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How to Prepare an Input File for Surface Calculations

September 11, 2013 Posted by Emre S. Tasci

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How to Prepare an Input File for Surface Calculations

Emre S. Tasci

19/07/2013

Abstract

This how-to is intended to guide the reader during the preparation of the input file for an intended surface calculation study. Mainly VESTA software is used to manipulate and transform initial structure file along with some home brewed scripts to do simple operations such as translations.

Table of Contents

Description

One of the questions I’m frequently being asked by the students is the preparation of input files for surface calculations (or more accurately, “a practical way” for the purpose). In this text, starting from a structural data file for bulk, the methods to obtain the input file fit for structure calculations will be dealt.

Tools

VESTA [A] is a free 3D visualization program for structural models that also allows editing to some extent. It is available for Windows, MacOS and Linux operating systems.
STRCONVERT [B] is a tool hosted on the Bilbao Crystallographic Server[, , ] (BCS) that offers visualization and also basic editing as well as conversion between various common structural data formats.
TRANSTRU [C] is an another BCS tool that transforms a given structure via the designated transformation matrix.

Procedure

Initial structural data

We start by obtaining the structural data for the unit cell of the material we are interested in. It should contain the unit cell dimensions and the atomic positions. Some common formats in use are: CIF[], VASP[], VESTA[] and BCS formats. Throughout this text, we will be operating on the Fm3mphase of the platinum carbide (PtC), reported by Zaoui & Ferhat[] as reproduced in BCS format in Table 1↓.

225
4.50 4.50 4.50 90. 90. 90.
2
C 1 4a 0 0 0
Pt 1 4b 0.5 0.5 0.5

Table 1 Structural data of Fm3m PtC
These data can be entered directly into VESTA from the form accessed by selecting the “File→New Structure” menu items and then filling the necessary information into the “Unit cell” and “Structure parameters” tabs, as shown in Figures 1↓ and 2↓.

figure VESTA_unitcell.png
Figure 1 VESTA’s “Unit cell” interface

figure VESTA_structureparameters.png
Figure 2 VESTA’s “Structure parameters” interface
As a result, we have now defined the Fm3m PtC (Figure 3↓).

figure VESTA_PtC_unitcell.png
Figure 3 Fm3m platinum carbide
An alternative way to introduce the structure would be to directly open its structural data file (obtained from various structure databases such as COD[], ICSD[] or Landolt-Bornstein[] into VESTA.

Constructing the Supercell

A supercell is a collection of unit cells extending in the a,b,c directions. It can easily be constructed using VESTA’s transformation tool. For demonstration purposes, let’s build a 3x3x3 supercell from our cubic PtC cell. Open the “Edit→Edit Data→Unit Cell…” dialog, then click the “Option…” button in the “Setting” row. Then define the transformation matrix as “3a,3b,3c” as shown in Figure 4↓.

figure VESTA_3x3x3.png

Figure 4 Transformation matrix for the supercell
Answer “yes” when the warning for the change of the volume of the unit cell appears, and again click “Yes” to search for additional atoms. After this, click “OK” to exit the dialog. This way we have constructed a supercell of 3x3x3 unit cells as shown in Figure 5↓.

figure VESTA_3x3x3.png

Figure 5 A 3x3x3 supercell

,figure PtC_3x3x3_supercell.png

Figure 6 Designating the boundaries for the construction of the supercell.
We have built the supercell for a general task but at this stage, it’s not convenient for preparing the surface since we first need to cleave with respect to the necessary lattice plane. So, revert to the conventional unit cell (either by reopening the data file or undoing (CTRL-z) our last action).
Now, populate the space with the conventional cell using the “Boundary” setting under the “Style” tool palette (Figure 6↑). In this case, we are building a 3x3x3 supercell by including the atoms within the -fractional- coordinate range of [0,3] along all the 3 directions (Figure 7↓).

figure unit_cells_single_and_super.png

Figure 7 (a)Single unit cell (conventional); (b) 3x3x3 supercell
At this stage, the supercell formed is just a mode of display – we haven’t made any solid changes yet. Also, if preferred one can choose how the unit cells will be displayed from the “Properties” setting’s “General” tab.

Lattice Planes

To visualize a given plane with respect to the designated hkl values, open the dialog shown in Figure 8↓ by selecting “Edit→Lattice Planes…” from the menu, then clicking on “New” and entering the intended hkl values, in our example (111) plane.

etasci@metu.edu.trfigure LatticePlane_111.png
Figure 8 Selecting the (111) lattice plane
The plane’s distance to the origin can be specified in terms of Å or interplane distance, as well as selecting 3 or more atoms lying on the plane (multiple selections can be realised by holding down SHIFT while clicking on the atoms) and then clicking on the “Calculate the best plane for the selected atoms” button to have the plane passing from those positions to appear. This is shown in Figure 9↓, done within a 3x3x3 supercell.

figure 3x3x3_Lattice_Plane_111.png
Figure 9 Selecting the (111) lattice plane in the 3x3x3 supercell
The reason we have oriented our system so that the normal to the (111) plane is pointing up is to designate this lattice plane as the surface, aligning it with the new c axis for convenience. Therefore we also need to reassign a and b axes, as well. This all comes down to defining a new unit cell. Preserving the periodicities, the new unit cell is traced out by introducing additional lattice planes. These new unit cells are not unique, as long as one preserves the periodicities in the 3 axial directions, they can be taken as large and in any convenient shape as allowed. A rectangular shaped such a unit cell, along with the outlying lattice plane parameters is presented in Figure 10↓ (the boundaries have been increased to [-2,5] in each direction for practicality).

figure new_unit_cell_complete.png
Figure 10 The new unit cell marked out by the lattice plane “boundaries”

