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Atomic-scale structure of nanosized materials by x

1. 16 802 A Then we the user tell s RAD to compute G r up to 25 Ain steps of 0 02 A and use a moderate damping factor 0 005 to outweigh higher q data points since those are very often somewhat noisy Then we the user hit s Apply A separate pop up window appears this time labeled G r g r in the upper left corner see below HL atr p exp 29 If the user decides to plot view the computed G r or and g r he she should check the corresponding box es on that window In this example we chose to plot G r see below This PDF RDF G r shows a strong unphysical ripple close to r 0 A but in general is of pretty good quality for a PDF RDF obtained on an in house XRD equipment back in 1988 The G r has its first physically sensible peak positioned at about 3 5 A Gd Gd first atom neighbor distance and shows physical oscillations up to about 15 20 A This real space distance may be viewed as a length of structural coherence in the bulk metallic glasses studied The high frequency ripples above 20 A are data noise Here the user has an option to compute the so called pair correlation function g r as well He she may use either the experimental value for the atomic number density p stored 30 in the RAD project file usually a more correct approach or use an estimate for po computed after the low r slope of G r 2 3 The latter option is activated by checking the corresponding box in the
2. F Q sinQr dO 1 with F Q O S Q 1 The total structure factor S Q is the normalized scattered intensity from the powder sample PDF analysis yields the real local structure whereas an analysis of the Bragg scattering alone yields the average crystal structure Determining the PDF has been the ap proach of choice for characterizing glasses liquids and amorphous materials for a long time 8 However its widespread application to study crystalline materials has been relatively recent 9 Very high real space resolu tion is required to differentiate the distinct Ga As and In As bond lengths present in Ga In As High real space resolution is obtained by measuring S Q to a very high value of Q Omax 40 A An indium neutron ab sorption resonance rules out neutron measurements in the 1999 The American Physical Society 4089 VOLUME 83 NUMBER 20 Ga In As system We therefore carried out x ray pow der diffraction measurements To access Q values in the vicinity of 40 50 T itis necessary to use x rays with energies 50 keV The experiments were carried out at the A2 24 pole wiggler beam line at Cornell High Energy Synchrotron Source CHESS which is capable of deliver ing intense x rays of energy 60 keV Six powder samples of Ga _ In As with x 0 0 0 17 0 5 0 67 0 83 and 1 0 were measured The samples were made by standard methods and the details of the sample preparation will be reported elsewh
3. J Appl Cryst 22 1989 387 BH AM Ng 1 Rad gtk 1 0 RAD a program for analysis of X ray diffraction data from amorphous materials for personal computers Copwraht 1989 2010 USER s Manual written by Valeri Petkov E mail petkov phy cmich edu url http www phy cmich edu people petkov Document created February 2011 Disclaimer RAD is distributed as is 1 e without any warranty assumed or implied The entire risk as to the quality and performance of the program is with the User The use of RAD for any commercial purpose is prohibited Acknowledging RAD Any publication and or presentation of results obtained using RAD should make clear that RAD has been used and contain the following reference V Petkov RAD a program for analysis of X ray diffraction data from amorphous materials for personal computers J Appl Crystallogr 22 1989 387 89 a copy of the paper is attached for user s convenience Installing RAD WINDOWS XP VISTA 7 Users Download amp Run Rad Setup Application Let it install the GTK run environment and libraries as well as the program The set up application will create a program folder group associated with RAD Gtk see below and place a RAD icon on the screen A subfolder with examples of RAD usage will also be created Click on the RAD icon to run the program Before installing a newer version of RAD always Un install the current version to do it just click on Unista
4. Radial distribution function calculation window HER i g r p exp Data 6 00 g r p exp 5 00 4 00 2 00 1 00 3 00 0 0 5 0 10 0 15 0 20 0 25 0 r A A g r computed using the experimental value of po 0 039200 atoms A we entered in the RAD project file is shown above This g r shows a strong unphysical peak at distances below 1 A but otherwise is of pretty good quality Note as it should be g r oscillates around one while the corresponding G r around zero see the plot below 31 E Save As 34 close Ctrl 2 00 1 00 0 50 0 50 1 00 0 0 5 0 10 0 15 0 20 0 25 0 r A By checking the Data option in the upper left corner in the G r data plot window see above the user can save the data shown in this window in x y ascii or x y 0 1 ascii format Note the only way to save RAD s computed quantities i e I q S q q S q 1 G r or g r is i first to display the respective quantity in the Data plotting window ii view it iii and if satisfied save it using the Save As option RAD will NOT save anything unless it 1s told to do so 32 Help option of RAD Gtk Provides general and contact info about RAD and its authors see below EE RAD Gtk Experiment amp Sample Info Edit Experiment Data Processing Magi E ENS I Help Ctr H EN 4 brief user s manual and best examples are posted at Athos f www phy c
5. VOLUME 83 NUMBER 20 PHYSICAL REVIEW LETTERS 15 NOVEMBER 1999 High Real Space Resolution Measurement of the Local Structure of Ga _ In As Using X Ray Diffraction V Petkov I K Jeong J S Chung M F Thorpe S Kycia and S J L Billinge Department of Physics and Astronomy and Center for Fundamental Materials Research Michigan State University East Lansing Michigan 48824 1116 Cornell High Energy Synchrotron Source Cornell University Ithaca New York 14853 Received 7 June 1999 High real space resolution atomic pair distribution functions PDFs from the alloy series Gai In As have been obtained using high energy x ray diffraction The first peak in the PDF is resolved as a doublet due to the presence of two nearest neighbor bond lengths Ga As and In As as previously observed using x ray absorption fine structure The widths of nearest and higher neighbor PDF peaks are analyzed by separating the broadening due to static atom displacements from the thermal motion The PDF peak width is 5 times larger for distant atomic neighbors than for nearest neighbors The results are in agreement with model calculations PACS numbers 61 66 Dk 61 43 Dq 61 72 Dd The average atomic arrangement of crystalline semicon ductor alloys is usually obtained from the position and in tensities of the Bragg peaks in a diffraction experiment 1 and the actual nearest neighbor and sometimes next near est neighbor distances for various
6. 1545 1997 39 4807 1999 M F Thorpe et al in Local Structure from Diffraction edited by S J L Billinge and M F Thorpe Plenum New York 1998 p 157 16 J S Chung R I Barabash and M F Thorpe unpublished 814 Chem Mater 2006 18 814 821 Atomic Scale Structure of Nanocrystalline Ba Sr TiO x 1 0 5 0 by X ray Diffraction and the Atomic Pair Distribution Function Technique Valeri Petkov Milen Gateshki Markus Niederberger and Yang Ren Department of Physics Central Michigan University Mt Pleasant Michigan 48859 Max Planck Institute of Colloids and Interfaces Potsdam D 14424 Germany and Advanced Photon Source Argonne National Laboratory Argonne Illinois 60439 Received September 23 2005 Revised Manuscript Received November 13 2005 The atomic scale structure of nanocrystalline Ba Sr TiOs x 1 0 5 0 powders has been studied using high energy X ray diffraction Rietveld refinement and the atomic pair distribution function technique The studies show that the materials are well ordered at nanometer length distances The three dimensional atomic ordering in BaosSro5TiOs and SrTiO nanopowders may well be described by a cubic structure of the perovskite type similar to that occurring in the corresponding bulk crystals The three dimensional atomic ordering in BaTiO is more complex It is cubic like on average but locally shows slight distortions of a tetragonal type The new struc
7. 387 389 387 JOHNSON C K 1971 ORTEPII Report ORNL 3794 revised Oak Ridge National Laboratory Tennessee USA LARSON A C 1969 Acta Cryst A25 51 LARSON A C amp GABE E J 1978 Computing in Crystal lography edited by H SCHENK R OLTHOF HAZEKAMP H VAN KONINGSVELD amp G C BASSI pp 81 89 Delft Univ Press LE PAGE Y 1982 J Appl Cryst 15 255 259 LE PAGE Y 1987 J Appl Cryst 20 264 269 LE PAGE Y 1988 J Appl Cryst 21 983 984 LE PAGE Y amp GABE E J 1979 J Appl Cryst 12 464 466 MAGUIRE M 1982 Int J Man Mach Stud 16 237 MAIN P FISKE S J HULL S E LESSINGER L GERMAIN G DECLERCQ J P amp WOOLFSON M M 1980 MULTANS80 A System of Computer Programs for the Automatic Solution of Crystal Structures from X ray Diffraction Data Univs of York England and Louvain Belgium MOTHERWELL W D S amp CLEGG W 1978 PLUTO Program for plotting molecular and crystal structures Univ of Cambridge England RAD a program for analysis of X ray diffraction data from amorphous materials for personal computers By V PETKOV Sofia University Department of Solid State Physics Sofia 1126 Bulgaria Received 21 May 1988 accepted 10 February 1989 Abstract RAD is an interactive computer program for radial distribu tion analysis of X ray diffraction data from amorphous materials RAD has been written in Fortran 77 and runs on IBM
8. M Hwang S J Park D H Ren Y Petkov V J Phys Chem B 2004 108 14956 22 Billinge S J L Kanatzidis M G Chem Commun 2004 7 749 23 Petkov V Trikalitis P N Bozin E S Billinge S J L Vogt T Kanatzidis M G J Am Chem Soc 2002 124 10157 24 a Niederberger M Pinna N Polleux J Antonietti M Angew Chem Int Ed 2004 45 2270 b Niederberger M Garnweitner G Pinna N Antoneitti M J Am Chem Soc 2004 126 9120 816 Chem Mater Vol 18 No 3 2006 1400 1200 1000 800 600 Intensity arb units 400 Da TiO cubic model 200 BaTiO tetragonal model En n aiT a ee Hber E LED en T EET EFE RSET l1 PEEEEt Vibrett teeter NEN 2 4 6 8 10 12 Bragg angle 20 deg Figure 2 Experimental powder diffraction patterns for nanocrystalline Ba Sri T103 symbols and calculated patterns solid lines obtained through Rietveld refinements The positions of the Bragg peaks of the cubic upper set of bars and tetragonal lower set of bars structures that were fitted to the diffraction pattern of BaTiO are given in the lower part of the plot Note the diffraction data for SrTiO and BaosSro TiOs are ap proximated with a cubic type structure only The corresponding goodness of fit factors Ry for each of the refinements are reported in Table 1 the present experiments The scattered radiation was collected with an imaging plate detector ma
9. a J La 52 45 Atom XL C 5e BAe Lae CE PR ND PHe 54 Eu GO T DY HO ER THE TE Lu HF TA W RE 05 IR PT AU HG TL PB BT Pe AT RN FRe RA AC TH PAS u HFa Pue AFH a CMe DE CFa Crkay Aft 11 31 0 46 10 53 ll T 28 96 T Bl T 0T 5 5T 6 13 5 81 5 59 5 34 5 16 5 04 92 4 82 wie Tb amp 71 d 65 63 4 61 bg h 55 h 1 4 79 4 89 5 02 ae ee 5 44 5 70 3 98 6 36 6 82 7 38 B 14 8 93 10 1 11 B1 12 70 215 38 14 13 215 16 145 78 15 75 217 14 Af 10 29 11 19 B Bl 3455 3 82 amp 0B8 E 4 60 4 92 5 19 5 49 5 82 bl 6 37 56 75 1 12 Te Tet 8 24 6 67 9 18 9 74 10 23 10477 11 31 12 02 12 76 13 39 14 19 14 99 15 79 16 66 17 56 18 56 19 50 065 21 65 22 69 21454 22 59 20 60 21 49 22 50 23 452 S4 LT Pekay A 3 23 4 04 5 14 T 09 B 67 a T 10 39 212 63 9 03 788 T 16 5 655 b 24 5 91 5 70 5 955 5 33 5 18 5 04 5 92 4 13 4 67 60 4 58 4 57 4 62 4 66 4 1T3 81 94 5 11 5 32 5 62 5 B8 b5 27 5 6B T 23 7 B2 6 57 09 57 Af 9 99 10 83 11 50 12 45 10 74 8 37 6 88 3 57 3 82 204 4 27 LEE T9 4 225 5 25 52 54 5 78 5 04 6 40 6 74 Tel TeSf 7495 B 3T 8 40 9 35 9 92 10 41 11 03 11 65 12 28 12 95 13 65 L4 43 15 16 16 06 15 83 17 54 18 40
10. 19 27 20 04 20 91 11 245 21 87 12 09 20 77 13 68 1 62 46 Cukoy Are Af 1 20 7 15 14 0 7 275 1 66 8 30 2 01 6 90 95 9 46 3 01 10 05 3 7T0 10 65 4 55 11 22 b 02 ll 9Z2 29 89 10 17 9 18 10 72 9 31 8 27 9 B831 B 13 1 04 3 51 9 27 3 73 B 05 3 93 T 3b 4 10 6 685 4 29 5 53 4 55 b 11 4 76 25 83 5 0f 5 506 5 37 5 39 5 64 Self 5 94 5 08B 6 24 4 93 6 63 4 80 7 04 4 72 1 39 7 863 57 BZT 4 51 8 f2 4 47 9 19 4 46 9 49 46 lQ 2 4 51 10 74 4 52 11 40 4 61 11 95 4 70 12 52 4 85 13 06 Lu 13 668 54 18 14 22 5 4 14 84 5 68 15 53 b5 00 16 17 5 38 16 83 MoKery AT 20 559 05 53 0 48 0a 44 0 42 D 40 0 39 f 38 O 38 0 39 D 40 0 2 0 45 0 51 0 56 D 62 D 70 D 80 J0 03 1 17 J4 5 l T7T7 22405 2 36 2 T6 3 21 3 79 ie 54 5 a TO 9 20 9 02 Te42 5 9B 5 H07 T 18 7 79 9 19 10 55 8 32 Te4e b AG 6 44 b 12 Afi 2 30 fe SD 2266 Za Bb 3 04 3 22 3 5 3 60 3 82 i 03 foot PLI amp T3 Gu 95 Sele 5 49 5 15 6 03 6 34 5 65 6 99 Ta 34 T 68 Ba 05 Bs 43 8 84 9 225 9 7L 10 18 10 55 11 15 9 47 9 92 034 T bT B 04 8 38 B Th 9 09 6 27 fi ee 4 64 4 85 5 05 Se Aghay Aft AI 0 07 1 62 D T9 1 76 0 73 1 88 0 65T7T 2 02 0 52 2 415 0 58 2 26 0 54 2 51 0 51 32 54 0 4B 2 70 D 46 2 B5 O 44 3 0
11. 73 0 33 1 54 0 37 1 03 GAs 0 84 l T4 214 03 1 37 1 45 0 978 0 29 1 75 0 38 lal GE 0 B81 1 486 0 97 Lle4 1 31 1 04 0 21 1 94 0 35 1 29 AS O T4 2 09 0 B88 1 64 1 17 1 17 0 12 2 17 0 35 1 45 SEs 0 59 2 33 0 81 1 83 1 06 Le 31 D 0Z fae 0 32 l 6l BR 0O 64 2 60 0 74 2404 0 95 lesb 0 21 2 68 D 27T 14719 KR 0 50 2 89 0 68 2 26 0 88 1 62 0 47 2 596 O 21 1496 RB 0 58 3 23 0 63 2 53 0 80 1 81 0 89 3 29 0 12 2 20 0 57 3 58 0 59 2 H0 73 2 00 1 5 3 63 0 01 25443 u 0 5T 3 95 0 56 3 08 0 57 feel is J 4 00 0 18 2 b5T ZR Q0 60 4 31 0 54 3 37 O 52 2 42 3 14 0 78 0 41 2 93 NB D 55 4 78 D 54 43 74 0 58 24 6B 2419 0 86 0 73 3 22 MO O f2 5 16 0 54 4 04 0 54 2 89 1 79 0 93 1 22 3 51 TCa 0 80 5 63 0 56 4 40 0 51 3 15 1 55 1 02 2401 3 282 RU D 95 5 06 O 62 4 74 0451 3 39 3T 1 049 5 B8B e 16 RH 1 14 5 55 0 69 35 12 0 51 3 67 1 24 1 1B T2456 0 84 PD 1 36 7 06 0 7B 5 52 D 52 3 95 1 13 1 27 2 05 0 90 AG 1 64 7 62 D 90 5 96 0 55 4227 1 03 1 37 1 73 0 97 CD 2 00 8 25 206 6 45 0 55 amp 62 0 95 1 49 1 51 1 05 2 55 8 589 1 25 6296 Da 65l 4 98 0 88 1 60 21 35 1 L3 N 3 02 9 60 1 4 5 T 50 0 68 BeAT 0 41 1 73 l zz l z2 SB 3 T 104 33 1 80 B 0B8 Dafi Ss TS 0 75 1486 1 11 1 31 TE 7T9 11 10 2a l18 6 68 0 88 5 21 0 869 2400 1 0272 le i I 5 35 11 94 2 66 9 33 1 03 5 58 66 22 15
12. SrTiO and BaosSro5T1O possess an atomic arrangement that Chem Mater Vol 18 No 3 2006 821 is cubic type at both short and longer range interatomic distances The local symmetry with nanocrystalline BaTiO is tetragonal but the slight tetragonal distortions seem to average out and the structure of the material is better described in terms of a cubic like ordering at longer range distances That is presumably the reason nanocrystalline BaTiO similarly to SrTiO and BaosSro5TiOs does not show spontaneous polarization at room temperature This study is another demonstration of the ability of the PDF technique to yield three dimensional structural informa tion for materials of limited structural coherence including nanocrystalline materials The technique succeeds because it relies on total scattering data obtained from the material and as a result is sensitive to its essential structural features regardless of crystalline periodicity and size Acknowledgment Thanks are due to M Beno from APS Argonne National Laboratory for the help with the synchrotron experiments The work was supported by NSF through grant DMR 0304391 NIRT The Advanced Photon Source is sup ported by DOE under Contract W 31 109 Eng 38 CM052145G
13. Table 1 are as low 718496 as could be achieved with a PDF refinement These results support the findings of the Rietveld refinements that the atomic ordering in nanocrystalline SrTiO and BaosSrosTiO3 can be well described in terms of the perovskite cubic type structure space group Pm3m depicted in Figure la Models based on the perovskite tetragonal type structure space group P4mm were also attempted with the PDF data for SrTiO and BagsSro TiOs These models although having more internal degrees of freedom did not give any significant Nanocrystalline Ba Sr TiO3 x 1 0 5 0 Structure e DD 9O 6 EOCKe o SED Atomic PDF G A S 0 oco cen Nes e p ae Bn AY b PAE DP ER P Oe TD Gub Jb DE OD D n6 E e 6 6 2 Iw CS Tu du CaA So tr e 2 d a a F rfc eoe e RS sick UD Measure c Se Tah eem atate Sens CERO 0 5 10 15 20 25 30 Radial distance r A Figure 6 Experimental symbols and model solid line PDFs for SrTi03 The model PDF is based on a structure of the cubic type shown in Figure la The parameters of the model are given in Table 1 The reliability factor Ry 1s reported in the lower part of the figure The peak at 6 7 Ais given an enlarged scale in the inset Its shape is well reproduced by a cubic type model improvement in the reliability factors nor in reproducing the important details in the experimental data This observation
