ = 2 t/tan Θ.

Figure 2.

Five cases for formulation of sample transparency aberration interrupted by an opaque specimen holder. Ω ₀ = W/2 + Ω/2 − τ, Ω ₁ = W/2 − Ω/2, Ω ₂ = W – τ, Ω ₃ = W/2 + Ω/2, t ₁ = (Ω ₁/2)tanΘ, t ₂ = (W/2)tanΘ, and t ₃ = (Ω ₃/2)tanΘ, as shown in the schematic illustrations.

In case (a), both the incident and diffracted beams are not interrupted by the side walls of the sample holder. The incident X-ray beam is attenuated by the sample, and attenuated beam can reach the backside face of the sample. In case (b), the incident beam is not interrupted by the upstream side wall of the specimen holder, but the diffracted beam is partly interrupted by the downstream side wall of the holder. In case (c), both the incident and diffracted beams are interfered by the side walls of the sample holder, and the incident beam partly reaches the backside face of the sample. In case (c′), the incident beam is not interfered by the upstream side wall of the sample holder, but cannot effectively reach the backside face, because the diffracted beam is fully interrupted by the downstream side wall of the sample holder. In case (d), both the incident and diffracted beam are interrupted by the side walls of the sample holder, and the incident beam cannot reach the backside face.

The sample transparency aberration function for the cases (a), (b), (c), (c’), and (d) can be formulated by a combination of two types of aberration functions, (I) formula for infinite width and finite thickness, ω _I(Δ2Θ, υ), and (II) formula for finite width and infinite thickness ω _II(Δ2Θ, υ), which are given by

(1) $$ {\omega}_{\mathrm{I}}\left(\Delta 2\varTheta, \upsilon \right)=\left\{\begin{array}{cc}\frac{1}{\gamma }{\mathrm{e}}^{\Delta 2\varTheta /\gamma }& \left[-\upsilon <\Delta 2\varTheta <0\right]\\ {}0& \left[\mathrm{otherwise}\right]\end{array}\right., $$

(2) $$ {\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, \upsilon \right)=\left\{\begin{array}{cc}\frac{1}{\gamma}\left(1+\frac{\Delta 2\varTheta }{v}\right){\mathrm{e}}^{\Delta 2\varTheta /\gamma }& \left[-\upsilon <\Delta 2\varTheta <0\right]\\ {}0& \left[\mathrm{otherwise}\right]\end{array}\right., $$

where

(3) $$ \gamma =\frac{\sin 2\varTheta }{2\mu R}, $$

is the decay parameter. The type I function ω _I(Δ2Θ, υ) has the formula of exponential function doubly truncated at Δ2Θ = −υ and Δ2Θ = 0. The type II function ω _II(Δ2Θ, υ) has the formula of the first-order Laguerre function truncated at the first node Δ2Θ = −υ.

The aberration function ω ^(a)(Δ2Θ) for the case (a) is simply given by the type I function,

(4) $$ {\omega}^{\left(\mathrm{a}\right)}\left(\Delta 2\varTheta \right)={\omega}_{\mathrm{I}}\left(\Delta 2\varTheta, u\right), $$

for u = 2 t cos Θ/R. The location of the truncation of the aberration function –u about Δ2Θ is equivalent with the peak shift of a specimen displaced by −t along the normal direction.

The aberration function ω ^(b)(Δ2Θ) for the case (b) is expressed by

(5) $$ {\omega}^{\left(\mathrm{b}\right)}\left(\Delta 2\varTheta \right)={\omega}^{\left(\mathrm{b}1\right)}\left(\Delta 2\varTheta \right)+{\omega}^{\left(\mathrm{b}2\mathrm{a}\right)}\left(\Delta 2\varTheta \right)-{\omega}^{\left(\mathrm{b}2\mathrm{b}\right)}\left(\Delta 2\varTheta \right), $$

(6) $$ {\omega}^{\left(\mathrm{b}1\right)}\left(\Delta 2\varTheta \right)=\frac{\Omega_0}{\Omega}{\omega}_{\mathrm{I}}\left(\Delta 2\varTheta, u\right), $$

(7) $$ {\omega}^{\left(\mathrm{b}2\mathrm{a}\right)}\left(\Delta 2\varTheta \right)=\frac{\tau }{\Omega}{\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, u\right), $$

(8) $$ {\omega}^{\left(\mathrm{b}2\mathrm{b}\right)}\left(\Delta 2\varTheta \right)=\frac{\Omega_1}{\Omega}{\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, {u}_1\right), $$

for Ω = B/sin Θ, Ω ₀ = W/2 + Ω/2, Ω ₁ = W/2 − Ω/2, τ = 2 t/tan Θ, and u ₁ = 2t ₁ cos Θ/R = Ω ₁ sin Θ/R. Figure 3 illustrates how the function ω ^(b)(Δ2Θ) is constructed.

