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ePDF tools, a processing and analysis package of the atomic pair distribution function for electron diffraction
Computer Physics Communications 238 (2019) 295–301
Contents lists available at ScienceDirect
Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc
ePDF tools, a processing and analysis package of the atomic pair distribution function for electron diffraction✩ ∗
Honglong Shi , Minting Luo, Wenzhong Wang School of Science, Minzu University of China, 27 Zhong guancun South Avenue, Haidian district, Beijing, 100081, People’s Republic of China
article
info
Article history: Received 6 September 2017 Received in revised form 19 November 2018 Accepted 26 November 2018 Available online 9 January 2019 Keywords: Atomic pair distribution function Electron diffraction PDF SAED ePDF tools
a b s t r a c t We present a new processing & analysis package of the atomic pair distribution function (PDF) for the electron diffraction (ED) pattern based on the total scattering theory which has been broadly used in the X-ray or neutron diffraction to reveal the local atomic structures, but PDF analysis based on the electron diffraction is still rarely used for the lack of an efficient PDF tool. The program was written as a package of DigitalMicrograph, a very popular software to record and analyze the data of transmission electron microscopes (TEM) in the most of TEM laboratories. The azimuthal rotation-average projection algorithm was implemented to enhance the signal-noise ratio of the intensity profile. In order to obtain the reduced structure function, the cubic spline fitting method was utilized to subtract the background scattering. The real-time displayed PDF makes data normalization easier and more accurate, and results of coordination numbers and averaged bond angles can be extracted from calibrated PDFs in real time. Program summary Program Title: ePDF tools Program Files doi: http://dx.doi.org/10.17632/ff94wp42wd.1 Licensing provisions: GPLv3 Programming language: DigitalMicrograph scripting language Nature of problem: The total scattering technique has been widely used in X-ray or neutron diffraction, but for the electron diffraction it still lacks an efficient analysis tool for the atomic pair distribution function. Solution method: Intensity profile is obtained by azimuthal rotation-average projection of 2D SAED pattern, the background scattering is subtracted by the cubic spline method, data normalization is realtime performed based on the total scattering theory, the atomic pair distribution functions obtained by Fourier transform will be used to quantitative analysis. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The atomic pair distribution function (PDF) technique based on the total-scattering method [1–3] which allows both the Bragg and diffuse scattering to be analyzed together without bias, has been broadly used in the local structure analysis of materials, revealing variations of the local chemical or ligand condition. e.g., it reveals local atomic features of ZnTe and ZnSe around 2.5 Å for the ZnSe1−x Tex crystals [4]. It provides a direct evidence that the Ti and Zr atoms do not occupy equivalent octahedral sites expected from the crystallographic cubic perovskite structure for the BaTi1−x Zrx O3 relaxors [5]. To get an accurate PDF with a good resolution, the coherent kinematic scattering must be collected over a wide range of Q-values (diffraction vector), typically, 30∼50 ✩ This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect. com/science/journal/00104655). ∗ Corresponding author. E-mail address: [email protected] (H.L. Shi). https://doi.org/10.1016/j.cpc.2018.11.019 0010-4655/© 2018 Elsevier B.V. All rights reserved.
Å−1 . However, the available maximum Q in the traditional X-ray diffractometer is about 8 Å−1 for Mo-Ka tubes and 16 Å−1 for Ag tubes, respectively. Transmission electron microscope (TEM) is a powerful technique to perform microstructural analysis at atomic scale both in the real & reciprocal space by a combination of high-resolution TEM (HRTEM) and the selected-area electron diffraction (SAED) techniques [6,7]. But it is still not widely used for the PDF analysis based on the SAED pattern [8] as that of X-ray or neutron diffraction [9,10] although the high-quality SAED patterns with a broad diffraction range can be recorded within several seconds. The major obstacle is the strong multiple-scattering effect of electrons [11], which makes the scattering intensity distorted and limits the accurate PDF analysis based on the SAED pattern. Recently, the approximate kinematic electron scattering can be collected as the development of the precession technique, and it is possible to perform the accurate PDF analysis based on the electron diffraction if the efficient PDF tools are available. ePDF tools, as a package of DigitalMicrograph software [12] which has been broadly used to perform TEM data record, image
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processing and analysis in most of TEM laboratories around world, provide a robust quantitative PDF analysis based on the SAED pattern, with the following novelties: (i) Intensity profile is obtained by azimuthal rotation-average projection of the SAED pattern to enhance its signal-noise ratio, as well as to weaken the effect of the preferred orientation; (ii) The cubic spline fitting method is implemented to subtract the background scattering; (iii) The real-time displayed PDF make normalization of the scattering intensities easier and more accurate; (iv) The quantitative results of the calibrated PDFs can be realtime displayed. The paper is organized as follows. In Section 2, we present the fundamental theory of the total scattering and PDF methods. The description of the provided software is given in Section 3, and the test cases are shown in Section 4. The main features of the software and the possible extensions are summarized in Section 5.
(ii) The pair distribution function g(r) is non-negative.
{
G (r) =
−4πρ0 r , r < r0 G(r)/G(r0 ), r ≥ r0
(6)
2.3. Other PDFs and relations Generally, we will frequently encounter different pair distribution functions, e.g., the reduced pair distribution function G(r), the pair distribution function g(r) and the radial distribution function R(r). g(r) will be derived from G(r) and the atomic number density ρ0 , G (r )
g (r ) =
4πρ0 r
+1
(7)
And RDF can be calculated from g(r), R(r) = 4πρ0 r 2 g (r )
(8)
2.4. PDF analysis
2. Fundamentals of the total scattering and PDF methods 2.1. The total scattering theory
The coordination number can be obtained by integral of the radial distribution function over r from r1 to r2 ,
The reduced pair distribution function, G(r), can be obtained by Fourier transformation of the diffraction profile according to [8],
N =
G (r ) =
2
Qmax
∫
π
Q [S (Q ) − 1] sin (Qr ) dQ
Where F (Q ) = Q [S (Q ) − 1] is the reduced structure function, and the structure function S (Q ) can be obtained by the appropriate normalization of the background-subtracted diffraction intensity, Ic (Q ), N ∗Ic (Q ) − ⟨fe2 (Q )⟩ + c
⟨fe (Q )⟩2
(2)
⟨fe2 (Q )⟩ and ⟨fe (Q )⟩2 , are the composition-averaged electron form factors, can be calculated according to the relationship between fe (Q ) and fx (Q ), fe (Q ) =
me e2 z − fx (Q ) ( ) 2ℏ2 Q2
(3)
The parameter N in Eq. (2), is a normalization factor which depends on the electron form factor and the background-subtracted experimental intensity (Tran et al., 2017),
∫ Qmax Qmin
( ) ⟨fe2 Q ′ ⟩dQ ′
N = ∫Q max Qmin
Ic (Q ′ ) dQ ′
(4)
The parameter c in Eq. (2) is an extra constant to restrict S (Q ) approaching to unity when Q = 4π sin θ/λ → ∞. Fourier transform S(Q) to get G(r) involves an integral over Q from zero to infinity, but the finite measured Q range will result in the termination error in G(r) [2,13]. Therefore, a Gaussian function is brought in ePDF tools to dampen F(Q) with increasing Q [2], 2 /2
G (Q ) = e−(DQ )
r2
R(r)dr
(9)
r1
And the average bond angle can be derived by r1 to r2 , (1)
Qmin
S (Q ) = 1 +
∫
(5)
Where the parameter D is an user-defined factor (damping factor, 0.08 by default), corresponding to ‘Qdamp ’ factor defined in PDFgui [14]. 2.2. PDF calibrations In order to obtain the accurate PDF, it should be calibrated as follows. (i) The reduced G(r) behaves like −4πρ0 r as r → 0, where ρ0 is the average atomic number density.
