A. Scale transform of abscissa

When the diffraction intensity profile is plotted on the abscissa scale of x = ln sinθ instead of the diffraction angle 2θ, whole powder diffraction data can be expressed as the convolution with the common spectroscopic profile function of source X-ray (Ida and Toraya, Reference Ida and Toraya2002). Figure 1 illustrates the core idea about the scale transform. Observed powder diffraction profile of LaB₆ is plotted on the diffraction angle 2θ in Figures 1(a) and 1(b) plots exactly the same intensity data on the natural logarithm of sinθ. Each of reflection peaks is composed of Kα ₁ and Kα ₂ peaks, and the separation of Kα ₁ and Kα ₂ peaks varies on the horizontal scale of 2θ, while it is constant on the horizontal scale of x = ln sinθ. In other words, the diffraction intensity profile is expressed by the convolution with the common instrumental function about spectroscopic profile of the source X-ray on the scale of x = ln sinθ, but it is not a convolution on the scale of 2θ.

Figure 1. Diffraction intensity data of LaB₆ plotted on (a) 2θ and (b) ln sin θ.

When the observed data are given by a set of diffraction angles and intensities (2θ, I), the abscissa and ordinate values are changed to (x, y) by the following equations,

(1)$$x = \ln \sin \theta, $$

(2)$$y = 2IC\,(2\theta )\tan \theta, $$

where C (2θ) is the geometric correction factor given by

(3)$$C\,(2\theta ) = \displaystyle{{2\sin \theta \sin 2\theta} \over {1 + {\cos} ^22\theta}}, $$

for a Bragg–Brentano powder diffractometer without a diffractive optical element.

After the scale-transformed data (x, y) are changed to (x′, y′) by the deconvolution–convolution process, the intensity will be treated by

(4)$${I}^{\prime} = \displaystyle{{{y}^{\prime}} \over {2C(2\theta )\tan \theta}}, $$

to obtain the modified intensity data on the original 2θ scale, (2θ, I′).

B. Emission of Cu-target X-ray

Assuming the peak energy and wavelength of the CuKα ₁ emission to be E _α1 = 8.04783 keV (Deutsch et al., Reference Deutsch, Förster, Hölzer, Härtwig, Hämäläinen, Kao, Huotari and Diamant2004) and λ_α1 = 1.5405929Å, we here apply the transformation given by

(5)$$x = \ln \displaystyle{{E_{\alpha 1}} \over E} = \ln \displaystyle{\lambda \over {\lambda _{\alpha 1}}},$$

for arbitrary photon energy E and wavelength λ. A model for spectroscopic profile of the source X-ray, f _X(x), is given by

(6)$$f_{\rm X}(x) = f_\alpha (x) + S_{\beta /\alpha} f_\beta \,(x) + S_{\rm B}(x),$$

where f _α(x) and f _β(x) are the normalized CuKα and Kβ emission peak profile functions, S _β/α the CuKβ/CuKα intensity ratio, and S _B(x) the contribution of the white X-ray caused by bremsstrahlung.

The CuKα emission peak profile is expressed by

(7)$$f_\alpha (x) = \mathop \sum \limits_{i = 1}^2 \mathop \sum \limits_{\,j = 1}^2 \displaystyle{{2I_{\alpha ij}} \over {\pi \Delta x_{\alpha ij}}}\left[ {1 + 4{\left( {\displaystyle{{x - x_{\alpha ij}} \over {\Delta x_{\alpha ij}}}} \right)}^2} \right]^{ - 1},$$

(8)$$x_{\alpha ij} = \ln \displaystyle{{E_{\alpha 1}} \over {E_{\alpha ij}}}\; \approx \displaystyle{{E_{\alpha 1} - E_{\alpha ij}} \over {E_{\alpha 1}}},$$

(9)$$\Delta x_{\alpha ij} \approx \displaystyle{{W_{\alpha ij}} \over {E_{\alpha ij}}},$$

where the values of intensity I _αij, location E _αij, and peak width W _αij of the component Lorentzian functions given in a literature (Deutsch et al., Reference Deutsch, Förster, Hölzer, Härtwig, Hämäläinen, Kao, Huotari and Diamant2004) are listed in Table I.

Table I. Peak energy E, full width at half maximum W and integrated intensity I for doublet and quartet models for CuKα emission and a quintet model for CuKβ emission (Deutsch et al., Reference Deutsch, Förster, Hölzer, Härtwig, Hämäläinen, Kao, Huotari and Diamant2004).