Figure 3.

Construction of the aberration function for the case (b). Case (b) is the sum of the components (b-1) and (b-2). The part (b-2) is equivalent with the difference of (b-2-a) and (b-2-b).

The aberration function ω ^(c)(Δ2Θ) for the case (c) is given by

(9) $$ {\omega}^{\left(\mathrm{c}\right)}\left(\Delta 2\varTheta \right)=\frac{\varOmega_2}{\varOmega }{\omega}_{\mathrm{I}}\left(\Delta 2\varTheta, u\right)+\frac{\tau }{\varOmega }{\omega}_{\mathrm{I}\mathrm{I}}\left(\Delta 2\varTheta, u\right), $$

for Ω₂ = W − τ.

The aberration function ω ^(c’)(Δ2Θ) for the case (c’) is expressed by

(10) $$ {\omega}^{\left({\mathrm{c}}^{\prime}\right)}\left(\Delta 2\varTheta \right)=\frac{\varOmega_3}{\varOmega }{\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, {u}_3\right)-\frac{\varOmega_1}{\varOmega }{\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, {u}_1\right), $$

for Ω ₃ = W/2 + Ω/2, Ω₁ = W/2 − Ω/2, u ₃ = Ω ₃ sin Θ/R, and u ₁ = Ω ₁ sin Θ/R. Figure 4 illustrates how the aberration function in the case (c’) is constructed.

Figure 4.

Construction of case (c’). Case (c’) is identical to the difference between the case (c’-1) and (c’-2).

The aberration function ω ^(d)(Δ2Θ) for the case (d) is simply given by the type II function:

(11) $$ {\omega}^{\left(\mathrm{d}\right)}\left(\Delta 2\varTheta \right)=\frac{W}{\varOmega }{\omega}_{\mathrm{II}}\left(\Delta 2\varTheta, {u}_2\right), $$

for u ₂ = W sin Θ/R .

The k-th order power average of Δ2Θ for the type-I function ω _I(Δ2Θ, υ), s_k ^(I)(υ) can be calculated by a recurrence formula:

(12) $$ {s}_k^{\left(\mathrm{I}\right)}\left(\upsilon \right)=-{\left(-\upsilon \right)}^k{\mathrm{e}}^{-\frac{\upsilon }{\gamma }}- k\gamma {s}_{k-1}^{\left(\mathrm{I}\right)}\left(\upsilon \right), $$

and the initial value of s ₀^(I)(υ) = 1 − e^−υ/γ . The k-th order power average of Δ2Θ for the type-II function ω _II(Δ2Θ, υ), s_k ^(II)(υ), can be calculated by

(13) $$ {s}_k^{\left(\mathrm{I}\mathrm{I}\right)}\left(\upsilon \right)={s}_k^{\left(\mathrm{I}\right)}\left(\upsilon \right)+\frac{s_{k+1}^{\left(\mathrm{I}\right)}\left(\upsilon \right)}{\upsilon }. $$

Cumulants of any orders of the aberration functions can then be calculated through the power averages of the function s_k ^(I)(υ).

B. Case of translucent specimen holder

It is assumed that the linear attenuation coefficient of the specimen holder is μ′. The sample transparency aberration function ω _ST(Δ2Θ) is given by an integral formula:

(14) $$ {\omega}_{\mathrm{ST}}\left(\Delta 2\varTheta \right)=\frac{2\mu }{B}\int_{Z_{\mathrm{L}}}^0\int_{X_L(z)}^{X_U(z)}\delta \left(\Delta 2\varTheta -\frac{2z\cos \varTheta }{R}\right)g\left(x,z\right)\;\mathrm{d}x\mathrm{d}z, $$

where δ(x) is the Dirac delta.