θ = 2 arcsin(r2 /2r1 )
(10)
3. Description of the package 3.1. Distribution and installation The package (Install files.rar) is available from the website: https://github.com/hlshi527214/ePDF-tools/tree/master/Instal ler files Unzip it to get four associated files: (i) ePDF Tools.gtk: a compiled package of ePDF tools. (ii) Scattering factor.txt: a text file for the scattering factors of electrons. (iii) Example pattern.dm3: an example pattern for test. (iv) ReadMe.txt: a text file to briefly describe ePDF tools. (v) Demo video of ePDF tools.wmv: a demonstration video for the use of ePDF tools. The following two steps are required to install ePDF tools in DigitalMicrograph software: (i) Copy the file ePDF Tools.gtk to ...\Gatan\DigitalMicrograph\ PlugIns, a new menu ePDF Tools will be built on the menu bar. Where DigitalMicrograph software (Gatan Microscopy Suite Software) can be available at http://www.gatan.com/products/temanalysis/gatan-microscopy-suite-software. (ii) Launch DigitalMicrograph, and Click the item ePDF Tools/ Scattering factors and then select the file Scattering factor.txt for importing the scattering factors for electrons. And now, only click the item ePDF Tools/ePDF calculator 2017 to enter the graphical user interface, as shown in Fig. 1(b). 3.2. Input files and parameters (i) SAED pattern: it can process the native DM files (dm3) or other common gray scale images (JPG, GIF, TIF, BMP etc.). (ii) Input parameters: the compositions (Symb. and At. %) and the acceleration voltage (kV) will be used to calculate the curves of the electron form factors for the given compositions. The material density (Dens., g/cm3 ) will be used to perform PDF calibration. And the other factors, r_max and Size define the size of PDFs. To get an accurate PDF we must set the correct parameters of the composition and the material density, which will be discussed in Section 4.
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Fig. 1. (a) Flowchart of the PDF analysis in ePDF tools, the majority steps are: (i) profile preparations, (ii) background subtractions, (iii) data normalization and other corrections, (iv) G(r) calibrations and quantitative analysis. (b) The graphical user interface of ePDF tools.
3.3. Software overview ePDF tools, as a package of DigitalMicrograph software, can process electron diffraction patterns included native DigitalMicrograph files (dm3), as well as all common gray scale images (JPG, GIF, TIF, BMP etc.). The operation of the package is controlled via a graphical user interface (Fig. 1(b)), where the composition box is used to set parameters, the PDF tools box is utilized to perform PDF processing and analysis, and the Image processing box is used to prepare the intensity profile and data saving. A complete process of PDF analysis based on the electron diffraction pattern in ePDF tools is illustrated in Fig. 1(a). Firstly, define the center of the SAED pattern and project it as 1D intensity profile I (Q ), and then subtract the background scattering to get the coherent part of the scattering intensity Ic (Q ). Next, perform data normalization and some corrections to get the reduced structure function F (Q ), and then yield the raw reduced pair distribution function G (r ) by Fourier transform F (Q ). Finally, calibrate and quantitatively analyze the pair distribution functions. (i) The composition box ‘Symb.’ and ‘At%’ define the symbols of elements and the corresponding atom ratios. For example, the composition is CaCO3 , so Ca, 1; C, 1; O, 3 should be entered in these two columns. From these, the electron form factors of ⟨fe2 (Q )⟩ and ⟨fe (Q )⟩2 will be calculated. ‘kV’ is the acceleration voltage of TEM, 200 kV by default. ‘Dens.’ is the material density (g/cm3 ) which will be used to perform PDF calibration. ‘r_max’ and ‘Size’ define the size of PDFs, e.g., r_max=20 Å and Size=2000 pixels. (ii) The PDF tools box ‘Calc.’ is used to calculate electron form factors according to Eq. (3). ‘Scatt.’ adds curves of electron form factors on the intensity profile. ‘BG’ subtracts the background scattering based on the cubic spline fitting method. ‘F(Q)’ is used to perform data normalization and some correction including Gaussian damping and data smoothing according to Eqs. (2), (4) and (5). ‘G(r)’ yields the reduced G(r) by Fourier transform F(Q) according to Eq. (1). ‘Calib.’ calibrates the reduced G(r) based on the specified material density according to Formula (6). ‘Data’ is used to perform quantitative PDF analysis according to Eqs. (9)–(10).
‘Prefer.’ sets the preference parameters. Details of other buttons will be listed in Readme.txt. (iii) The image processing box ‘Center’ finds the center of the transmitted spot in the SAED pattern. ‘Profile’ yields the intensity profile of SAED pattern by azimuthal rotation-average projection. ‘Calib.’ calibrates the intensity profile by a known diffraction peak. ‘SaveGr’ saves the reduced G(r) as PDFgui format. ‘SaveXY ’ exports PDFs as two columns data. And details of other buttons will be listed in Readme.txt. 4. Illustrative examples We focus now on the use of ePDF tools as an example of polycrystalline gold (Example pattern.dm3), details are shown in demo video. The material density is 19.3 g/cm3 and the acceleration voltage is 200 kV. The size of the expected PDF is 2000 pixels, and the scale is 20 Å. All of these parameters should be set in composition box. 4.1. The profile preparation (i) Find the center of the transmitted spot. Move the cursor to the desired position and then press the space bar to define the initial center (x0 , y0 ) after clicking Center button in Image processing box, it will automatically refine it based on the 2D Gaussian fitting algorithm. In this example, the refined center is (718.942, 480.212), as shown in Fig. 2(a). (ii) And then, 2D SAED pattern will be transformed into 1D intensity profile based on the azimuthal rotation-average algorithm after clicking the ‘Profile’ button. If the camera length of TEM was not well calibrated using the standard specimen, or the processed pattern is an external image without calibration, it should be carefully calibrated by a known diffraction peak, e.g., 2.355 Å for (111) peak of the gold crystal. Carefully drag an ROI (region of interest) across the desired peak and then hit the ‘Calib.’ button in the image processing box to calibrate it, a dialog is prompted to set the d-spacing of the selected peak. In this example, we input 2.355 and the center of the ROI will be calibrated by the entered value, as shown in Fig. 2(b). (iii) Calculate and add curves of the electron form factors on the intensity profile by subsequently clicking ‘Calc.’ and ‘Scatt.’ buttons.
