ARTICLE IN PRESS Ultramicroscopy 108 (2008) 1540– 1545
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QtUCP—A program for determining unit-cell parameters in electron diffraction experiments using double-tilt and rotation-tilt holders Hongsheng Zhao a,b,c,, Deqi Wu a,b,c, Jincheng Yao b, Aimin Chang b a b c
Xinjiang Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Urumqi 830011, PR China Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, PR China Graduate School of Chinese Academy of Sciences, Beijing 100049, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 November 2007 Received in revised form 30 March 2008 Accepted 6 May 2008
A computer program, QtUCP, has been developed based on several well-established algorithms using GCC 4.0 and Qts 4.0 (Open Source Edition) under Debian GNU/Linux 4.0r0. It can determine the unitcell parameters from an electron diffraction tilt series obtained from both double-tilt and rotation-tilt holders. In this approach, two or more primitive cells of the reciprocal lattice are determined from experimental data, in the meantime, the measurement errors of the tilt angles are checked and minimized. Subsequently, the derived primitive cells are converted into the reduced form and then transformed into the reduced direct primitive cell. Finally all the patterns are indexed and the leastsquares refinement is employed to obtain the optimized results of the lattice parameters. Finally, two examples are given to show the application of the program, one is based on the experiment, the other is from the simulation. & 2008 Elsevier B.V. All rights reserved.
PACS: 61.05.J 61.05.a Keywords: Electron diffraction Unit-cell determination Niggli-reduced cell Computer program
1. Introduction Determination of the unit-cell for unknown crystalline phase is a basic requirement for materials characterization and the first step of ab initio structure determination. Electron diffraction technique, for example, transmission electron microscopy (TEM) and low energy electron diffraction (LEED), as a counterpart of X-ray and neutron diffraction techniques have been extensively used in material characterization and structure determination. Due to the feature of electron diffraction technique, it is natural to determine the unit-cell for unknown crystalline phases by the method of reciprocal lattice reconstruction from an electron diffraction tilt series. A simple reconstruction method was shown in the book by Vanishtein [1]: a two-dimensional (2D) lattice was constructed from an electron diffraction tilt series. This method is troublesome in the application to crystalline phases belonging to monoclinic or triclinic systems. A general 3D reciprocal lattice reconstruction method was discussed by Fraundorf [2] and a program reconstructing the 3D reciprocal lattice from an electron
Corresponding author at: Xinjiang Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Urumqi 830011, PR China. Tel.: +86 991 384 5404; fax: +86 0991 383 8957. E-mail address:
[email protected] (H. Zhao).
0304-3991/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2008.05.001
diffraction tilt series was recently developed by Zou et al. [3]. But Zou’s method has two main disadvantages: it uses two quite complicated formulae to calculate the overall tilt angles; besides, Zou’s TRICE program must require his another program—ELD [4,5] to pretreat the electron diffraction patterns. Based on the reduced ternary quadratic form of mathematics, in 1928, Niggli proved that a crystal lattice could be characterized by a unique choice of reduced cell, that is, Niggli’s reduced cell [6], and there are 44 primitive reduced (Niggli) cells corresponding to 14 Bravais lattices. The core idea of this method is: the determination of a Niggli’s reduced cell and the transformation of it to a conventional cell. Kuo [7,8] applied the concept of the Niggli’s reduced cell and cell reduction technique on the unit-cell determination in electron diffraction experiments. But Kuo’s method is laborious due to its direct comparison between the 44 Niggli cells and the corresponding 14 Bravais lattices. In this paper, an approach for the determination of the unitcell of an unknown crystalline phase in electron diffraction experiments is described. The method by Kelly [9] is first used to compute the tilt angles between every two of these tilt series for both double-tilt and rotation-tilt holders. The modified procedure by Kuo [7,8] is then used to calculate the direct primitive cell by using the Niggli reduction algorithm. The procedure by Clegg [10] is used subsequently to overcome the problem in directly comparing the reduced cell to 44 forms
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of the Niggli cell. A program to perform the above steps has been developed using GCC 4.01 and Qts 4.0 (Open Source Edition)2 under Debian GNU/Linux 4.0r03 and two examples are given to show the application of the program, the program is available to academic users (only binary distribution presently, shipped with minimal glibc and Qt libraries) and those interested should contact the author directly.4
every two consecutive patters
2. Summary of the approach
Determine two or more primitive cells
Calculation of the overall Tilt Angles of
of the reciprocal lattice from experimental data The approach for unit-cell determination by using the Niggli’s reduced cell is shown in Fig. 1. Some details are described here for its application to electron diffraction data. In the following sections of this work, an electron diffraction pattern is characterized by the lengths of two basic vectors, r 1 ¼ jr1 j, r 2 ¼ jr2 j and the angle between them: cos ðrd 1 ; r2 Þ, or by lengths of these three qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vectors: r 1 , r 2 and r 3 ¼ jr3 j ¼ jr1 j2 þ jr2 j2 2jr1 jjr2 j cos ðrd 1 ; r 2 Þ.
