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56
Appendix A
Calculation of the powder diffraction
pattern
This appendix describes the calculation of the powder diffraction pattern used in the
penalty function.
A.1 Formulation of the powder diffraction pattern
We calculate the powder diffraction pattern according to the following formula,
J(θ, R) = |S(θ, R)|2 L(θ)P (θ),
(A.1)
where θ is the diffraction angle, R is the crystal structure, S is the structure factor, L
is the Lorentz factor, and P is the polarization factor. In order to bring the theoretical
value closer to the experimental one, it is necessary to multiply this formula by additional
factors, such as the absorption factor and the temperature (Debye-Waller) factor. Since
these factors are complicated to calculate during the simulation, we omit them in the
calculation of the penalty function. On the other hand, the Lorentz factor and the polarization factor are expressed by the simple following formulas that depend only on the
diffraction angle,
L(θ) =
(A.2)
sin θ cos θ
P (θ) =
1 + cos2 (2θ)
(A.3)
The structure factor is the main part that depends on the atomic coordinates and is
expressed by the following formula,
S(θ, R) =
j=1
Fj (θ) exp(2πih · r˜j ),
(A.4)
where subscript j is the index of atoms, N is the number of atoms, r˜ is the atomic
fractional coordinate, h is the vector of the Miller index, and F is the atomic form factor.
From Bragg’s law, the diffraction angle corresponding to the Miller index h is as follows
57
A.1 Formulation of the powder diffraction pattern
(see Fig. A.1),
θh = arcsin
2dh
= arcsin
λ|Bh|
4π
(A.5)
where λ is the wavelength, d is the inter-planar spacing, and B is the matrix consisting
of the reciprocal lattice vectors (b1 , b2 , b3 ).
Fig. A.1. Schematic diagram of Bragg’s law. The white circles are atoms arranged with
the inter-planar spacing d, and red arrows are rays with the wave length λ and
the diffraction angle θ. The path difference between the two rays is equal to
2d sin θ.
The atomic form factor depends on atomic species, and in the case of the X-ray diffraction, it is expressed by the following formula,
F (θ) =
i=1
ai exp −bi
sin θ
#2 +
+ c.
(A.6)
We use the database [41] for the parameters (a, b, c), which depend on atomic species. A
similar equation holds in the case of the neutron diffraction, and we also use the database
[42]. In the X-ray diffraction, the atomic form factor of hydrogen is much smaller than
that of a heavy atom, whereas in the neutron diffraction, it is relatively large and hydrogen
atoms can be detected.
58
Appendix A Calculation of the powder diffraction pattern
A.2 Diffraction peak smearing
In the equation A.1, the diffraction peak is a delta function at a diffraction angle θ, but
in the experiment, it has a finite peak width. We implement two smearing functions,
Gaussian smearing and Lorentzian smearing. The powder diffraction pattern with the
finite peak width is expressed by the following formula,
I(θ) =
J(θh )f (|θ − θh |),
(A.7)
where f is the smearing function. In our calculations, in order to reduce the computational
costs, f is regarded as zero at large |θ − θh |. Gaussian and Lorentzian smearing function
are as follows,
(θ − θh )2
fgaussian = √
exp −
(A.8)
2σ 2
2πσ
florentzian =
π (θ − θh )2 + γ 2
(A.9)
where σ is the standard deviation, and γ is the scale parameter. σ in Gaussian smearing
and γ in Lorentzian smearing correspond to the peak width, respectively.
In the Rietveld analysis [3], a smearing function that combines Gaussian and Lorentzian
smearings is used to reproduce the experimental result. In the structure prediction method
based on data assimilation, it is not necessary to accurately refine the peak width, but in
order to improve the search efficiency, it is better to set the peak width close to that in
the experiment.
A.3 Range of the Miller indices
The Miller index is an arbitrary integer, but the range of it to be considered is limited by
the range of the diffraction angle. Assuming that the range of the diffraction angle is [θm ,
θM ], from Eq. A.5, the range of the Miller index h is as follows,
4π sin θm
4π sin θM
≤ |Bh| ≤
(A.10)
Let r be the right side of Eq. A.10, and consider the upper limit of the Miller index.