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Fig. 2. Profile preparations: (a) find the center of the transmitted spot, (b) the obtained intensity profile and the profile calibration.
Fig. 3. Snap captures of (a) background subtractions, (b) data normalization and some corrections of F(Q).
4.2. The background subtraction
4.4. G(r) calibrations
Drag an ROI at the first sampling point and then click ‘BG’ button to fit the sampling points (or ROIs) based on the cubic spline algorithm. Curves of the background, I(Q) and F(Q) will be real-time updated when you add, move, change, or delete ROIs, as seen in Fig. 3(a). Generally, for crystalline materials the sampling points are just located at the troughs between diffraction peaks. If the fitted background curve looks good or the displayed parameters N and c converge to a constant when ROIs are subsequently added to the profile, delete the ‘BG’ ROI for the next step. In this example, the refined parameters are N=0.160582 and c=2.40094, respectively.
The reduced pair distribution function, the raw G(r) will be obtained by clicking ‘G(r)’ button when the ‘Damped’ F(Q) looks good. Note that intensities of the obtained G(r) in this step is not so accurate, which depends on the parameters of N, c and the damping factor. So, it should be calibrated before quantitative PDF analysis according to Eq. (6). Calibration of the reduced G(r) involved in ePDF tools is that the left side of the user-defined ROI spanning the first physical density peak will linearly go through zero, and the right side (r = r0 ) of ROI defines G (r0 ) = −4πρ0 r0 by which G(r) curve is re-scaled when you press down ‘Calib.’ button in the PDF tools box, as shown in Fig. 4(a). In this example, the ROI is ranging from 2.56 to 3.48 Å, and intensities of G(r) at r0 = 3.48 Å before and after calibration are −15.1618 and −2.58051 Å−2 , respectively. When you delete the ROI, the calibrated PDFs will be updated.
4.3. The data normalization and corrections Click ‘F(Q)’ button after background subtraction, the intensity profile with ROIs labeled as ‘N’, ‘c’, ‘Damp’ and ‘Smooth’ is automatically updated, as shown in Fig. 3(b). Move ROIs to adjust parameters of N, c, D and the smoothing parameter, the ‘Damped’ F(Q) slice and the ‘Live G(r)’ profile will be real-time displayed. A good normalization will be that F(Q) is smoothly decayed out and fluctuates around zero as increasing Q. If the examined profile behaves bad signal-noise ratio and there will be some sparks at high-Q region, you can move or change the ‘Smooth’ ROI to filter the intensity profile over the right part of the ROI in which the ROI width defines the smooth strength. If F(Q) at high-Q region shows strong oscillations due to the distorted scattering, we can move or change the ‘Damp’ ROI to make it falling out as increasing Q. In this example, the ‘Damped’ F(Q) curve falls out with increasing Q and it oscillates around zero as Q → Qmax when parameters of N, c and D are 0.164732, 2.15733 and 0.08, respectively.
4.5. The PDF analysis Any one of three types of calibrated PDFs, g(r), G(r) or R(r), can be used to perform quantitative analysis in ePDF tools, the results (e.g., positions r1 and r2 , the averaged bond angle, and the coordination number) will be real-time displayed as changing or moving the ROI after pressing down ‘Data’ button, as illustrated in Fig. 4(b). In this example, the first peak located at r=2.903 Å corresponds to the bond length of the nearest neighboring atoms from the cubic corner to the face center. Therefore, √ the lattice parameter of the examined specimen is 2.903 Å × 2 = 4.105 Å, which is well consistent with the gold crystal. The measured coordination number from 2.56 to 3.48 Å is 11.51 since the first shell of the facecentered cubic structure contains 12 atoms. The calculated bond
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Fig. 4. Snap captures of (a) PDF calibrations and (b) the quantitative PDF analysis.
Fig. 5. Quantitative PDF analysis based on the SAED pattern of anatase crystallites: (a) Intensity profile and the background curve of the inset SAED pattern, (b) the extracted F(Q), and (c) the anatase structure model fit to the resulting PDF curve.
angle between the first two peaks is 88.1◦ , just corresponding to the averaged angle among (100) and (110) lattice planes of the gold crystal. Details are listed in Table 1, it indicates that the calibrated PDF from the polycrystalline gold is well agreement with the gold crystal. In addition, more crystallographic information can be extracted from the obtained PDFs by using external software such as PDFgui [14], RMCProfile [15], or RMC++ [16]. Fig. 5 shows an example of performing the quantitative PDF analysis of polycrystalline anatase in PDFgui. The scattering type is X-ray (PDF obtained from electron diffraction is approximately equivalent to that of X-ray diffraction), the fitting range is 3.3∼20 Å−1 for avoiding the correlated motion of neighboring atoms. Qmax = 15.26 Å−1 , Qdamp = 0.102, and Qbroad = 0.123. The refined parameters are scale factor, Qbroad , unit cell, atomic coordination of the oxygen atom zO , and isotropic thermal displacements of atoms, the agreement factor Rw is 0.195, as shown in Fig. 5(c). Details of the refined parameters are listed in Table 2. The refined unit cell and the oxygen site are a×c=3.7971(39)×9.517(13) Å, and zO = 0.208(1), respectively, which are well consistent with two reference structures (ICSD #9852 and 9853). And the refined isotropic thermal displacement factors of atoms are reasonable, although some reports [17] suggested the thermal displacement factors extracted from electron
Table 1 Results of PDF analysis based on the SAED pattern from the polycrystal gold: The calculated peak positions (rCalc ) and coordination numbers (NCalc. ) of the standard gold specimen (a = 4.0786 Å), and the measured peak positions (rExp. ) and coordination numbers (NExp. ) within sub-sections. Where ‘–’ shows the unidentified peak from the measured RDF. rCalc. (Å)
rExp. (Å)
NCalc.
NExp.