14. a project file the user should select the Save option see below 13 EE RAD Gtk Experiment Melee Edi Experiment Data Processing Help Ctrl h Chro Ctrl 5 Ctrl g Again RAD project files are text files once saved on disk they may be opened viewed with any text editor As an example below is a print out of the RAD project file for Gdo 57Alo 43 we are discussing here amp xml version 1 6 encoding UTF 8 gt amp t RadGTK v1 XML file lt rad xml gt amp t Chemistry information gt lt chemistry gt species number 2 gt 2 id 0 gt 1347 Z gt 2 1d 1 gt 64 lt 2 gt lt species gt lt element symbol al gt lt name gt Aluminum lt name gt amp z2 13 amp 22 amp cancentration BO h54388884 concentration Fp 8 856888 fp lt fpp gt 6 651 080 lt Fpp gt lt element gt lt element symbol Gd gt lt name gt Gadolinium lt name gt lt z764 2 gt lt concentration gt 6 57 66006 lt concentration gt lt fp gt 8 560080 lt Fp gt lt fpp gt 3 988080 lt Fpp gt lt element gt lt chemistry gt amp t Experiment information amp experiment lt wavelength gt 6 769688 wavelength gt Cabsorption sample gt 6 675606 lt absorption sample gt Cabsorption substrate 6 666068 lt absorption substrate gt lt density gt 6 639200 lt density gt amp deadtime 88888824 deadtime lt geometry gt Reflection lt geometry gt lt monochromator gt In hou
15. and Ag Ka radiation after D Cromer Acta Cryst 18 1965 p 17 Note Af and Af are much smaller than f which is pretty close to Z the atomic number of the respective atomic species CrK a Fek ory Cust ii MCF c AqgE cn Atom Af Af Af Age Aft AI Aft fin Aft Ag d a 0 19 OQ 12 O 14 0 09 0 10 0 C3 0 92 0 02 e Ce His 20 0 27 0 15 0 21 0 12 0 14 0 0 0 0 U S O 0e Hos DO 25 039 ag z20 0 29 0 15 a 19 0 05 O US 0 045 0 03 AL O29 0 53 0 25 0 39 0 19 0 27 O Q0T O 0F 0 05 U 05 S1 Q 33 0 70 0 29 04 53 Q 23 0 36 0 09 0 0979 0 06 0 06 Pe O 35 0 390 0 327 O468 Q 27T Q 4 65 0 11 O 12 0 08 0 08 5 O 36 1 15 D 35 06 86 0 31 0 58 0 13 O 16 0 09 2 10 CL Oed la 0 36 1 07 0 33 O f2 0 15 0 19 ll 0 13 ARa O T 1 76 0 35 1 32 0 3656 0 89 D 1B O z24 0413 O 16 Ke O Li 2 19 0 31 1 64 O37 1 11 0 21 0 30 0 15 0 20 CAS O l6 2 64 0 21 1 98 0 36 1 434 0 24 0 36 D IB 0 245 5Cs 0 55 3 416 0 02 2 38 04 33 1 642 OQ 26 0 43 0 20 0 429 TI 14 65 3 73 0 29 2 680 Da 1 90 0 429 04 51 0 23 D 34 Ve A T2 O 62 84 3 29 0 10 2 423 0 31 0 60 0 75 0 40 CR 2 28 O TL 1 91 3 83 O L3 oe OF 34 0 69 O 2T Oe 46 AM 1 75 0 82 3 76 0 64 0 50 3 00 0 36 0 80 O 30 Du 54 FE 1 47 0 94 2 21 0 74 Laka 3 45 Q 37 Q 9Z 0 32 62 CO 1 28 1 07 1 74 0 64 2 51 3495 0 37 1 06 0 34 O0 Tl Ni 1 15 1 19 46 0 93 23 20 0 6T D a7 1 20 D 35 D BO cu 1 05 1 33 1431 1 204 2 15 0 T5 04 36 1 36 0 35 0 91 IN 0 975 1 49 1 17 1 17 1
16. available from the author on a A upon request am indebted to Mr N Zotov IPM BAN Sofia for valuable discussions and advice References CROMER D T 1965 Acta Cryst 18 17 23 CROMER D T amp MANN J B 1968 Acta Cryst A24 312 323 J Appl Cryst 1989 22 389 393 COMPUTER PROGRAMS 389 KAPLOW R STRONG S amp AVERBACH B 1965 Phys Rev Sect A 138 1336 1345 KLUG H P amp ALEXANDER L E 1954 X ray Diffrac tion Procedures p 791 New York Wiley KONNERT J H amp KARLE J 1973 Acta Cryst A29 702 710 PETKOV V APOSTOLOV A amp SKUMRYEV V 1989 J Non Cryst Solids 108 75 79 PiNGS C J amp WASER J 1968 J Chem Phys 48 3016 3018 RULAND W 1964 Br J Appl Phys 15 1301 1307 SAVITZKY A amp GOLAY M J E 1964 Anal Chem 36 1627 1640 THUSSE B J 1984 J Appl Cryst 17 61 76 WAGNER C N J 1969a J Vac Sci Tech 6 650 657 WAGNER C N J 1969b Adv X ray Anal 12 50 70 WAGNER C N J 1978 J Non Cryst Solids 31 1 40 WARREN B E amp Mozzi R L 1970 J Appl Cryst 3 59 65 SIR88 a direct methods program for the automatic solution of crystal structures By M C BURLA Dipartimento di Scienze della Terra Universit 06100 Perugia Italy M CAMALLI Istituto G Giacomello Area della Ricerca CNR 00016 Montelibretti Roma Italy G CASCARANO and C GIACOVAZZO Dipartimento Geomi
17. bond and the closed symbols for the In As bond Solid lines theory See text for details The mean square static and thermal distortions are added Here Me represents both the metals Ga and In which behave in the same way Note that the scale in the lower panel is expanded by a factor of 10 compared to the upper panel 4091 VOLUME 83 NUMBER 20 G r o Distance r A FIG 5 Experimental open circles and theoretical solid line PDFs for Ga In As for concentrations x 0 5 and x 0 33 previous paragraph We therefore expect the squared width A to be a sum of the two parts The thermal part o is almost independent of the concentration and we fit o by a linear function of the composition x between the two end points in Fig 4 To better understand the strain it is convenient to assume that all the force constants are the same and independent of chemical species Then it can be shown 16 for any such model that Aj oj Boxe Roc cR 5 x95 where the subscripts ij refer to the two atoms that lead to a given peak in the reduced PDF For the Kirkwood model the A are functions of the ratio of force constants 8 only It further turns out that the Aij are independent of whether a site in one sublattice is Ga or In so we will just refer to that as the metal site By taking mean values from the force constants used in the simulation we find that B a 0 83 and that for nearest neighbor pairs A 0 0712 For mor
18. carrection may be performed by means of repeated Fourier transfor mations Kaplow Strong amp Averbach 1965 and a new extended file up to 2s with corrected i s values may be created This file may be used as an input data file in the subroutine CALCRD RAD has been tested using published X ray diffraction data for silica glass Konnert amp Karle 1973 Fig 1 shows the agreement between si s values computed by RAD and by RADILS Konnert amp Karle 1973 The program RAD proved to be useful for analysis of X ray diffraction data from metallic glasses Petkov Apostolov amp Skumryev 1989 3 Implementation of RAD RAD consists of 1500 statements written in Microsoft Fortran 77 occupying about 110 Kbyte of core memory It runs on IBM PC XT AT under PC DOS 3 1 DOS utilities B T 2 0 E dat ee ee p ccm 0 5 10 15 s A Fig 1 Interference functions si s for silica glass Full line after RAD multiplied by a proper constant factor because of the different sharpening factors used broken line after RADILS with permission from J Karle and commands can be easily performed without leaving RAD During the data processing the raw data the corrected data the independent coherent and the incoherent scatter ing the interference function i s the reduced RDF and the RDF can be displayed by means of built in graphic routines if the computer is equipped with a CGA graphics card rtran source code is
19. for BaTiO3 The model PDF is based on the tetragonal type structure shown in Figure 1b The parameters of the model are given in Table 2 The reliability factor Ry is reported in the lower part of the figure A portion of the experimental data open circles is compared to model ones based on cubic type solid line and tetragonal type structures solid symbols in the inset on an enlarged scale The experimental data are better reproduced by the tetragonal type model BaTiO orthorhombic 0 8 0 6 2 A Atomic PDF G o o e N 0 2 0 0 2 5 5 0 7 5 10 0 Radial distance r Figure 8 Experimental symbols and model solid line PDFs for BaTiO3 The model PDF is based on the orthorhombic type structure shown in Figure lc The reliability factor Rw is reported in the lower part of the figure An arrow marks the position of the first PDF peak where the model and experimental data show a strong disagreement The same peak is given in the inset on an enlarged scale distances of 2 0 A and two Ti O distances of 2 16 A with the orthorhombic model three Ti O distances of 1 87 A and three Ti O distances of 2 13 A with the rhombohedral model resulting in a split first PDF peak a feature the experimental data do not show On the other hand the model based on the tetragonal type structure considerably improves the reliability factor compare the PDF based R values reported in Tables and 2 calculated over the whole rang
20. normalization constant and q S q 1 data Here we find that a small additional correction 3 to the corrected data and a normalization constant of 0 0864 give a satisfactory result The respective q S q 1 is given in the plot below This q S q 1 behaves well up to q values of about 22 but then shows an unphysical upturn likely due to an improper Compton scattering correction Users often may see similar unphysical behavior of their data Possible solutions re do the data corrections from the very beginning look into the sample thickness absorption correction background scattering collect the data one more using another detector etc If nothing works still save the q S Q 1 data 39 EE O S Q 1 aan Data 2 50 Q S Q 1 0 50 1 00 00 20 40 60 80 100 120 140 16 0 18 0 200 220 240 2 q A 1 Then open the x y ascii file with any text editor and chop off the higher q part of the q S Q 1 that misbehave In our case we chopped off the data above q 22 5 At User s do NOT use distorted misbehaving not oscillating around zero etc q S q 1 data sets If you still have to do it be aware that the quality of the respective PDF RDF data may be compromised very seriously Do extra PDF RDF quality checks e g see 8 In this example we decide to use q S Q 1 data up to about 22 5 A only see below 40 li O1S Q 1 BEK Jata 1 50 1 00 0 50 0 00 0 50 1 00 L 00 20 40 60 8
21. pairs of atoms by XAFS measurements 2 In this Letter we show how high energy x ray diffraction and the resulting high resolution atomic pair distribution functions PDFs can be used for studying the local atomic arrangements in Ga _ In As al loys We show that the first peak in the PDFs can be re solved as a doublet and hence the mean position and also the widths of the Ga As and In As bond length distribu tions determined The detailed structure in the PDF can be followed out to very large distances and the widths of the various peaks obtained We use the concentration de pendence of the peak widths to separate the broadening due to static atom displacements from the thermal broad ening At large distances the static part of the broadening is shown to be up to 5 times larger than it is for nearest neighbor pairs Using a simple valence force field model we get good agreement with the experimental results Ternary semiconductor alloys in particular Ga _ In As have technological significance because they allow important properties such as band gaps to be tuned continuously between the two end points by varying the composition x Surprisingly there is no complete experi mental determination of the microscopic structure of these alloys On average both GaAs and InAs form in the zinc blende structure where Ga or In and As atoms occupy two interpenetrating face centered cubic lattices and are tetrahedrally coordinated to each othe
22. ray Diffraction Procedures for Polycrystalline Materials Wiley New York 1974 28 Petkov V Jeong I K Chung J S Thorpe M F Kycia S Billinge S J L Phys Rev Lett 1999 85 4089 29 Petkov V Billinge S J L Sashtri S D Himmel B Phys Rev Lett 2000 55 3436 30 Petkov V Yunchov G J Phys Condens Matter 1996 8 6145 31 Petkov V Zavalij P Y Lutta S Whittingham M S Parvanov V Shastri S D Phys Rev B 2004 69 085410 32 Dmowski W Egami T Swider Lyons K E Love C T Rolison D R J Phys Chem B 2002 106 12677 33 Gateshki M Petkov V Williams G Pradhan S K Ren Y Phys Rev B 2005 71 224107 34 Petkov V J Appl Crystallogr 1989 22 387 Nanocrystalline Ba Sr TiO3 x 1 0 5 0 Structure 3 N Atomic PDF G 10 15 20 25 30 Radial distance r A Figure 3 Experimental PDFs for nanocrystalline Ba Sr TiO3 extracted from the corresponding powder diffraction patterns using data extending to wave vectors as high as 28 The position of the first PDF peak reflecting Ti O first neighbor atomic pairs is marked with an arrow Note the intensity of the first PDF peak changes with the relative Sr Ba content due to the different scattering power for X rays of Ba and Sr A broken line runs as a guide to the eye through the shoulder of the PDF peak centered at approximately 6 7 A The shoulder diminishes with Sr conte
23. reduced RDFs G r are shown re spectively in figs 3 and 4 For checking the influence of the high angle XRD data upon the final G r values the upper V Petkov et al Radial distribution functions for RE Al metallic glasses ih iade i n3 Emm os 5 10 15 S UT Fig 1 The umcoczecsed reduced interference function F s for Gd Al aed tte cesdual difference after subtraction of the Fi s comad sccording to the method of Kaplow et al 8 n 0 5 10 15 20 R A Fig 2 The uncorrected reduced radial distribution function G r for Gd4Al and the residual difference after the subtrac tion of the G r corrected according to the method of Kaplow je Aij 1 et al 8 8 YM eA Dy Al 0 3 Li 0 Tb Al i Gd AL P i i r Al 0 h 3 Fig 3 Total reduced interference functions for RE Al metallic glasses with displaced zeroes R 0 5 10 20 15 R L Fig 4 Total reduced radial distribution functions for RE4Al4 metallic glasses with displaced zeroes limit of integration Smax was changed up to s 2 In addition a complementary XRD experiment was performed for the Gd Al sample using Co K radiation and Ross balanced Fe Mn filters However it produced no qualitative changes in the G r Therefore a conclusion can be drawn that all details of the G r including the weak shoulder on the second peak near r 7 A see fig 4 have physical meaning and the atom atom sep
24. reinforced our conclusion that nanocrystalline SrTiO and BaosSro TiOs studied by us possess a cubic type structure at room temperature as their crystalline analogues do The atomic ordering in BaTiO however is not so well described in terms of the perovskite cubic type structure as the relatively high value 25 see Table 1 of the corresponding reliability factor Ry shows That is why we attempted three more structural models based on the other three structural modifications tetragonal space group P4mm orthorhombic space group Amm2 and rhombohe dral space group R3m occurring with bulk crystalline BaTiO3 Results from PDF analyses of the experimental data in terms of these three structural models are presented in Figures 7 8 and 9 respectively As the results in Figures 8 and 9 show the orthorhombic and rhombohedral type models see Figures 1c and 1d may be unambiguously ruled out because they feature substantially distorted Ti Og octahedral units i e a broad distribution of first neighbor Ti O distances 2 Ti O distances at 1 86 A two Ti O 45 Here it may be noted that the agreement factors achieved with the PDF refinements appear somewhat higher when compared to those resulted from the Rietveld refinement of diffraction data in reciprocal space see Tables 1 and 2 This reflects the fact that an atomic PDF differs from the corresponding XRD pattern and is a quantity much more sensitive to the local atomic orde
25. 0 0 4 3411 0 42 3234 0 42 3 50 0 42 3 68 42 3 48 0 44 4 06 0 46 He 2b 0 48 4 44 0 51 4 T0 0 55 4 93 0 60 5 418 20 55 5 43 0 TZ 5 69 0 81 5 95 D0D 90 6 24 1 00 6 55 l1 13 6 85 1 28 7 18 1 455 Ta Se 1 65 7 88 1 B9 B 245 2 16 B 52 2249 9 03 2 87 9 42 3 31 9 BT7 3 95 10 29 4 85 10 7 6 10 11 18 7 61 49 52 B 04 6 89 6 T1 Telt 5 233 T 6 healt Ta15 5 23 8 0 Appendix 3 Atomic Number density definition ATOMIC NUMBER DENSITY Number of Atoms n and Number Density N The number of atoms or molecules n in a mass m of a pure material having atomic or molecular weight M is easily computed from the following equation using Avogadro s number N4 6 022x 10 atoms or molecules per gram mole mN n 4 1 M In some situations the atomic number density N which is the concentration of atoms or molecules per unit volume V is an easier quantity to find when the density p is given vat PN av V M 2 Number Density for Compounds For a chemical compound mixture Z which is composed of elements X and Y the number atom density of the compound is calculated from EM Nz N Eni Av 3 mix In some cases the desired quantity is the number density of the compound constituents Specifically if Z X Y then there are p atoms of X and q atoms of Y for every molecule of Z hence Ny pNz 4 Ny qNz Example Calculate the number den
26. 0 100 120 140 160 180 200 22 q 1 We compute the respective G r using the parameters shown below d Radial distribution function calculation RDF calculation i 1246 9 points G Sro 1 data File Program Files rade p Q 22 410000 max PH 40 000000 Br step 0 020000 Damping Factor 0 002500 usually 0 005 Number density after air slope Atoms LE show f hide curves toolbox And obtain pretty good results see below i Gir Data 1 00 G r 0 75 0 50 0 50 0 0 5 0 10 0 15 0 20 0 29 0 30 0 35 0 r A The examples above are just a glimpse of what RAD users may face in real life Indeed atomic PDF RDF analysis is not so difficult as it looks in first sight Sure some time and persistence are necessary to become a successful RAD user But isn t the same with any research effort Good luck 42 5 f References 1 C Giacovazzo et al in Fundamentals of x ray crystallography Oxford University Press 1998 2 H P Klug and L E Alexander in X ray diffraction procedures for polycrystalline and amorphous materials John Wiley amp Sons 1974 T Egami and S J L Billinge in Underneath the Bragg peaks Pergamon Press Amsterdam 2003 V Petkov Maerials Today 11 2008 28 3 D A Keen J Appl Cryst 34 2001 172 4 B Toby and T Egami Acta Cryst A 48 1992 336 5 P J Chupas et al J Appl Cryst 40 2007 463 6 V Petkov et a