Then, Eq. A.10 can be interpreted that the grid point h is included in the curved surface
S " (see Fig. A.2). S " is obtained on a linear transformation of the sphere S with radius
r by the inverse matrix B −1 = 2π
AT . A is the matrix consisting of the lattice vectors
(a1 , a2 , a3 ). It is sufficient to find the maximum value of the (x, y, z)-coordinates on the
curved surface S " .
59
A.3 Range of the Miller indices
Fig. A.2. Schematic diagram for finding the upper limit of the Miller index. The left
and right figures are transferred to each other on linear transformations by
the reciprocal lattice vectors B and the lattice vector A, respectively. The
white circles are the grid points corresponding to the Miller indices h. The
curved surface S is the sphere with radius r, and S ! is obtained on a linear
transformation of the sphere S. The maximum value of the x-coordinate on S !
is the x-coordinate of the tangent point with the normal vector (1, 0, 0) on S ! .
This point corresponds to the tangent point with the normal vector a1 on S.
The tangent plane at the point where the x-coordinate is maximum on S " is orthogonal
to x-axis and contains (0, 1, 0) and (0, 0, 1) vectors. The plane obtained on a linear
transformation of this tangent plane by the matrix B is the tangent plane of S and
contains the reciprocal lattice vectors b2 and b3 , that is, orthogonal to the lattice vector
a1 , b2 × b3 . Therefore, the tangent point on S is |ar1 | a1 , and the tangent point on S " is
−1
a1 . Finally, the upper limit of the x-coordinate is expressed as follows,
|a1 | B
r % −1 &
r % T &
B a1 x =
A a1 x =
|a1 |.
|a1 |
2π|a1 |
2π
(A.11)
Since the same applies to the lower limit of the x-coodinate and the limits of the y and
z-coordinates, the upper and lower limits of the Miller index h are as follows.
|a1 |, ± |a2 |, ± |a3 | .
2π
2π
2π
(A.12)
60
Appendix A Calculation of the powder diffraction pattern
A.4 Derivative with respect to atomic coordinates
In order to calculate the force applied to each atom from the penalty function, it is
necessary to differentiate the diffraction pattern with respect to the atomic coordinates.
From Appendix A.1, only the structure factor depends on the atomic coordinates. The
derivative of the structural factor with respect to the atomic coordinates can be calculated
as follows,
∂S
∂ !
Fj (θ) exp(2πih · r˜j )
∂r
∂r j=1
Fj (θ)
j=1
j=1
j=1
(A.13)
exp(2πih · A−1 rj )
∂rj
Fj (θ) exp(2πih · r˜j ) × 2πi
1 T
(h ·
B rj )
∂rj
2π
Fj (θ) exp(2πih · r˜j ) × iBh
= iSBh.
(A.14)
(A.15)
(A.16)
(A.17)
The stress tensor can be calculated by differentiating with respect to the cell parameters.
As Eq. A.5 shows, the diffraction angle θ depends on the cell parameters, so it is sufficient
to differentiate with respect to θ. However, as seen in Eq. A.7, the smearing function also
depend on θ, and its derivative is very large near the peak position. In order to optimize
the cell parameters by the structure prediction method based on data assimilation, it is
necessary to define a penalty function which is smooth for changes in the peak position.
61
Appendix B
Crystallographic data in this study
This appendix shows the crystallographic data in this study.
B.1 Coesite
Table. B.1. The correct structure of the target material coesite used in Section 3.1 [38].