Trimming the new unit cell

The next procedure is simple and direct: remove all the atoms outside the new unit cell. You can use the “Select” tool from the icons on the left (the 2nd from top) or the shortcut key “s”. The “trimmed” new unit cell is shown in Figure 11↓, take note of the periodicity in each section. It should be noted that, this procedure is only for deducing the transformation matrix, which we’ll be deriving in the next subsection: other than that, since it’s still defined by the “old” lattice vectors, it’s not stackable to yield the correct periodicity so we’ll be fixing that.

figure new_unit_cell__3_angles.png
Figure 11 Filtered new unit cell with the original cell’s lattice vectors

Transforming into the new unit cell

Now that we have the new unit cell, it is time to transform the unit cell vectors to comply with the new unit cell. For the purpose of obtaining the transformation matrix (which actually is the matrix that relates the new ones with the old ones), we will need the coordinates of the 4 atoms on the edges. Such a set of 4 selected atoms are shown in Figure 12↓. These atoms’ fractional coordinates are as listed in Table 2↓ (upon selecting an atom, VESTA displays its coordinates in the information panel below the visualization).

figure new_axes.png
Figure 12 The reference points for the new axe1s

Label a’ b’ c’ o’
a 0 1/2 2 1
b 1 -1/2 1 0
c -1/2 1/2 1/2 -1/2
Table 2 Reference atoms’ fractional coordinates
Taking o’ as our new origin, the new lattice vectors in terms of the previous ones become (as we subtract the “origin’s” coordinates):
a’ = -a+b
b’ = -1/2a-1/2b+c
c’ = a+b+c
Therefore our transformation matrix is:
⎡⎢⎢⎢⎣ − 1 − 121 1 − 121 011⎤⎥⎥⎥⎦
(i.e., “-a+b,-1/2a-1/2b+c,a+b+c” — the coefficients are read in columns)
Transforming the initial cell with respect to this matrix is pretty straight forward, in the same manner we exercised in subsection 3.2↑, “Constructing the supercell”, i.e., “Edit→Edit Data→Unit Cell..”, then “Remove symmetry” and “Option…” and introducing the transformation matrix. A further translation (origin shift) of (0,0,1/2) is necessary (the transformation matrix being “-a+b,-1/2a-1/2b+c,a+b+c+1/2”) if you want the transformed structure look exactly like the one shown in Figure 11↑.

A Word of caution

Due to a bug, present in VESTA, you might end with a transformed matrix with missing atoms (as shown in Figure – compare it to the middle and rightmost segments of Figure 11↑). For this reason, always make sure you check your resulting unit cell thoroughly! (This bug has recently been fixed and will be patched in the next version (3.1.7)-personal communication with K.Momma)

figure new_unit_cell__3_angles__vesta_trd.png
Figure 13 The transformed structure with missing atoms
To overcome this problem, one can conduct the transformation via the TRANSTRU [E] tool of the Bilbao Crystallographic Server. Enter your structural data (either as a CIF or by definition into the text box), select “Transform structure to a subgroup basis”, in the next window enter “1” as the “Low symmetry Space Group ITA number” (1 corresponding to the P1 symmetry group) and the transformation matrix either in abc notation (“-a+b,-1/2a-1/2b+c,a+b+c+1/2”) or by directly entering the matrix form (both are filled in the screenshot taken in Figure 14↓).

figure TRANSTRU_screenshot.png
Figure 14 TRANSTRU input form
In the result page export the transformed structure data (“Low symmetry structure data”) to CIF format by clicking on the corresponding button and open it in VESTA. This way all the atoms in the new unit cell will have been included.

Final Touch

The last thing remains is introducing a vacuum area above the surface. For this purpose we switch from fractional coordinates to Cartesian coordinates, so when the cell boundaries are altered, the atomic positions will not change as a side result. VASP format supports both notations so we export our structural data into VASP format (“File→Export Data…”, select “VASP” as the file format, then opt for “Write atomic coordinates as: Cartesian coordinates”).
After the VASP file with the atomic coordinates written in Cartesian format is formed, open it with an editor (the contents are displayed in Table3↓).

generated_by_bilbao_crystallographic_server


1.0


6.3639612198 0.0000000000 0.0000000000


0.0000000000 5.5113520622 0.0000000000


0.0000000000 0.0000000000 7.7942290306


C Pt


12 12


Cartesian


0.000000000 3.674253214 6.495164688


0.000000000 1.837099013 1.299064110


3.181980610 1.837099013 1.299064110


1.590990305 0.918577018 6.495164688


4.772970915 4.592774880 1.299064110


1.590990305 4.592774880 1.299064110


4.772970915 0.918577018 6.495164688


0.000000000 0.000000000 3.897114515


3.181980610 0.000000000 3.897114515


4.772970915 2.755676031 3.897114515


1.590990305 2.755676031 3.897114515


3.181980610 3.674253214 6.495164688


0.000000000 3.674253214 2.598050405


1.590990305 0.918577018 2.598050405


4.772970915 0.918577018 2.598050405


1.590990305 4.592774880 5.196178858


3.181980610 1.837099013 5.196178858


0.000000000 1.837099013 5.196178858


4.772970915 4.592774880 5.196178858


0.000000000 0.000000000 0.000000000


3.181980610 3.674253214 2.598050405


3.181980610 0.000000000 0.000000000


4.772970915 2.755676031 0.000000000


1.590990305 2.755676031 0.000000000
Table 3 The constructed (111) oriented VASP file’s contents
Directly edit and increase the c-lattice length from the given value (7.7942 Å in our case) to a sufficiently high value (e.g., 25.0000 Å) and save it. Now when you open it back in VESTA, the surface with its vacuum should be there as in Figure 15↓.

figure surface_w_vacuum.png
Figure 15 Prepared surface with vacuum
At this point, there might be a couple of questions that come to mind: What are those Pt atoms doing at the top of the unit cell? Why are the C atoms positioned above the batch instead of the Pt atoms?
The answer to these kind of questions is simple: Periodicity. By tiling the unit cell in the c-direction using the “Boundary…” option and designating our range of coordinates for the {x,y,z}-directions from 0 to 3, we obtain the system shown in Figure 16↓.