27. 00 Dispersion correction F 2 3 oo 3 900000 1 D Wwaasmaier et al Acta Cryst A51 1995 p 416 2 http aa nist gav physlab dataJFFastJindex cfr 3 D Cromer Acta Cryst 18 1965 p 17 soo Next by clicking on the Experiment Description button left upper corner the user is allowed to supply information about the particular XRD experiment set up as shown below 11 4 Pro ject settings Sample Description Experiment Description Experiment Description ray wavelength 0 703000 A Linear absorption sample 0 675000 Ht Linear absorption substratre cell 0 000000 HE Sample s density 0 039200 atoms A gt Debector s dead time 0 000000 5 Experimental geometry Polarization type In house circular Es m a m m T m zi lt Compton scattering Here we specify that x rays with a wavelength A 0 709 A i e Mo Ka has been used the sample was measured in flat plate reflection geometry the detector was fast enough for the signal coming from the sample 1 e 0 2 us dead time and the x ray radiation was circularly polarized which is typical for sealed x ray tube sources There has been no x ray energy discrimination during the XRD data collection and hence the XRD data contain a contribution coming from Compton inelastic scattering 1 e Compton scattering 12 is All in the collected XRD data The liner absorption factor yt for the particular Gdo 57Alo 43 samp
28. 056 and f 20 051 for Al when Mo Ka radiation is used Information for f and f may be obtained from the National Institute of Standards amp Technology www site http www nist gov physlab data xray_gammaray cim or from literature sources e g the f and f values for Mo and Ag radiation as computed by D Cromer are given in Appendix 2 for RAD users convenience Information about the atomic scattering factors f q for atomic species with Z 1 to Z 98 comes with RAD 10 so the user should not worry about it Here is the place to note that RAD can handle XRD data sets from materials composed of up to 10 atomic species with atomic numbers Z from 1 to 98 Selecting Atomic Species 2 from the drop down list allows the user to supply information about Gd as exemplified below 4 Pro ject settings Sample Description Sample description Experiment Description E Number of chemical species max 10 p Z Species information Atomic species 2 v Atomic number 54 Element ad Gadolinium i Atomic concentration 0 570000 Parameters of the 5 Gaussian Fit 1 to the x ray atomic scattering Factors al 24 598117 bi 2 435028 az 17 104952 E 246961 aa 13 222581 b3 13 996325 a 3 266152 b4 110 563091 a5 48 995213 b5 0 001383 c 43 505684 Dispersion corection F 2 3 0 5600
29. 1 molar equiv of titanium isopropoxide and the reaction mixture was transferred to a steel autoclave and heated at 574 K for 48 h The heating took place under subsolvothermal conditions since the boiling point of benzyl alcohol is about 478 K Representative TEM images of thus obtained nanocrystalline materials are shown in Figures 2 3 and 4 in ref 24 The TEM images reveal that the samples consist of individual particles with an average size of about 5 nm Analyses based on the width of the peaks in the X ray diffraction patterns of Ba Sr TiO performed by us yielded very similar estimates for the average nanocrystallites size The TEM studies also show that the nanocrystals are uniform in size and mostly spherical No large particles or agglomerates are observed Thus obtained nanocrys talline Ba Sr TiO x 1 0 5 0 powders were loaded into glass capillaries and subjected to synchrotron radiation scattering experi ments 2 2 Synchrotron Radiation Scattering Experiments Syn chrotron radiation scattering experiments were carried out at the beamline 11 ID C Advanced Photon Source Argonne National Laboratory using X rays of energy 115 232 keV A 0 1076 at room temperature X rays of higher energy were used to obtain diffraction data to higher values of the wave vector Q which is important for the success of PDF analysis Qmax 28 with 20 Toby B H Egami T Acta Crystallogr A 1992 48 336 21 Gateshki
30. 4 and A is a modified wavelength are here calculated The corrected X ray diffraction data are scaled into elec tron units by the so called high angle method Wagner 1978 Bux j Y xis wc ds je Fin 5 mas f Uris nG EPG AGYT as 6 and the interference function i s is calculated As an COMPUTER PROGRAMS independent check the integral J Si s ds 2 20 po 7 is computed according to the so called sum rule Wagner 1978 When a satisfactory normalization is reached by varying the low limit of integration Smin 5444 2 Smin Smax the i s data are stored as a new file The subroutine CALCR D uses as input an interference function data file and again the parameter data file The reduced RDF G r is calculated properly damped Klug amp Alexander 1954 as a Fourier transform of the i s G r 2 7 f si s sin sr exp bs ds 8 where b is the damping factor A straight line is fitted to the G r in the range r 0 1 A to obtain an estimate for the average atomic density po on the basis of the expression G r 47por 9 which holds only for small values of r The RDF is calcu lated and the reduced RDF or optionally the RDF is stored as a new data file Residual errors in the i 5 due to incorrect normalization of the data may introduce spurious oscillations in the corresponding G r in the region between the origin and the first peak In order to avoid such errors a
31. 5 3 6 6112 j83 9672 395 1506 409 4646 27 5286 449 3842 476 0588 511 2541 552 3105 602 57851 658 5657 17 2819 3 2806 918 5352 846 5656 851 1745 829 06508 T87 047 29 4600 665 7761 685 3624 550 0443 501 5684 459 2458 421 5414 389 0881 361 3227 339 1941 321 6118 385 7858 297 8141 286 2086 The number of data points per file experimental or RAD computed may not be more than 30 000 Also a single experimental XRD intensity value and therefore computed 27 I q S q and G r value should not be larger than 10 Users please try not to exceed count rates of 10 cps per single detector do not saturate detectors and so go in the non linear detector dead time correction regime Keep the number of data points per pattern below 30 000 and normalize the XRD intensities in your patterns per second per the number of detectors per the number of angular sectors of integration if using an area detector etc 1 e take care that no single intensity data point in the experimental XRD files has a value greater than 10 Then the XRD data corrections implemented in RAD may work Indeed why such limitations Users RAD is just like a car A car is designed to tow a particular load carry a particular number of passengers run with a particular maximum speed etc Attempts to run a car beyond its build in capabilities will blow its engine out The same will happen to RAD No good for anybody Also the author VP af
32. Commun 2005 135 290 9 Thongrueng J Nishio K Watanable Y Nagata K Tsuchiya T J Aust Ceram Soc 2001 37 51 10 Josef J Vimala T M Raju J Murthy V R K J Appl Phys 1999 32 1049 11 Liu R S Cheng Y C Chen J M Liu R G Wang J L Tsai J C Hsu M Y Mater Lett 1998 37 285 12 Uchino K Sadanaga E Hirose T J Am Ceram Soc 1989 72 1151 13 Begg B D Vance E R Nowotny J J Am Ceram Soc 1994 77 3186 14 Yashimura M Hoshina T Ishimura D Kobayashi S Nakamura W Tsurumi T Wada S J Appl Phys 2005 98 014313 15 Matsuda H Kobayashi N Kobayashi T Miyazawa K Kuwabara N J Non Cryst Solids 2000 271 162 16 Zhang J Yin Z Zhang M S Phys Lett A 2003 310 479 17 Takeuchi T Tabuchi M Ado K Honjo K Nakamura O Kageyama H Suyama Y Ohtori N Nagasawa M J Mater Sci 1997 32 4053 18 Soten I Miguez H Yang S M Petrov S Coombs N Tetreault N Matsuura N Ruda H E Ozin G Adv Funct Mater 2002 12 19 Frey M H Payne D A Phys Rev B 1996 54 3158 Chem Mater Vol 18 No 3 2006 815 scale structure Usually the structure of materials is deter mined from the Bragg peaks in their diffraction patterns However nanocrystalline materials lack the extended order of the usual crystals and show diffraction patterns with a pronounced diffuse c
33. PC XT AT or compatible computers i Introduction The radial distribution function 47r p r RDF is used to characterize amorphous structures It represents the number of atoms in spherical shell of radius r and unit thickness The function is zero for values of r less than the hard sphere diameter of the atoms and modulates about 4mr p for larger values of r where p is the average atomic density of the amorphous material Peaks in the RDF r indicate frequently occurring atom atom distances the area under a peak is equal to the average number of atom pairs within a particular range of distances Klug amp Alexander 1954 The reduced RDF G r 2 4zr p r po is associated by a Fourier transformation with the interference function i s which is the structure dependent part of the experi mental X ray diffraction data In the program RAD the interference function is defined following Pings amp Waser 1968 as i s eco to 5 zn 1 where s 4s sin 8 A 20 is the scattering angle A the 0021 8898 89 040387 03 03 00 wavelength are the atomic scattering factors x the molar fractions of components n the number of atomic species and 7 s is the coherently scattered intensity in electron units i The aim of the radial distribution analysis is to evaluate the RDF r as the Fourier transform of the i s which in turn can be obtained from the experimental intensities I s The total scattered
34. actor Ry is reported in the lower part of the figure 1 0 0 8 e D sE OOD 64 66 68 e I CIE RO OD emer aam cz occ o N ns maa e Atomic PDF G S O D T d qi y 1 D 1 i q ib E q d 1 i D q q q q 1 OP o 4 b 9 D q q OL D 4 cQ l D p Ce P SSeS rm e NA NASA ER A ecece DOO oec au A itt PP 8 ur espressi Doaa 2 e RI RD Cerea Sane e CS ri e fud Coe dk BO OO er Oe ao COs SS emce TS eer Ow Cog COME OU Cr qoc 7 AR fads Ae OO a an 79 q d CO 0 5 10 15 20 25 30 Radial distance r Figure 5 Experimental symbols and model solid line PDFs for Bao Sro 5TiOs The model PDF is based on a structure of the cubic type shown in Figure la The parameters of the model are given in Table 1 The reliability factor Rw is reported in the lower part of the figure The peak at 6 7 is given an enlarged scale in the inset Its shape is well reproduced by a cubic type model refined values of the structural parameters from the analysis are summarized in Table 1 In the case of SrTiO and Bag 5 Sro s TiO the PDF based fit yielded structural parameters that are in good agreement with the present Rietveld results see Table 1 The agreement documents well the fact that the atomic PDF provides a firm quantitative basis for structure determination The reliability factors defined by eq 5 also reported in
35. analyses of the experimental data for Ba Sr TiOs x 1 0 5 0 in terms of the cubic structure are presented in Figures 4 5 and 6 Structure data from literature sources were used as initial values in the PDF refinements The PDF refinements were done with the help of the program PDFFIT To mimic the presence of limited structural disorder in the nanocrystalline materials we multiplied the model PDF data with a decaying exponent of the type exp arv as originally suggested by Ergun and later on implemented in a similar manner by Gilbert et al Typical values for a used were of the order of 0 1 The 5 40 Buttner R H Maslen E N Acta Crystallogr 1983 59 7764 41 Evans H T Acta Crystallogr 1967 1 1948 42 Hewat A W Ferroelectrics 1974 6 215 43 Proffen T Billinge S J L J Appl Crystallogr 1999 52 572 44 Ergun S Schehl R R Carbon 1973 11 127 Gilbert B Huang F Zhang H Z Waychunas G A Banfield J F Science 2004 305 65 Petkov et al BaTiO cubic ee OCD ja mm Z ar a es OS 9 rer AP TOR T D D D q D DI q q D b D q D D O b D p D D Decet OQ e BOA 0 5 10 15 20 25 30 Radial distance r A Figure 4 Experimental symbols and model solid line PDFs for BaTiO The model PDF is based on the cubic type structure shown in Figure 1a The parameters of the model are given in Table 1 The reliability f
36. and corrections are done without any problem RAD notices problems reports about them a separate pop up window appears labeled I q in the up left corner see below It allows the user to view the experimental sample amp background XRD data sets and the corrected I q data Note occasionally this and other smaller size RAD program windows may appear pop up behind the main RAD program window and so remain hidden from the user s sight Users feel free to shift RAD windows around on the computer screen to be able to see access them in the most convenient for you way By checking the boxes Raw data and Background see below Corrected data 28 the user can plot view the experimental data he she is analyzing They appear in yet another Data plotting window see below 17 lili Raw data 26 Background 28 e E Data 900 00 Raw data 26 Background 26 800 00 700 00 600 00 500 00 400 00 300 00 200 00 100 00 0 0 29 0 50 0 15 0 100 0 125 0 28 If when the user checks the Corrected data box see below Raw data 28 Background 2B ell scattering zB he she can plot view the corrected and extrapolated to q 0 data I q see below 18 tea Corrected data 28 Data If the user is satisfied with the experimental XRD data corrections he she could check the Data option in the upper left corner of the Data plot wind
37. arations and coordination numbers derived from RDFs are within the error limits stated below All of the above mentioned calculations and corrections were performed on an IBM PC XT with use of the program RAD 9 3 Discussion As shown in fig 3 the F s curves for all of the samples except Pr Al are composed of a sharp first maximum second maximum with a shoulder on its large s side and subsequent small oscilla tions due to the poor XRD data statistics The shoulder of the second F s peak for Pr Al is not very strong but is still clearly visible The positions of the maxima are listed in table 1 The values obtained for Gd Al correspond with those reported by Buschow 10 Figure 4 shows that the features of the reduced RDFs do not vary substantially with the con stituent RE elements For all samples the correla tions between the atomic positions decrease with increase of the atom atom separations and vanish 78 V Petkov et al Radial distribution functions for RE Al metallic glasses Table 1 Positions of the maxima of the interference functions for REAL in A Pr4Al Gd Al Tb Al Dy Al First 2 25 2 30 2 39 2 39 Second 3 95 3 80 3 75 3 75 Shoulder 4 50 4 45 4 50 Third 5 60 5 65 5 70 5 70 completely at about 15 A thus indicating the existence of a short range topological order only It is commonly known that for multicomponent alloys the total G r derived from a single dif fraction experiment is a su
38. at couple to the change in the angle between adjacent nearest neigh bor bonds We choose these parameters to fit the end members 3 with ag44 96 N m Q in as 97 N m D Ga As Ga DAs Ga As 10 N m and VOLUME 83 NUMBER 20 65 604 99 50 45 40 0 1 23 4 5 6 7 8 9 10 o Distance r FIG 2 The reduced PDF G r for Ga In As measured at 10 K for various concentrations Bin As In DAs In As 6 N m The additional an gular force constant required in the alloy is taken to be the geometrical mean so that jfgG4AsIn Ji Bossa DIR eda We have constructed a series of cubic 512 atom periodic supercells in which the Ga and In atoms are distributed randomly according to the composition x The system is relaxed using the Kirkwood potential to find the displacements from the virtual crystal positions The volume of the supercell is also adjusted to find the minimum energy Using this strained static structure a dynamical matrix has been constructed and the eigenvalues and eigenvectors found numerically From this the Debye Waller factors for all the individual atoms in the supercell can be found and hence the PDF of the model by including the Gaussian broadening of all the subpeaks which is the correct procedure within the harmonic approximation 15 The model PDF is plotted with the data in the inset to Fig 3 and in Fig 5 The agreement at higher r is comparable to that in the r range shown All the individ
39. cause of the inherently diffuse nature of the respective XRD patterns However the reciprocal space resolution of the experimental set up including that of the detector should not be too low either As an example XRD patterns for polycrystalline Si NIST powder standard collected with two different types of detectors a single point detector and an Image Plate mar345 detector are shown in Fig 1 The lower resolution of the XRD data collected with an IP detector leads to an extra broadening of the peaks in the XRD pattern and hence to a loss of information in the higher r region of the corresponding atomic PDF This loss may be critical or not depending on the degree of periodicity crystallinity of the material studied 400 350 300 250 200 150 Intensity arb u 100 50 0 5 10 15 20 A n i p p P it Kp A TO Wh YD N D D Atomic PDF G r 0 10 20 30 40 50 Radial distance r Figure Experimental XRD patterns for Si powder collected with a point solid line in red and area IP mar345 symbols detectors while the rest of the experimental set up was kept unchanged The corresponding atomic PDFs are shown in the lower part of the plot Background signal treatment Air sample holder etc background type scattering should be kept to a minimum since atomic PDFs are based on only the coherent elastic component of the XRD data see Eg 3 Remember weaker background s