Conversion the reciprocal primitive reduced cell to its corresponding real primitive reduced cell
2.1. Calculation of the overall tilt angles For analysis with the aid of electron diffraction technique of crystal construction, it is necessary to make adjustments to the specimen in order to obtain the corresponding diffraction patterns belonging to different zones. Several methods [9,11–13] have been shown for the computation of the tilt angles by using the doubletilt or rotation-tilt holders. Among the methods mentioned above, Kelly’s solution [9] is simple and exact for the computation of the tilt angle eij using spherical trigonometry, which is used in our work. With the conventional side entry, double tilt specimen stage, the tilt axis paralleling to the axis of the holder is defined as the X axis, and the other,which is perpendicular to the axis of the holder, is defined as the Y axis. In the present paper the X axis tilt is denoted by f and the Y axis tilt by x. In order to determine the orientation of the electron beam relative to the specimen in terms of the tilt angles conveniently, it is necessary to define two axial systems, as shown in Fig. 2. The first is fixed relative to the diffractometer and is defined by X m, Y m , and Z m , where X m , is parallel to the double tilt holder axis (X m ¼ X), Z m is antiparallel to the electron beam, and Y m is colinear with the cross product Z m X m , according to right-handed coordinate system. Note that Y m is the same as Y when there is no tilt about the X axis (Y m ¼ Y when f ¼ 0). The other axial system is fixed relative to the specimen and is defined as X s , Y s , and Z s . Z s is the normal to the specimen, Y s is the direction in the plane of the specimen corresponding to the second tilt axis Y, and X s lies in the direction of cross product Y s Z s . Note that X s is parallel to X when there is no tilt about the Y axis (X s ¼ X when x ¼ 0). When the specimen is tilted by angles fi and xi about the X and Y axes, respectively, the relative orientation of the two axial systems is shown as a stereographic projection centered about the electron beam direction Z m in Fig. 3. If the specimen is tilted to another position defined by the angles fj and xj , the angle eij between these two positions of the specimen, i.e., the tilt angle between the two positions of the electron beam, is given by the following formula [9]: cos eij ðj; xÞ ¼ cosðxj xi Þ cos fi cos fj þ sin fi sin fj
1 2 3 4
http://gcc.gnu.org/gcc-4.0/. http://www.trolltech.com/download/opensource.html. http://www.debian.org/. Please contact me through E-mail:
[email protected].
(1)
Transformation the real primitive reduced cell to the conventional cell
Obtain the bravais lattice parameters and index all experimental reflections Fig. 1. The flowchart diagram of the unit-cell determination by the cell reduction method.
Zs
Zm (Beam)
is)
lt Ax (ξ Ti
φi
Ys
O ) xis
(φ
lt A
Ti
Ym
ξi
Xm
Xs Fig. 2. Schematic diagram showing the relative positions of the axes in the two axial systems.