Lattice parameters
(˚
A, degree)
a = 7.1367
b = 12.3695
c = 7.1190
α = 90.00
β = 119.57
γ = 90.00
Atomic coordinates
(fractional)
Si (0.18197, 0.14169, 0.07227)
Si (0.81803, 0.85831, 0.92773)
Si (0.81803, 0.14169, 0.42773)
Si (0.18197, 0.85831, 0.57227)
Si (0.68197, 0.64169, 0.07227)
Si (0.31803, 0.35831, 0.92773)
Si (0.31803, 0.64169, 0.42773)
Si (0.68197, 0.35831, 0.57227)
Si (0.28394, 0.09194, 0.54066)
Si (0.71606, 0.90806, 0.45934)
Si (0.71606, 0.09194, 0.95934)
Si (0.28394, 0.90806, 0.04066)
Si (0.78394, 0.59194, 0.54066)
Si (0.21606, 0.40806, 0.45934)
Si (0.21606, 0.59194, 0.95934)
Si (0.78394, 0.40806, 0.04066)
O (0.07598, 0.12685, 0.55943)
O (0.92402, 0.87315, 0.44057)
O (0.92402, 0.12685, 0.94057)
O (0.07598, 0.87315, 0.05943)
O (0.57598, 0.62685, 0.55943)
O (0.42402, 0.37315, 0.44057)
O (0.42402, 0.62685, 0.94057)
O (0.57598, 0.37315, 0.05943)
(0.26683,
(0.73317,
(0.73317,
(0.26683,
(0.76683,
(0.23317,
(0.23317,
(0.76683,
(0.28918,
(0.71082,
(0.71082,
(0.28918,
(0.78918,
(0.21082,
(0.21082,
(0.78918,
(0.00000,
(0.00000,
(0.50000,
(0.50000,
(0.25000,
(0.75000,
(0.75000,
(0.25000,
0.14625,
0.85375,
0.14625,
0.85375,
0.64625,
0.35375,
0.64625,
0.35375,
0.03805,
0.96195,
0.03805,
0.96195,
0.53805,
0.46195,
0.53805,
0.46195,
0.36631,
0.63369,
0.86631,
0.13369,
0.25000,
0.75000,
0.25000,
0.75000,
0.32787)
0.67213)
0.17213)
0.82787)
0.32787)
0.67213)
0.17213)
0.82787)
0.02165)
0.97835)
0.47835)
0.52165)
0.02165)
0.97835)
0.47835)
0.52165)
0.25000)
0.75000)
0.25000)
0.75000)
0.00000)
0.00000)
0.50000)
0.50000)
62
Appendix B Crystallographic data in this study
B.2 !-Zn(OH)2
Table. B.2. The correct structure of the target material &-Zn(OH)2 used in Section 3.2
[39].
Lattice parameters
(˚
A, degree)
a = 4.87176
b = 5.06348
c = 8.75226
α = 90.00
β = 90.00
γ = 90.00
Atomic coordinates
(fractional)
Zn (0.067570, 0.642226, 0.123390)
Zn (0.432430, 0.357774, 0.623390)
Zn (0.932430, 0.142226, 0.376610)
Zn (0.567570, 0.857774, 0.876610)
H (0.030919, 0.641940, 0.847283)
H (0.469081, 0.358060, 0.347283)
H (0.969081, 0.141940, 0.652717)
H (0.530919, 0.858060, 0.152717)
H (0.242591, 0.824785, 0.658357)
H (0.257409, 0.175215, 0.158357)
(0.757409,
(0.742591,
(0.120869,
(0.379131,
(0.879131,
(0.620869,
(0.188286,
(0.311714,
(0.811714,
(0.688286,
0.324785,
0.675215,
0.111773,
0.888227,
0.611773,
0.388227,
0.316113,
0.683887,
0.816113,
0.183887,
0.841643)
0.341643)
0.577116)
0.077116)
0.922884)
0.422884)
0.228961)
0.728961)
0.271039)
0.771039)
B.3 Al-Ca-H
Table. B.3. The new Al12 Ca20 H76 structure proposed in Section 3.3.
Lattice parameters
(˚
A, degree)
a = 12.809756
b = 12.809756
c = 6.814734
α = 90.00
β = 90.00
γ = 90.00
Atomic coordinates
(fractional)
Al (0.500011, 0.005163,
Al (0.003018, 0.505047,
Al (0.502574, 0.005254,
Al (0.331902, 0.335357,
Al (0.168770, 0.836904,
Al (0.839311, 0.163375,
Al (0.663408, 0.663518,
Al (0.831443, 0.834761,
Al (0.668759, 0.336545,
Al (0.338640, 0.663467,
Al (0.163486, 0.163750,
Al (1.000437, 0.505635,
0.537170)
0.537047)
0.037029)
0.224586)
0.222126)
0.215090)
0.214548)
0.727048)
0.722520)
0.715482)
0.712352)
0.036693)
(0.393267,
(0.429963,
(0.247695,