40. ck on the RAD icon The following 3 program windows will open see below One labeled RAD in the upper left corner is used to run test RAD in an expert mode display system program error messages etc The second labeled Curve toolboxes 1s used to display results computed by RAD namely the corrected experimental data I q the structure function S q and the atomic PDF RDF G r Usually these two program windows have a limited usage and so may be collapsed minimized in size but still kept active when RAD is running RAD Gtk Experiment amp Sample Info Edit Experiment Data Processing Help EE Curve toolboxes E The third window labeled RAD Gtk 1n the upper left corner see above 1s the RAD s action control window It too should be kept active while RAD is running This window allows the user to get access to the following actions options activated by clicking on them Experiment amp Sample Info Entry Edit Experiment Info Data Processing and Help Below we explain each of these options Experiment amp Sample Info entry option of RAD Gtk When activated this option allows the user to start a new open an existing or save a current RAD project or Quit see below RAD projects are text xml format files saved with an extension rpf The files may be opened viewed with any text editor lli RAD Gtk BEES mEn nmmrws mii Edi Experiment Data Processing Help Mew Ctrl M I Open Chrl 0 el Save Ctrl 5 A Quit Ct
41. cor in our example see below RAD reads the data file and reports that 1n this case it has 543 data points the last one being at Qmax 16 802 A Since Compton scattering is present in the XRD data 52 Structure factor calculation Structure factor calculation 243 Q points cog 16 802000 Corrected data File Ci Papers_ 1 HRADIT _ H Additive correction High angle method 12 000000 Integration limit usually 0 75 x Q J Estimated normalization constant 4 042723 Normalization constant value show I hide curves toolbox 24 since no x ray energy sensitive detector has been used during data collection the former has to be computed and subtracted from the latter Here the user should decide about the value for the so called Breit Dirac recoil factor 2 RAD offers values of 1 2 or 3 The user selects the way the I q data 1s to be normalized into absolute electron units At first the so called high angle method may be selected by checking the relevant box see above and a value for the limit of integration provided 12 in our example The user hits Apply RAD computes a normalization constant 4 048 in the example above converts the I q data in absolute units subtracts the computed Compton scattering and computes the structure factor S q and the so called reduced structure factor q S q 1 2 3 the attached JAC paper describing RAD gives description of the normalization process S q calcu
42. d out up to now In the present paper we report the results of a structural study of RE Al RE Pr Gd Tb Dy metallic glasses The glassy structure is usually expressed in terms of a radial distribution function RDF r which represents the number of atoms in a spheri cal shell of radius r and unity thickness This characterization is however a one dimensional picture of the three dimensional atomic structure The complete structural characterization with de termination of the structural building units and their arrangement can be derived only from a structural model with RDF r and average atomic density p in correspondence with the experimental RDF r and p The aim of our structural study is to obtain the RDFs for RE4Al4 metallic glasses by means of the X ray diffraction XRD method to analyse 0022 3093 89 03 50 Elsevier Science Publishers B V North Holland Physics Publishing Division them in detail and to propose an appropriate structural model on this basis Quasi crystalline structural models based on randomly oriented microcrystalline regions of close packed RE atoms and non crystalline structural models based on a dense random packing of hard spheres will be used as trial structural models since the RE AlI metallic glasses have no crystalline counterparts 2 Experimental part 2 1 Sample preparation and characterization The specimens were alloyed from 99 9 pure RE and Al by arc melting under an a
43. duce XRD patterns that are very diffuse in nature rendering traditional Bragg peaks based crystallography very difficult to apply A combination of higher energy x ray diffraction and atomic PDF RDF data analysis 2 has proven to be very useful in cases like this The frequently used reduced atomic PDF RDF G r gives the number of atoms in a spherical shell of unit thickness at a distance r from a reference atom as follows 2 3 G r 4nrp p r po 1 1 where p r and p are the local and average atomic number densities respectively and r is the radial distance As defined the PDF RDF G r is a one dimensional function that oscillates around zero showing positive peaks at distances separating pairs of atoms 1 e where the local atomic density exceeds the average one The negative valleys in the PDF RDF G r correspond to real space vectors not having atoms at either of their ends With this respect the atomic PDF resembles the so called Patterson function that is widely applied in traditional x ray crystallography 1 However while the Paterson function is discrete and peaks at interatomic distances within the unit cell of a crystal the atomic PDF RDF 1s a continuous function reflecting all interatomic distances occurring in a material This is a great advantage when studying materials whose structure is difficult to be described in terms of extended periodic lattices The PDF G r is the Fourier transform of the experimentally observable to
44. e of PDF data from 1 to 28 A and better describes the fine features in the experimental PDF data appearing at low r values in particular the intensity distribution of the two subcomponents of the split PDF peak at 10 A and the position and intensity of the shoulder of the peak at 6 9 A see the inset in Figure 7 The shoulder reflects mostly correlations between oxygen atoms from neighboring Ti Og octahedra and its pronounced presence in the PDF for BaTiO and 820 Chem Mater Vol 18 No 3 2006 BaTiO rhombohedral Atomic PDF G R 27 96 7 5 10 0 0 0 2 5 5 0 Radial distance r Figure 9 Experimental symbols and model solid line PDFs for BaTiO3 The model PDF is based on the rhombohedral type structure shown in Figure Id The reliability factor Rw is reported in the lower part of the figure An arrow marks the position of the first PDF peak where the model and experimental data show a strong disagreement The same peak is given in the inset on an enlarged scale almost disappearance in the PDFs for SrTiO and BaosSro TiO see Figure 3 indicates that those octahedral units are somewhat distorted rotated in the former material and almost perfectly lined up in the materials containing Sr The results suggest that the atomic ordering in nanocrystalline BaTiO studied by us is likely to exhibit slight distortions similar to those occurring in bulk tetragonal BaTiO crystal The tetragonal structure too feat
45. e Folder Places Name Modified Recently Used Examples Unknown 5 petkolvg pixmaps Unknown Desktop gd riF Unknown Local Disk C gd rpF Unknown wee HP RECOVER praject ica Unknown D DVD RAM Driv rad exe Unknown Removable Di rad gui Unknown ES My Book 3 rad ico Unknown e PCA pps on C setup ica Unknown lll Se phyusers on fF unins000 dat Unknown Se petkolvgt on uninsD 0 exe Unknown WebPeople n Courses an Fr Examples x y ASCII file iw x v1 ASCII File x v 1 ASCII file Experimental XRD data format see files GdAI th and Backgr_GdAI th for example RAD assumes that the experimental XRD data are presented as two columns 21 20 or q value vs intensity in ascii format RAD saves and uses the computed I q S q and G r in the same x y ascii format For example below is what GdAI th looks like when opened viewed with any text editor just two columns of numbers The first is the Brag angle in 20 the second XRD intensity in counts per second 11 8808 11 280808 11 4060 11 6668 11 8000 127 8886 12 2000 12 4060 12 6668 12 8000 13 0000 13 2000 13 40600 13 6668 13 8000 14 8886 15 2008 14 4686 14 6668 14 8000 15 8686 15 2008 15 4686 15 6668 15 8000 16 8686 16 2086 16 4688 16 6668 16 8886 17 8888 17 2000 17 4000 17 6668 17 8000 18 8686 363 72H5 365 8581 366 9811 37 1 54H85
46. e distant pairs the motion of the two atoms becomes uncoupled so that A Aj Aj and we find that for the metal site A 0 375 and for the As site A 1 134 The validity of the approximation of using mean values for the force constants was shown to be accurate by calculating the model PDF for all compositions as described above and comparing to the prediction of Eq 2 16 Equation 2 shows good agreement with the data for near and far neighbor PDF peaks and for the different sublattices over the whole alloy range as shown in Fig 4 using only parameters taken from fits to the end members There 4092 PHYSICAL REVIEW LETTERS 15 NOVEMBER 1999 is a considerably larger width associated with the As As peak in Fig 4 when compared to the Me Me peak because the As atom is surrounded by four metal cations providing five distinct first neighbor environments 4 5 The theoretical curve in the lower panel of Fig 4 is predicted to be the same for the Ga As and the In As bond length distribution using the simplified approach The Kirkwood model seems adequate to describe the experimental data at this time although further refinement of the error bars may require the use of a better potential containing more parameters We would like to thank Rosa Barabash for discus sions and help with the analysis of the static atom displacements and Andrea Perez and the support staff at CHESS for help with data collection and analy sis Thi
47. e scientists untrained in direct methods or experienced people prepared to trust the SIR88 default mode can often solve crystal structures without personal intervention However a large range of options is available to experienced crystallographers for choosing their own way of solving crystal structures 1989 International Union of Crystallography Journal of Non Crystalline Solids 108 1989 75 79 North Holland Amsterdam BA 75 RADIAL DISTRIBUTION FUNCTIONS FOR RE Al METALLIC GLASSES RE Pr Gd Tb Dy V PETKOV A APOSTOLOV and V SKUMRYEV Sofia University Department of Solid State Physics Sofia 1126 Bulgaria Received 29 July 1988 Revised manuscript received 13 October 1988 Using X ray diffraction the radial distribution functions for RE Al RE Pr Gd Tb Dy metallic glasses have been determined The RE RE atomic distances and first coordination numbers have also been estimated A structural model based on spherically arranged dense packed tetrahedra is proposed 1 Introduction In recent years rare earths RE aluminum AI amorphous alloys have been of considerable inter est because of their peculiar magnetic and electri cal properties 1 2 and refs therein For elucida tion of these properties a detailed knowledge of the structure is needed as physical properties are correlated to a great extent with the local atomic arrangement However structure investigations have not been carrie
48. e very likely to be local in nature up to 10 15 and coexist with a cubic type arrangement at longer range interatomic distances The coexistence of a lower symmetry local and a higher symmetry average atomic arrangements is not an unusual picture and has even been observed with perfectly crystalline materials such as In Ga As semiconductors for example These are single phase Petkov et al o BaTiO cubic 2 ooooo O N A A o oc 0 2 12 14 BaTiO cubic Atomic PDF G o NO T N GF Ru mm 99 Oe qe Ne Stig o BaTiO tetragonal 0 0 16 18 20 22 24 26 Radial distance r A Figure 10 Low r 0 15 A upper panel and higher r part 15 28 A lower panel of the experimental symbols and model solid line PDFs for BaTiO3 The model PDFs are based on the cubic and tetragonal type structures shown in Figure la and 1b respectively The corresponding structural parameters are summarized in Table 1 and Table 2 respectively The reported in the figure reliability factors Ry are calculated over the corresponding range of distances materials possessing a long range cubic structure and sub stantially distorted local atomic ordering Nanocrystalline ZrO has also shown a distorted local and cubic type longer range atomic structure Furthermore recent NMR studies have suggested that even bulk cubic BaTiO crystal may be viewed as an assembly of a large number of small and randomly
49. en in C but has bindings to many other popular programming languages such as C Python and others 3 What is XRD and atomic PDF RDF analysis XRD is used to determine the atomic scale structure of materials The technique is based on the fact that the wavelength of x rays 1s comparable in size to the distances between atoms in condensed matter So when a material exhibiting a long i e at least um range periodic atomic order such as a single crystal or polycrystalline powder is irradiated with x rays it acts as an extended periodic grating and produces a diffraction pattern showing numerous sharp spots called Bragg diffraction peaks By measuring and analyzing the positions and intensities of the Bragg peaks it is possible to determine the spatial characteristics of the grating 1 e to determine the three dimensional 3D atomic arrangement in the crystalline material under study This 1s the essence of the so called crystal structure determination by XRD 1 Over the years the technique has been perfected and applied successfully to a variety of crystalline materials from simple metals to complex proteins X ray diffraction can also be applied to study the structure of materials where atoms are ordered only at short 1 e less than a nm to intermediate tens of nm range distances such as liquids glasses fine nano sized powders polymers etc When irradiated with x rays these materials act as quite imperfect gratings and pro
50. ere 10 All measurements were done in symmetric transmission geometry at 10 K The relative intensities of the Bragg peaks compare well with those expected from the crystal structure suggesting that the samples are free of any significant texture Low tempera ture was used to minimize thermal vibration in the samples and hence to increase the sensitivity to static atomic dis placements A double crystal Si 111 monochromator was used Scattered radiation was collected with an intrinsic germanium detector connected to a multichannel analyzer The elastic component was separated from the incoher ent Compton scattering before data analysis 10 Sev eral diffraction runs were conducted with each sample and the resulting spectra averaged to improve the statistical ac curacy The data were normalized for flux corrected for background scattering detector deadtime and absorption and divided by the average form factor to obtain the to tal structure factor S Q 7 8 11 using the program RAD 10 12 Experimental reduced structure factors F Q are shown in Fig 1 The corresponding reduced PDFs G r are shown in Fig 2 The data for the Fourier transform were terminated at Omax 45 T beyond which the sig nal to noise ratio became unfavorable This is a very high momentum transfer for x ray diffraction measurements For comparison Qmax from a Cu K x ray tube is less than 8 f Significant Bragg scattering well defined peaks are
51. he correlation length between polar units in 46 Zalar B Lebar A Selinger J Blinc R Phys Rev B 2005 71 064107 47 Lines M E Glass A M In Principles and Applications of Ferroelectrics and Related Materials Clarendon Oxford 1977 Nanocrystalline Ba Sr TiO3 x 1 0 5 0 Structure ferroelectric materials are of the order of at least 10 and 2 nm in directions parallel and perpendicular to the polarization vector respectively With nanocrystalline BaTiO the polar units slightly distorted rotated Ti Og octahedra are cor related over distances of about 1 5 nm only and may not become a driving force strong enough to transform the longer range structure into an asymmetric tetragonal one even at room temperature As a result the material does not show macroscopic spontaneous polarization as observed in practice P 5 Conclusions The atomic arrangement in nanocrystalline Ba Sr Ti03 x 1 0 5 0 has been studied by synchrotron radiation scattering experiments and Rietveld and atomic PDF tech niques The materials have been found to possess an atomic arrangement well defined over 2 2 5 nm distances and resembling the one occurring in the crystalline perovskites Although the structural coherence length in the nanostruc tured materials is greatly reduced their structure still may be described in terms of a repetitive unit cell containing only a few atoms as the data summarized in Tables 1 and 2 show
52. he usage of a Ge detector the Compton scattering has been eliminated from the 3 Pro ject settings Sample Description Experiment Description Experiment Description sray wavelength Linear absorption sample uk Linear absorption substratre cell Lil Sample s density Atoms A Detector s dead time 5 Experimental geometry Polarization type Compton scattering 34 XRD data during data collection see below that the option to be selected in such cases 1s Compton All out Users may start RAD open the existing project file InGaAs rpf and look over the sample experimental set up information provide in it Some of this info is shown in the screen shot above In this example the step Raw data corrections 1s trivial and we skip it Here we will demonstrate the next step the derivation of q S q see below We the user read s the corrected data InGaAs cor and applies a small correction to them Structure factor calculation Structure factor calculation Corrected data File Qu Pee Additive correction Breit Dirac recoil Factor High angle method Integration limit usually 0 75 x Eu e Estimated normalization constant Mormalization constant value Trial and error method 66 269600 show hide curves toolbox The corresponding q S q 1 looks quite good to very high wave vectors of about 40 Al see below 35 O S Q 1 When an atomic PDF RDF is computed from it in the way