Integral range (Å)
2.884 4.079 4.995 5.768 6.449 7.064 7.630 8.157 8.652 9.120 9.565 9.9905 10.3984
2.903 4.043 4.983 5.807 6.430 – 7.637 – 8.747 – – – 10.3533
12 6 24 12 24 8 48 6 36 24 24 24 72
12.07 6.02 23.42 12.94 27.44
2.520∼3.480 3.480∼4.453 4.453∼5.453 5.453∼6.067 6.073∼7.033
55.13
7.033∼8.173
83.22
8.173∼9.673
99.34
9.667∼10.860
diffraction will be two or three times larger than those of neutron experiments, indicating that ePDF tools is an effective package to extract quantitative PDFs from SAED patterns.
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H.L. Shi, M.T. Luo and W.Z. Wang / Computer Physics Communications 238 (2019) 295–301 Table 2 Comparison of the refined parameters of the PDF extracted from ePDF tools with two typical ICSD structures.
a
Parameters
Refined
Ref. (9852-ICSD)
Ref. (9853-ICSD)
Unit cell
a = b = 3.7971(39) Å c = 9.517(13) Å
a = b = 3.7842(13) Å c = 9.5146(15) Å
a = b = 3.7892(4) Å c = 9.537(1) Å
zO
0.208(1)
0.2081(2)
0.2079(2)
UTi
0.0100(12)
0.0049(8) (B = 0.39)a
0.0081(8) (B = 0.64)a
UO
0.0103(18)
0.0077(11) (B = 0.61)a
0.0120(11) (B = 0.95)a
Note: Isotropic thermal displacement factors of atoms are calculated by Debye–Waller factor U = B/8π.2
Table A.1 Software metadata. Nr
(executable) Software metadata description
Please fill in this column
S1
Current software version
v1.0
S2
Permanent link to executables of this version
https://github.com/hlshi527214/ePDF-tools/tree/master/Installer files
S3
Legal Software License
GNU Public License GPLv3
S4
Computing platform / Operating System
Microsoft Windows
S5
Installation requirements & dependencies
Any computer with Gatan Microscopy Suite software: http://www.gatan.com/ products/tem-analysis/gatan-microscopy-suite-software
S6
If available Link to user manual if formally published include a reference to the publication in the reference list
https://github.com/hlshi527214/ePDF-tools/tree/master/Example
S6
Support email for questions
[email protected]
Nr
Code metadata description
Please fill in this column
C1
Current Code version
v1.0
C2
Permanent link to code / repository used of this code version
https://github.com/hlshi527214/ePDF-tools/tree/master/Sourcecode
C3
Legal Code License
GNU Public License GPLv3
C4
Code Versioning system used
DigitalMicrograph
C5
Software Code Language used
DigitalMicrograph script language
C6
Compilation requirements, Operating environments & dependencies
Any computer with Gatan Microscopy Suite software: http://www.gatan.com/ products/tem-analysis/gatan-microscopy-suite-software
C7
If available Link to developer documentation / manual
https://github.com/hlshi527214/ePDF-tools/tree/master/Example
C8
Support email for questions
[email protected]
Table A.2 Code metadata.
5. Conclusions
Appendix A. Required metadata
ePDF tools are a versatile package to perform PDF analysis for electron diffraction patterns including the profile preparation, data normalization and corrections of F(Q), and the quantitative analysis of PDFs. The obtained PDFs will be further imported into PDFgui, RMCProfile or RMC++ to extract the other local structures by the PDF refinement or the reverse Monte Carlo simulation. To get an accurate PDF from SAED patterns, keep your mind in the TEM experiments that: (i) Select a thin sample or using the precession device to collect the kinematic electron scattering since the cubic spline fitting method cannot filter the dynamic scatter. (ii) Choose a small camera length to collect scattering signals over large Q range. (iii) Choose an exposure time as long as possible to collect the high-quality pattern.
A.1. Current executable software version
Acknowledgments This work was supported by the National Natural Science Foundation of China [Grant Number 11604394], Undergraduate Research and Innovative Undertaking Program of Beijing [Grant Numbers BEIJ2016110008, BEIJ2015110022].
See Table A.1. A.2. Current code version See Table A.2. Appendix B. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.cpc.2018.11.019. References [1] B.E. Warren, X-Ray Diffraction, Dover Publications, 1990. [2] T. Egami, S.J.L. Billinge, Underneath the Bragg Peaks Structural Analysis of Complex Materials, Pergamon, 2003. [3] P. Juhás, T. Davis, C.L. Farrow, S.J.L. Billinge, J. Appl. Crystallogr. 46 (2012) 560–566. [4] P.F. Peterson, T. Proffen, I.K. Jeong, S.J.L. Billinge, K.S. Choi, M.G. Kanatzidis, P.G. Radaelli, Phys. Rev. B 63 (2001) 165211. [5] C. Laulhé, F. Hippert, R. Bellissent, A. Simon, G.J. Cuello, Phys. Rev. B 79 (2009) 7715–7722.
H.L. Shi, M.T. Luo and W.Z. Wang / Computer Physics Communications 238 (2019) 295–301 [6] X. Zou, Y. Sukharev, S. Hovmöller, Ultramicroscopy 52 (1993) 436–444. [7] J.H. Chen, H.W. Zandbergen, D.V. Dyck, Ultramicroscopy 98 (2004) 81. [8] M. Abeykoon, C.D. Malliakas, P. Juhas, E.S. Bozin, M.G. Kanatzidis, S.J.L. Billinge, Z. Kristallogr. 227 (2012) 248–256. [9] F. Li, J.S. Lannin, Phys. Rev. Lett. 65 (1990) 1905–1908. [10] S.J.L. Billinge, Z. Kristallograp. Int. J. Struct. Phys. Chem. Aspects Crystal. Mater. 219 (2004) 117–121. [11] J.M. Cowley, Diffraction Physics, third ed., 1995. [12] DigitalMicrograph, Gatan Inc., Pleasanton, United States, 2017.
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[13] P.F. Peterson, E.S. Bozin, T. Proffen, S.J.L. Billinge, J. Appl. Crystallogr. 36 (2003) 53–64. [14] C.L. Farrow, P. Juhas, J.W. Liu, D. Bryndin, E.S. Božin, J. Bloch, T. Proffen, S.J. Billinge, J. Phys. Condens. Matter Inst. Phys. J. 19 (2007) 335219. [15] M.G. Tucker, D.A. Keen, M.T. Dove, A.L. Goodwin, Q. Hui, J. Phys. Condens. Matter Inst. Phys. J. 19 (2007) 335218. [16] G. Evrard, L. Pusztai, J. Phys. Condens. Matter 17 (2005) S1. [17] A.M.M. Abeykoon, H. Hu, L. Wu, Y. Zhu, S.J.L. Billinge, J. Appl. Crystallogr. 48 (2015) 244–251.