figure surface_3x3x3.png
Figure 16 New unit cell with periodicity applied
From the periodicity, in fractional coordinates, f|z = 0 = f|z = 1, meaning the atoms at the base and at the very top of the unit cell are the same (or “equivalent” from the interpretational side). If we had wanted the Pt atoms to be the ones forming the surface, then we would have made the transformation with the “-a+b,-1/2a-1/2b+c,a+b+c” matrix instead of “-a+b,-1/2a-1/2b+c,a+b+c+1/2”, so the half shift would amount the change in the order, and then increase the c-lattice parameter size in the Cartesian coordinates notation (i.e., in the VASP file). The result of such a transformation is given in Figure 17↓.

figure surface_alternative_3_phases.png
Figure 17 Alternative surface with the Pt atoms on the top (Left: prior to vacuum introduction; Center: with vacuum; Right: tiled in periodicity)

Alternative Ways

In this work, a complete treatment of preparing a surface in silico is proposed. There are surely more direct and automated tools that achieve the same task. It has been brought to our attention that the commercial software package Accelrys Materials Studio [F] has an integrated tool called “Cleave Surface” which exactly serves for the same purpose discussed in this work. Since it is propriety software, it will not be further described here and we encourage the user to benefit from freely accessible software whenever possible.
Also, it is always to compare your resulting surface with that of an alternate one obtained using another tool. In this sense, Surface Explorer [G] is very useful in visualizing how the surface will look, eventhough its export capabilities are limited.
Chapter 4 (“DFT Calculations for Solid Surfaces”) of Zhang’s Ph.D. thesis[] contains very fruitful discussions on possible ways to incorporate the surfaces in computations.
It is the author’s intention to soon implement such an automated tool for constructing surfaces from bulk data, integrated within the Bilbao Crystallographic Server’s framework of tools.

Conclusion

Preparing a surface fit for atomic & molecular calculations can be tedious and tiresome. In this work, we have tried to guide the reader in a step-by-step procedure, sometimes wandering off from the direct path to show the mechanisms and reasons behind the usually taken on an “as-it-is” basis, without being pondered upon. This way, we hope that the techniques acquired throughout the text will be used in conjunction with other related problems in the future. A “walkthru” for the case studied ((111) PtC surface preparation) is included as Appendix to provide a quick consultation in the future.

Acknowledgements

This work is the result of the inquiries by the Ph.D. students M. Gökhan Şensoy and O. Karaca Orhan. If they hadn’t asked a “practical way” to construct surfaces, I wouldn’t have sought a way. Since I don’t have much experience with surfaces, I turned to my dear friends Rengin Peköz and O. Barış Malcıoğlu for help, whom enlightened me with their experiences on the issue.

Appendix: Walkthru for PtC (111) Surface

  1. Open VESTA, (“File→New Structure”)
    1. “Unit Cell”:
      1. Space Group: 225 (F m -3 m)
      2. Lattice parameter a: 4.5 Å
    2. “Structure Parameters”:
      1. New→Symbol:C; Label:C; x,y,z=0.0
      2. New→Symbol:Pt; Label:Pt; x,y,z=0.5
    3. “Boundary..”
      1. x(min)=y(min)=z(min)=-2
      2. x(max)=y(max)=z(max)=5
    4. “Edit→Lattice Planes…”
      1. (hkl)=(111); d(Å)=9.09327
      2. (hkl)=(111); d(Å)=1.29904
      3. (hkl)=(-110); d(Å)=3.18198
      4. (hkl)=(-110); d(Å)=-3.18198
      5. (hkl)=(11-2); d(Å)=3.67423
      6. (hkl)=(11-2); d(Å)=-4.59279
      (compare with Figure 10↑)
    5. Select (“Select tool” – shortcut “s” key) and delete (shortcut “Del” key) all the atoms lying out of the area designated by the lattice planes.
    6. Select an “origin-atom” of a corner and the 3 “axe-atoms” in the edge of each direction of the unit cell, write down their fractional coordinates
      1. o(1,0,-1/2)
      2. a(0,1,-1/2)
      3. b(1/2,-1/2,1/2)
      4. c(2,1,1/2)
    7. Subtract the origin-atom coordinates from the rest and construct the transformation matrix accordingly
      P = ⎡⎢⎢⎢⎣ − 1 − 121 1 − 121 011⎤⎥⎥⎥⎦
    8. Transform the initial unit cell with this matrix via TRANSTRU (http://www.cryst.ehu.es/cryst/transtru.html) (Figure 14↑)
    9. Save the resulting (low symmetry) structure as CIF, open it in VESTA, export it to VASP, select “Cartesian coordinates”
    10. Open the VASP file in an editor, increase the c-lattice parameter to a big value (e.g. 25.000) to introduce vacuum. Save it and open it in VESTA.

References

[1] Mois Ilia Aroyo, Juan Manuel Perez-Mato, Cesar Capillas, Eli Kroumova, Svetoslav Ivantchev, Gotzon Madariaga, Asen Kirov, and Hans Wondratschek. Bilbao crystallographic server: I. databases and crystallographic computing programs. Zeitschrift für Kristallographie, 221(1):1527, 01 2006. doi: 10.1524/zkri.2006.221.1.15.

[2] Mois I. Aroyo, Asen Kirov, Cesar Capillas, J. M. Perez-Mato, and Hans Wondratschek. Bilbao crystallographic server. ii. representations of crystallographic point groups and space groups. Acta Crystallographica Section A, 62(2):115128, Mar 2006. http://dx.doi.org/10.1107/S0108767305040286

[3] M I Aroyo, J M Perez-Mato, D Orobengoa, E Tasci, G De La Flor, and A Kirov. Crystallography online: Bilbao crystallographic server. Bulg Chem Commun, 43(2):183197, 2011.