53. herence First as eqs 2 and 3 imply the total scattering including Bragg scattering as well as diffuse scattering contributes to the PDF In this way both the average longer range atomic structure manifested in the Bragg peaks and the local structural distortions manifested in the diffuse component of the diffraction pattern are reflected in the PDF And second the atomic PDFs do not imply any periodicity and can be used to study the atomic ordering in materials showing any degree of structural coherence ranging from crystals to glasses and even liquids Recently the atomic PDF approach has been successfully applied to nanocrystalline materials as well Experimental PDFs for the samples studied were obtained as follows First the coherently scattered intensities were extracted from the corresponding XRD patterns by applying appropriate corrections for flux background Compton scattering and sample absorption The intensities were normalized in absolute electron units reduced to structure functions Q S Q 1 and Fourier transformed to atomic PDFs Thus obtained experimental atomic PDFs are shown in Figure 3 All data processing was done with the help of the program RAD As can be seen in Figure 3 the experimental PDFs are rich in structural features but they vanish at interatomic distances of 2 2 5 nm which are much shorter than the average size of the nanocrystals 5 nm 27 Klug H P Alexander L E In X
54. hree years and has contributed to the determination of several difficult crystal structures hitherto no published description of SIR has been available except for the abstract presented at the IX ECM Meeting 0021 8898 89 040389 05 03 00 Cascarano Giacovazzo Burla Nunzi Polidori Camalli Spagna amp Viterbo 1985 The theoretical basis of SIR is the theory of representa tions Giacovazzo 1977 1980a b according to which for any structure invariant or seminvariant the set of diffrac tion magnitudes may be arranged in a sequence of subsets in order of their expected effectiveness in the statistical sense for the estimation of Thus different formulas can be used for estimating a given each exploiting a different subset of diffraction magnitudes and therefore a different amount of prior information faster access to this subset is guaranteed by storing all the reciprocal lattice in the central memory In SIR88 one and two phase seminvariants and three and four phase invariants are estimated according to various formulas and subsequently used in the phasing process Symmetry is used by SIRS88 in a general way so allowing the use of phase relationships in all space groups A further goal of SIRS88 is to minimize the amount of expertise the user has to bring to solving a crystal structure To this end all decisions concerning data manipulations or appropriate choice of parameters can be assumed by the program Therefor
55. hs by about 0 012 A since our data were measured at 10 K whereas the XAFS experiments were at room temperature The nearest neighbor peak is the only peak which is sharp in the experimental PDFs as can be seen in Fig 2 From the second neighbor onwards the signi ficant atomic displacements in the alloy samples result in broad atom pair distributions without any resolvable split ting Model calculations show that this broadening is in trinsic and not due to any experimental limitations The PDF peak widths in Ga In As were quantified by fit ting individual peaks using Gaussians convoluted with sinc functions with FWHM 0 086 A to account for the experi mental resolution coming from the finite Qmax At low r this was accomplished by fitting individual peaks At high r where there is significant peak overlap a profile fitting regression program was used 13 The resulting mean square Gaussian standard deviations are shown in Fig 4 One can see that the static contribution to the PDF peak width on 2nd and higher neighbors is up to 5 times larger than on the near neighbors The peak width has a maxi mum at a composition x 0 5 and affects the common As more than the mixed metal sublattice In order to better understand these results we have modeled to the static and thermal disorder in the alloy by using a Kirkwood potential 14 The potential contains nearest neighbor bond stretching force con stants and force constants 9 th
56. ignal is much easier to correct for Sample related but unwanted signal X rays are both scattered from and absorbed inside materials via various processes 1 2 The absorption of higher energy x rays is relatively low and usually does not pose much of a problem in the XRD data reduction process The same is true for multiple scattering 2 of higher energy x rays Inelastic Compton scattering however may be very strong especially at high wave vectors 2 If possible it should be eliminated from the experimental XRD data during data collection by using energy sensitive detectors 6 Note analytical procedures for computing Compton scattering and subtracting it from experimental XRD data are inevitably based on various approximations and therefore may or may not work well for every material Fluorescent scattering from the sample should also be kept to a minimum This could be achieved either by using an energy sensitive detector and or employing x rays of energy below the absorption edge of the most strongly scattering atomic species in the material under study In summary a successful atomic PDF RDF study requires an XRD experiment done with a due care Software like RAD can reduce virtually any XRD data set to an atomic PDF RDF but whether this PDF RDF is a physically meaningful representation of the atomic scale structure of the material studied very much depends on the quality of experimental XRD data 5 Using RAD Cli
57. illary scattering absorption and polarization smooth the data if necessary 15 extend extrapolate it linearly to q 0 wave vectors and put it in equidistant 4q steps The result is saved as a file in x y ascii format of corrected XRD data Such a file is needed to compute the respective S q To do the correction the user activates the step Raw data corrections from the Data processing RAD Gtk option and supplies the names of the files containing the XRD intensities scattered from the sample file GdAI th in our example see below and background air Backgr GdAl th in our example see below Note in this example the sample was free standing and no support cell scattering was present Also in this example we decided to smooth the XRD data slightly Then we hit the Apply button lower right corner see below RAD will read the experimental XRD data files and report the number of data points in them as well as the value of the Bragg angle wave vector for the last XRD data point collected see the right hand side of the example below Data correction Data Format 2H vs Intensity 28 vs Intensity vw 429 2B points Sample scattering data File name iadaAl th H 145 000000 max 429 24 points Background correction Backgr_tdal th ze E 143 000000 Substrate cell scattering pm Data smoothing Slight show l hide curves toolbox When the experimental XRD data sets are entered
58. im mediately evident in Fig 1 up to Q 35 T in the end members GaAs and InAs This implies that the samples have long range order and that there is little positional dis order dynamic or static on the atomic scale The Bragg peaks disappear at much lower Q values in the alloy data the samples are still long range ordered but they have sig nificant local positional disorder At high Q values oscil lating diffuse scattering is evident This has a period of 27 2 5 T and contains information about the shortest atomic distances in Ga In As alloys seen as a sharp first peak in G r at 2 5 see Fig 2 In the alloys this peak is split into a doublet as is clearly evident in Fig 2 with a shorter Ga As bond and a longer In As bond This peak is shown on an expanded scale in the inset to Fig 3 for all the compositions measured We determined the po sitions of the two subcomponents of the first PDF peak i e the mean Ga As and In As bond lengths and the re sults are shown in Fig 3 Also shown is the room tem perature result previously obtained in the XAFS study of Mikkelson and Boyce 2 There is clearly good agree 4090 PHYSICAL REVIEW LETTERS 15 NOVEMBER 1999 65 60 55 50 45 40 5 7 30 0 5 10 15 20 25 30 35 40 45 Wavevector Q A FIG 1 The reduced structure factor F Q for Ga _ In As measured at 10 K for various concentrations ment The PDF based bond lengths are shifted to smaller lengt
59. intensity J s is related to 2 s Wagner 19694 neglecting the small angle and multiple scattering by I2 s BIG I G V EPG AG M s Ta 8 where B is an unknown normalization constant fals is the background scattering from air substrate etc Pls and A s are the appropriate polarization and absorption factors respectively M s is the monochromator attenu ation function and J s is the incoherent scattering in electron units scaled to one structural unit 2 Outline of the program The program consists of a main part and four subroutines SETUPD DATRED NORM CALCRD where separate steps in processing the experimental intensities are per formed The main program consists of input output and control statements corresponding to the different options C 1989 International Union of Crystallograjhy 388 The subroutine SETUPD creates the X ray diffraction data file which includes scattering angles and up to 750 intensity values and a parameter data file supplied by the user The parameter data file includes the number of atomic species up to ten constituting the sample the correspond ing molar fractions x and the atomic numbers Z the atomic scattering factor coefficients Cromer amp Mann 1968 anomalous dispersion corrections Cromer 1965 the wavelength A the linear absorption factor of the sample the average atomic density of the amorphous material the counter dead time some c
60. ion The corrected data were spline smoothed recalculated in steps of As 0 05 from s 0 to Sma 4c sin 6 A 16 8 and scaled into electron units by the so called high angle method Only the coherently scatter ed intensity 77 was extracted after removing the incoherent Compton scattering The interference functions n o je Ese se i 1 and the reduced RDFs G r 2 n f sts sin sr ds where all symbols have the usual meaning have been computed according to the Pings and Waser method 7 Some residual errors in the s mainly occur ring as a result of the incorrect normalization of the XRD data introduce large spurious oscilla tions in the region between the origin and the first peak in the corresponding G r For preventing such errors a correction was performed by means of repeated Fourier transforms in the way pro posed by Kaplow Strong and Averbach 8 Figure 1 shows the uncorrected reduced inter ference function F s sI s for Gd AI and the residual difference after the subtraction of the F s corrected according to the method of Kaplow et al Figure 2 shows the results of the same procedure but for the corresponding reduced RDF G r As can be seen from these figures the change for all values of F s is small of the order of the data noise at high s values but is enough to remove the ripples in the G r at small values of r The final interference functions F s and the corresponding
61. ion studies and Raman scattering experiments have shown that fine BaTiO powders with crystallite size as small as 40 nm show a structure with tetragonal distortions and exhibit somewhat reduced but still measurable spontaneous polariza tion Recently the attention has shifted to even finer powders with crystallites as small as only a few nanometers The reason is that having Ba Sr J103 in a nanocrystalline state is a key requirement for producing defect free thin films 6 Furthermore nanosize powders provide good sinterability which is an essential property for the fabrication of advanced ceramic materials When in a nanocrystalline state Ba Sr T103 ceramics are non ferroelectric resulting in stable dielectric properties 118 Several explanations for the disappearance of ferroelectricity have been put forward They point to the absence of long range cooperative structural distortions as one of the main reasons that could lead to a suppression of the thermody namically stable tetragonal phase and the related to it ferroelectric behavior in nanostructured Ba Sr TiOs A thorough understanding of this so called size effect and the properties of nanocrystalline barium strontium based materi als obviously requires a detailed knowledge of their atomic 6 Lytle F W J Appl Phys 1964 55 2212 7 Salje E K Gallardo M C Jimenes J Cerro J J Phys Condens Matter 1998 10 5535 8 Wang Y X Solid State
62. ity so the conditions are suitable for performing XRD measurements in symmetrical transmission geometry As the transmission geometry has ad vantages in the small angle region 3 4 but is not efficient in the high angle region because of the slower counting rate in comparison with the re flection the following experimental procedure was adopted First the samples were mounted in a symmetri cal reflection geometry and scattered intensity col lected in A28 steps of 0 2 from 20 12 to 60 and A20 0 4 from 20 60 to 145 Second the samples were positioned in sym metrical transmission geometry and measurements performed in the range from 20 7 to 70 in steps of 0 2 Third a consistency check between the trans mission and the reflection data was performed 5 If the test result were satisfactory a unique set of XRD data from 20 7 up to 145 was com posed combining the low angle transmission data with the high angle reflection data otherwise the XRD data set was rejected and the measurements repeated This procedure ensures XRD data free from systematic errors 5 6 The background scattering was measured with only the substrate mounted on the goniometer axis 2 3 X ray diffraction data processing The raw XRD data were corrected for back ground scattering counter dead time polarization and absorption The missing values between 20 0 and 20 7 have been derived by means of a linear extrapolat
63. l Phys Rev Lett 83 1999 4089 7 E Curis and S Benazeth J Synchrotron Rad 7 2000 262 8 V Petkov and R Danev J Appl Cryst 31 1998 609 43 Appendix 1 Periodic Table of the Elements Puriodic Table of the Flamants Periodic Table of the Elements Hy rogen f r 6 aso now es meimioids Metals sma TG Basin th mata a I T nsiton metals Name Carbon ter meas croup 1 Noble gases Other norenetals 6 3 J c UN zm Eam 118 710 Bi Bismuth a Group 3 Group 4 Group 5 Group amp Group 7 Group 8 Group 3 Group 10 Groupii Group 12 2072 208 580 40 114 Uuq Ununquadium 289 The discoveries of elements with atomic numbers 112 114 and 116 have Deen reported but not fully confirmed lere aer ort o LS AS HLON HER ME SEONCEECAUEAR NES 82 Pb Lead 44 Appendix 2 Dispersion corrections to the atomic scattering factors for x rays Definition The history of the description of the scattering of an atom when dlummated with X rays 13 that initially wavelength dependencies were wnored This was intially referred to as normal scattering The wavelength dependencies were then corrections to the normal scattering and also called anomalous These had to describe changes in amplitude and phase respectively initially given the symbols Af and Af Thus the X ray scattering factor of an atom is described by the equatiotr f2htAM FM Below are Af and Af anomalous dispersion corrections for Mo Ka
64. lable devices such as thin film and multilayer capaci tors To whom correspondence should be addressed E mail petkov phy cmich edu t Central Michigan University Max Planck Institute of Colloids and Interfaces Argonne National Laboratory 1 Chandler C D Roger C Hampden Smith M J Chem Rev 1993 93 1205 2 Bhalla A S Gou R Roy R Mater Res Innovations 2000 4 3 3 Kwei G Billinge S J L Cheong S W Saxton J G Ferroelectrics 1995 64 57 4 Hennings D Klee M Waser R Adv Mater 1991 5 334 10 1021 cm052145g CCC 33 50 Figure 1 Fragments of the cubic a tetragonal b orthorhombic c and rhombohedral type d structures occurring with bulk BaTiO crystals All structure types feature a 3D network of corner sharing Ti Og octahedra with Ba atoms solid circles occupying the open space between them as shown in a Note the octahedral units are perfect in the cubic type structure The Ti O octahedra shown are somewhat distorted with the non cubic type structure types due to the off center displacement directions shown with arrows of Ti atoms solid circles at the center of the octahedra as depicted in b c and d The octahedra are centered by Ti atoms small solid circles and coordinated by oxygens open cicles The unit cell in the case of the cubic type perovskite structure is outlined with thin solid lines SrTiO is a typical perovskite possessi
