Bayesian estimation to identify crystalline phase structures for X-ray diffraction pattern analysis

Figures & data

Figure 1. Observation process of XRD data and a conceptual diagram of the proposed method, that is, Bayesian inverse estimation to identify the crystalline phase structure and their known structures for XRD analysis [9].

Figure 2. Supplementary diagram of the similarity calculation procedure. [(a) Observed XRD data $\mathcal{D}$ and (b) Crystal structure factor ${\mathcal{F}}_k$].

Table 1. Fifty candidate crystal structures prepared from the AtomWork [17], which is the inorganic material database. Two $\mathrm{Mn}{\mathrm{O}}_2$ structure types were obtained from the literature [19,20], respectively.

chemical composition	crystal structure	chemical composition	crystal structure
TiO2	Rutile	TiO2	Anatase
TiO2	Brookite	O2	O2
Ti3O5	Ta3N5	Ti	Mg
Ti	W	TiO2	Fe2N0.94
Ti2O5	a	TiO2	CdI2
TiO2	Al2O3	TiO0.2	Mg
Ti5O9	Ti5O9	Ti7O13	Ti7O13
Ti9O17	Ti9O17	Ti4O7	a
Ti4O7	Ti4O7	Ti4O7	b
TiO2	Fe2N0.94	TiO	NaCl
Ti4O5	Ti4O5	TiO2	CdI2
Ti4O7	Ti4O7	Ti4O7	a
Ti4O7	b	Ti5O9	Ti5O9
Ti9O17	Ti9O17	Ti6O11	Ti6O11
Ti7O13	Ti7O13	Ti8O15	Ti8O15
Ti6O11	Ti6O11	Ti3O	Ti3O
TiO	TiO	Ti0.84O0.84	TiO
Ti4O5	Ti4O5	Ti	Ti
Ti6O	Ti6O	Ti6O	Ti6O
TiO2	Mg	TiO2	ZrO2-b
TiO2	MnO2	Ti2O5	b
Ti3O5	V3O5	TiO	WC
TiO2	VO2-b	TiO2	MnO2
Ti2O5	b	Ti2O5	a
TiO2	VO2-b	Ti3O5	V3O5

Figure 3. Selection results from 50 candidates for each temperature in the REMC method. The x- and the y-axes denote the candidate crystal structures and index of inverse temperature $\mathit{\tau }$, respectively. This figure shows a visualization of the indicator probability $\mathit{P}(\mathit{g}|\mathit{Y};\mathit{\beta }={\beta }_{\tau })$ [%]. The large index corresponds to lower temperatures. The color scale denotes the sampling frequency of $g_k=1$ on a log scale. The dark red indicates the presence of crystal structure in the measured sample.

Figure 4. Cosine similarity between the measurement XRD data $\mathcal{D}$ and the crystal structure factors $\mathcal{F}$ for prescreening. In this figure, the red line denotes the threshold value set at 0.5.

Figure 5. Selection result from 12 candidates for each temperature in the REMC method. The x- and the y-axes denote the candidate crystal structures and the index of the inverse temperature $\mathit{\tau }$. This figure shows a visualization of the indicator probability $\mathit{P}(\mathit{g}|\mathit{Y};\mathit{\beta }={\beta }_{\tau })$ [%]. The large index corresponds to lower temperatures. The color scale denotes the sampling frequency of $g_k=1$ on a log scale. The dark red color indicates the presence of crystal structures in the measured sample.

Figure 6. Result of profile analysis in the measurement XRD data using our method [(a): Fitting result via profile function in the measurement XRD data. In this figure, the black and the red lines indicate the measurement XRD data and the fitting profile functions, respectively. (b): Peak components in three crystal structures of $\mathrm{Ti}{\mathrm{O}}_2$; Anatase, Brookite, and rutile. The red, green, and blue lines indicate the peaks of Anatase, Brookite, and rutile, respectively].

Figure 7. Posterior distribution of the peak area ratio on the triangular diagram when analyzing the measurement XRD data using the proposed method. The units for the axes are percentages [%].

Figure 8. Expanded view of the posterior distribution of the peak area ratio when analyzing the measurement XRD data using the proposed method. The units for the axes are percentages [%].

Figure 9. Posterior distribution of the profile parameters in the measurement XRD data. The red, green, and blue histograms correspond to the posterior distribution of Anatase, Brookite, and rutile, respectively. [(a): Peak height $\mathit{h}$, (b): Peak shift $\mathit{\mu }$, (c): Gauss-Lorentz ratio $\mathit{r}$, and (d): asymmetry parameter $\mathit{\alpha }$]. (e) and (f) are Gauss width $\mathrm{\Sigma }(x_i;u_k,v_k,w_k,{\alpha }_k)$ and Lorentz width $\mathrm{\Omega }(x_i;u_k,v_k,w_k,{\alpha }_k)$, where $x_i$ is $2\mathit{\theta }=60$ [${ }^{\circ }$]. The black dot-dash line is a true parameter of the profile function.