[4] S. R. Hall, F. H. Allen, and I. D. Brown. The crystallographic information le (cif): a new standard archive file for crystallography. Acta Crystallographica Section A, 47(6):655685, Nov 1991.

[5] G. Kresse and J. Furthmüller. Ecient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 54:1116911186, Oct 1996.

[6] Koichi Momma and Fujio Izumi. VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data. Journal of Applied Crystallography, 44(6):12721276, Dec 2011.

[7] A Zaoui and M Ferhat. Dynamical stability and high pressure phases of platinum carbide. Solid State Communications, 151(12):867869, 6 2011.

[8] Saulius Grazulis, Adriana Daskevic, Andrius Merkys, Daniel Chateigner, Luca Lutterotti, Miguel Quirós, Nadezhda R. Serebryanaya, Peter Moeck, Robert T. Downs, and Armel Le Bail. Crystallography open database (cod): an open-access collection of crystal structures and platform for world-wide collaboration. Nucleic Acids Research, 40(D1):D420D427, 2012.

[9] Alec Belsky, Mariette Hellenbrandt, Vicky Lynn Karen, and Peter Luksch. New developments in the inorganic crystal structure database (icsd): accessibility in support of materials research and design. Acta Crystallographica Section B, 58(3 Part 1):364369, Jun 2002.

[10] Springer, editor. The Landolt-Boernstein Database (http://www.springermaterials.com/navigation/). Springer Materials.

[11] Yongsheng Zhang. First-principles statistical mechanics approach to step decoration at solid surfaces . PhD thesis, Frei Universitat Berlin, 2008.

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✶✻
❬✺❪ ●✳ ❑r❡ss❡ ❛♥❞ ❏✳ ❋✉rt❤♠ü❧❧❡r✳ ❊✣❝✐❡♥t ✐t❡r❛t✐✈❡ s❝❤❡♠❡s ❢♦r ❛❜ ✐♥✐t✐♦ t♦t❛❧✲❡♥❡r❣2 ❝❛❧❝✉❧❛t✐♦♥s ✉s✐♥❣ ❛ ♣❧❛♥❡✲✇❛✈❡
❜❛s✐s s❡t✳ P❤2s✳ ❘❡✈✳ ❇✱ ✺✹✿✶✶✶✻✾✕✶✶✶✽✻✱ ❖❝t ✶✾✾✻✳
❬✻❪ ❑♦✐❝❤✐ ▼♦♠♠❛ ❛♥❞ ❋✉❥✐♦ ■3✉♠✐✳ ❱❊❙❚❆✸ ❢♦r t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ✈✐s✉❛❧✐3❛t✐♦♥ ♦❢ ❝r2st❛❧✱ ✈♦❧✉♠❡tr✐❝ ❛♥❞ ♠♦r♣❤♦❧✲
♦❣2 ❞❛t❛✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ ❈r2st❛❧❧♦❣r❛♣❤2✱ ✹✹✭✻✮✿✶✷✼✷✕✶✷✼✻✱ ❉❡❝ ✷✵✶✶✳
❬✼❪ ❆ ❩❛♦✉✐ ❛♥❞ ▼ ❋❡r❤❛t✳ ❉2♥❛♠✐❝❛❧ st❛❜✐❧✐t2 ❛♥❞ ❤✐❣❤ ♣r❡ss✉r❡ ♣❤❛s❡s ♦❢ ♣❧❛t✐♥✉♠ ❝❛r❜✐❞❡✳ ❙♦❧✐❞ ❙t❛t❡ ❈♦♠♠✉♥✐✲
❝❛t✐♦♥s✱ ✶✺✶✭✶✷✮✿✽✻✼✕✽✻✾✱ ✻ ✷✵✶✶✳
❬✽❪ ❙❛✉❧✐✉s ●r❛➸✉❧✐s✱ ❆❞r✐❛♥❛ ❉❛➨❦❡✈✐↔✱ ❆♥❞r✐✉s ▼❡r❦2s✱ ❉❛♥✐❡❧ ❈❤❛t❡✐❣♥❡r✱ ▲✉❝❛ ▲✉tt❡r♦tt✐✱ ▼✐❣✉❡❧ ◗✉✐rós✱
◆❛❞❡3❤❞❛ ❘✳ ❙❡r❡❜r2❛♥❛2❛✱ P❡t❡r ▼♦❡❝❦✱ ❘♦❜❡rt ❚✳ ❉♦✇♥s✱ ❛♥❞ ❆r♠❡❧ ▲❡ ❇❛✐❧✳ ❈r2st❛❧❧♦❣r❛♣❤2 ♦♣❡♥ ❞❛t❛✲
❜❛s❡ ✭❝♦❞✮✿ ❛♥ ♦♣❡♥✲❛❝❝❡ss ❝♦❧❧❡❝t✐♦♥ ♦❢ ❝r2st❛❧ str✉❝t✉r❡s ❛♥❞ ♣❧❛t❢♦r♠ ❢♦r ✇♦r❧❞✲✇✐❞❡ ❝♦❧❧❛❜♦r❛t✐♦♥✳ ◆✉❝❧❡✐❝
❆❝✐❞s ❘❡s❡❛r❝❤✱ ✹✵✭❉✶✮✿❉✹✷✵✕❉✹✷✼✱ ✷✵✶✷✳
❬✾❪ ❆❧❡❝ ❇❡❧s❦2✱ ▼❛r✐❡tt❡ ❍❡❧❧❡♥❜r❛♥❞t✱ ❱✐❝❦2 ▲2♥♥ ❑❛r❡♥✱ ❛♥❞ P❡t❡r ▲✉❦s❝❤✳ ◆❡✇ ❞❡✈❡❧♦♣♠❡♥ts ✐♥ t❤❡ ✐♥♦r❣❛♥✐❝
❝r2st❛❧ str✉❝t✉r❡ ❞❛t❛❜❛s❡ ✭✐❝s❞✮✿ ❛❝❝❡ss✐❜✐❧✐t2 ✐♥ s✉♣♣♦rt ♦❢ ♠❛t❡r✐❛❧s r❡s❡❛r❝❤ ❛♥❞ ❞❡s✐❣♥✳ ❆❝t❛ ❈r2st❛❧❧♦❣r❛♣❤✐❝❛
❙❡❝t✐♦♥ ❇✱ ✺✽✭✸ P❛rt ✶✮✿✸✻✹✕✸✻✾✱ ❏✉♥ ✷✵✵✷✳
❬✶✵❪ ❙♣r✐♥❣❡r✱ ❡❞✐t♦r✳ ❚❤❡ ▲❛♥❞♦❧t✲❇♦❡r♥st❡✐♥ ❉❛t❛❜❛s❡ ✭❤tt♣✿✴✴✇✇✇✳s♣r✐♥❣❡r♠❛t❡r✐❛❧s✳❝♦♠✴♥❛✈✐❣❛t✐♦♥✴✮✳ ❙♣r✐♥❣❡r
▼❛t❡r✐❛❧s✳
❬✶✶❪ ❨♦♥❣s❤❡♥❣ ❩❤❛♥❣✳ ❋✐rst✲♣r✐♥❝✐♣❧❡s st❛t✐st✐❝❛❧ ♠❡❝❤❛♥✐❝s ❛♣♣r♦❛❝❤ t♦ st❡♣ ❞❡❝♦r❛t✐♦♥ ❛t s♦❧✐❞ s✉r❢❛❝❡s✳ P❤❉ t❤❡s✐s✱
❋r❡✐ ❯♥✐✈❡rs✐t❛t ❇❡r❧✐♥✱ ✷✵✵✽✳

