computer programs
Niggli reduction and
determination^{a}School of Science, Minzu University, 27 Zhang Guancun South Avenue, Haidian District, Beijing, 100081, People's Republic of China, and ^{b}School of Physical Science and Technology, Guangxi University, No. 100 Daxuedong Road, Xixiangtang District, Nanning, Guangxi, 530004, People's Republic of China
^{*}Correspondence email: honglongshi@outlook.com
A new algorithm has been developed and coded in DigitalMicrograph (DM) to reduce a threedimensional to the Niggli cell and further convert to the Bravaislattice The core of this algorithm is the calculation of the three shortest noncoplanar vectors to compose the The is converted into the realspace and then to the Bravaislattice The symmetryconstrained is, in turn, converted back into the realspace the reciprocal and the reciprocal The DM package demonstrates superior numerical stability and can tolerate large uncertainties in the experimentally measured input making it especially suitable for electron Additionally, the DM package can be used to calculate various crystallographic parameters including Bravaislattice plane indices, zoneaxis indices, tilt angles and the radius of the highorder Laue zone ring, thus facilitating the correct determination of the Niggli cell and the Bravais lattice.
Keywords: Niggli reduction; reduced cells; symmetry constraints; Bravais lattices; unit cells.
1. Introduction
The fundamental step for ab initio new determination (Putz et al., 1999; Le Bail et al., 1988; Young, 1993) is determining the Bravaistype unitcell parameters of a crystal. The can be uniquely obtained once the that can be constructed from Xray, neutron or electron diffraction measurements (Pecharsky & Zavalij, 2003; Fultz & Howe, 2013; Williams & Carter, 2009) is reduced to the Niggli cell (Santoro & Mighell, 1970; Gruber, 1973; de Wolff, 2006), because the Niggli cell provides a unique description of a lattice and is defined independently of the lattice symmetry. This is well documented in International Tables for Crystallography, Vol. A (de Wolff, 2006).
In 1928, Niggli put forward a set of conditions that produce a unique choice of basis vectors of a lattice. Subsequently, Křivý & Gruber (1976) presented a numerical algorithm for the Niggli reduction, and later the number of iterations of this algorithm was optimized by Zuo et al. (1995). Křivý and Gruber use the following notation in the description of the reduction algorithms:
where a, b and c are the basis vectors of a cell with parameters a, b, c, α, β and γ; the parameters A, B, C, D, E and F represent the Nigglimatrix elements S_{ij}.
In realworld applications, two major problems exist during the reduction procedure. The first problem is the treatment of rounding errors, since algorithms implemented using finiteprecision floatingpoint algebra can result in infinite loops when the rounding errors are improperly treated (e.g. irrational numbers). This first problem was addressed by GrosseKunstleve et al. (2004) by introducing a tolerance factor to improve the numerical stability of algorithms. The second problem is the treatment of the measurement errors of the experimentally measured input cell (Yang et al., 2017). These cells with errors generate matrix elements S_{ij} with uncertainties that can be further magnified and propagated to the successive steps of reduction, finally resulting in a with a large uncertainty that makes correct determination of the very difficult.
Electron diffraction is a powerful technique to determine et al., 2014; Wen et al., 2018; Sheng et al., 2016), and the 3D reciprocal cell can be determined directly from electron diffraction patterns (Shi & Li, 2021; Jiang et al., 2011; Li, 2019; Zhao et al., 2008; Zou et al., 2004). However, the measured from an electron diffraction pattern suffers from a poor measurement precision because the measured d spacings exhibit around 1% error, resulting from the diffraction distortions of the electromagnetic lens and the uncertainty of the camera length of the transmission electron microscope (Mitchell & Van den Berg, 2016; Hou & Li, 2008). Moreover, the measurement error of interzonal angles of a tilt series can be up to a few degrees. Although the relation between the Niggli cell and the is definitive, errors in the experimentally observed cell will propagate to the Niggli cell, making the conversion of the Niggli cell into the very difficult.
at the nanoscale (ZhengHere, we present an alternative algorithm and a corresponding DigitalMicrograph (DM; Gatan, 2019) package, Niggli Reduction Tools, to calculate the Niggli cell and convert the Niggli cell into the In the new algorithm, parameters ɛ_{1–3} are introduced as tolerance factors for the measurement errors of the basisvector lengths. Such treatment can tolerate large experimental error and achieve better numerical stability. The parameters ɛ_{d} and ɛ_{A} define errors on the lengths and angles of the Bravaislattice Additionally, symmetry constraints on the determined Bravais cell are used to evaluate the measurement errors of the observed cell in the electron diffraction pattern. The DM package can also be used to calculate the Bravaislattice plane indices, zoneaxis indices and tilt angles, as well as the radius of the highorder Laue zone (HOLZ) ring of the electron diffraction pattern. The robustness of the new algorithm for converting the Niggli cell to the will be demonstrated and discussed.