The CuKβ emission peak profile is similarly expressed by

(10)$$f_\beta (x) = \mathop \sum \limits_{i = 1}^5 \displaystyle{{2I_{\beta i}} \over {\pi \Delta x_{\beta i}}}\left[ {1 + 4{\left( {\displaystyle{{x - x_{\beta ij}} \over {\Delta x_{\alpha ij}}}} \right)}^2} \right]^{ - 1},$$

(11)$$x_{\beta i} = \ln \displaystyle{{E_{\alpha 1}} \over {E_{\beta i}}},$$

(12)$$\Delta x_{\beta i} \approx \displaystyle{{W_{\beta i}} \over {E_{\beta i}}},$$

where the values of component intensity I _βi, location E _βi, and peak width W _βi are also listed in Table I.

The contribution of white X-ray, S _B(x) in Eq. (6), should be dependent on (i) the acceleration voltage and target material of the X-ray tube, (ii) absorption and scattering of beryllium window of the sealed tube and atmosphere through the X-ray beam path from the X-ray source to the detector, (iii) energy-dependent sensitivity of the detector, and also (iv) the settings of the pulse-height discriminator or analyzer.

Trincavelli et al. (Reference Trincavelli, Castellano and Riveros1998) have proposed a formula for the spectroscopic distribution of the bremsstrahlung given by

(13)$$\eqalign{\displaystyle{{\Delta I} \over {\Delta E}} & = \sqrt Z \displaystyle{{E_0 - E} \over E} \cr & \left[ { - 54.86 - 1.072E + 0.2835E_0 + 30.4\ln Z + \displaystyle{{875} \over {Z^2\; E_0^{0.08} \;}}} \right],} $$

where Z is the atomic number of the target material, E ₀ the upper limit of the photon energy in the unit of keV, and E the photon energy of the X-ray. The intensity distribution on the scale of x = ln (E _α1/E) is calculated from ΔI/ΔE by

(14)$$\displaystyle{{\Delta I} \over {\Delta x}} = \displaystyle{{E\Delta I} \over {\Delta E}}.$$

Figure 2 shows the intensity distribution of the bremsstrahlung calculated by Eq. (13) for Cu target and $E_0 = 45{\kern 1pt} {\rm keV}$, multiplied by the transmittance of air (4:1 mixture of nitrogen and oxygen) through the path of 480 mm in length, which is calculated by the Cromer–Liberman algorithm (Cromer and Liberman, Reference Cromer and Liberman1981; Cromer, Reference Cromer1983).

Figure 2. White X-ray (bremsstrahlung) spectra calculated by the formula of Trincavelli et al. (Reference Trincavelli, Castellano and Riveros1998) and Eqs (15)–(17).

Here we apply a simplified model for the white X-ray intensities, the normalized formula of which is given by

(15)$$S_{\rm B}(x) = \left\{ {\matrix{ {\displaystyle{{I_{\rm B}} \over N}\left( {\displaystyle{{x - x_0} \over {\Delta x_{\rm B}}}} \right){\rm erfc}\left( {\displaystyle{{x - x_{\rm B}} \over {\Delta x_{\rm B}}}} \right)} & {\left[ {x_0 \lt x} \right]} \cr 0 & {\left[ {x \le x_0} \right]} \cr}} \right.,$$

(16)$$N = \displaystyle{{\Delta x_{\rm B}} \over 4}\left[ {(1 + 2t_0^2 )\,{\rm erfc}\,(t_0) - \displaystyle{{2t_0} \over {\sqrt \pi}} \exp \,( - t_0^2 )} \right]\; \;, $$

(17)$$t_0 = \displaystyle{{x_0 - x_{\rm B}} \over {\Delta x_{\rm B}}},$$

where the integrated intensity of white X-ray I _B, and the profile parameters x _B and Δx _B are treated as adjustable parameters. The lower limit on the scale of x, x ₀ = ln (E _α1/E ₀), which corresponds to the upper limit of the photon energy, may appear to be treated as a fixed parameter determined by the acceleration voltage, but the value can be changed in practical application, because it is likely that the upper limit of the photon energy of the white X-ray to be detected is restricted by the energy-dependent sensitivity of the detector rather than the acceleration voltage. The intensity profile calculated by Eqs (15)–(17) is also shown in Figure 2. Note that almost linear dependence of the peak profile of the function S _B (x) on the side of smaller x is dominantly determined by the bremsstrahlung characterized by the parameter x ₀, but the intensity for larger x is assumed to be restricted by the extinction by the air, and simulated by adjusting the values of the parameters x _B and Δx _B.