The limits of the integrals, Z _L, X _L(z), and X _U(z) are given by

(15) $$ {Z}_{\mathrm{L}}=\max \left\{-t,-\frac{B}{2\cos \varTheta }-\frac{W\tan \varTheta }{2}\right\} $$

(16) $$ {X}_{\mathrm{L}}(z)=\max \left\{-\frac{W}{2},-\frac{z}{2\tan \varTheta }-\frac{B}{2\sin \varTheta}\right\} $$

(17) $$ {X}_{\mathrm{U}}(z)=\min \left\{\frac{W}{2},-\frac{z}{2\tan \varTheta }+\frac{B}{2\sin \varTheta}\right\}. $$

The function g(x,z) represents the effective transmittance of the incident and diffracted beams, reflected at the point (x,z), which is given by the product of the transmittance for the incident and diffracted (emitting) beams g ⁽ⁱ⁾(x,z) and g ^(e)(x,z), as g(x,z) = g ⁽ⁱ⁾(x,z) g ^(e)(x,z).

The transmittance for the incident beam g ⁽ⁱ⁾(x,z) is given by

(18) $$ {g}^{\left(\mathrm{i}\right)}\left(x,z\right)=\exp \left[-\mu {l}^{\left(\mathrm{i}\right)}\left(x,z\right)-\mu^{\prime }l{\prime}^{\left(\mathrm{i}\right)}\left(x,z\right)\right], $$

where the function l ⁽ⁱ⁾(x,z) represents the path length of the incident beam inside the sample body, which is given by

(19) $$ {l}^{\left(\mathrm{i}\right)}\left(x,z\right)=\left\{\begin{array}{cc}-z/\sin \varTheta & \left[-W/2\le {x}^{\left(\mathrm{i}\right)}\right]\\ {}\left(W+2x\right)/2\cos \varTheta & \left[{x}^{\left(\mathrm{i}\right)}<-W/2\right]\end{array}\right., $$

where x ⁽ⁱ⁾ is the horizontal location of the incident point, given by x ⁽ⁱ⁾ = x + z/tan Θ. The function l′⁽ⁱ⁾(x,z) represents the path length of the incident beam in the sample holder, which is given by

(20) $$ l{\prime}^{\left(\mathrm{i}\right)}\left(x,z\right)=\left\{\begin{array}{cc}0& \left[-W/2\le {x}^{\left(\mathrm{i}\right)}\right]\\ {}-\left(W+2{x}^{\left(\mathrm{i}\right)}\right)/2\cos \varTheta & \left[-{x}^{\left(\mathrm{i}\right)}<-W/2\right]\end{array}\right.. $$

Figure 5 illustrates the geometry about the reflection interfered by the upstream side part of the sample holder.

Figure 5.

Interference by the upstream side part of the sample holder for the reflection at the point (x, z).

The transmittance for the diffracted (emitting) beam g ^(e)(x,z) is given by

(21) $$ {g}^{\left(\mathrm{e}\right)}\left(x,z\right)=\exp \left[-\mu {l}^{\left(\mathrm{e}\right)}\left(x,z\right)-\mu^{\prime }l{\prime}^{\left(\mathrm{e}\right)}\left(x,z\right)\right]. $$

The function l ^(e)(x,z) represents the path length in the sample for the diffracted beam, which is given by

(22) $$ {l}^{\left(\mathrm{e}\right)}\left(x,z\right)=\left\{\begin{array}{cc}-z/\sin \varTheta & \left[{x}^{\left(\mathrm{e}\right)}\le W/2\right]\\ {}\left(W-2x\right)/2\cos \varTheta & \left[W/2<{x}^{\left(\mathrm{e}\right)}\right]\end{array}\right., $$

where x ^(e) is the horizontal location of the emission point, given by x ^(e) = x – z/tan Θ. The function l′^(e)(x,z) represents the path length in the sample holder for the diffracted beam, which is given by

(23) $$ l{\prime}^{\left(\mathrm{e}\right)}\left(x,z\right)=\left\{\begin{array}{cc}0& \left[{x}^{\left(\mathrm{e}\right)}\le W/2\right]\\ {}\left(2{x}^{\left(\mathrm{e}\right)}-W\right)/2\cos \varTheta & \left[W/2<{x}^{\left(\mathrm{e}\right)}\right]\end{array}\right.. $$