2. Algorithms
2.1. Fundamentals of the Niggli reduction and the unitcell determination
Since a
has lattice points only at its vertices, all lattice points appear at the vertices of the collection of the identical primitive cells of the lattice. This means that each point of the lattice can be retrieved from any other lattice point by a vector sum of cell edges of a primitive cell.Let vectors a, b and c be the edges of a The translation vector t of the lattice point at (u, v, w) can be described as . If we start with an arbitrary with edges a, b and c, and wish to obtain a Niggli with edges t_{1}, t_{2} and t_{3}, the three edges of the new cell can be expressed in terms of the old ones as , where the subscript i = 1, 2, 3. The indices (u_{i}, v_{i}, w_{i}) are small integers. The edge length of the new cell can be determined by the scalar product with itself, ; and the angles between two edges of the new cell can be calculated by using the relation . In this way, the three shortest nonplanar vectors compose the Niggli with parameters a_{0}^{*}, b_{0}^{*}, c_{0}^{*}, , and .
The obtained Niggli a_{0}, b_{0}, c_{0}, , and ) through the relationship between the real and The realspace can then be transformed into the Bravaislattice (unit cell; a, b, c, , and γ) based on the conditions of the Niggli matrix elements S_{ij} as reported in International Tables for Crystallography, Vol. A.
described in the can be transformed into a realspace (Errors of the Niggli
propagated from the reciprocal (or the input cell) can propagate to the Bravaislattice These errors can be evaluated and corrected by applying symmetry constraints on the standard (1) Edges and angles of the obtained can be constrained as the symmetryconstrained (abbreviated as the sym. unit cell) to satisfy the symmetry of the (2) In turn, the symmetryconstrained is inversely transformed into the realspace the reciprocal and the reciprocal The obtained is a symmetryconstrained cell that can be used to evaluate and correct the errors of the input cell, or even to perform of the lattice parameters from experimental electron diffraction data.2.2. Description of the program
The package Niggli Reduction Tools is based on the DigitalMicrograph software, whose language is similar to C or C++. The package grants permission to copy, use or modify the code for any purpose under a license.
2.2.1. Distribution and installation
The package is freely available by email (honglongshi@outlook.com) and as supporting information to this article, and the DigitalMicrograph software is available at https://www.gatan.com/products/temanalysis/gatanmicroscopysuitesoftware.
There are two files in this package:
(i) NiggliReduction.gtk: a compiled package of Niggli Reduction Tools.
(ii) Tutorial.pdf: a concise help file.
To install the package, the file NiggliReduction.gtk should be copied to ...\Gatan\DigitalMicrograph\PlugIns. A new menu `ED Tools / Niggli Reduction Tools' will be built on the menu bar of DigitalMicrograph. Clicking the menu `ED Tools / Niggli Reduction Tools' will launch the graphical user interface (GUI) [Fig. 1(b)].
2.2.2. Software overview
Niggli Reduction Tools can reduce any 3D reciprocal to the Niggli cell and determine the Bravaislattice as shown in Fig. 1(b). The GUI has four sections: (1) the `Parameters' box defines the input cell, the index range `N', and the tolerance factors `eps 1~3' and `eps d/A'; (2) the `Reduced Cell List' box lists the three shortest nonplanar vectors within the tolerance factors eps 1~3; (3) the `Unit Cell List' box lists the possible Bravais lattices within the tolerance of eps d/A; and (4) the `Results' box outputs a concise list of derived cells and other useful parameters.
The parameters of the package are defined as follows:
Input cell: defines the measured reciprocal ^{−1} or nm^{−1}. If a realspace cell is input, it can be converted into a reciprocal cell by simultaneously clicking the `Alt' key and the `Calc.' button.