For the case of rotation-tilt holder, if the tilt about the axis of the holder is defined by the angle f as before, and the second tilt, in this case about the normal to the specimen, is denoted by o, the corresponding equation is [9]:
cos eij ðj; oÞ ¼ cosðoj oi Þ sin fi sin fj þ cos fi cos fj
(2)
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i
-Xm
-Xs
A
φi
φi
Zm Ys
i
experiments, e.g., always tilting the specimens in the same tilt directions (never the reverse), make sure that the precisions of the measurements reach 0:2 mm in length and 0:5 in degree, tilt the double-tilt or rotation-tilt holders as slow as possible insures the common vector of the patters does not disappear from the screen, finally, the diffraction spots used should be taken as close as possible to the pattern’s center.
Ym 2.3. Transformation the real primitive reduced cell to the conventional cell
Zs φi
Xm Fig. 3. Stereographic projection showing the relative position of the electron beam and the specimen axes after a double tilt.
2.2. Cell reduction procedure When a series of electron diffraction tilt patterns are taken for the determination of a unit-cell parameters, large errors can result from inaccurate measurement of tilt angles, which are reputed to be limited to an accuracy of 0:5 . These errors can be calibrated by using the Kikuchi maps [11,14]. However, there are a number of problems when using the Kikuchi map method. First, the specimen area of interest is often observed to be too thin to produce Kikuchi line pairs; second, for a severely strained specimen the visible Kikuchi line pairs are poorly defined, and third, in case of a small grained specimen, tilting of the specimen may cause the disappearance of the corresponding diffraction pattern, due to specimen shift. In these situations it is difficult, or may not be possible to adjust the crystal orientation to follow the Kikuchi map [15]. Referring to the procedure in the approach by Kuo [7,8] and Li [16], the following procedure has been adopted here to check and minimize the tilt errors. The reduction of the Niggli’s cells are then carried out by the fully elaborated reduction algorithms [17,18]. Two basic reflections are selected to construct the primitive cell for each 2D electron pattern and described as vectors r1i and r2i for the ith pattern, where r1i is the common vector of the tilt series. For any two diffraction patterns ðr1i ; r2i Þ and ðr1j ; r2j Þ with a tilt angle eij between them, by changing the value of the angle eij within the range of experimental errors, a series of primitive reciprocal cells are then calculated by the three non-coplanar vectors r1i ð¼ r1j Þ, r2i , r2j and reduced. Assuming that three consecutively tilted electron diffraction patterns have been obtained, two such series of reduced primitive reciprocal cells can be calculated from these electron diffraction patterns, e.g., one series from patterns 1 and 2 while the other series from patterns 2 and 3. The closest lattice parameters can be found from the two reciprocal cells belong to the corresponding tilt series and regarded as the common vector. The corresponding tilt angles e12 and e23 are then used for the calculation of the direct primitive cells and then the average one of them is adopted for the next step. In many literatures, the details on converting the reciprocal primitive reduced cells to their corresponding real primitive reduced cells can be found, e.g., International Tables for Crystallography: Reciprocal Space, Vol. B [19]. The procedure described here can be used to check and minimize the measurement errors of tilt angles. Still and all, the efforts to minimize the errors should be carried out in