It is not too complicated to evaluate the values of the k-th power average and the cumulants of the aberration function by a numerical method. It should be noted that it will be better to convert the integral variable z as e^2μz/sinΘ  = u on the numerical evaluation of the integral because e^2μz/sinΘ approximates the function g(x,z) for a sufficiently large value of W. The k-th order power average of the sample transparency aberration function is then given by

(24) $$ \left\langle {\left(\Delta 2\varTheta \right)}^k\right\rangle =\frac{\sin \varTheta }{B}\int_{U_{\mathrm{L}}}^1\int_{X_{\mathrm{L}}(z)}^{X_{\mathrm{U}}(z)}{\left(\frac{2z\cos \varTheta }{R}\right)}^kg\left(x,z\right)\mathrm{d}x\frac{\mathrm{d}u}{u}, $$

where the lower limit of the integral about u is given by

(25) $$ {U}_{\mathrm{L}}={e}^{2\mu {Z}_L/\sin \Theta}, $$

C. Comparison of models for opaque and translucent sample holders

The k-th power average of the sample transparency aberration function for a translucent sample holder is evaluated by a Gauss–Legendre quadrature with the number of sampling points n along the u (z)-direction and m along the x-direction. The zeroth order power average s ₀, which represents the integrated intensity of the aberration function, is normalized by multiplying B/W sin Θ for the diffraction angles B/sin Θ > W, to avoid doubly counting the spill-over (finite specimen width) effect, when the automatic recovery of the intensity lost by finite width of the sample is applied on the deconvolutional process about equatorial aberration. The values expressed by W sin Θ/B may be regarded as the relative intensities for a hypothetical transparent sample holder, where the interference of the sample holder is completely neglected.

Figure 6 shows the values of effective transmittance s’₀ of sample transparency about opaque and translucent sample holders, calculated for the goniometer radius R = 150 mm, divergent slit open angle Φ _DS = 1.25°, specimen width W = 20 mm, specimen thickness t = 0.618 mm, penetration depth of the sample powder μ ⁻¹ = 0.218 mm, and penetration depth of the sample holder μ′⁻¹ = 0.138 mm, as will be described in the experimental section of this study. No significant difference has been detected by increasing the number of sampling points of Gauss–Legendre quadrature more than 20 × 20.

Figure 6.

Values of effective transmittance evaluated for an opaque sample holder, and a translucent sample holder on variation of number of sampling points of Gauss–Legendre integrals. Vertical lines indicate the locations of the boundaries between the cases (a), (b), (c), and (d), for an opaque sample holder.

The values of the effective transmittance s′₀ for an opaque holder approach to 1 − μ ⁻¹/W = 0.9891 for 2Θ → 0, as expected, while the values numerically calculated for the translucent holder show almost constant values s′₀ = 0.9976 in the low angle region.

Figure 7 shows a magnified plot about the irregular behavior of the normalized effective transmittance about the results of calculation about opaque and translucent sample holders. The behavior of the normalized transmittance calculated for a translucent sample holder shows slight irregularity, which is clearly caused by the normalization, where the intensities calculated for the translucent sample holder are divided by the intensity expected for a transparent sample holder. As the amount of irregular change caused by the formula for a translucent sample holder is less than 0.2%, it will not cause severe problem on analysis of experimental data.

Figure 7.

Magnified plot about irregular behaviors in the normalized effective transmittance calculated for opaque and translucent sample holders.

Figure 8 shows the values of the first-order cumulant (average peak shift) κ ₁ = ⟨Δ2Θ⟩ and the reduced third-order cumulant (asymmetry parameter), defined by κ ₃^(1/3) = sign(⟨(Δ2Θ)³⟩)|⟨(Δ2Θ)³⟩|^1/3, of the sample transparency aberration function, for opaque and translucent sample holders. The values on a conventional assumption for an infinitely wide and thick specimen (W = ∞, t = ∞) are also shown in Figure 8. The difference in the results for opaque and translucent sample holders are less significant than that found in the effective transmittance, while the difference from the conventional assumption appears at higher angles, where the effect of finite thickness of the specimen becomes significant. It is suggested that the finite thickness of the sample should properly be accounted for simulation or DCT of XRD data of a typical inorganic material with the penetration depth of μ ⁻¹ = 0.218 mm, while the effects of transparency of the sample holder may be negligible.

Figure 8.

The first-order and the reduced third-order cumulants of the sample transparency aberration function for opaque and translucent sample holders. Values calculated based on a conventional assumption are also plotted in the figure.