Contents lists available at ScienceDirect
Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc
ePDF tools, a processing and analysis package of the atomic pair distribution function for electron diffraction✩ ∗
Honglong Shi , Minting Luo, Wenzhong Wang School of Science, Minzu University of China, 27 Zhong guancun South Avenue, Haidian district, Beijing, 100081, People’s Republic of China
article
info
Article history: Received 6 September 2017 Received in revised form 19 November 2018 Accepted 26 November 2018 Available online 9 January 2019 Keywords: Atomic pair distribution function Electron diffraction PDF SAED ePDF tools
a b s t r a c t We present a new processing & analysis package of the atomic pair distribution function (PDF) for the electron diffraction (ED) pattern based on the total scattering theory which has been broadly used in the X-ray or neutron diffraction to reveal the local atomic structures, but PDF analysis based on the electron diffraction is still rarely used for the lack of an efficient PDF tool. The program was written as a package of DigitalMicrograph, a very popular software to record and analyze the data of transmission electron microscopes (TEM) in the most of TEM laboratories. The azimuthal rotation-average projection algorithm was implemented to enhance the signal-noise ratio of the intensity profile. In order to obtain the reduced structure function, the cubic spline fitting method was utilized to subtract the background scattering. The real-time displayed PDF makes data normalization easier and more accurate, and results of coordination numbers and averaged bond angles can be extracted from calibrated PDFs in real time. Program summary Program Title: ePDF tools Program Files doi: http://dx.doi.org/10.17632/ff94wp42wd.1 Licensing provisions: GPLv3 Programming language: DigitalMicrograph scripting language Nature of problem: The total scattering technique has been widely used in X-ray or neutron diffraction, but for the electron diffraction it still lacks an efficient analysis tool for the atomic pair distribution function. Solution method: Intensity profile is obtained by azimuthal rotation-average projection of 2D SAED pattern, the background scattering is subtracted by the cubic spline method, data normalization is realtime performed based on the total scattering theory, the atomic pair distribution functions obtained by Fourier transform will be used to quantitative analysis. © 2018 Elsevier B.V. All rights reserved.
1. Introduction The atomic pair distribution function (PDF) technique based on the total-scattering method [1–3] which allows both the Bragg and diffuse scattering to be analyzed together without bias, has been broadly used in the local structure analysis of materials, revealing variations of the local chemical or ligand condition. e.g., it reveals local atomic features of ZnTe and ZnSe around 2.5 Å for the ZnSe1−x Tex crystals [4]. It provides a direct evidence that the Ti and Zr atoms do not occupy equivalent octahedral sites expected from the crystallographic cubic perovskite structure for the BaTi1−x Zrx O3 relaxors [5]. To get an accurate PDF with a good resolution, the coherent kinematic scattering must be collected over a wide range of Q-values (diffraction vector), typically, 30∼50 ✩ This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect. com/science/journal/00104655). ∗ Corresponding author. E-mail address: [email protected] (H.L. Shi). https://doi.org/10.1016/j.cpc.2018.11.019 0010-4655/© 2018 Elsevier B.V. All rights reserved.
Å−1 . However, the available maximum Q in the traditional X-ray diffractometer is about 8 Å−1 for Mo-Ka tubes and 16 Å−1 for Ag tubes, respectively. Transmission electron microscope (TEM) is a powerful technique to perform microstructural analysis at atomic scale both in the real & reciprocal space by a combination of high-resolution TEM (HRTEM) and the selected-area electron diffraction (SAED) techniques [6,7]. But it is still not widely used for the PDF analysis based on the SAED pattern [8] as that of X-ray or neutron diffraction [9,10] although the high-quality SAED patterns with a broad diffraction range can be recorded within several seconds. The major obstacle is the strong multiple-scattering effect of electrons [11], which makes the scattering intensity distorted and limits the accurate PDF analysis based on the SAED pattern. Recently, the approximate kinematic electron scattering can be collected as the development of the precession technique, and it is possible to perform the accurate PDF analysis based on the electron diffraction if the efficient PDF tools are available. ePDF tools, as a package of DigitalMicrograph software [12] which has been broadly used to perform TEM data record, image
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processing and analysis in most of TEM laboratories around world, provide a robust quantitative PDF analysis based on the SAED pattern, with the following novelties: (i) Intensity profile is obtained by azimuthal rotation-average projection of the SAED pattern to enhance its signal-noise ratio, as well as to weaken the effect of the preferred orientation; (ii) The cubic spline fitting method is implemented to subtract the background scattering; (iii) The real-time displayed PDF make normalization of the scattering intensities easier and more accurate; (iv) The quantitative results of the calibrated PDFs can be realtime displayed. The paper is organized as follows. In Section 2, we present the fundamental theory of the total scattering and PDF methods. The description of the provided software is given in Section 3, and the test cases are shown in Section 4. The main features of the software and the possible extensions are summarized in Section 5.
(ii) The pair distribution function g(r) is non-negative.
{
G (r) =
−4πρ0 r , r < r0 G(r)/G(r0 ), r ≥ r0
(6)
2.3. Other PDFs and relations Generally, we will frequently encounter different pair distribution functions, e.g., the reduced pair distribution function G(r), the pair distribution function g(r) and the radial distribution function R(r). g(r) will be derived from G(r) and the atomic number density ρ0 , G (r )
g (r ) =
4πρ0 r
+1
(7)
And RDF can be calculated from g(r), R(r) = 4πρ0 r 2 g (r )
(8)
2.4. PDF analysis
2. Fundamentals of the total scattering and PDF methods 2.1. The total scattering theory
The coordination number can be obtained by integral of the radial distribution function over r from r1 to r2 ,
The reduced pair distribution function, G(r), can be obtained by Fourier transformation of the diffraction profile according to [8],
N =
G (r ) =
2
Qmax
∫
π
Q [S (Q ) − 1] sin (Qr ) dQ
Where F (Q ) = Q [S (Q ) − 1] is the reduced structure function, and the structure function S (Q ) can be obtained by the appropriate normalization of the background-subtracted diffraction intensity, Ic (Q ), N ∗Ic (Q ) − ⟨fe2 (Q )⟩ + c
⟨fe (Q )⟩2
(2)
⟨fe2 (Q )⟩ and ⟨fe (Q )⟩2 , are the composition-averaged electron form factors, can be calculated according to the relationship between fe (Q ) and fx (Q ), fe (Q ) =
me e2 z − fx (Q ) ( ) 2ℏ2 Q2
(3)
The parameter N in Eq. (2), is a normalization factor which depends on the electron form factor and the background-subtracted experimental intensity (Tran et al., 2017),
∫ Qmax Qmin
( ) ⟨fe2 Q ′ ⟩dQ ′
N = ∫Q max Qmin
Ic (Q ′ ) dQ ′
(4)
The parameter c in Eq. (2) is an extra constant to restrict S (Q ) approaching to unity when Q = 4π sin θ/λ → ∞. Fourier transform S(Q) to get G(r) involves an integral over Q from zero to infinity, but the finite measured Q range will result in the termination error in G(r) [2,13]. Therefore, a Gaussian function is brought in ePDF tools to dampen F(Q) with increasing Q [2], 2 /2
G (Q ) = e−(DQ )
r2
R(r)dr
(9)
r1
And the average bond angle can be derived by r1 to r2 , (1)
Qmin
S (Q ) = 1 +
∫
(5)
Where the parameter D is an user-defined factor (damping factor, 0.08 by default), corresponding to ‘Qdamp ’ factor defined in PDFgui [14]. 2.2. PDF calibrations In order to obtain the accurate PDF, it should be calibrated as follows. (i) The reduced G(r) behaves like −4πρ0 r as r → 0, where ρ0 is the average atomic number density.