The unit of the input cell can be Åeps 1~3: defines the factors ɛ_{1–3} to give the tolerance lengths of the shortest vectors t_{1}, t_{2} and t_{3}. This mainly depends on the measurement error Δp (typically, 1–5 pixels) and the resolution r of the examined electron diffraction pattern (or the image scale, e.g. nm^{−1} per pixel), and ɛ_{1–3} = Δpr.
eps d/A: defines the tolerance factors ɛ_{d} (the unit is ångström) and ɛ_{A} (the unit is degree) of the in matching the Bravais lattice.
N: defines the range of indices (u, v, w) used in searching for the shortest vectors.
The algorithm of Niggli Reduction Tools is as follows:
S1. Input a reciprocal a^{*}, b^{*}, c^{*}, , , .
S2. For −N ≤ (u, v, w) ≤ N, calculate the length of the vector t_{u,v,w}.
S3. Find the first three minima t_{10}, t_{20} t_{30}.
S4. If t_{10} < t_{i} < t_{10} + ɛ_{1}, find the collection of the first minima and create the vector t_{1}.
S5. If t_{20} < t_{i} < t_{20} + ɛ_{2}, find the collection of the second minima (noncollinear with t_{1}) and create the vector t_{2}.
S6. If t_{30} < t_{i} < t_{30} + ɛ_{3}, find the collection of the third minima (noncoplanar with t_{1} and t_{2}) and create the vector t_{3}. The three vectors t_{1}, t_{2} and t_{3} compose the reciprocal reduced cell.
S7. Convert the reciprocal
to the real reduced cell.S8. Convert the real ɛ_{d} and ɛ_{A} and determine the symmetryconstrained unit cell.
into the Bravaislattice within the tolerance ofS9. Convert the constrained
to the real reduced cell.S10. Convert the real
to the reciprocal reduced cell.S11. Convert the reciprocal
to the reciprocal and calculate other parameters.Fig. 1(a) shows the workflow of the package Niggli Reduction Tools. After the parameters of the reciprocal cell have been input, the three basis vectors of the input cell are created in the orthogonal coordinate system as follows (note: if the input cell is a realspace cell, press the `Alt' key and click the `Calc.' button to convert it to the reciprocal cell):
where V^{*} is the volume of the input cell. The vector length t of each index (u,v,w ) within the index range of ±N is then calculated; and the three minima (t_{10}, t_{20}, t_{30} ) of the t_{i} list can be determined.
Next, the first minimum t_{1} within the range is found and the vector t_{1} is created; the second minimum t_{2} (nonlinear with the vector ) within the range is found and the vector created; and the third minimum t_{3} (noncoplanar with vectors and ) within the range is found and the vector created. Three lists of t_{1}, t_{2} and t_{3} that compose the are found in this way and displayed in the `Reduced Cell List` box [Fig. 1(b)].
After the t_{1}, t_{2} and t_{3}, it will be converted into the real ( a_{0}, b_{0}, c_{0}, , and ). The basis vectors of the are created in the orthogonal coordinate system, written as the matrix A in equation (3); and the transformation matrix M can be found in International Tables for Crystallography, Vol. A (de Wolff, 2006). In this way, the real is transformed into 44 Bravaislattice unit cells U by matrix multiplication of the matrices M and A:
has been chosen by successively clicking the listThe lattice parameters are determined from the elements of the matrix U as follows:
Only those cells whose edge and angle errors fall in the range of ɛ_{d} and ɛ_{A} are listed in the `Unit Cell List' box [Fig. 1(b)]. When one cell is chosen from the `Unit Cell List' box, the selected will be constrained according to the symmetry of the The symmetryconstrained is further inversely converted into the real the reciprocal and the reciprocal A concise result list of cells is displayed in the `Results' box; more details of the results are output in the `Results' window of the Digital Micrograph software.
Some additional useful parameters are calculated to help choose the correct Niggli cell and S matrix and the Bravaislattice criteria are live displayed when choosing one cell in the `Unit Cell List' box; (3) the Bravaislattice plane indices, the zoneaxis indices and tilt angles are calculated; and (4) the radius of the HOLZ ring is derived.
(1) errors between the observed cell and the symmetryconstrained cell are evaluated; (2) the3. Illustrative examples
To demonstrate the use of the Niggli Reduction Tools package, two examples are given here. The first is to reduce a cell with small errors constructed from a simulated electron diffraction pattern of an La_{2}(Ti_{2}O_{7}) crystal oriented at . The second is to reduce a cell with large uncertainties determined from the experimental electron diffraction pattern of a silicon crystal.