65. lation in some more detail Again a separate pop up window appears this time labeled S q 1n the upper left corner see below It allows the user to plot view the computed S q or q S q 1 data In this example we decide to plot q S q 1 The result is shown below 25 tilt Q S Q 1 ox Data 4 00 Q S Q 1 3 00 2 00 1 00 0 EMI IDEE UU UUEEUUmmUUUUUUUUUUUUIUMUIUIIMRMEEEREEmEEEEEEEEMDTIT IG LG r T 1 00 0 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0 q A 1 This q S q 1 data is not of very good quality since it does not oscillate around zero for user s convenience zero appears as a fine dotted line Obviously some XRD data corrections and or I q data normalization have not been done in a precise enough way RAD provides an option to correct for some small residual errors in the I q data by adding subtracting a constant 26 Structure factor calculation Structure factor calculation ots oinks Corrected data File CiPapers 111RADAIT 3 z z 15 802000 max Additive correction 7 000000 High angle method EO y y Integration limit usually 0 75 x G zu Marmalizakian constant value Estimated normalization constant Trial and error method 4 608783 show hide curves toolbox By trial and error yes human intervention 1s still important in PDF RDF data analysis we decide to subtract a small number 7 from the originally corrected I q data and t
66. le has turned out to be 0 675 Note ut can be measured more precise approach see Appendix 4 or estimated 1 2 from the material s mass absorption coefficient u p density p and thickness less precise approach Also here we supply information about the material s atom number density p in this case 0 0392 atoms No sample cell substrate has been used and so the data entry for substrate s ut is zero Note RAD can also process data collected in flat plate transmission geometry or when the sample has been inside a capillary Also x rays could have been completely linearly polarized as with synchrotron sources and Compton scattering could have been eliminated during data collection option Compton scattering All out The user can select between those options by clicking on the respective drop down menus in the Experiment description window When all the necessary sample and experiment related information has been supplied the user should click the Apply button lower right corner see above RAD will perform some checks on the consistency of the Sample amp Experiment Info provided by the user and may issue a warning message if a problem is identified Users are advised to pay full attention to RAD s warning error messages Experiment amp Sample Info that has been entered and found error free 1s considered validated and so can be saved as a RAD project file in xml format with an extension rpf To save
67. ll see below ayy oe ee fay Rad Gtk LJ Riad Gtk on Ehe Web RAR Uninstall Rad Gtk Mac OSX Users First install the GTK working environment GTK Framework then install RAD for Mac OSX Intel 10 4 or Mac OSX Intel 10 5 or Mac OSX Intel 10 6 RAD icon will be put in Applications folder To run RAD just click on the icon Examples coming with RAD will be put in folder usr share rad examples Those may be moved to any other folder if the user decides so Note RAD requires a screen resolution size of at least 1000 x 800 pixels If it is less than that some of the RAD functionalities described below may be lost Also occasionally RAD may not wish to start just after being installed As with other applications restart your computer and try again 1 What is RAD RAD takes x ray diffraction XRD data and reduces it to an atomic pair radial distribution function PDF RDF It has been used by the author VP and many research groups all over the world for more than 20 years It is based on rigorous XRD data reduction procedures established back when XRD was born Yet RAD is under constant modification following the on going development of XRD data collection procedures and instrumentation The author thanks Sebastien Le Roux for the invaluable help in developing a user friendly GTK based interface for RAD 2 What is GTK GTK is a toolkit for creating graphical user interfaces GTK is writt
68. llow one to draw a definitive conclusion in favor of either of the two different structure models attempted either Moreover the Rietveld analysis of the XRD data for the nanocrystalline BaTiO yielded negative values for the mean square atomic displacements of oxygen atoms also known as thermal factors see Tables 1 and 2 Such unphysical results are often obtained with Rietveld analyses of powder diffraction patterns for materials with considerably reduced structural coherence The problems stem from the inability of the Rietveld analysis to handle properly diffrac tion patterns showing both broad Bragg peaks and pro nounced diffuse scattering As we demonstrate below the difficulties are greatly reduced when the diffraction data are analyzed in terms of the corresponding atomic PDFs Similarly to the Rietveld technique the PDF technique employs a least squares procedure to compare experimental and model data PDF calculated from a plausible structural model The structural parameters of the model unit cell constants atomic coordinates and thermal factors are adjusted until the best possible fit to the experimental data is achieved The progress of the refinement is assessed by computing a reliability factor Ry gt wa _ Gy 1 2 R _ ji where G 9 and G are the experimental and calculated PDFs respectively and w are weighting factors reflecting the statistical quality of the individual data points Results from the PDF
69. m of the partial re duced RDFs G 7 Gr s 2 w G r where the weighting factor w is an appropriate average value of the weighting function w s which varies slowly with s 3 5 The weighting factor of the RE RE pairs is predominant more than 75 for the samples under investigation On the contrary the contribu tion of the Al Al pairs to the experimental RDF is negligible only about 1 Furthermore the individual contributions of the RE RE and the RE AI pairs to the first RDFs maxima are not resolved because of the insufficient resolution of the XRD experiment Smax 16 8 ATD There fore the area under the first peak of the RDFs is approximately equal to the average number of the nearest neighbours of the RE atoms and the positions of the first maxima are strongly de termined by the RE RE separation According to these considerations the number of the RE nearest neighbours and the RE RE atomic separation can be estimated on the basis of the total RDFs only The estimated number of RE nearest neighbours is as follows CNp 13 5 F 0 5 CNga 14 0 0 5 CNy 14 04 0 5 CNp 14 0 40 5 Not surprisingly these values are high because coordi nation numbers higher than 12 are well known in the complex crystal structures of RE AI alloys 11 The estimated distances between the RE atoms are as follows Ry p 3 55 0 03 A Roa_ca 3 50 F 0 03 A Rm 3 45 F 0 03 A Rpy_py 3 45 0 03 A These values are
70. mich edu peaple petkavIsafEware htl Questions suggestions and bug reports may be direcked to V Petkov petkov iphv cmich edu 5 Le Roux s le rouxicmich edu Analyzing multiple data sets i e temperature pressure etc series Once a RAD project file 1s read created users may process one after another several data sets from the same sample re using the RAD control 1 e Data processing parameters they entered used for the first data set Any subsequent data set be it raw XRD data corrected I q data or q S q 1 data however should have the same number of data points including the same starting and ending 20 Q values and Aq steps as the very first data set If not RAD may behave unpredictably 33 More examples of RAD s usage The current RAD distribution comes with two more examples One describes processing of synchrotron XRD data for Ino 33Gao 67As semiconductor The study has been published in Petkov et al PRL 83 1999 p 4089 a copy of the paper is attached for user s convenience Here we provide the raw experimental data InGaAs th the RAD project file InGaAs rpf the corrected XRD data InGaAs cor the reduced structure function data InGaAs qsq and the computed G r InGaAs rdf An energy sensitive Ge solid state detector has been used in this study which had resulted in negligible background scattering so no data for background air scattering are provided Also thanks to t
71. mploying the Rietveld and PDF techniques respectively 4 Discussion At first we approached the experimental XRD patterns with the widely employed Rietveld technique The Rietveld technique is used for crystal structure determination and refinement from powder diffraction data The method employs a least squares procedure to compare experimental Bragg intensities with those calculated from a plausible structural model The parameters of the model are then adjusted until the best fit to the experimental diffraction data is achieved The progress of the fit is assessed by computing various goodness of fit factors with the most frequently used being 4 w ye _ 3 1 2 where y and y are the observed and calculated data points and w are weighting factors taking into account the statistical accuracy of the diffraction experiment The Ri etveld analyses were carried out with the help of the program FullProf The XRD patterns of nanocrystalline SrTiO and BaosSro5TiO were fit with a cubic structure of a perovskite type that is found with the corresponding bulk crystals at room temperature The XRD pattern of BaTiO was approached with both the cubic and tetragonal structures see Figures la and 1b occurring with the corresponding bulk crystal Results from the Rietveld refinements are presented in Figure 2 and the values of the refined structural parameters in Tables 1 and 2 As can be seen in Figure 2 the XRD pattern of S
72. mputer model Modelling is presently under way The authors are indebted to Mr L Bozukov and Mr V Kostadinov for their help in producing the RE4 Al4 samples References 1 B Giessen A Hines and L Kobacoff IEEE Trans Magn 16 1980 1203 2 K Shirakawa K Aoki and T Masumoto J Non Cryst Solids 61 amp 62 1984 1371 3 H Klug and L Alexander in X ray Diffraction Proce dures Wiley New York 1954 p 721 4 C Wagner J Non Cryst Solids 42 1980 3 5 B Thijsse J Appl Cryst 17 1984 61 6 B Thijsse and J Sietsma J Non Cryst Solids 61 amp 62 1984 361 7 C Pings and J Waser J Chem Phys 48 1968 3016 8 R Kaplow S Strong and B Averbach Phys Rev A138 1965 1336 9 V Petkov J Appl Cryst in press 10 K Buschow Sol St Commun 27 1978 275 11 W Pearson in Crystal Chemistry and Physics of Metals and Alloys Wiley New York 1972 12 T Egami and Y Waseda J Non Cryst Solids 64 1984 113 13 M Maurer J Friedt and G Krill J Phys F 13 1983 2389 14 S Nandra and P Grundy J Phys F 7 1977 207 15 K Fukuda S Katayama T Katayama A Nukui and A Makisima Jap J Appl Phys 25 1986 1640 16 T Ichikawa Phys Stat Sol A29 1975 293 17 J Sadoc J Dixmier and A Guinier J Non Cryst Solids 12 1973 46 18 C Briant and J Burton Phys Stat Sol B85 1978 393 19 M Laridjani and J Sadoc J de Phys 42 1981 1293
73. n G r 1s defined as G r 4arlp r pol 1 25 a Chupas P J Qiu X Lee P Grey C P Billinge S J L J Appl Crystallogr 2003 36 1342 b Petkov V Qadir D Shastri S D Solid State Commun 2004 129 239 26 Hammersley A P Hanfland M Hausermann D High Pressure Res 1996 14 235 Petkov et al where p r and po are the local and average atomic number densities respectively and r is the radial distance It peaks at characteristic distances separating pairs of atoms and thus reflects the atomic scale structure The PDF G r is the Fourier transform of the experimentally observable total structure function S Q i e G r 2 7 pes Q S Q 1 sin Qr dQ 2 where Q is the magnitude of the wave vector Q 47 sin 0 4 20 is the angle between the incoming and outgoing radiation beams and 4 is the wavelength of the radiation used The structure function is related to the coherent part of the total scattered intensity as S O 1 O 9 c fKQU VI AO 6 where Q is the coherent scattering intensity per atom in electron units and c and f are the atomic concentration and X ray scattering factor respectively for the atomic species of type i As can be seen from eqs 1 3 the PDF is simply another representation of the powder XRD data However exploring the diffraction data in real space is advantageous especially in the case of materials of limited structural co
74. neralogico Campus Universitario 70124 Bari Italy G POLIDORI Dipartimento di Scienze della Terra Universita 06100 Perugia Italy R SPAGNA Istituto G Giacomello Area della Ricerca CNR 00016 Montelibretti Roma Italy and D VITERBO Dipartimenio di Chimica Universit della Calabria 87030 Rende Cosenza Italy Received 20 January 1989 accepted 28 March 1989 Abstract SIR88 is an integrated package of computer programs for the solution of crystal structures The package is based on the estimation of one and two phase structure semin variants and three and four phase structure invariants according to the theory of representations Giacovazzo 1977 Acta Cryst A33 933 944 1980 Acta Cryst A36 362 372 The program works in all the space groups and in most cases it is able to provide the correct solution without user intervention Some prior information like the availability of a partial structure or of pseudotranslational symmetry is easily exploited to obtain the structure solution Introduction The SIR semi invariants representations package has been developed for solving crystal structures by direct methods Its establishment was initiated some years ago The present release SI R88 is the second which we consider suitable for distribution and includes new features with respect to the previous version SIR85 Even though SIRS85 has been distributed by the authors to many laboratories worldwide over the last t
75. ng a cubic structure see Figure la at room temperature Although BaTiO and SrTiO have structures of a similar perovskite type they show very different transition behavior It is not until SrT103 is cooled to 110 K when its cubic structure distorts and 5 Frey M H Payne D A Appl Phys Lett 1993 65 2753 2006 American Chemical Society Published on Web 01 13 2006 Nanocrystalline Ba Sr TiO3 x 1 0 5 0 Structure becomes tetragonal Thus SrTiO exhibits paraelectric behavior at room temperature although recent studies suggest that the material is indeed an incipient ferroelectric whose ferroelectricity is suppressed by quantum fluctuations Barium based mixed oxides have also attracted much attention In particular Ba Sr TiO3 has shown excellent dielectric properties especially as thin films At room temperature and low concentrations of Sr x lt 0 5 these mixed oxides adopt a tetragonal type structure featuring slightly distorted Ti Og octahedra see Figure 1b At higher concentrations of Sr the structure is of the cubic type shown in Figure 1a It has been discovered that many of the useful properties of perovskite materials are critically dependent on the crystallite size For example it has been found that at room temperature the structure of BaTiO transforms to cubic like when the crystallite size becomes smaller than 100 nm On the other hand high resolution synchrotron radiat
76. ngs Sample Description Experiment Description Experiment Description 2 ray wavelength Linear absorption sample Linear absorption substratre cell Sample s density Detector s dead time Experimental geometry Polarization type Compton scattering 0 107200 i 0 050000 ut D OOOO HE 0 078000 Atoms gododd 5 Transmission Synchrotron linear All in Users may start RAD open the existing project file SrT103 rpf and look over the sample experimental set up information provide in it Some of this info is shown in the screen shot above The step Raw data corrections 1s trivial and we again skip it Here we will demonstrate the next step the derivation of q S q 1 see below We the user read s the corrected data SrT1Os cor and applies a small correction 3 to them 38 Structure factor calculation Structure factor calculation 1447 Q points Corrected data File C Program Filestrad Q ae max i Additive correction 3 000000 Breit Dirac recoil Factor 1 vi High angle method Integration limit usually 0 75 x d m Estimated normalization constant Normalization constant value Trial and error method 0 086400 show f hide curves toolbox Then runs the high angle normalization method first obtains a good estimated for the normalization constant and continues in a trial and error mode to obtain the best
77. nt This observation shows that the nanocrystalline Ba Sr TiO3 not only lacks the extended order of usual crystals but shows some structural distortions that further reduce their structural coherence Such structural distortions are frequently observed with nanoparticles of sizes from 5 10 nm and are often ascribed to surface relaxation effects The distortions are more pronounced with nanocrystalline BaTiO than with the samples containing Sr since its PDF decays to zero faster than those of the other two materials see Figure 3 The first peak in the three experimental PDFs shown in Figure 3 is positioned at approximately 1 98 2 A which is close to the average Ti O first neighbor distance observed in the corresponding crystalline bulk perovskites The area under the peak yields 5 8 2 oxygen neighbors for each titanium atom reflecting the presence of Ti Og octahedral units in the nanomaterials The peak is quite sharp and appears with almost the same shape and full width at half maximum 70 18 A in the PDFs for the three samples showing that they all are built of well defined Ti Og octahedra Also the three experimental PDFs exhibit a similar oscillatory behav ior at longer interatomic distances indicating that nanoc rystalline Ba Sr TiOs x 1 0 5 0 share common structural features those of an extended network of Ti Og octahedra However some fine but clearly noticeable dif ferences in the experimental PDFs are al