θ = 2 arcsin(r2 /2r1 )
(10)
3. Description of the package 3.1. Distribution and installation The package (Install files.rar) is available from the website: https://github.com/hlshi527214/ePDF-tools/tree/master/Instal ler files Unzip it to get four associated files: (i) ePDF Tools.gtk: a compiled package of ePDF tools. (ii) Scattering factor.txt: a text file for the scattering factors of electrons. (iii) Example pattern.dm3: an example pattern for test. (iv) ReadMe.txt: a text file to briefly describe ePDF tools. (v) Demo video of ePDF tools.wmv: a demonstration video for the use of ePDF tools. The following two steps are required to install ePDF tools in DigitalMicrograph software: (i) Copy the file ePDF Tools.gtk to ...\Gatan\DigitalMicrograph\ PlugIns, a new menu ePDF Tools will be built on the menu bar. Where DigitalMicrograph software (Gatan Microscopy Suite Software) can be available at http://www.gatan.com/products/temanalysis/gatan-microscopy-suite-software. (ii) Launch DigitalMicrograph, and Click the item ePDF Tools/ Scattering factors and then select the file Scattering factor.txt for importing the scattering factors for electrons. And now, only click the item ePDF Tools/ePDF calculator 2017 to enter the graphical user interface, as shown in Fig. 1(b). 3.2. Input files and parameters (i) SAED pattern: it can process the native DM files (dm3) or other common gray scale images (JPG, GIF, TIF, BMP etc.). (ii) Input parameters: the compositions (Symb. and At. %) and the acceleration voltage (kV) will be used to calculate the curves of the electron form factors for the given compositions. The material density (Dens., g/cm3 ) will be used to perform PDF calibration. And the other factors, r_max and Size define the size of PDFs. To get an accurate PDF we must set the correct parameters of the composition and the material density, which will be discussed in Section 4.
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Fig. 1. (a) Flowchart of the PDF analysis in ePDF tools, the majority steps are: (i) profile preparations, (ii) background subtractions, (iii) data normalization and other corrections, (iv) G(r) calibrations and quantitative analysis. (b) The graphical user interface of ePDF tools.
3.3. Software overview ePDF tools, as a package of DigitalMicrograph software, can process electron diffraction patterns included native DigitalMicrograph files (dm3), as well as all common gray scale images (JPG, GIF, TIF, BMP etc.). The operation of the package is controlled via a graphical user interface (Fig. 1(b)), where the composition box is used to set parameters, the PDF tools box is utilized to perform PDF processing and analysis, and the Image processing box is used to prepare the intensity profile and data saving. A complete process of PDF analysis based on the electron diffraction pattern in ePDF tools is illustrated in Fig. 1(a). Firstly, define the center of the SAED pattern and project it as 1D intensity profile I (Q ), and then subtract the background scattering to get the coherent part of the scattering intensity Ic (Q ). Next, perform data normalization and some corrections to get the reduced structure function F (Q ), and then yield the raw reduced pair distribution function G (r ) by Fourier transform F (Q ). Finally, calibrate and quantitatively analyze the pair distribution functions. (i) The composition box ‘Symb.’ and ‘At%’ define the symbols of elements and the corresponding atom ratios. For example, the composition is CaCO3 , so Ca, 1; C, 1; O, 3 should be entered in these two columns. From these, the electron form factors of ⟨fe2 (Q )⟩ and ⟨fe (Q )⟩2 will be calculated. ‘kV’ is the acceleration voltage of TEM, 200 kV by default. ‘Dens.’ is the material density (g/cm3 ) which will be used to perform PDF calibration. ‘r_max’ and ‘Size’ define the size of PDFs, e.g., r_max=20 Å and Size=2000 pixels. (ii) The PDF tools box ‘Calc.’ is used to calculate electron form factors according to Eq. (3). ‘Scatt.’ adds curves of electron form factors on the intensity profile. ‘BG’ subtracts the background scattering based on the cubic spline fitting method. ‘F(Q)’ is used to perform data normalization and some correction including Gaussian damping and data smoothing according to Eqs. (2), (4) and (5). ‘G(r)’ yields the reduced G(r) by Fourier transform F(Q) according to Eq. (1). ‘Calib.’ calibrates the reduced G(r) based on the specified material density according to Formula (6). ‘Data’ is used to perform quantitative PDF analysis according to Eqs. (9)–(10).
‘Prefer.’ sets the preference parameters. Details of other buttons will be listed in Readme.txt. (iii) The image processing box ‘Center’ finds the center of the transmitted spot in the SAED pattern. ‘Profile’ yields the intensity profile of SAED pattern by azimuthal rotation-average projection. ‘Calib.’ calibrates the intensity profile by a known diffraction peak. ‘SaveGr’ saves the reduced G(r) as PDFgui format. ‘SaveXY ’ exports PDFs as two columns data. And details of other buttons will be listed in Readme.txt. 4. Illustrative examples We focus now on the use of ePDF tools as an example of polycrystalline gold (Example pattern.dm3), details are shown in demo video. The material density is 19.3 g/cm3 and the acceleration voltage is 200 kV. The size of the expected PDF is 2000 pixels, and the scale is 20 Å. All of these parameters should be set in composition box. 4.1. The profile preparation (i) Find the center of the transmitted spot. Move the cursor to the desired position and then press the space bar to define the initial center (x0 , y0 ) after clicking Center button in Image processing box, it will automatically refine it based on the 2D Gaussian fitting algorithm. In this example, the refined center is (718.942, 480.212), as shown in Fig. 2(a). (ii) And then, 2D SAED pattern will be transformed into 1D intensity profile based on the azimuthal rotation-average algorithm after clicking the ‘Profile’ button. If the camera length of TEM was not well calibrated using the standard specimen, or the processed pattern is an external image without calibration, it should be carefully calibrated by a known diffraction peak, e.g., 2.355 Å for (111) peak of the gold crystal. Carefully drag an ROI (region of interest) across the desired peak and then hit the ‘Calib.’ button in the image processing box to calibrate it, a dialog is prompted to set the d-spacing of the selected peak. In this example, we input 2.355 and the center of the ROI will be calibrated by the entered value, as shown in Fig. 2(b). (iii) Calculate and add curves of the electron form factors on the intensity profile by subsequently clicking ‘Calc.’ and ‘Scatt.’ buttons.