A primitive basis a, b, c is called a reduced basis if it is righthanded and if the components of the metric tensor G, which bb cc has the form: aa bc ca ab , satisfy the corresponding conditions shown in International Tables for Crystallography: Space-group symmetry, Vol. A [20]. The matrix G for the reduced basis is called the Niggli matrix. The above approach is the traditional way of assessing the metric symmetry of a lattice, which was widely used [6–8,18,21]. But, as it is indicated by Li [16]: the recognition and interpretation of the reduced form are difficult due to the effects of the errors in the cell parameters, rounding errors in calculation, and the equality or inequality conditions in calculation. Several methods have been proposed to solve the problem [10,22–24]. The practical procedure proposed by GrosseKunstleve [24], is used in the present paper for the determination of the Niggli reduced unit cell. Subsequently, the method by Bucksch [25] is used to convert the reduced cell to conventional cell. In this procedure, the second stable algorithm presented by Grosse-Kunstleve [24], the minimum reduction, is used and that does not require using a tolerance. It produces a cell with minimum lengths and all angles acute or obtuse. The algorithm is a simplified and modified version of the Buerger-reduction algorithm of Ref. [26]. The results of the above procedure is then used to obtain the parameters of the conventional cell by vector algebraic method [25]. Though the above reduction procedure, the ultimate bravais lattice parameters can be obtained: a, b, c, a, b, g. 2.4. Indexing all experimental reflections and least-squares refinement (LSR) It was pointed out [8,10] that an overall transformation matrix can be obtained by successive multiplications of the transformation matrices corresponding to each step of the cell reduction algorithm. This overall transformation matrix can be used for indexing of the reflections under the lattice parameters of the reduced primitive cell [8]. The general procedure [27] for indexing electron diffraction patterns with known lattice parameters is to find the indices of the two basic reflections r1 , r2 by matching their vector lengths jr1 j, jr2 j and the angle cos ðrd 1 ; r2 Þ between them (or r 1, r 2 , r 3 ¼ jr1 r2 j) to experimental results. Basic reflections in each pattern are indexed using the conventional cell under a certain tolerated value. In addition, computer programs for the application of stereographic projection to specimen orientation adjustment in electron diffraction experiments, e.g., CaRine s TM Crystallography [28] and SingleCrystal [29] can be used to confirm the angle relationship of zone axes among the electron diffraction tilt series. In order to fixing the conventional cell’s dimensions obtained as above, the LSR [16,30–32] was then carried out in reciprocal space by using the three basic lengths in each electron pattern. Finally, the more concise real space lattice parameters were
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reduced, the details of the conversion can be found in many crystallographic literatures, such as, International Tables for Crystallography: Reciprocal Space, Vol. B [19].
3. Computer implementation According to the approach described in Section 2, the computer implementation is not so complex. One program, i.e., QtUCP, has been written and compiled by using GCC 4.0, and the GUI interface of it has been performed by Qts 4.0 (Open Source Edition) under Debian GNU/Linux 4.0r0, see Fig. 4. Since all the algorithms used here were mentioned above and also given in literatures, only a brief description of implementation is given below and two examples are given in Section 4. Firstly, input parameters, including the camera constant and the values of r1i ð¼ r1j Þ, r2i , r2j of the two selected electron diffraction patterns numbered with i and j as well as the tilt angle eij ðj; xÞ (for the double-tilt holder) or eij ðj; oÞ (for the rotation-tilt holder) between them, are used by QtUCP to generate a series of the Niggli matrices with the value of eij ðj; xÞ for the double-tilt holder or eij ðj; oÞ for the rotation-tilt holder. All of these matrices are changed in a small step within a range of estimated error (e.g., bb cc 11) and have the form as follows: aa bc ca ab . Secondly, the QtUCP generate the reduced Niggli cell represented by lengths and all angles acute or obtuse, and the averaged one of them is then used to obtain the lattice parameters of a reduced primitive cell. Finally, all experimental reflections are indexed and the unit-cell dimensions are fixed though LSR by QtUCP. So, the ultimate parameters of Bravais lattice of the corresponding conventional cell is obtained.