3.1. Determination of the and of a lowsymmetry lattice (small errors)
The input cell for the Niggli reduction is obtained by reconstructing the 3D reciprocal cell from a simulated electron diffraction pattern of the monoclinic structure La_{2}(Ti_{2}O_{7}) with subpixel measurement error (scale = 0.019424 nm^{−1} per pixel). The cell parameters described in the are a^{*} = 2.2204, b^{*} = 2.2872, c^{*} = 1.8037 nm^{−1}, = 37.94, = 35.65, = 70.11°, as shown in Fig. 2(a). After inputting the cell parameters and clicking the `Calc.' button, we obtain the lists of t_{1}, t_{2} and t_{3} in the `Reduced Cell List' box (ɛ_{1–3} = 0.1 nm^{−1} and N = 10), as shown in Fig. 1(b). Here, we select the with the shortest edges (the topmost one, a_{0}^{*} = 0.7738, b_{0}^{*} = 1.2941, c_{0}^{*} = 1.8037 nm^{−1}, = 89.97, = 89.95, = 81.50°); according to the real–reciprocal relationship of the lattice, the real is calculated to be a_{0} = 13.0674, b_{0} = 7.8130, c_{0} = 5.5442 Å, = 90.02, = 90.05, = 98.50° with the Nigglimatrix elements of A = 170.76, B = 61.04, C = 30.74, D = −0.01, E = −0.06 and F = −15.09. Subsequently, the real is converted into the Bravais lattices that are listed in the `Unit Cell List' box only when the differences of lengths and angles of the cell fall in the range of ɛ_{d} and ɛ_{A}. In this example, two Bravais lattices, mP and aP, are listed. Requiring that the matrix elements of the meet the conditions A ≠ B ≠ C, D = E = 0 and F ≠ 0 (No. 34, II25), the monoclinic structure mP is selected. The obtained is mP, i.e. it must satisfy the symmetry of the monoclinic (a ≠ b ≠ c, α = γ = 90°). After applying the symmetry constraints to the selected the parameters of the cell become a = 13.0674, b = 5.5442, c = 7.8130 Å, α = 90.00, β = 98.50, γ = 90.00°. In turn, the symmetryconstrained is inversely converted into the real the reciprocal and the reciprocal (detailed parameters are listed in Table 1). The symmetryconstrained reciprocal is a^{*} = 2.2199, b^{*} = 2.2871, c^{*} = 1.8037 nm^{−1}, = 37.94, = 35.66, = 70.12°. The errors between the input cell and the symmetryconstrained cell are evaluated to be = 0.0005, = 0.0001, = 0.0000 nm^{−1}, = 0.00, = −0.01, = −0.01°, respectively.

For convenience to evaluate the candidate Niggli cell and the Niggli Reduction Tools provides additional useful parameters:
(1) `S matrix' and `Criteria': the S matrix of the and the Bravais criteria for transforming the to the unit cell.
(2) `Indices' and `Angles': the latticeplane indices of the diffraction spots indicated by the three basis vectors of the input cell, and the angles between the vectors. The latticeplane indices inherit the symmetry of the
and hence can be used to select the and the in the `Reduced Cell List' and the `Unit Cell List'; the derived angles between the vectors can be compared with the measured values in the tilt series experiment.(3) `Zones' and `Tilt Angles': the zoneaxis indices of the diffraction patterns for constructing the threedimensional reciprocal
and the tilt angles between zone axes. The tilt angles can be directly compared with the angles calculated from `Tilt X' and `Tilt Y' of the transmission electron microscope.(4) `HOLZ Ring': the radii of the HOLZ rings of three zoneaxis patterns which can be compared with the measured ones.
In this illustration, the S matrix of the (170.76, 61.04, 30.74, −0.01, −0.06, −15.08), fully satisfies the criteria (A ≠ B ≠ C, D = E = 0, F ≠ 0). The latticeplane indices of the three diffraction spots indicated by the basis vectors are (011 )_{a*}, and (010 )_{c*}, respectively. Therefore, the interplanar angles are 70.12, 37.94 and 35.66° between the vectors and , and , and and , respectively (versus 70.11°, the measured angle based on the onepattern method). The of the examined pattern is , and the radius of the HOLZ ring is calculated to be 17.2712 nm^{−1} versus 17.2728 nm^{−1} for the measured one. The small differences between the observed cell and the determined cell, as well as the comparison between the measured values and calculated ones based on the symmetryconstrained suggest that the determined Niggli cell and the Bravaislattice are valid.