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Fig. 2. Profile preparations: (a) find the center of the transmitted spot, (b) the obtained intensity profile and the profile calibration.
Fig. 3. Snap captures of (a) background subtractions, (b) data normalization and some corrections of F(Q).
4.2. The background subtraction
4.4. G(r) calibrations
Drag an ROI at the first sampling point and then click ‘BG’ button to fit the sampling points (or ROIs) based on the cubic spline algorithm. Curves of the background, I(Q) and F(Q) will be real-time updated when you add, move, change, or delete ROIs, as seen in Fig. 3(a). Generally, for crystalline materials the sampling points are just located at the troughs between diffraction peaks. If the fitted background curve looks good or the displayed parameters N and c converge to a constant when ROIs are subsequently added to the profile, delete the ‘BG’ ROI for the next step. In this example, the refined parameters are N=0.160582 and c=2.40094, respectively.
The reduced pair distribution function, the raw G(r) will be obtained by clicking ‘G(r)’ button when the ‘Damped’ F(Q) looks good. Note that intensities of the obtained G(r) in this step is not so accurate, which depends on the parameters of N, c and the damping factor. So, it should be calibrated before quantitative PDF analysis according to Eq. (6). Calibration of the reduced G(r) involved in ePDF tools is that the left side of the user-defined ROI spanning the first physical density peak will linearly go through zero, and the right side (r = r0 ) of ROI defines G (r0 ) = −4πρ0 r0 by which G(r) curve is re-scaled when you press down ‘Calib.’ button in the PDF tools box, as shown in Fig. 4(a). In this example, the ROI is ranging from 2.56 to 3.48 Å, and intensities of G(r) at r0 = 3.48 Å before and after calibration are −15.1618 and −2.58051 Å−2 , respectively. When you delete the ROI, the calibrated PDFs will be updated.
4.3. The data normalization and corrections Click ‘F(Q)’ button after background subtraction, the intensity profile with ROIs labeled as ‘N’, ‘c’, ‘Damp’ and ‘Smooth’ is automatically updated, as shown in Fig. 3(b). Move ROIs to adjust parameters of N, c, D and the smoothing parameter, the ‘Damped’ F(Q) slice and the ‘Live G(r)’ profile will be real-time displayed. A good normalization will be that F(Q) is smoothly decayed out and fluctuates around zero as increasing Q. If the examined profile behaves bad signal-noise ratio and there will be some sparks at high-Q region, you can move or change the ‘Smooth’ ROI to filter the intensity profile over the right part of the ROI in which the ROI width defines the smooth strength. If F(Q) at high-Q region shows strong oscillations due to the distorted scattering, we can move or change the ‘Damp’ ROI to make it falling out as increasing Q. In this example, the ‘Damped’ F(Q) curve falls out with increasing Q and it oscillates around zero as Q → Qmax when parameters of N, c and D are 0.164732, 2.15733 and 0.08, respectively.
4.5. The PDF analysis Any one of three types of calibrated PDFs, g(r), G(r) or R(r), can be used to perform quantitative analysis in ePDF tools, the results (e.g., positions r1 and r2 , the averaged bond angle, and the coordination number) will be real-time displayed as changing or moving the ROI after pressing down ‘Data’ button, as illustrated in Fig. 4(b). In this example, the first peak located at r=2.903 Å corresponds to the bond length of the nearest neighboring atoms from the cubic corner to the face center. Therefore, √ the lattice parameter of the examined specimen is 2.903 Å × 2 = 4.105 Å, which is well consistent with the gold crystal. The measured coordination number from 2.56 to 3.48 Å is 11.51 since the first shell of the facecentered cubic structure contains 12 atoms. The calculated bond
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299
Fig. 4. Snap captures of (a) PDF calibrations and (b) the quantitative PDF analysis.
Fig. 5. Quantitative PDF analysis based on the SAED pattern of anatase crystallites: (a) Intensity profile and the background curve of the inset SAED pattern, (b) the extracted F(Q), and (c) the anatase structure model fit to the resulting PDF curve.
angle between the first two peaks is 88.1◦ , just corresponding to the averaged angle among (100) and (110) lattice planes of the gold crystal. Details are listed in Table 1, it indicates that the calibrated PDF from the polycrystalline gold is well agreement with the gold crystal. In addition, more crystallographic information can be extracted from the obtained PDFs by using external software such as PDFgui [14], RMCProfile [15], or RMC++ [16]. Fig. 5 shows an example of performing the quantitative PDF analysis of polycrystalline anatase in PDFgui. The scattering type is X-ray (PDF obtained from electron diffraction is approximately equivalent to that of X-ray diffraction), the fitting range is 3.3∼20 Å−1 for avoiding the correlated motion of neighboring atoms. Qmax = 15.26 Å−1 , Qdamp = 0.102, and Qbroad = 0.123. The refined parameters are scale factor, Qbroad , unit cell, atomic coordination of the oxygen atom zO , and isotropic thermal displacements of atoms, the agreement factor Rw is 0.195, as shown in Fig. 5(c). Details of the refined parameters are listed in Table 2. The refined unit cell and the oxygen site are a×c=3.7971(39)×9.517(13) Å, and zO = 0.208(1), respectively, which are well consistent with two reference structures (ICSD #9852 and 9853). And the refined isotropic thermal displacement factors of atoms are reasonable, although some reports [17] suggested the thermal displacement factors extracted from electron
Table 1 Results of PDF analysis based on the SAED pattern from the polycrystal gold: The calculated peak positions (rCalc ) and coordination numbers (NCalc. ) of the standard gold specimen (a = 4.0786 Å), and the measured peak positions (rExp. ) and coordination numbers (NExp. ) within sub-sections. Where ‘–’ shows the unidentified peak from the measured RDF. rCalc. (Å)
rExp. (Å)
NCalc.
NExp.