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4. Two examples In order to show the usage of the procedure described above, two examples for the determination of unit-cell from a series of tilt electron diffraction patterns are given here. The first example is based on the experimental data of Al5Co2 which is a primitive hexagonal phase with space group number 194, the short international Hermann–Mauguin space group symbol ˚ c¼ P63/mmc and the lattice parameters of a ¼ 7:6715 A, ˚ [33]. The second example is based on the simulated 7:6085 A data of the H15B3Mg1O13 phase. H15B3Mg1O13 is a primitive monoclinic phase with space group number 14, the short international Hermann–Mauguin space group symbol P21 =C and ˚ b ¼ 13:1145 A, ˚ c¼ the lattice parameters of a ¼ 6:8221 A, ˚ and b ¼ 104:55 [34]. 12:0351 A 4.1. Unit cell determination of Al5 Co2 This example is used by Li [16], and it is recomputed here for a comparison. Fig. 5 shows three consecutive electron diffraction patterns with a common vector (jr1 j) taken from the Al5Co2 phase. Table 1 shows the basic reflections in the three consecutive electron diffraction patterns of the Al5Co2 phase and the comparison of measured and calculated values of eij ðj; xÞ of these patterns are listed in Table 2. Reduced direct primitive cells were then calculated for two pair patterns Fig. 5 (1, 2) and (2, 3), the results together with the average ones are listed in Table 3. The average ones of the above were then used by QtUCP to generate the ultimate Bravais lattice of the corresponding conventional cell and the indices of basic reflections and zone axis of each pattern are
Fig. 4. Snapshot of the graphic interface of the program QtUCP. The data in the input fields were used in the first sample of Section 4.
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Fig. 5. Experimental electron diffraction patterns of the Al5Co2 phase, which are used as an example for the unit-cell determination. Basic reflections are marked in each diffraction patterns, courtesy of Li [16].
˚ and c ¼ 7:760 A, ˚ which is as a ¼ ð7:668 þ 7:674Þ=2 ¼ 7:671 A comparable to the results of Li [16].
Table 1 Basic reflections in the electron diffraction patterns of the Al5Co2 phase Patterns no. (i)
jr1 j
jr2 j
jr3 j
1 2 3
2.815 2.815 2.815
5.476 3.725 5.676
6.917 3.778 5.725
Length unit is millimeter (mm) and the camera constant is 18.79 (mm A˚), which was calibrated using the ring pattern of an evaporated aluminum standard sample [16].
Table 2 Comparison of measured and calculated values of eij ðj; xÞ, angle unit is degree (1) (15.90, 7.90) (1.50, 2.20) 18.28 18.30
Initial tilts ðfi ; xi Þ Final tilts ðfj ; xj Þ Measured eij ðj; xÞ Calculated eij ðj; xÞ
(15.90, 7.90) (17.50, 5.70) 36.66 35.99
(1.50, 2.20) (17.50, 5.70) 18.38 17.78
Table 3 The direct primitive lattices derived from the electron diffraction patterns Patterns no. (i)
a
b
c
a a
b
g
1 and 2 2 and 3
7.680 7.656
7.673 7.675
7.750 7.770
88.61 90.41
90.21 89.55a
119.68 119.28
Avg.
7.668
7.674
7.760
90.90
90.33
119.48
Length unit is angstrom (A˚) and angle unit is degree (1). a In reduced primitive unit-cell generated by the algorithm of GrosseKunstleve et al. [24], all angles either less than 901 or large/equal 901. To calculate the average value we take supplementary angles of all these three angles.