78. o apply a normalization constant of 4 608 note it 1s close but not the same as the high angle method estimate of 4 048 The corresponding q S q 1 shows a much better behavior see below and we decide to save it To do it we check the Data option in the q S q 1 data plot window see below A drop down menu appears and again we use the Save As option 2 Q S Q 1 4 00 3 00 2 00 1 00 0 0 2 0 4 0 6 0 8 0 10 0 120 14 0 16 0 q A 1 The user 1s free to select any file name extension for the q S q 1 data We saved it under the name GdAl qsq in x y ascii format 3 Compute an atomic PDF RDF from a reduced factor q S q 1 EE RAD Gtk Experiment amp Sample Info Edit Experiment aE REX amp Raw data corrections S q calculation 49 RDF Gir calculation 3 Show curve toolboxes 28 To do it the user should activate the option RDF G r calculation from the drop down menu list of Data processing RAD Gtk window see above and enter the name of the file containing the q S q 1 data file GdAl gsq in our example see below Im Radial distribution function calculation OG 5rQq 1 data File jum value Or step Damping Factor Number density after air slope RDF calculation S43 Q points 0 020000 0 005000 usually 0 005 Atoms J Ae RAD reads the data file and reports that it has 543 data points the last one being at Qu
79. omponent and a few broad Bragg like features This renders the traditional diffraction techniques for structure determination very difficult to apply That is why structural studies on nanocrystals are scarce and the atomic arrangement in Ba Sr T103 nanopowders has not been determined in detail yet Recently it has been shown that the three dimensional 3D structure of materials with reduced structural coherence including nanocrystals can be determined using the so called atomic pair distribution function PDF technique Here we employ the tradi tional Rietveld and the nontraditional PDF technique to determine the 3D structure of Ba Sr T1O x 1 0 5 0 nanoparticles with crystallites having size as small as 5 nm We find that these nanostructured materials possess a well defined atomic arrangement that may be described in terms of the perovskite type structure depicted in Figure 1 The new structural information helps one understand better the dielectric properties of the nanomaterials 2 Experimental Section 2 1 Sample Preparation Nanocrystalline Ba Sr TiO3 samples were obtained through a recently discovered approach employing a nonhydrolytic and halide free procedure In the first step of the procedure metallic barium and or strontium were dissolved in anhydrous benzyl alcohol at elevated temperatures 343 373 K Generally Sr needed a higher temperature to dissolve than Ba The resulted solutions were mixed with
80. onal close packed hcp and face centered cubic fcc crystalline structures Ichikawa s dense random packing of hard spheres 16 the 1cosa hedral structural model 17 18 and the investi gated RE AI glasses are collected together in table 2 The survey of table 2 shows that the structural model based on the crystalline fcc structure of Pr or the hcp structure of Gd Tb and Dy cannot account for the experimental data They are only in agreement with the icosahedral and Ichikawa s non crystalline structural models based on a ran dom dense hard spheres packing with a high de gree of tetrahedrality the hard spheres are packed in such a way that all holes have tetrahedral symmetry This means that the metal atoms in V Petkov et al Radial distribution functions for RE Al metallic glasses 79 Table 2 Relative radii of coordination spheres for some close packed structures and RE4Al metallic glasses Coordination sphere 2nd 3rd 4th fcc 1 41 1 73 2 00 hcp 1 41 1 63 1 73 Ichikawa 1 74 1 95 2 70 Ichikawa 1 66 1 99 2 22 Icosahedral 1 67 2 00 2 53 Pr Al 1 70 2 00 2 51 Gd Al 1 67 2 01 2 53 Tb Al 1 68 2 01 2 33 Dy Al 1 67 2 00 2 49 With a low degree of tetrahedrality With a high degree of tetrahedrality RE Al glasses are well approximated by an as sembly of tetrahedral structural units Probable atomic arrangement accounting for the high coordination numbers observed is the assembly of slightly distor
81. ontrol flags determined from the experimental configuration geometry and monochromatiz ation type and the monochromator attenuation function Ruland 1964 Some of the numbers density p for instance are not crucial in the data processing and can be neglected in the data file The subroutine DATRED uses as input data the X ray diffraction data file and a parameter data file The experi mental data undergo the following treatments i Subtraction of background scattering according to Warren amp Mozzi 1970 and Wagner 19695 ii Correction for counter dead time Klug amp Alexander 1954 iii Correction for polarization Wagner 1978 Thijsse 1984 depending on the type of monochromatization iv Correction for absorption Wagner 1969a depend ing on the geometry v The missing values between 20 0 and the first experimental data are derived by linear extrapolation to the origin vi The corrected XRD data are smoothed Savitzky amp Golay 1964 calculated in steps of As 0 05 from s 0 to Sax AT SiN 6 A by means of a cubic spline interpola tion and stored as a new data file The subroutine NORM uses a corrected data file from DATRED and the parameter data file as input The independent coherent scattering n X xfi 3 the sharpening factor 2 P so 4 and the incoherent scattering Te A A E xZi is C bis 5 1 where a b are semi empirical expressions Thijsse 198
82. oriented tetragonal nanosize domains with dynamically elongated unit cells which transform into a phase with macroscopic tetragonal structure only when cooled below 393 K With nanocrystalline BaTiO such a transfor mation of the local tetragonal type distortions into a mac roscopic tetragonal type structure that is thermodynamically stable at room temperature may not occur because of the very limited structural coherence length 2 nm in the material In summary the results of our structural studies show that nanocrystalline SrTiO and Bao 5Sro s T103 possess a structure of a perovskite type exhibiting almost perfect Ti Og units arranged in a long range pattern with cubic symmetry The atomic arrangement in nanocrystalline BaTiO is also of a perovskite type but exhibits slight tetragonal distortions that show up at short range interatomic distances only This new structural information helps one understand the dielectric properties of Ba Sr TiOs x 1 0 5 0 nano ceramics as follows The longer range cubic centrosym metric structures of SrTiO and BaosSro 5T1O are incom patible with the appearance of ferroelectricity and the materials do not show spontaneous polarization as experi mentally observed The situation with BaTiO is more complex The material shows local tetragonal distortions but they seem to be confined to distances as short as 10 15 A only As the theoretical estimates of Lines and Glass suggest t
83. ot be terminated too early 1 e gmax should be at least 15 20 A Diffraction data at such high wave vectors can be obtained using x rays of a shorter wavelength 1 e of higher energy X rays of higher than usual energy can be delivered by synchrotron or laboratory sources such as sealed x ray tubes with a Mo energy 17 keV or Ag energy 22 keV anodes Note the energy of Cu Ka radiation is only about 8 keV and hence gmax may not get higher than 8 A or so Therefore Cu Ka radiation is not suitable for higher energy XRD aimed at atomic PDF RDF data analysis XRD data statistics and collection time Whatever source of higher energy x rays is used the XRD data should be collected with a very good statistical accuracy That may mean having at least 10 000 counts collected at any data point diffraction angle 4 To achieve it XRD data may need to be collected much longer than in the case of more usual applications such as Rietveld analysis Thus when a sealed x ray tube source and a single point e g scintillation detector are employed the data collection may take tens of hours More powerful sources of x rays rotating anode generators and synchrotrons and or large area detectors may reduce the XRD data collection time to minutes 5 Experimental set up q space resolution In general structure studies on poorly or completely non periodic materials do not require experimental set ups of very high reciprocal q space space resolution be
84. ow see below 19 ran Corrected data 26 Data Eil Save As M close Ctrl 900 00 800 00 700 00 600 00 500 00 400 00 200 00 200 00 100 00 0 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0 20 A drop down menu see above will appear allowing the user to save the corrected data I q or close the data plot window If the Save As option is selected a new window appears see below It allows the user to save the I q data in x y ascii format see the screen shot below Note this data format has been chosen to be the working format of RAD In this example we save the corrected XRD data in a file GdAl cor The user 1s free to select any file name extension for the corrected XRD data An option is provided to save the data in x y 0 1 ascii format This is a 4 column ascii file where the third and forth columns are zeros and ones This format is not used by RAD but by other programs and is introduced just to make RAD s output portable to third party 20 software applications such as DISCUS and PDFgui So again RAD users show save their results in the x y ascii format when those results are to be used in other calculations to be performed by RAD Use the x y 0 1 ascii format if RAD s output is to be ported to DISCUS http discus sourceforge net or PDFgui http www diffpy org liii Save File i Marne Corrected data 2H i Browse For other Folders Program Files Creat
85. py of the paper 1s attached for user s convenience In this example see above we first enter the number of chemical species 2 in our case in the Number of chemical species data field upper right corner see above and hit Enter This is the only time the user should hit the Enter key This tells RAD to open a drop down list menu button second line from the top in the example window shown above that lists consecutively the chemical species in the material studied Then we enter the atomic number Z for species 1 it is 13 for Al see Appendix 1 and the atomic concentration of Al 0 43 Note the chemical composition of the metallic glass in this example could be entered as Gd4Als Gds7Al43 Gdo 57Alo 43 etc Here we opt for the last chemical formulation The others would have been equally good Users however should be aware that depending on the choice of the chemical formula unit the computed q S q 1 and PDF RDF G r would differ times a constant factor which scales with the total number of atoms in the respective chemical formula unit used 2 3 RAD users be consistent in your choices of the chemical formula unit for the material you study Stick to the same formula unit e g Gdo 57Alo 43 when you use the respective PDF RDF data to extract atomic coordination numbers and or fit compare the experimental PDF RDFs with structure models In this example we the user should also supply information about f 20
86. r 1 However both extended x ray absorption fine structure XAFS experiments 2 and theory 3 have shown that Ga As and In As bonds do not take some average value but remain close to their natural lengths of ox As 2 437 A and D As 2 610 A in the alloy Because of the two different bond lengths present the zinc blende structure of Ga In As alloys becomes locally distorted A number 0031 9007 99 83 20 4089 4 15 00 of authors 2 5 have proposed distorted local structures but there have been limited experimental data available to date The fully distorted structure is a prerequisite as an input for accurate band structure and phonon dispersion calculations 6 The technique of choice for studying the local structure of semiconductor alloys has been XAFS 2 5 However XAFS provides information only about the immediate atomic ordering first and sometimes second coordination shells and all longer ranged structural features remain hidden To remedy this shortcoming we have taken the alternative experimental approach of obtaining high resolution PDFs of these alloys from high energy x ray diffraction data The PDF is the instantaneous atomic density density correlation function which describes the local arrangement of atoms in a material 7 The PDF G r is the sine Fourier transform of the experimentally observable total structure factor S Q where Q is the magnitude of the wave vector given by 2 Q max G r
87. r345 The use of an imaging plate detector greatly reduces the data collection time and improves the statistical accuracy of the diffraction data as demonstrated by recent experiments on materials with reduced structural coherence Up to 10 images were taken for each of the samples The exposure time was 10 s image The corresponding images were combined subjected to geometrical corrections integrated and reduced to one dimensional X ray diffraction XRD patterns using the computer program FIT2D 6 Thus obtained XRD patterns for the three samples studied are presented in Figure 2 3 Results As can be seen in Figure 2 the experimental XRD patterns of nanocrystalline Ba Sr T103 powders show only a few broad Bragg like peaks that merge into a slowly oscillating diffuse component already at Bragg angles as low as 6 As our subsequent analyses show the diffraction patterns of SrTiO and BaosSro5T1Os can be indexed in a cubic unit cell while that of BaTiO can be indexed in both cubic and tetragonal unit cells of the perovsike type structure type shown in Figure 1 Such diffraction patterns are typical for materials of limited structural coherence and are obviously difficult to be tackled by traditional techniques for structure determination However when reduced to the corresponding atomic PDFs they become a structure sensitive quantity lending itself to structure determination The frequently used atomic Pair Distribution Functio
88. rTiO and BaosSro sTiO3 are very well reproduced by a model based on the cubic structure of a perovskite type shown in Figure la The results show that even when in the nanocrystalline state SrTiO and BaosSro5TiOs adopt the structure type of the corresponding bulk crystals The XRD pattern for nanocrystalline BaTiO is almost equally well reproduced by the cubic and tetragonal perovskite type structure as the data presented in Figure 2 shows The values of the corresponding goodness of fit factors Rw see Tables S Cc 36 Rietveld H M J Appl Crystallogr 1969 2 65 37 Young R A In The Rietveld Method Oxford University Press New York 1996 38 Rodriguez Carvajal J Physica B 1993 192 55 39 Abramov Y A Tsirel son V G Zavodnik V E Ivanov S A Brown I D Acta Crystallogr B 1983 59 942 818 Chem Mater Vol 18 No 3 2006 Table 2 Structure Data for Nanocrystalline BaTiO as Obtained by the Present Rietveld and PDF Refinements Rietveld PDF a 3 987 6 3 997 6 c A 4 091 7 4 0851 7 2 Ti 0 467 5 0 470 3 z O1 0 138 5 0 130 7 z O2 0 495 6 0 490 3 Upa A 0 027 2 0 010 2 Ur A 0 006 2 0 018 2 Uo A 0 020 2 0 021 2 Ry 248 2 The refinements are based on the tetragonal type structure presented in Figure 1b The goodness of fit Rietveld see eq 4 and reliability PDF see eq 5 factors Ry are reported for each of the refinements and 2 does not a
89. rgon atmo sphere A number of ribbons 30 um thick and 2 mm wide were obtained from each master alloy through the melt spinning technique A number of pieces were cut from adjacent sections of the ribbons and carefully arranged onto a rectangular plastic frame using adhesive tape as a substrate to form a single layer The layers prepared in this way were used for the XRD measurements The chemical composition of all samples corre sponds to the overall formula RE A1 Electron microprobe measurements in selected areas show there are slight fluctuations in the relative RE and 76 V Petkov et al Radial distribution functions for RE 44l metallic glasses Al concentration in the wheel quenched and air quenched ribbon sides only The density of the samples p Pr Al 5 9 F 0 1 g cm p Gd Al 6 6 Y 0 1 g cm p Tb Al 6 9 F 0 1 g cm p Dy Al 7 1 F 0 1 g cm was measured using the Archimedes method 2 2 X ray diffraction measurements The XRD data were collected by a semiauto matic powder diffractometer using filtered Mo k radiation and scintillation registration Different collecting sequences were used for averaging the systematic effect of the apparatus instability The data were collected with the same sample several times so that at least 5000 counts were accu mulated for each Bragg angle The linear absorption factors u measured by simple attenuation experiments are of order of un