On the conventional assumption, both the first-order cumulant κ ₁ and the reduced third-order cumulant κ ₃^(1/3) of the sample transparency aberration function should be proportional to sin 2Θ, and the ratio κ ₃^(1/3)/κ ₁ should be 2^1/3 ≈ 1.26, corresponding with that the aberration function of the conventional model is expressed by an exponential function truncated at the origin, Δ2Θ = 0.

The author would like to note that implementation of the formulas for a translucent sample holder to evaluate the effective transmittance and cumulants of the sample transparency aberration is rather easier than that for an opaque holder, while the cost for computation may become slightly more expensive.

A. DCT

Spectroscopic profile of the source X-ray, based on the model proposed by Deutsch et al. (Reference Deutsch, Förster, Hölzer, Härtwig, Hämäläinen, Kao, Huotari and Diamant2004), has been deconvolved, and hypothetical singlet Lorentzian located at Cu Kα₁ (K-L3) wavelength (λ = 1.54059 Å) with the full width at half maximum (FWHM) of Δλ/λ = 0.00035 has been convolved on the logarithmic sine scale, χ _X = ln sin Θ. Axial divergence and equatorial aberrations have been treated by a method previously proposed (Ida, Reference Ida2021a).

The cumulants of sample transparency aberration for the goniometer radius R = 150 mm, divergent slit open angle Φ _DS = 1.25°, specimen width W = 20 mm, specimen thickness t = 0.618 mm, the penetration depth of the sample powder μ ⁻¹ = 0.218 mm, and penetration depth of the sample holder μ′⁻¹ = 0.138 mm are calculated by 20 × 20-sampling point Gauss–Legendre quadrature. The “root_legendre()” method in “special” module in SciPy library is used to find the sample points and weights of Gauss–Legendre quadrature.

Sample transparency aberration function is modeled by the double convolution of a truncated exponential function f ^(TE)(x) and a rectangular function f ^(R)(x), defined by

(26) $$ {f}^{\left(\mathrm{TE}\right)}(x)\equiv \left\{\begin{array}{cc}{e}^x& \left[x<0\right]\\ {}0& \left[0\le x\right]\end{array}\right. $$

(27) $$ {f}^{\left(\mathrm{R}\right)}(x)\equiv \left\{\begin{array}{cc}1& \left[-1<x<0\right]\\ {}0& \left[\mathrm{elsewhere}\right]\end{array}\right., $$

The first-order and the third-order cumulants of the functions are k ₁^(TE) = −1, k ₃^(TE) = −2, k ₁^(R) = −0.5, and k ₃^(R) = 0, respectively. When the third-order cumulant of the sample transparency aberration is given by κ ₃(2Θ), the scale transform χ ^(TE)(2Θ) suitable for the DCT about the truncated exponential function should be given by

(28) $$ {\chi}^{\left(\mathrm{TE}\right)}\left(2\varTheta \right)={k}_3^{\left(\mathrm{TE}\right)1/3}\int \frac{\mathrm{d}2\varTheta }{\kappa_3^{1/3}\left(2\varTheta \right)}, $$

to satisfy the requirement for scale invariance of the cumulant, expressed by κ ₁ ∝ κ ₂^1/2 ∝ κ ₃^1/3 ∝ …It is assumed that the first-order cumulant of the sample transparency aberration is given by κ ₁(2Θ). The scale transform χ ^(R)(2Θ) for the rectangular function is then evaluated by solving the differential equation:

(29) $$ \frac{\mathrm{d}{\chi}^{\left(\mathrm{R}\right)}\left(2\varTheta \right)}{\mathrm{d}2\varTheta }=\frac{k_1^{\left(\mathrm{R}\right)}}{\kappa_1\left(2\varTheta \right)-{k}_1^{\left(\mathrm{TE}\right)}/\left[\mathrm{d}{\chi}^{\left(\mathrm{TE}\right)}\left(2\varTheta \right)/\mathrm{d}2\varTheta \right]}. $$

to satisfy the requirement for additivity of cumulants on convolution.