Table 4 The indices of reflection in the electron diffraction patterns Patterns no. (i)
r1i ðh k lÞ
r2i ðh k lÞ
½u v w
1 2 3
ð1 1¯ 0Þ ð1 1¯ 0Þ ð1 1¯ 0Þ
¯ ð1 1 1Þ ¯ ð1 0 1Þ ¯ ð1 0 2Þ
[11 2] [111] [2 2 1]
obtained and listed in Table 4. Considering that within the tolerance used by the program (0.01), a b, a b 90 and g 120 in Table 3, so the unit-cell favorites a hexagonal unitcell. Through the above procedure, the unit-cell was determined as a hexagonal unit-cell and the lattice parameters were assigned
4.2. Unit cell determination of H15 B3 Mg1 O13 Fig. 6 shows three electron diffraction patterns simulated using TM SingleCrystal [29]. Lengths of basic reflection vectors in each diffraction patterns were calculated and truncated at the precision of 0.001 mm. The camera length is set to 1500 mm, and the electron wavelength is set to 0.037 A˚, which is corresponding to 100 keV electron. so, the camera constant is assigned to 55.5 mm A˚. This example simulates the case when highly accurate data (i.e., the lengths of the basic reflections) are obtained from experiments. The lengths of r1 , r2 , r3 and the tilt angles of these patterns are listed in Table 5. Reduced direct primitive cells were then calculated for two pair patterns Fig. 6 (1, 2) and (2, 3), the results together with the average ones were listed in Table 6. The average ones of the above were then used by QtUCP to generate the ultimate Bravais lattice of the corresponding conventional cell and the indices of basic reflections and zone axis of each pattern are obtained and listed in Table 7. Considering that within the tolerance used by the program (0.01), aabac, a g 90 and b 104:42 in Table 6, so the unit-cell favorites a primitive monoclinic unit-cell. Through the above procedure, the lattice ˚ parameters were assigned as a ¼ ð6:818 þ 6:836Þ=2 ¼ 6:827 A, ˚ b ¼ ð13:100 þ 13:125Þ=2 ¼ 13:113 A, c ¼ ð12:041 þ 12:038Þ=2 ¼ ˚ and b ¼ ð103:89 þ 104:95Þ=2 ¼ 104:42 . 12:040 A
5. Conclusions A program, QtUCP, has been developed, which allows the determination of unit-cell parameters in electron diffraction experiments using double-tilt and rotation-tilt holders. The program is compiled by using GCC 4.0, and the GUI interface of it has been performed by Qts 4.0 (Open Source Edition) under Debian GNU/Linux 4.0r0. As shown in the examples given in the work, it is a feasible way to determine the unit-cell of a crystalline phase from electron diffraction data by this procedure. In our approach, the overall tilt angle has been computed first and the errors of it can be minimized during the determination of the lattice parameters of initial reciprocal primitive unit-cell and the diffraction patterns are indexed with the initial unit-cell under a given tolerance value. The accuracy of the lattice parameters depends on the accuracy of the measurement of the lengths of the reflection vectors. If the convergent-beam electron diffraction (CBED) patterns in higher-order Laue zone (HOLZ) lines are also
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r2
r2
r2 r1
1545
r1
r1
Fig. 6. Simulated electron diffraction patterns of the H15B3Mg1O13 phase, which are used as an example for the unit-cell determination. Basic reflections are marked in each diffraction patterns.
Table 5 Basic reflections in the electron diffraction patterns of the Ce5Cu19P12 phase and the tilt angles of these patterns Patterns no. (i)
jr1 j
jr2 j
jr3 j
ei
1 2 3
15.404 11.229 11.229
7.756 8.514 6.365
17.246 12.754 11.432
0.00 47.46 42.54
Length unit is millimeter (mm), angle unit is degree (1) and the camera constant is 55.5 mm A˚.
Table 6 The direct primitive lattices derived from the electron diffraction patterns Patterns no. (i)
a
b
c
a
b
g
1 and 2 2 and 3
6.818 6.836
13.100 13.125
12.041 12.038
90.49 89.61a
103.89 104.95
89.92a 90.38
Avg.
6.827
13.113
12.040
90.44
104.42
90.23
Length unit is angstrom (A˚) and angle unit is degree (1). a In reduced primitive unit-cell generated by the algorithm of GrosseKunstleve et al. [24], all angles either less than 901 or large/equal 901. To calculate the average value we take supplementary angles for these two angles.
Table 7 The indices of reflection in the electron diffraction patterns Patterns no. (i)
r1i ðh k lÞ
r2i ðh k lÞ
½u v w
1 2 3
(1 0 0) (1 0 0) (1 0 0)
(0 1 0) ¯ ð0 1 1Þ (1 0 0)
[0 0 1] [0 11] [0 1 0]
recorded and fitted, more accurate lattice parameters should be obtained.
Acknowledgments This work was supported by the National Nature Science Foundation of China (no. O311011301) and the Knowledge
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