C. Effect of Ni filter for CuK-emission X-ray source

When a Ni foil is used as a Kβ filter, the virtual spectroscopic profile of the source X-ray, f _X:Ni(x), should be given by

(18)$$f_{{\rm X:Ni}}(x) = T_{{\rm Ni}}(x)f_{\rm X}(x),$$

where T _Ni(x) is the transmittance spectrum of the Ni foil, given by

(19)$$T_{{\rm Ni}}(x) = \exp \left[ { - {(\mu /\rho )}_{{\rm Ni}}(x)\rho _{{\rm Ni}}t_{{\rm Ni}}} \right],$$

where ρ _Ni and t _Ni are the density and thickness of the Ni foil, respectively, and (μ/ρ)_Ni(x) is the mass absorption coefficient spectrum of Ni. The spectrum of (μ/ρ)_Ni(x) calculated by the method of Cromer and Liberman (Cromer and Liberman, Reference Cromer and Liberman1981; Cromer, Reference Cromer1983) is further simplified here by the following equation,

(20)$$\eqalign{(\mu /\rho )_{{\rm Ni}}(x) = \left\{ {\matrix{ {337.58\exp \left[ {2.7130\,(x - x_{{\rm Ni} - K})} \right]} & {\left[ {x \lt x_{{\rm Ni} - K}} \right]} \cr {41.145\exp \left[ {2.8097\,(x - x_{{\rm Ni} - K})} \right]} & {\left[ {x_{{\rm Ni} - K} \le x} \right]} \cr}} \right.,}$$

where x _Ni−K = −0.03478 is the location of NiK-absorption edge ($E_{{\rm Ni} - K} = 8.3327{\kern 1pt} {\rm keV}$) relative to the CuKα ₁ peak position.

Figure 3 compares the mass absorption coefficient spectrum calculated by the Cromer–Liberman method and the simplified formula given by Eq. (20). No significant difference between the Cromer–Liberman profile and the profile calculated by Eq. (20) can be detected in the range displayed in Figure 3.

Figure 3. Comparison of the mass absorption coefficient spectra of Ni calculated by the Cromer–Liberman method (thick broken line), a simplified model (thin solid line), and the hypothetical spectrum without K-absorption edge (thin broken line).

D. Hypothetical CuKα ₁ singlet spectrum without NiK-absorption edge

Removal of CuKα ₂, Kβ sub-peaks and NiK-edge structure can be achieved by deconvolution of the realistic spectroscopic profile modeled in the Sections II.B and II.C, and convolution with a hypothetical spectroscopic profile of a CuKα ₁ singlet, the model which will be described in this section.

Hypothetical CuKα ₁ singlet spectrum without NiK-absorption edge is expressed by

(21)$$f_{{\rm X*:Ni*}}(x) = T_{{\rm Ni*}}(x)f_{{\rm X*}}(x),$$

where T _Ni*(x) is the transmittance spectrum given by

(22)$$T_{{\rm Ni*}}(x) = \exp \left[ { - {(\mu /\rho )}_{{\rm Ni*}}(x)\rho _{{\rm Ni}}t_{{\rm Ni}}} \right],$$

where (μ/ρ)_Ni*(x) is the hypothetical mass absorption coefficient spectrum of Ni without K-absorption edge, given by

(23)$$(\mu /\rho )_{{\rm Ni*}}(x) = 41.145\exp \left[ {2.8097(x - x_{{\rm Ni} - K})} \right]\;. $$

As can be seen in the spectrum calculated by Eq. (23) plotted in Figure 3, it is just extrapolation of the spectroscopic profile of Ni in the lower energy (larger x) region.

The function f _X*(x) in Eq. (21) is a CuKα ₁ singlet spectrum, which we here assume to be

(24)$$f_{{\rm X*}}(x) = f_{\alpha 1}(x) + f_{\rm B}S_{\rm B}(x),$$

where f _α1(x) is given by

(25)$$f_{\alpha 1}(x) = \displaystyle{{2I_{\alpha 1}} \over {\pi \Delta x_{\alpha 1}}}\left[ {1 + 4{\left( {\displaystyle{{x - x_{\alpha 1}} \over {\Delta x_{\alpha 1}}}} \right)}^2} \right]^{ - 1},$$

which is identical to the Kα ₁ component in the Kα ₁, Kα ₂ doublet model (see Table I).