(0.304244,
(0.104170,
(0.199422,
(0.275273,
(0.251859,
(0.090047,
(0.110123,
(0.389815,
(0.302659,
0.766778,
0.705195,
0.602437,
0.565146,
0.894427,
0.934485,
0.889934,
0.746446,
0.932491,
0.736419,
0.232646,
0.434728,
0.863840)
0.535184)
0.873155)
0.542758)
0.710286)
0.043467)
0.362151)
0.129567)
0.312873)
0.369379)
0.368334)
0.048838)
B.3 Al-Ca-H
Ca (0.015420, -0.004752, 0.039557)
Ca (0.484803, 0.491280, 0.040281)
Ca (0.514707, 0.494307, 0.541170)
Ca (0.982685, 0.989151, 0.537713)
Ca (0.932544, 0.727437, 0.229506)
Ca (0.432343, 0.773718, 0.220318)
Ca (0.567838, 0.226603, 0.228757)
Ca (0.067889, 0.273081, 0.217527)
Ca (0.220985, 0.572058, 0.218955)
Ca (0.279747, 0.072355, 0.214524)
Ca (0.725710, 0.928484, 0.226622)
Ca (0.776591, 0.427751, 0.223853)
Ca (0.432544, 0.226399, 0.729145)
Ca (0.933172, 0.272773, 0.720397)
Ca (0.067933, 0.725746, 0.728927)
Ca (0.567350, 0.772103, 0.721009)
Ca (0.720780, 0.072674, 0.718421)
Ca (0.779762, 0.572221, 0.715348)
Ca (0.226267, 0.429268, 0.725852)
Ca (0.275707, 0.928949, 0.723158)
H (0.962768, 0.518215, 0.793265)
H (0.041755, 0.496431, 0.293066)
H (0.948064, 0.627964, 0.523266)
H (0.055494, 0.385916, 0.584412)
H (0.120078, 0.562359, 0.594489)
H (0.880794, 0.451961, 0.513059)
H (0.885366, 0.560309, 0.108509)
H (0.122494, 0.454243, 1.004958)
H (0.057008, 0.628784, 0.042721)
H (0.947454, 0.383660, 0.055655)
H (0.461099, 0.008106, 0.293226)
H (0.541870, 0.005978, 0.793358)
H (0.384702, 0.061043, 0.604221)
H (0.622056, 0.953058, 0.508453)
H (0.556149, 0.128408, 0.534006)
H (0.446979, 0.884057, 0.568232)
H (0.380434, 0.953085, 0.007598)
H (0.619000, 0.061626, 0.101415)
H (0.555263, 0.884589, 0.072071)
H (0.446988, 0.128357, 0.031924)
H (0.243024, 0.751148, 0.647358)
H (0.442122, 0.587476, 0.788658)
(0.415507,
(0.435968,
(0.227379,
(0.247598,
(0.073280,
(0.254947,
(0.258935,
(0.202937,
(0.107362,
(0.062637,
(0.755229,
(0.572095,
(0.758564,
(0.702528,
(0.607314,
(0.562899,
(0.801804,
(0.889878,
(0.916010,
(0.934911,
(0.727701,
(0.746668,
(0.699092,
(0.610534,
(0.774165,
(0.752661,
(0.587929,
(0.564135,
(0.883523,
(0.929624,
(0.893547,
(0.944753,
(0.742902,
(0.750116,
(0.605151,
(0.383282,
(0.618348,
(0.398676,
(0.899486,
(0.119240,
(0.804579,
(0.063647,
0.426118,
0.308499,
0.389938,
0.247117,
0.208127,
0.103135,
0.253491,
0.068335,
0.264613,
0.083103,
0.603578,
0.705949,
0.753375,
0.567688,
0.765620,
0.583327,
0.933755,
0.731794,
0.924802,
0.807369,
0.890272,
0.746553,
0.434587,
0.234624,
0.390176,
0.246713,
0.429989,
0.309860,
0.112002,
0.205285,
0.268716,
0.090128,
0.250292,
0.101437,
0.394932,
0.613126,
0.613711,
0.397375,
0.899628,
0.115050,
0.064654,
0.810769,
63
0.316167)
0.052714)
0.366369)
0.130386)
0.533699)
0.870501)
0.650720)
0.538267)
0.864283)
0.779928)
0.372325)
0.035581)
0.150522)
0.040467)
0.364382)
0.284326)
0.550899)
0.869815)
0.817192)
0.553836)
0.868660)
0.634524)
0.544864)
0.868264)
0.863575)
0.630062)
0.813679)
0.552110)
0.676588)
0.033478)
0.361043)
0.288546)
0.147886)
0.374681)
0.211249)
0.176826)
0.671557)
0.721478)
0.226555)
0.167075)
0.043351)
0.052614)
...
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