Document generated by eLyXer 1.2.3 (2011-08-31) on 2013-09-11T11:44:36.446114

>> pdf version

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Meanwhile…

August 7, 2012 Posted by Emre S. Tasci

… the Imperial Tie-Fighters kept on coming… (aka Structure Data Converter & Editor + Visualizer with full support for magnetic space groups)

imperial_fleet_magnetic_structures

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Exploring through the paths of subgroups, collecting indexes on the way…

June 6, 2011 Posted by Emre S. Tasci

Having a personal wiki has its pros and cons with the pros being obvious but, on the other hand, one doesn’t feel like reposting what he has already committed into the corresponding wikipedia entry (and since reposting means filtering out the sensitive information and generalizing it, it is double the effort). So this fact, more or less, explains my absence of activity in this blog for the last 1.5 years. But then, there is one negative aspect of personal wikipedia “blogging” – it’s like putting everything under the rug : you classify the data/information and then you forget that it is there. That’s when “actual” blogging (like this one) comes handy. So, sorry & welcome back..

Today, I’ll present a code that I’ve just written to parse out the compatible paths from a tree. In my case, the three is the list of subgroups with indexes that one can acquire using Bilbao Crystallographic Server’s marvelous SUBGROUPGRAPH tool. For low indexes, you can already have SUBGROUPGRAPH do this for you by specifying your supergroup G, subgroup H and the index [G:H]. For example, for G=136, H=14 and [G:H] = 8, you can have the following paths drawn:

subgroupgraph_tempfile963

But as I said, things can get rough when you have a high index. In that case, we can (and we will) try to parse and analyze the paths tree which is outputted by SUBGROUPGRAPH when no index is designated, e.g., for G=136, H=14:

maxsubs_subggraph

Where you see the maximal subgroups for the involved groups and their corresponding indexes. So, to go from P4_2/mnm (#136) to P2_1/c (#14) with index 8, we can follow the path:

P4_2/mnm (#136) –[2]–> Cmmm (#65) –[2]–> Pmna (#53) –[2]–> P2_1/c (#14)

with the related indexes given in the brackets totaling to 2x2x2 = 8.

The problem I had today was to find possible paths going from G=Fd-3m (#227) to H=Cc (#9) with index [G:H] = 192.

I first parsed the subgroup list into an array with their indexes, then followed the possible paths.

<?PHP
/* Emre S. Tasci <exxx.txxxx@ehu.es>                    *
 * By parsing the possible subgroup paths obtained from 
 * SUBGROUPGRAPH, finds the paths that are compatible 
 * with the given terminal super & sub groups and the 
 * designated index.
 *
 *                                              06/06/11 */

# Input Data === 0 ====================================================
$start_sup = 227;
$crit_index = 192;
$end_sub = 9;

$arr = file("data.txt"); # A sample of data.txt for the case of #227->#9
                         # is included at the end of the code.

# Input Data === 1 ====================================================

$arr_subs = Array();
$arr_labels = Array();
foreach($arr as $line)
{
    $auxarr = preg_split("/[ \t]+/",trim($line));
    #echo $auxarr[2]."\n";
    $sup = $auxarr[1];
    $suplabel = $auxarr[2];
    $arrsubs = split(";",$auxarr[3]);
    $arr_labels[$sup] = $suplabel;
    foreach($arrsubs as $subind)
    {
        $auxarr2 = Array();
        preg_match("/([0-9]+)\[([0-9]+)\]/",$subind,$auxarr2);
        #print_r($auxarr2);
        $sub = $auxarr2[1];
        $index = $auxarr2[2];
        $arr_subs[$sup][$index][] = $sub;
    }
}
#print_r($arr_subs);