Integral range (Å)
2.884 4.079 4.995 5.768 6.449 7.064 7.630 8.157 8.652 9.120 9.565 9.9905 10.3984
2.903 4.043 4.983 5.807 6.430 – 7.637 – 8.747 – – – 10.3533
12 6 24 12 24 8 48 6 36 24 24 24 72
12.07 6.02 23.42 12.94 27.44
2.520∼3.480 3.480∼4.453 4.453∼5.453 5.453∼6.067 6.073∼7.033
55.13
7.033∼8.173
83.22
8.173∼9.673
99.34
9.667∼10.860
diffraction will be two or three times larger than those of neutron experiments, indicating that ePDF tools is an effective package to extract quantitative PDFs from SAED patterns.
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H.L. Shi, M.T. Luo and W.Z. Wang / Computer Physics Communications 238 (2019) 295–301 Table 2 Comparison of the refined parameters of the PDF extracted from ePDF tools with two typical ICSD structures.
a
Parameters
Refined
Ref. (9852-ICSD)
Ref. (9853-ICSD)
Unit cell
a = b = 3.7971(39) Å c = 9.517(13) Å
a = b = 3.7842(13) Å c = 9.5146(15) Å
a = b = 3.7892(4) Å c = 9.537(1) Å
zO
0.208(1)
0.2081(2)
0.2079(2)
UTi
0.0100(12)
0.0049(8) (B = 0.39)a
0.0081(8) (B = 0.64)a
UO
0.0103(18)
0.0077(11) (B = 0.61)a
0.0120(11) (B = 0.95)a
Note: Isotropic thermal displacement factors of atoms are calculated by Debye–Waller factor U = B/8π.2
Table A.1 Software metadata. Nr
(executable) Software metadata description
Please fill in this column
S1
Current software version
v1.0
S2
Permanent link to executables of this version
https://github.com/hlshi527214/ePDF-tools/tree/master/Installer files
S3
Legal Software License
GNU Public License GPLv3
S4
Computing platform / Operating System
Microsoft Windows
S5
Installation requirements & dependencies
Any computer with Gatan Microscopy Suite software: http://www.gatan.com/ products/tem-analysis/gatan-microscopy-suite-software
S6
If available Link to user manual if formally published include a reference to the publication in the reference list
https://github.com/hlshi527214/ePDF-tools/tree/master/Example
S6
Support email for questions
[email protected]
Nr
Code metadata description
Please fill in this column
C1
Current Code version
v1.0
C2
Permanent link to code / repository used of this code version
https://github.com/hlshi527214/ePDF-tools/tree/master/Sourcecode
C3
Legal Code License
GNU Public License GPLv3
C4
Code Versioning system used
DigitalMicrograph
C5
Software Code Language used
DigitalMicrograph script language
C6
Compilation requirements, Operating environments & dependencies
Any computer with Gatan Microscopy Suite software: http://www.gatan.com/ products/tem-analysis/gatan-microscopy-suite-software
C7
If available Link to developer documentation / manual
https://github.com/hlshi527214/ePDF-tools/tree/master/Example
C8
Support email for questions
[email protected]
Table A.2 Code metadata.
5. Conclusions
Appendix A. Required metadata
ePDF tools are a versatile package to perform PDF analysis for electron diffraction patterns including the profile preparation, data normalization and corrections of F(Q), and the quantitative analysis of PDFs. The obtained PDFs will be further imported into PDFgui, RMCProfile or RMC++ to extract the other local structures by the PDF refinement or the reverse Monte Carlo simulation. To get an accurate PDF from SAED patterns, keep your mind in the TEM experiments that: (i) Select a thin sample or using the precession device to collect the kinematic electron scattering since the cubic spline fitting method cannot filter the dynamic scatter. (ii) Choose a small camera length to collect scattering signals over large Q range. (iii) Choose an exposure time as long as possible to collect the high-quality pattern.
A.1. Current executable software version
Acknowledgments This work was supported by the National Natural Science Foundation of China [Grant Number 11604394], Undergraduate Research and Innovative Undertaking Program of Beijing [Grant Numbers BEIJ2016110008, BEIJ2015110022].
See Table A.1. A.2. Current code version See Table A.2. Appendix B. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.cpc.2018.11.019. References [1] B.E. Warren, X-Ray Diffraction, Dover Publications, 1990. [2] T. Egami, S.J.L. Billinge, Underneath the Bragg Peaks Structural Analysis of Complex Materials, Pergamon, 2003. [3] P. Juhás, T. Davis, C.L. Farrow, S.J.L. Billinge, J. Appl. Crystallogr. 46 (2012) 560–566. [4] P.F. Peterson, T. Proffen, I.K. Jeong, S.J.L. Billinge, K.S. Choi, M.G. Kanatzidis, P.G. Radaelli, Phys. Rev. B 63 (2001) 165211. [5] C. Laulhé, F. Hippert, R. Bellissent, A. Simon, G.J. Cuello, Phys. Rev. B 79 (2009) 7715–7722.
H.L. Shi, M.T. Luo and W.Z. Wang / Computer Physics Communications 238 (2019) 295–301 [6] X. Zou, Y. Sukharev, S. Hovmöller, Ultramicroscopy 52 (1993) 436–444. [7] J.H. Chen, H.W. Zandbergen, D.V. Dyck, Ultramicroscopy 98 (2004) 81. [8] M. Abeykoon, C.D. Malliakas, P. Juhas, E.S. Bozin, M.G. Kanatzidis, S.J.L. Billinge, Z. Kristallogr. 227 (2012) 248–256. [9] F. Li, J.S. Lannin, Phys. Rev. Lett. 65 (1990) 1905–1908. [10] S.J.L. Billinge, Z. Kristallograp. Int. J. Struct. Phys. Chem. Aspects Crystal. Mater. 219 (2004) 117–121. [11] J.M. Cowley, Diffraction Physics, third ed., 1995. [12] DigitalMicrograph, Gatan Inc., Pleasanton, United States, 2017.
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[13] P.F. Peterson, E.S. Bozin, T. Proffen, S.J.L. Billinge, J. Appl. Crystallogr. 36 (2003) 53–64. [14] C.L. Farrow, P. Juhas, J.W. Liu, D. Bryndin, E.S. Božin, J. Bloch, T. Proffen, S.J. Billinge, J. Phys. Condens. Matter Inst. Phys. J. 19 (2007) 335219. [15] M.G. Tucker, D.A. Keen, M.T. Dove, A.L. Goodwin, Q. Hui, J. Phys. Condens. Matter Inst. Phys. J. 19 (2007) 335218. [16] G. Evrard, L. Pusztai, J. Phys. Condens. Matter 17 (2005) S1. [17] A.M.M. Abeykoon, H. Hu, L. Wu, Y. Zhu, S.J.L. Billinge, J. Appl. Crystallogr. 48 (2015) 244–251.