In this example, a lowsymmetry anisotropic crystal La_{2}(Ti_{2}O_{7}) is examined, and the top three shortest vectors in the list of t_{1}, t_{2}, and t_{3} always compose the Niggli However, a highsymmetry crystal often generates multiple equivalent vectors; and the errors of the input cell make these equivalent vectors different, which complicates the procedure of choosing the reduced cell.
3.2. Determination of the and the of a highsymmetry lattice (large errors)
Here, we will discuss a highsymmetry case and an input cell with large uncertainties (e.g. 1–2 pixels, 0.057783 nm^{−1} per pixel). The was extracted from the experimental electron diffraction pattern of a silicon single crystal [Fig. 2(b)]. The parameters of the input cell are a^{*} = 5.2083, b^{*} = 7.9618, c^{*} = 5.1259 nm^{−1}, = 13.30, = 60.94, = 71.93°. After the Niggli reduction, the three nonplanar shortest vectors , and produce the candidate with parameters a_{0}^{*} = 3.1020, b_{0}^{*} = 3.1987, c_{0}^{*} = 3.2412 nm^{−1}, = 110.94, = 107.13, = 108.88°. Subsequently, the is converted into the real The Smatrix elements of the (A = 14.7961, B = 14.5689, C = 13.9112, D = 7.1280, E = 6.6602, F = 7.0548) approximately meet the conditions A = B = C and D = E = F = A/2 (cF, 1, I1); thus, the Bravaislattice is converted to a = 5.3133, b = 5.4598, c = 5.4866 Å, α = 91.56, β = 92.64, γ = 88.87°. After applying the symmetry constraints of the cubic the symmetryconstrained cell is a = b = c = 5.4199 Å, α = β = γ = 90.00°, which in turn is inversely converted to the symmetryconstrained real direct cell, the reciprocal and the reciprocal (see detailed parameters in Table 1). The symmetryconstrained is a^{*} = 5.2186, b^{*} = 8.0424, c^{*} = 5.2186 Å^{−1}, = 13.26, = 60.00, = 71.07°. The errors between the symmetryconstrained cell and the input cell are evaluated to be = −0.0103, = −0.0806, = −0.0927 nm^{−1}, = 0.04, = 0.94, = 0.86°, respectively. These evaluated errors are entirely consistent with the measurement errors (0.05–0.1 nm^{−1} in length and 1° in angle of the input cell).
Strictly speaking, the determined et al., 2009; Hou & Li, 2008). Hence, in this work the userdefined parameters ɛ_{1–3} are introduced to accommodate the measurement uncertainties of the observed cell (or the input cell). Meanwhile, the parameters ɛ_{1–3} will also result in diverse choices for the Niggli cell. For a highsymmetry lattice and an input cell with large uncertainties, this problem will become more severe. For instance, the measurement errors of the input cell in this example can produce four approximately equivalent reduced cells within the tolerance range of ɛ_{1–3} = 0.1 nm^{−1} (details are listed in Table 2). Although the listed reduced cells exhibit slight differences and so are converted into different Bravais lattices, these Bravais lattices possess the same symmetryconstrained with parameters a = b = c = 5.4199 Å, α = β = γ = 90.00°. Moreover, the latticeplane indices corresponding to the reciprocal vectors (listed in Table 2), {220}, {313} and {202}, indicate that these four Niggli cells are indeed equivalent cells. Therefore, to eliminate the ambiguity of choosing the for the highsymmetry lattice, we choose each of these approximately equivalent cells in the `Reduced Cell List' box and check the latticeplane indices and the symmetryconstrained as well as other derived parameters in the `Results' box.