# Start exploring
$start_sup = sprintf("%03d",$start_sup);
$end_sub = sprintf("%03d",$end_sub);
$arr_paths = Array();
$arr_paths[$start_sup] = 1;

findpath($start_sup);

function findpath($current_sup)
{
    global $arr_paths,$arr_subs,$crit_index,$end_sub;
    #echo "*".$current_sup."*\n";
    if(strrpos($current_sup,"-") !== FALSE)
    {
        $current_sup_last_sup = substr($current_sup,strrpos($current_sup,"-")+1);
        $current_sup_last_sup = substr($current_sup_last_sup,0,strpos($current_sup_last_sup,"["));
    }
        
    else
        $current_sup_last_sup = $current_sup;
    #echo "*".$current_sup_last_sup."*\n";
    $subindexes = array_keys($arr_subs[$current_sup_last_sup]);
    foreach($subindexes as $subindex)
    {
        $subs = $arr_subs[$current_sup_last_sup][$subindex];
        echo $subindex.": ".join(",",$subs)."\n";
        foreach($subs as $sub)
        {
            echo $current_sup."-".$sub."[".$subindex."]"."\n";
            $arr_paths[$current_sup."-".$sub."[".$subindex."]"] = $arr_paths[$current_sup] * $subindex;
            if($arr_paths[$current_sup."-".$sub."[".$subindex."]"]<$crit_index)
                findpath($current_sup."-".$sub."[".$subindex."]");
            elseif($arr_paths[$current_sup."-".$sub."[".$subindex."]"] == $crit_index && $sub == $end_sub)
                echo "\nPath Found: ".$current_sup."-".$sub."[".$subindex."]"."\n\n";
        }
    }
    #print_r($subindexes);
}
print_r($arr_paths);

/*
    data.txt:
1   227   Fd-3m   141[3];166[4];203[2];216[2]
2   220   I-43d   122[3];161[4]
3   219   F-43c   120[3];161[4];218[4]
4   218   P-43n   112[3];161[4];220[4]
5   217   I-43m   121[3];160[4];215[2];218[2]
6   216   F-43m   119[3];160[4];215[4]
7   215   P-43m   111[3];160[4];216[2];217[4];219[2]
8   203   Fd-3   070[3]
9   167   R-3c   015[3];161[2];165[3];167[4];167[5];167[7]
10   166   R-3m   012[3];160[2];164[3];166[2];166[4];166[5];166[7];167[2]
11   165   P-3c1   015[3];158[2];163[3];165[3];165[4];165[5];165[7]
12   164   P-3m1   012[3];156[2];162[3];164[2];164[3];164[4];164[5];164[7];165[2]
13   163   P-31c   015[3];159[2];163[3];163[4];163[5];163[7];165[3];167[3]
14   162   P-31m   012[3];157[2];162[2];162[3];162[4];162[5];162[7];163[2];164[3];166[3]
15   161   R3c   009[3];158[3];161[4];161[5];161[7]
16   160   R3m   008[3];156[3];160[2];160[4];160[5];160[7];161[2]
17   159   P31c   009[3];158[3];159[3];159[4];159[5];159[7];161[3]
18   158   P3c1   009[3];158[3];158[4];158[5];158[7];159[3]
19   157   P31m   008[3];156[3];157[2];157[3];157[4];157[5];157[7];159[2];160[3]
20   156   P3m1   008[3];156[2];156[3];156[4];156[5];156[7];157[3];158[2]
21   141   I41/amd   070[2];074[2];088[2];109[2];119[2];122[2];141[3];141[5];141[7];141[9]
22   122   I-42d   043[2];122[3];122[5];122[7];122[9]
23   121   I-42m   042[2];111[2];112[2];113[2];114[2];121[3];121[5];121[7];121[9]
24   120   I-4c2   045[2];116[2];117[2];120[3];120[5];120[7];120[9]
25   119   I-4m2   044[2];115[2];118[2];119[3];119[5];119[7];119[9]
26   118   P-4n2   034[2];118[3];118[5];118[7];118[9];122[2]
27   117   P-4b2   032[2];117[2];117[3];117[5];117[7];117[9];118[2]
28   116   P-4c2   027[2];112[2];114[2];116[3];116[5];116[7];116[9]
29   115   P-4m2   025[2];111[2];113[2];115[2];115[3];115[5];115[7];115[9];116[2];121[2]
30   114   P-421c   037[2];114[3];114[5];114[7];114[9]
31   113   P-421m   035[2];113[2];113[3];113[5];113[7];113[9];114[2]
32   112   P-42c   037[2];112[3];112[5];112[7];112[9];116[2];118[2]
33   111   P-42m   035[2];111[2];111[3];111[5];111[7];111[9];112[2];115[2];117[2];119[2];120[2]
34   109   I41md   043[2];044[2];109[3];109[5];109[7];109[9]
35   088   I41/a   015[2];088[3];088[5];088[7];088[9]