The parameter f _B in Eq. (24) is a factor to adjust the background in the output data. When the white background is fully removed from the input data and not added to the output (f _B = 0), it is likely that negative intensities will be created in the background region because of the statistical variation of the observed intensities. Although there will be no reason not to allow negative intensities when the statistical errors are explicitly assumed in subsequent analysis of data, it may be difficult for some users of traditional codes for Rietveld analysis (Rietveld, Reference Rietveld1969) to include explicit error values in their analyses. On the other hand, when the white background is fully added (f _B = 1), the background intensities in the output data should naturally be enhanced, because we here assume higher transmittance of Ni without K-absorption as defined by Eq. (23).

E. Deconvolution and convolution treatments

In our previous report (Ida and Toraya, Reference Ida and Toraya2002), we have applied the analytical formula for the Fourier transform of the spectroscopic distribution of the source X-ray. In this study, the Fourier transforms of the model functions are numerically calculated applying the following formulas,

(26)$$F_{{\rm X}:{\rm Ni}}(k) = \mathop \sum \limits_{\,j = - n/2}^{n/2 - 1} f_{{\rm X}:{\rm Ni}}(\,j{\rm \Delta} x)\exp \left( {\displaystyle{{2\pi ikj} \over n}} \right),$$

(27)$$F_{{\rm X*}:{\rm Ni*}}(k) = \mathop \sum \limits_{j = - n/2}^{n/2 - 1} f_{{\rm X*}:{\rm Ni*}}(j{\rm \Delta} x)\exp \left( {\displaystyle{{2\pi ikj} \over n}} \right),$$

where Δx is the step interval of the sampling points. The values of Δx and n should be determined not to lose information included in the source data and to keep sufficient width of marginal region to reduce the end effects (Ida and Toraya, Reference Ida and Toraya2002).

The deconvolution–convolution process about intensities $y_j \to {y}^{\prime}_j$ is expressed by

(28)$${y}^{\prime}_j = \displaystyle{1 \over n}\mathop \sum \limits_{k = - n/2}^{n/2} \displaystyle{{F_{{\rm X*}:{\rm Ni*}}(k)} \over {F_{{\rm X:Ni}}(k)}}Y_k\exp \left( { - \displaystyle{{2\pi ikj} \over n}} \right),$$

(29)$$Y_k = \mathop \sum \limits_{j = 0}^{n - 1} y_j\exp \left( {\displaystyle{{2\pi ikj} \over n}} \right),$$

where the values of y _j in Eq. (29) are intensity values for the sampling points of x _j = x ₀ + jΔx, evaluated by cubic-spline interpolation of the source data. Marginal data in the range x ₀ + mΔx <x _j <x ₀ + nΔx for x ₀ + mΔx ≈ ln sinθ _max were filled with the values given by the following equation,

(30)$$y_j = \left\{ {\matrix{ {y_m} & {\left[ {m \lt j \lt \displaystyle{{2m + n} \over 3}} \right]} \cr {\displaystyle{{(2n + m - 3j)y_m + (3j - 2m - n)y_0} \over {n - m}}} & {\left[ {\displaystyle{{2m + n} \over 3} \lt j \lt \displaystyle{{m + 2n} \over 3}} \right]} \cr {y_0} & {\left[ {\displaystyle{{m + 2n} \over 3} \lt j \lt n} \right]} \cr}} \right..$$

The fast Fourier transform algorithm can be used for all the calculations of Eqs (26)–(29).

A. Sample

Standard LaB₆ powder (NIST SRM660a) was used as received. The powder was filled into the cavity of a flat glass holder with the capacity of 0.1152 mL and the rectangular cross-section of 20 mm × 15 mm (average depth of 0.384 mm). The filling factor of the powder was estimated at 26.7% by weight measurement, and the penetration depth of the CuKα X-ray into the powder sample was estimated at $\mu ^{ - 1} = 0.044{\kern 1pt} {\rm mm}$.

B. Data collection

Diffraction data were collected with a powder diffractometer (PANAlytical, X'Pert PRO MPD) with a Cu-target sealed tube, θ–θ type goniometer (PANAlytical, PW3050/60) equipped with a 1D Si strip detector (PANAlytical X'Celerator). The X-ray source and the detector are symmetrically located at the distance of 240 mm from the rotation axis of the goniometer. The X-ray tube was operated at the voltage of 45 kV and the current of 40 mA. Fixed-angle divergence slit of 0.5° and a couple of Soller slits with the open angle of 0.04 rad were used. Ni foil of 0.02 mm in thickness was used as a Kβ filter. The 1D powder diffraction intensity data of 8079 sampling points were created by an automatic measurement/data processing program (PANAlytical, Data Collector) from the integration of five iterations of continuous scan for the diffraction angles ranging from 10° to 145° with the nominal step interval of 0.0167° and nominal measurement time of 10.16 s per step. The total measurement time for data collection was about 1 h.