in this example is not a cubic phase but a triclinic structure. A similar case is often encountered during electron because of the large measurement uncertainties of the pattern, which often suffers from image distortions and the unreliable camera length (Mugnaioli

The other problem in practice is how to choose the correct ɛ_{d} and ɛ_{A} will be displayed in the `Unit Cell List'. Large values of ɛ_{d} and ɛ_{A} may cause the symmetry of the crystals to be overestimated. Generally, for the electron diffraction technique, the experimental error of the observed cell is larger than that determined in the Xray or neutron diffraction case; and the deviations of the edges and angles (ɛ_{d} and ɛ_{A}) of the determined from the standard cell will be up to 1–2 Å and 5–15°, respectively. To ensure that the correct is selected in the `Unit Cell List' box, we suggest choosing the cell with the highest possible symmetry and reasonable evaluated errors, while also checking the Bravais criteria, plane indices, zoneaxis indices, tilt angles and HOLZ ring (or symmetry in the HOLZ pattern). In this example, the `cF' cell with the highest symmetry and reasonable evaluated errors ( = −0.01, = −0.08, = −0.09 nm^{−1}, = 0.04, = 0.94, = 0.86° in Table S2 versus the measurement error of 0.05–0.1 nm^{−1} in length and 1° in angle of the input cell) is the best choice because the other cells, e.g. `hR', `oI' and `mC', belong to a subcell of `cF', although it possesses smaller figures of merit (FOM_{a} = 0.407 and FOM_{α} = 0.531 in `mC') than those (FOM_{a} = 0.992 and FOM_{α} = 1.020) of the cubic structure.
from the `Unit Cell List' box. After a has been chosen, it will be converted into 44 Bravaislattice unit cells based on the relationship between the and the and only those that match the symmetry of within the tolerance of4. Conclusions
We present a new DigitalMicrograph package to calculate the Niggli and to determine the Bravaislattice The package can tolerate large uncertainties of the observed cell while obtaining better numerical stability by introducing the factors ɛ_{1–3}. Some derived characteristic parameters including Bravaislatticeplane indices, zoneaxis indices, tilt angles, the radius of the HOLZ ring and the evaluated errors can be used to facilitate selection of the correct Niggli and determine the In order to make full use of these parameters to check or verify the determined cell, we suggest to record a highorder Laue pattern [simply focus the electron beam on the specimen and then record the pattern with a short camera length in the parallelbeam mode, or in the convergent beam electron diffraction (CBED) or nanobeam diffraction mode] when you record the electron diffraction pattern for reconstructing the reciprocal cell.
For convenience when choosing the correct
we summarize three typical cases of reciprocalcell reconstruction in electron diffraction analysis:(1) Singlepattern method. The measured parameters in the method are a 2D cell ( a^{*}, b^{*}, ) and the radius of the HOLZ ring, which can be compared with the derived value R_{1} in `HOLZ Ring'.
(2) Twopattern method. The measured parameters of the method are two 2D cells ( a^{*}, b^{*}, c^{*}, , ) and the tilt angle, which can be compared with the derived value in `Tilt Angles'. If the HOLZ rings of patterns are measured, they can be compared with the calculated parameters R_{1} and R_{2} in `HOLZ Ring'.
(3) Threepattern method. The measured parameters of the method are three 2D cells ( a^{*}, b^{*}, c^{*}, , , ). If the tilt angles are available, the derived values (, and in `Tilt Angles') can be compared with the measured ones; if the HOLZ rings of the patterns are measured, the calculated parameters ( R_{1}, R_{2} and R_{3} in `HOLZ Ring') can assist in choosing the Bravais cell. Moreover, the symmetry of the CBED pattern or the HOLZ pattern can be used to check the determined cell.
Note that the input cell constructed from electron diffraction patterns carries large uncertainties, and the determined e.g. the highorder Laue pattern and/or CBED. We suggest to improve the accuracy of the input cell, for example, by strictly calibrating the camera length of the transmission electron microscope and improving the accuracy of the diffraction spot measurement and the reciprocal cell reconstruction.
and the must be checked by other techniques,Supporting information
The package of Niggli reduction tools (*.gtk). DOI: https://doi.org/10.1107/S1600576721013212/te5085sup1.zip
A simple tutorial to use the package. DOI: https://doi.org/10.1107/S1600576721013212/te5085sup2.wmv
Supporting information. DOI: https://doi.org/10.1107/S1600576721013212/te5085sup3.pdf
Acknowledgements
We thank Minting Luo of the Institute of Process Engineering, Chinese Academy of Sciences, for useful discussions.
Funding information
This work was supported by the Fundamental Research Funds for the Central Universities (No. 2020QNPY101) and the Natural Science Foundation of China (grant No. 11974019; grant. No. 11774403).
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