36   074   Imma   012[2];015[2];044[2];046[2];051[2];052[2];053[2];062[2];074[3];074[5];074[7]
37   070   Fddd   015[2];043[2];070[3];070[5];070[7]
38   064   Cmce   012[2];014[2];015[2];036[2];039[2];041[2];053[2];054[2];055[2];056[2];057[2];060[2];061[2];062[2];064[3];064[5];064[7]
39   063   Cmcm   011[2];012[2];015[2];036[2];038[2];040[2];051[2];052[2];057[2];058[2];059[2];060[2];062[2];063[3];063[5];063[7]
40   062   Pnma   011[2];014[2];026[2];031[2];033[2];062[3];062[5];062[7]
41   061   Pbca   014[2];029[2];061[3];061[5];061[7]
42   060   Pbcn   013[2];014[2];029[2];030[2];033[2];060[3];060[5];060[7]
43   059   Pmmn   011[2];013[2];025[2];031[2];056[2];059[2];059[3];059[5];059[7];062[2]
44   058   Pnnm   010[2];014[2];031[2];034[2];058[3];058[5];058[7]
45   057   Pbcm   011[2];013[2];014[2];026[2];028[2];029[2];057[2];057[3];057[5];057[7];060[2];061[2];062[2]
46   056   Pccn   013[2];014[2];027[2];033[2];056[3];056[5];056[7]
47   055   Pbam   010[2];014[2];026[2];032[2];055[2];055[3];055[5];055[7];058[2];062[2]
48   054   Pcca   013[2];014[2];027[2];029[2];032[2];052[2];054[2];054[3];054[5];054[7];056[2];060[2]
49   053   Pmna   010[2];013[2];014[2];028[2];030[2];031[2];052[2];053[2];053[3];053[5];053[7];058[2];060[2]
50   052   Pnna   013[2];014[2];030[2];033[2];034[2];052[3];052[5];052[7]
51   051   Pmma   010[2];011[2];013[2];025[2];026[2];028[2];051[2];051[3];051[5];051[7];053[2];054[2];055[2];057[2];059[2];063[2];064[2]
52   046   Ima2   008[2];009[2];026[2];028[2];030[2];033[2];046[3];046[5];046[7]
53   045   Iba2   009[2];027[2];029[2];032[2];045[3];045[5];045[7]
54   044   Imm2   008[2];025[2];031[2];034[2];044[3];044[5];044[7]
55   043   Fdd2   009[2];043[3];043[5];043[7]
56   042   Fmm2   008[2];035[2];036[2];037[2];038[2];039[2];040[2];041[2];042[3];042[5];042[7]
57   041   Aea2   007[2];009[2];029[2];030[2];032[2];033[2];041[3];041[5];041[7]
58   040   Ama2   006[2];009[2];028[2];031[2];033[2];034[2];040[3];040[5];040[7]
59   039   Aem2   007[2];008[2];026[2];027[2];028[2];029[2];039[2];039[3];039[5];039[7];041[2];045[2];046[2]
60   038   Amm2   006[2];008[2];025[2];026[2];030[2];031[2];038[2];038[3];038[5];038[7];040[2];044[2];046[2]
61   037   Ccc2   009[2];027[2];030[2];034[2];037[3];037[5];037[7]
62   036   Cmc21   008[2];009[2];026[2];029[2];031[2];033[2];036[3];036[5];036[7]
63   035   Cmm2   008[2];025[2];028[2];032[2];035[2];035[3];035[5];035[7];036[2];037[2];044[2];045[2];046[2]
64   034   Pnn2   007[2];034[3];034[5];034[7];043[2]
65   033   Pna21   007[2];033[3];033[5];033[7]
66   032   Pba2   007[2];032[2];032[3];032[5];032[7];033[2];034[2]
67   031   Pmn21   006[2];007[2];031[2];031[3];031[5];031[7];033[2]
68   030   Pnc2   007[2];030[2];030[3];030[5];030[7];034[2]
69   029   Pca21   007[2];029[2];029[3];029[5];029[7];033[2]
70   028   Pma2   006[2];007[2];028[2];028[3];028[5];028[7];029[2];030[2];031[2];032[2];040[2];041[2]
71   027   Pcc2   007[2];027[2];027[3];027[5];027[7];030[2];037[2]
72   026   Pmc21   006[2];007[2];026[2];026[3];026[5];026[7];029[2];031[2];036[2]
73   025   Pmm2   006[2];025[2];025[3];025[5];025[7];026[2];027[2];028[2];035[2];038[2];039[2];042[2]
74   015   C2/c   009[2];013[2];014[2];015[3];015[5];015[7]
75   014   P21/c   007[2];014[2];014[3];014[5];014[7]
76   013   P2/c   007[2];013[2];013[3];013[5];013[7];014[2];015[2]
77   012   C2/m   008[2];010[2];011[2];012[2];012[3];012[5];012[7];013[2];014[2];015[2]
78   011   P21/m   006[2];011[2];011[3];011[5];011[7];014[2]
79   010   P2/m   006[2];010[2];010[3];010[5];010[7];011[2];012[2];013[2]
80   008   Cm   006[2];007[2];008[2];008[3];008[5];008[7];009[2]
81   007   Pc   007[2];007[3];007[5];007[7];009[2]
82   006   Pm   006[2];006[3];006[5];006[7];007[2];008[2]
83   009   Cc   007[2];009[3];009[5];009[7]
 */