The expression given by Eq. (29) is numerically unstable because the denominator on the right side of the Eq. (29) easily approaches to zero, for a thick specimen case, where the conventional assumption of infinitely thick sample (t → ∞) works well. An alternative expression with an arbitrary small number ε

(30) $$ \frac{\mathrm{d}{\chi}^{\left(\mathrm{R}\right)}\left(2\varTheta \right)}{\mathrm{d}2\varTheta }=\min \left\{\frac{1}{\varepsilon },\frac{k_1^{\left(\mathrm{R}\right)}}{\kappa_1\left(2\varTheta \right)-{k}_1^{\left(\mathrm{TE}\right)}/\left[\mathrm{d}{\chi}^{\left(\mathrm{TE}\right)}\left(2\varTheta \right)/\mathrm{d}2\varTheta \right]}\right\}, $$

is used for implementation of the DCT about the finite thickness of a sample. The value of the small number selected in this study is ε = 0.0001°. It would force additional and artificial DCT about the rectangular function with the width of 0.0001°, but it is not likely that it will cause a serious problem for practical use.

The DCT about spectroscopic profile, axial divergence and equatorial aberrations, and sample transparency aberration for a translucent sample holder are implemented in Python codes for handling NumPy (Harris et al., Reference Harris, Millman, van der Walt, Gommers, Virtanen, Cournapeau, Wieser, Taylor, Berg, Smith, Kern, Picus, Hoyer, van Kerkwijk, Brett, Haldane, Fernández del Río, Wiebe, Peterson, Gérard-Marchant, Sheppard, Reddy, Weckesser, Abbasi, Gohlke and Oliphant2020) and SciPy (Virtanen et al., Reference Virtanen, Gommers, Oliphant, Haberland, Tyler Reddy, Cournapeau, Burovski, Peterson, Weckesser, Bright, van der Walt, Brett, Wilson, Millman, Mayorov, Nelson, Jones, Kern, Larson, Carey, Polat, Feng, Moore, VanderPlas, Laxalde, Perktold, Cimrman, Henriksen, Quintero, Harris, Archibald, Ribeiro, Pedregosa and van Mulbregt2020) libraries, and applied to the XRD data of Si powder through Jupyter Notebook 7.0.8/Anaconda 3 on Apple macOS 14.3.1/Apple Macbook Pro 2020 edition. Total CPU time for the DCT reported by the user interface of Jupyter Notebook was 4.29 s.

B. Peak profile fitting analysis

Individual peak profile fitting (IPPF) analysis is applied to the data treated by the deconvolutional method. A symmetric model peak profile function defined by the convolution of the Lorentzian function and a mathematical model for a symmetrized instrumental function determined by the second and fourth-order cumulants (standard deviation and kurtosis) (Ida, Reference Ida2021b) is used for the IPPF analysis. Constant background intensity b, integrated peak intensity I, peak position 2Θ _peak, and the half width at half maximum (HWHM) w of the Lorentzian component are optimized to fit the peak intensity profile in the DCT data. The standard deviation σ and kurtosis k of the symmetrized instrumental function were treated as fixed parameters.

The contributions of the instrumental broadening caused by the finite widths of the source X-ray, w _X = 0.01 mm, and the interval of the detector strip w _D = 0.01 mm are accounted as σ _X = (w _X/R)/12^1/2 = 0.0011° and σ _D = (w _D/R)/12^1/2 = 0.0011° for the standard deviations, and κ _X^(1/4) = −(w _X/R)/120^1/4 = −0.0012° and κ _D^(1/4) = −(w _X/R)/120^1/4 = −0.0012° as the reduced fourth order cumulants, on assumption that the intensity distribution is modeled by the continuous uniform distribution.

The “leastsq()” method, which is based on the Levenberg–Marquardt algorithm (e.g. Press et al., Reference Press, Teukolsky, Vetterling and Flannery1992), of the module “optimize” in SciPy library (Virtanen et al., Reference Virtanen, Gommers, Oliphant, Haberland, Tyler Reddy, Cournapeau, Burovski, Peterson, Weckesser, Bright, van der Walt, Brett, Wilson, Millman, Mayorov, Nelson, Jones, Kern, Larson, Carey, Polat, Feng, Moore, VanderPlas, Laxalde, Perktold, Cimrman, Henriksen, Quintero, Harris, Archibald, Ribeiro, Pedregosa and van Mulbregt2020) is used with default settings for the IPPF analysis. It is assumed that the statistical errors of the deconvolved data are equal to the square root of the intensity in the unit of photon counts, though it may not fully be justified.