Figure 10. Selection result for each temperature in the REMC method in case-01. The x- and the y-axes denote the candidate crystal structures and the index of the inverse temperature $\mathit{\tau }$. This figure shows a visualization of the indicator probability $\mathit{P}(\mathit{g}|\mathit{Y};\mathit{\beta }={\beta }_{\tau })$ [%]. The large index corresponds to lower temperatures. The color scale denotes the sampling frequency of $g_k=1$ on a log scale. The dark red color indicates the presence of crystalline phase structures in the measured sample.

Figure 11. Selection result for each temperature in the REMC method in case-02. The x- and the y-axes denote the candidate crystal structures and the index of inverse temperature $\mathit{\tau }$. This figure shows a visualization of the indicator probability $\mathit{P}(\mathit{g}|\mathit{Y};\mathit{\beta }={\beta }_{\tau })$ [%]. The large index corresponds to lower temperatures. The color scale denotes the sampling frequency of $g_k=1$ on a log scale. The dark red color indicates the presence of crystalline phase structures in the measured sample.

Table 2. Cell lattice parameters of $\mathrm{Mn}{\mathrm{O}}_2$ and rutile crystal structure types in $\mathrm{Ti}{\mathrm{O}}_2$ [19,21].

Structure	Chemical	Length [nm]			Angle [degree]
Type	Composition	$\mathit{a}$	$\mathit{b}$	$\mathit{c}$	$<\mathit{\alpha }$	$<\mathit{\beta }$	$<\mathit{\gamma }$
Rutile	$\mathrm{Ti}{\mathrm{O}}_2$	0.30	0.46	0.46	90	90	90
$\mathrm{Mn}{\mathrm{O}}_2$	$\mathrm{Ti}{\mathrm{O}}_2$	0.29	0.49	0.94	90	90	90

Figure B1. Result of profile analysis in case-01 using our method [(a): Fitting result via profile function in the measurement XRD data. The black and the red lines represent the measurement XRD data and the fitting profile function, respectively. (b): Peak components in two crystal structures of $\mathrm{Ti}{\mathrm{O}}_2$; Anatase and rutile. The red and blue lines denote peaks of Anatase and rutile, respectively].

Table B1. True parameter of profile function in artificial data. We assumed that the XRD data were measured on a sample containing three crystalline phase structures of $\mathrm{Ti}{\mathrm{O}}_2$: Anatase, Brookite, and rutile.

${\mathrm{\Theta }}_{\mathrm{S}}$	Rutile	Brookite	Anatase
$h_k$	5000.0	5000.0	5000.0
${\mu }_k$	0.05	0.05	0.05
${\alpha }_k$	0.80	0.80	0.80.
$r_k$	0.50	0.50	0.50
$u_k$	0.005	0.005	0.005
$v_k$	0.005	0.005	0.005
$w_k$	0.05	0.10	0.15
$s_k$	0.10	0.15	0.20
$t_k$	0.005	0.005	0.005

Figure B2. Selection result from 50 candidates for each temperature using the REMC method for artificial XRD data. The x- and the y-axes denote the candidate crystal structures and the index of the inverse temperature $\mathit{\tau }$. This figure shows a visualization of the indicator probability $\mathit{P}(\mathit{g}|\mathit{Y};\mathit{\beta }={\beta }_{\tau })$ [%]. The large index corresponds to lower temperatures. The color scale denotes the sampling frequency of $g_k=1$ on a log scale. The dark red indicates the presence of the crystal structure in the measured sample.

Figure B3. Result of profile analysis in the artificial data using our method [(a): Fitting result via profile function in the artificial data. In this figure, the black and the red lines indicate the artificial data and the fitting profile function, respectively. (b): Peak components in three crystal structures of $\mathrm{Ti}{\mathrm{O}}_2$: Anatase, Brookite, and rutile. The red, green, and blue lines indicate peaks of Anatase, Brookite, and rutile, respectively].

Figure B4. Posterior distribution of the profile parameters in the artificial data. The red, green, and blue histograms are the posterior distribution of Anatase, Brookite, and rutile, respectively. [(a): Peak height $\mathit{h}$, (b): peak shift $\mathit{\mu }$, (c): Gauss-Lorentz ratio $\mathit{r}$, and (d): asymmetry parameter $\mathit{\alpha }$]. (e) and (f) are Gauss width $\mathrm{\Sigma }(x_i;u_k,v_k,w_k,{\alpha }_k)$ and Lorentz width $\mathrm{\Omega }(x_i;u_k,v_k,w_k,{\alpha }_k)$, where $x_i$ is $2\mathit{\theta }=60$ [degree]. The black dot-dashed line denotes the true parameter of the profile function.

Figure B5. Result of profile analysis in case-02 using our method [(a): Fitting result via profile function in the measurement XRD data. The black and red lines denote the measurement XRD data and the fitting profile function, respectively. (b): Peak components in two crystal structures of $\mathrm{Ti}{\mathrm{O}}_2$; Anatase and Brookite. The red and green lines denote peaks of Anatase and Brookite, respectively].

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