When run, the code lists all the paths with the matching ones preceded by “Path Found”, so if you grep wrt it, you’ll have, for this example:

sururi@husniya:/xxx$ php find_trpath_for_index.php |grep Path|sed "s:Path Found\: ::"
227-141[3]-070[2]-015[2]-009[2]-007[2]-007[2]-009[2]
227-141[3]-070[2]-015[2]-013[2]-007[2]-007[2]-009[2]
227-141[3]-070[2]-015[2]-013[2]-013[2]-007[2]-009[2]
227-141[3]-070[2]-015[2]-013[2]-013[2]-015[2]-009[2]
227-141[3]-070[2]-015[2]-013[2]-014[2]-007[2]-009[2]
227-141[3]-070[2]-015[2]-014[2]-007[2]-007[2]-009[2]
227-141[3]-070[2]-015[2]-014[2]-014[2]-007[2]-009[2]
227-141[3]-070[2]-043[2]-009[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-008[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-008[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-008[2]-009[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-012[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-012[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-010[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-011[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-011[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-011[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-008[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-008[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-012[2]-008[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-012[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-012[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-013[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-013[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-013[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-013[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-014[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-014[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-015[2]-009[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-015[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-012[2]-015[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-012[2]-015[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-009[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-013[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-013[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-013[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-015[2]-013[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-014[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-015[2]-014[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-008[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-008[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-008[2]-009[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-027[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-027[2]-037[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-035[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-035[2]-036[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-035[2]-037[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-035[2]-045[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-035[2]-046[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-038[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-038[2]-040[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-038[2]-046[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-039[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-039[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-039[2]-041[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-039[2]-045[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-039[2]-046[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-042[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-042[2]-036[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-042[2]-037[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-042[2]-040[2]-009[2]
227-141[3]-074[2]-044[2]-025[2]-042[2]-041[2]-009[2]
227-141[3]-074[2]-044[2]-031[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-031[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-044[2]-031[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-031[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-031[2]-033[2]-007[2]-009[2]
227-141[3]-074[2]-044[2]-034[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-008[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-008[2]-008[2]-009[2]
227-141[3]-074[2]-046[2]-008[2]-009[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-009[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-026[2]-036[2]-008[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-030[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-032[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-028[2]-041[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-030[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-030[2]-030[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-030[2]-034[2]-007[2]-009[2]
227-141[3]-074[2]-046[2]-030[2]-034[2]-043[2]-009[2]
227-141[3]-074[2]-046[2]-033[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-012[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-012[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-010[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-011[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-011[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-011[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-013[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-013[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-013[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-013[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-027[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-027[2]-037[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-035[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-035[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-035[2]-037[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-035[2]-045[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-035[2]-046[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-038[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-038[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-038[2]-046[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-039[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-039[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-039[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-039[2]-045[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-039[2]-046[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-042[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-042[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-042[2]-037[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-042[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-025[2]-042[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-026[2]-036[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-006[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-006[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-030[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-032[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-028[2]-041[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-063[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-063[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-063[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-064[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-064[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-051[2]-064[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-030[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-053[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-027[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-027[2]-037[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-054[2]-032[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-055[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-055[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-055[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-055[2]-032[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-026[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-026[2]-036[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-028[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-028[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-028[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-057[2]-029[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-059[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-059[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-059[2]-031[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-012[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-012[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-036[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-038[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-038[2]-040[2]-009[2]
227-141[3]-074[2]-051[2]-063[2]-038[2]-046[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-012[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-012[2]-015[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-036[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-039[2]-007[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-039[2]-008[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-039[2]-041[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-039[2]-045[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-039[2]-046[2]-009[2]
227-141[3]-074[2]-051[2]-064[2]-041[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-013[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-013[2]-013[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-013[2]-013[2]-015[2]-009[2]
227-141[3]-074[2]-052[2]-013[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-014[2]-007[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-014[2]-014[2]-007[2]-009[2]
227-141[3]-074[2]-052[2]-030[2]-007[2]-007[2]-009[2]
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227-166[4]-160[2]-160[4]-008[3]-009[2]
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227-166[4]-166[2]-012[3]-008[2]-007[2]-009[2]
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227-166[4]-166[2]-012[3]-012[2]-008[2]-009[2]
227-166[4]-166[2]-012[3]-012[2]-015[2]-009[2]
227-166[4]-166[2]-012[3]-013[2]-007[2]-009[2]
227-166[4]-166[2]-012[3]-013[2]-015[2]-009[2]
227-166[4]-166[2]-012[3]-014[2]-007[2]-009[2]
227-166[4]-166[2]-160[2]-008[3]-007[2]-009[2]
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227-166[4]-166[2]-166[2]-012[3]-008[2]-009[2]
227-166[4]-166[2]-166[2]-012[3]-015[2]-009[2]
227-166[4]-166[2]-166[2]-160[2]-008[3]-009[2]
227-166[4]-166[2]-166[2]-160[2]-161[2]-009[3]
227-166[4]-166[2]-166[2]-167[2]-015[3]-009[2]
227-166[4]-166[2]-166[2]-167[2]-161[2]-009[3]
227-166[4]-167[2]-015[3]-009[2]-007[2]-009[2]
227-166[4]-167[2]-015[3]-013[2]-007[2]-009[2]
227-166[4]-167[2]-015[3]-013[2]-015[2]-009[2]
227-166[4]-167[2]-015[3]-014[2]-007[2]-009[2]
227-166[4]-167[2]-161[2]-009[3]-007[2]-009[2]
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227-166[4]-167[2]-167[4]-015[3]-009[2]
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227-166[4]-166[4]-012[3]-008[2]-009[2]
227-166[4]-166[4]-012[3]-015[2]-009[2]
227-166[4]-166[4]-160[2]-008[3]-009[2]
227-166[4]-166[4]-160[2]-161[2]-009[3]
227-166[4]-166[4]-167[2]-015[3]-009[2]
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227-203[2]-070[3]-015[2]-009[2]-007[2]-007[2]-009[2]
227-203[2]-070[3]-015[2]-013[2]-007[2]-007[2]-009[2]
227-203[2]-070[3]-015[2]-013[2]-013[2]-007[2]-009[2]
227-203[2]-070[3]-015[2]-013[2]-013[2]-015[2]-009[2]
227-203[2]-070[3]-015[2]-013[2]-014[2]-007[2]-009[2]
227-203[2]-070[3]-015[2]-014[2]-007[2]-007[2]-009[2]
227-203[2]-070[3]-015[2]-014[2]-014[2]-007[2]-009[2]
227-203[2]-070[3]-043[2]-009[2]-007[2]-007[2]-009[2]
227-216[2]-119[3]-044[2]-008[2]-006[2]-007[2]-009[2]
227-216[2]-119[3]-044[2]-008[2]-006[2]-008[2]-009[2]
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227-216[2]-119[3]-115[2]-121[2]-042[2]-008[2]-009[2]
227-216[2]-119[3]-115[2]-121[2]-042[2]-036[2]-009[2]
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227-216[2]-119[3]-115[2]-121[2]-042[2]-040[2]-009[2]
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227-216[2]-119[3]-115[2]-121[2]-112[2]-037[2]-009[2]
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227-216[2]-119[3]-118[2]-034[2]-007[2]-007[2]-009[2]
227-216[2]-160[4]-008[3]-006[2]-007[2]-009[2]
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227-216[2]-215[4]-111[3]-035[2]-046[2]-009[2]
227-216[2]-215[4]-111[3]-112[2]-037[2]-009[2]
227-216[2]-215[4]-111[3]-120[2]-045[2]-009[2]
227-216[2]-215[4]-160[4]-008[3]-009[2]
227-216[2]-215[4]-160[4]-161[2]-009[3]
227-216[2]-215[4]-219[2]-120[3]-045[2]-009[2]
227-216[2]-215[4]-219[2]-161[4]-009[3]

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