90. ring in materials Furthermore the PDF G r is very sensitive to the effects of imperfect data correction and systematic errors As a result Ry s close to 20 are common for PDF refinements even of well crystallized materials 5 The inher ently higher absolute value of the reliability factors resulting from PDF based refinements does not affect their functional purpose as a residuals quantity that must be minimized to find the best fit and as a quantity allowing differentiation between competing structural models It may also be noted that when the atomic pair correlation function g r defined as g r p r po is used to guide a refinement of a structural model the resulting reliability factors Ry are signifi cantly lower than those reported from a refinement based on the corresponding PDF G r and very close to the values of the goodness of fit indicators reported from Rietveld analyses We however prefer to work with the PDF G r and not g r since the former scales with the radial distance r see the multiplicative factor in the definition of G r eq 1 and is thus more sensitive to the longer range atomic correlations Chem Mater Vol 18 No 3 2006 819 BaTiO tetragonal 64 66 68 r A Oe T Atomic PDF G c RII ED d cae ra Cu t Yu ACE ER TZ aN o aCe COMPS DOO CEP EP 0 5 10 15 20 25 30 Radial distance r A Figure 7 Experimental symbols and model solid line PDFs
91. rl g RAD project files contain information about the material studied and the experimental set up used This information is needed to run the option Data Processing Information about the material include the number of chemical species the atomic number of each of the species the species concentration and the anomalous dispersion terms f and f for that species and the radiation used An example Sample Description entry is given below j n Project settings a e Sample Description Sample description Experiment Descriptions Number of chemical species max 10 Atomic species Atomic number 13 Species information Hd Element Al Aluminum tomic concentration 0 430000 Parameters of the 5 Gaussian Fit 1 bo the X ray atomic scatkering Factors al 4 730796 bi 3 620931 az 2 313951 bz 43 0511867 ad 1 541980 b3 0 095960 at 1 117564 b4 106 952558 as 3 154754 b5 1 555918 c 0 139509 Dispersion corection F 2 3 0 056000 Dispersion correction F 2 3 0 051000 1 D Waasmaier et al Acta Cryst A51 1995 p 416 2 htkp iwe nist goviphyslab data fFastfindex cfr 3 D Cromer Acta Cryst 18 1965 p 17 It features one of the first applications of RAD on in house Mo Ka XRD data for Gd4Al5 metallic glass Results from this study are published in V Petkov et al Radial distribution functions for RE 4Al metallic glasses RE Pr Gd Tb Dy J Non Cryst Sol 108 1989 75 a co
92. s work was supported by DOE through Grant No DE FG02 97ER45651 CHESS is operated by NSF through Grant No DMR97 13424 1 R W G Wyckoff Crystal Structures Wiley New York 1967 Vol 1 2nd ed 2 J C Mikkelson and J B Boyce Phys Rev Lett 49 1412 1982 J C Mikkelsen and J B Boyce Phys Rev B 28 7130 1983 3 Y Cai and M F Thorpe Phys Rev B 46 15 879 1992 4 J L Martins and A Zunger Phys Rev B 30 6217 1984 M C Schabel and J L Martins Phys Rev B 43 11 873 1991 5 A Balzarotti et aL Phys Rev B 31 7526 1985 H Oyanagi et al Solid State Commun 67 453 1988 6 A Zunger et al Phys Rev Lett 65 353 1990 7 B E Warren X Ray Diffraction Dover New York 1990 8 Y Waseda The Structure of Non Crystalline Materials McGraw Hill New York 1980 9 T Egami Mater Trans 31 163 1990 T Egami in Local Structure from Diffraction edited by S J L Billinge and M F Thorpe Plenum New York 1998 p 1 10 I K Jeong F Mohiuddin Jacobs V Petkov and S J L Billinge unpublished 11 H P Klug and L E Alexander X ray Diffraction Pro ceedures for Polycrystalline Materials Wiley New York 1974 2nd ed 12 V Petkov J Appl Crystallogr 22 387 1989 13 Th Proffen and S J L Billinge J Appl Crystallogr 32 572 1999 14 J G Kirkwood J Chem Phys 7 506 1939 15 Jean S Chung and M F Thorpe Phys Rev B 55
93. se circular lt monochromator gt lt compton gt All in amp compton amp experiment amp t Apply project lt projyect gt TRUE lt project gt lt rad xml gt 14 Experiment amp Sample Info that has just been entered by the user or read from an old existing project file can be modified by selecting the Edit Experiment option see below of the RAD Gtk window RE RAD Gtk SEE Experiment amp Sample Info sss sense Data Processing Help rt y Sample settings rt Experiment settings Edit Experiment option of RAD Gtk When selected see above it will allow the user to walk through the Experiment amp Sample Info forms make changes validate and eventually save the modified Info Note Experiment amp Sample Info changes are accepted and taken into account in the XRD data processing PDF calculations only after the Apply button has been hit and the validation procedures completed with success 1 e no RAD error warning messages Data Processing option of RAD Gtk can be activated only if a RAD project file has been created from scratch or read from disk and validated iil RAD Gtk Experiment amp Sample Info Edit Experiment MAS Irea Help Raw data corrections 49 Siq calculation RDF Gir calculation 76 Show curve toolboxes This option allows the user to 1 Correct an experimental XRD data set for background e g air amp sample cell e g empty cap
94. shown below 2 Radial distribution function calculation G 5 Q 1 data File e value Or step Damping Factor Number density after Gtr slope show f hide curves toolbox RDF calculation _ 2061 Q points Ci Program Files rad E gt eere 2 599995 20 000000 i 0 020000 0 000200 usualy 0 005 Atoms Ai Cancel Apply 36 very fine structural features of Ino 33Gao 67As can be revealed In particular the first PDF RDF peak is split into two subcomponents positioned at approx 2 4 A and 2 6 A respectively see below They correspond to the presence of distinct Ga As and In As bonds in this material 0 0 2 0 4 0 6 0 8 0 10 0 12 0 14 0 16 0 18 0 r A The other example highlights a processing of synchrotron XRD data for SrTiO nanosized particles The study has been published in V Petkov et al Chem Mat 18 2006 p 814 a copy of the paper is attached for user s convenience Here we provide the raw experimental data SrT1Os th the background scattering data Kapton th the RAD project file SrT103 rpf the corrected XRD data SrT103 cor the reduced structure function data SrT103 qsq and the computed G r SrTiOs rdf A large area mar345 detector has been used in this study Since large area detectors are not x ray energy sensitive the raw XRD data contain a considerable contribution from Compton scattering option Compton All in see below 37 A Pro ject setti
95. sity of natural uranium in UO with pyo 10 5 g cm N 3 23 C PUO Av 10 5 g cm 6 022 x 10 atoms mol 2234x102 atoms cm Muo 238 0289 2 15 9994 g mol 47 Appendix 4 Linear absorption factor ut definition When passing through a material with a linear absorption coefficient u and thickness t x rays get attenuated absorbed The process can be quantified as follows I passed through sample I before sample exp t By measuring I and I see below the linear absorption factor can be obtained from ut In I I exposure lo Attachments VP s papers featuring RAD and examples of RAD usage illustrated in this User s manual 48 COMPUTER PROGRAMS Machine a b i b 6 iti c Computation FMLS 2 53 1 25 4 15 4 05 12 20 BDLS 1 17 0 46 2 10 2 21 5 02 DMAP 0 55 0 43 2 17 2 10 4 52 The Weitek 3167 coprocessor is twice as fast as the 80387 and the IBM version on a 3090 200 is more than 20 times as fast as the VAX version References BROWN I D 1983 Acta Cryst A39 216 224 GABE E J 1988 Crystallographic Computing 4 edited by N W ISAACS amp M R TAYLOR pp 381 392 Oxford Univ Press IUCr GABE E J amp LEE F L 1981 Acta Cryst A37 C339 GABE E J LEE F L amp LE PAGE Y 1985 Crystallo graphic Computing 3 edited by G M SHELDRICK C KR GER amp R GODDARD pp 167 174 Oxford Clarendon Press J Appl Cryst 1989 22
96. slightly smaller than the values of the corresponding close packed metallic Goldschmidt diameters dp 3 65 A dag 3 60 A dy 3 56 A dp 3 54 A 11 The observed reduction of the RE RE atomic separation cannot be explained in terms of a charge transfer because the electronic structure of the investigated RE is considered to be a stable one normally and furthermore our preliminary and Giessen s 1 magnetic susceptibility data are con sistent with the magnetic moments of RE ions only It should be mentioned that there is a dif ference between the Goldschmidt diameters de termined from close packed crystalline structures and the atomic diameters determined from RDFs for metallic glasses In the latter the local atomic arrangement is mainly determined by the minimi zation of the atomic level potential energy rather than the toplogical and symmetrical requirements as in the close packed crystalline structures 12 Therefore the experimentally obtained RE RE separations could be considered as effective diam eters of the RE atoms in RE AI glasses Simi larly compressed RE RE separations as com pared to the Goldschmidt diameters are found in many other RE metal glasses 13 15 For selection of a suitable structural model our considerations are directed to the radii of the coordination spheres estimated by the positions of the maxima in the RDFs In this aspect the relative radii of the coordination spheres for hexag
97. so observed For example the peak at approximately 6 7 A appears with a well pronounced shoulder in the PDF for BaTiO3 The same peak in the PDF for SrTiO has no such pronounced shoulder In general the initial analysis of the experimental PDFs suggests that the atomic ordering in Ba Sr TiO4 nanopo wders is likely to be of the same type but differ in some fine details for different concentrations of Ba and Sr To reveal the fine features in the atomic ordering in nano crystalline Ba Sr T103 we tested several structural models 35 Gilbert B Huang F Zhang H Z Waychunas G A Banfield J F Science 2004 305 65 Chem Mater Vol 18 No 3 2006 817 Table 1 Structure Data for Nanocrystalline Ba Sr TiO x 1 0 5 0 as Obtained by the Present Rietveld and PDF Refinements BaTiO Bao 5Sro sT1i O3 SrT103 Rietveld PDF Rietveld PDF Rietveld PDF a A 4 016 6 4 021 5 3 985 4 3 979 6 3 930 3 3 927 5 Usus 0 022 2 0 009 2 0 014 2 0 016 2 0 013 2 0 009 2 Uri A 0 016 2 0 015 2 0 013 2 0 006 2 0 008 2 0 010 2 Uo A 0 001 2 0 022 2 0 001 2 0 030 2 0 012 2 0 031 2 Rw 96 2 81 25 2 37 19 2 34 18 The refinements are based on the cubic type structure presented in Figure la The goodness of fit Rietveld see eq 4 and reliability PDF see eq 5 factors Ry are reported for each of the refinements analyzing the experimental diffraction data both in reciprocal and real space e
98. tal structure function S Q 1 e dmax G r 2 7 latsCa lsin gr dq 2 q 0 where q is the magnitude of the wave vector q 4rs1n0 20 is the angle between the incoming and outgoing x rays and A is the wavelength of the x rays used X ray diffraction usually employs the so called Faber Ziman type structure function S q related to only the coherent part of the diffraction pattern eq as follows 2 3 Sq 1 17 9 Ye 3 2 gt f ef Gn where c and f g are the atomic concentration and x ray scattering factor respectively for the atomic species of type i Note f q is a function both of x rays energy E and wave vector q i e f q fo q f q E 1f q E where f and f are the so called anomalous dispersion correction terms 1 2 Also note as defined S q oscillates around one and q S q 1 around zero It should also be noted that for a material comprising n atomic species a single diffraction experiment yields a total PDF RDF G r which is a weighted sum of n n J 2 partial PDFs RDFs G ri 1 e G r gt Wi Gi r 4 ij Here wi are weighting factors depending on the concentration and scattering power of the atomic species as follows wi cchi CiS ce f OI 5 For practical purposes wi s are often evaluated for q 0 2 3 4 How XRD data suitable for atomic PDF RDF analysis are collected Source of radiation The Fourier transformation see Eq 2 should n
99. ted tetrahedra arranged around a common vertex as it is well known that close filling of space through repetition only of regular tetrahedra is impossible It is worth men tioning that similar structural units Al centered icosahedra occur in the REAI type crystal struc ture 11 When heated the RE Al metallic glasses crystallize in this type of structure A similar structural model including a definite amount of octahedra has been proposed for Gd Y metallic glasses by Laridjani and Sadoc 19 There are only a few traces of octahedral structural units on the low r side of the second RDF peaks of the investigated RE Al metallic glasses see fig 4 At least qualitatively our structural model is capable of accounting for the peculiarities of the experimental RDFs yet much work remains to be done in estimating the fraction of the spherically packed tetrahedra and the degree of their dis tortion for statistical analysis of the coordination polyhedra formed and in checking the uniqueness of the model These open questions will be answered by subsequent computer modelling of the RE AI structure 4 Conclusion The structure of RE Al RE Pr Gd Tb Dy metallic glasses can be described as a tetra hedral dense packing of RE and Al atoms The proposed structural model based on a local spherical assembling of the tetrahedra could be confirmed by a complete analysis of the coordina tion polyhedra of a properly constructed co
100. ter some careful thoughts decided that it does not worth the effort to try propagating experimental counting statistical etc errors from the raw XRD data to I q then to S q and finally to PDF RDF G r Error propagation via a Fourier transformation is tricky Instead the author suggests RAD users consider that all data points of the derived S q s and G r s are with the same random type error i e assume use uniform errors weights in all S Q PDF RDF data By the way studies 7 showed that this assumption 1s quite appropriate Within this assumption each S QPDF RDF data point is assigned uniform i e 1 error The S Q PDF RDF data then can be saved in the x y 0 1 ascii format and directly used by PDFgui and DUSCUS in structure refinements The standard deviations of the structural parameters resulted from those refinements will be based on the one unit errors in each of the S QPDF RDF data points against which the structure model has been refined 23 2 Compute a total structure factor S q from corrected XRD I q data i RAD Gtk Experiment amp Sample Info Edit Experiment EE 49 Raw data corrections Sig calculation RDF Gir calculation s Show curve toolboxes To do it the user should activate the option S q calculation from the drop down menu list of Data processing RAD Gtk window see above and enter the name of the file containing the corrected XRD data I q file GdAl
101. tural information helps one to understand better the dielectric properties of these nanomaterials 1 Introduction Crystalline perovskite type oxides show many useful properties and are widely used as catalysts and in piezo electrics and ferroelectrics A prime example is the family of Ba Sr T103 oxides in particular BaTiO3 The material exists in several crystallographic modifications each showing a particular dielectric behavior At high temperature BaTiO has a centrosymmetric cubic structure and is paraelectric Between room temperature and 393 K the material possesses a tetragonal type structure below 278 K the structure is orthorhombic and below 183 K it is rhombohedral Frag ments of the four polymorphs occurring with BaTiO are presented in Figure 1 The picture is characteristic of perovskites all crystalline modifications of Ba Sr TiO3 feature a three dimensional network of Ti Og octahedra with Ba Sr atoms occupying the network channels The asym metry of the low temperature crystallographic modifications arises from a displacement of the Ti cations with respect to the oxygen octahedra as depicted in Figures 1b 1d and gives rise to spontaneous polarization As a result BaTiO becomes ferroelectric below 393 K The high permittivity of the tetragonal modification of BaTiO and the ability to switch the direction of polarization in response to external electric fields have found application in commercially avai
102. ual peaks shown in the figures consist of many Gaussian subpeaks The overall fit to the experimental G r is excellent and the small discrepancies in Fig 5 between theory and experiment are probably due to small residual experimental errors Note that in comparing with experiment the theoretical PDF has been convoluted with a sinc function to incorporate the truncation of the experimental data at Qmax 45 A The technique discussed above could be extended using a better force constant model with more parameters but does not seem necessary at this time The contributions from static displacements and ther mal motion to the widths of the individual peaks in PHYSICAL REVIEW LETTERS 15 NOVEMBER 1999 0 0 0 2 0 4 0 6 0 8 1 0 Composition x FIG 3 Solid symbols Ga As and In As bond lengths vs composition as extracted from the present PDFs Open symbols room temperature XAFS results from Ref 2 Inset Split nearest neighbor PDF peak from the data symbols and the model solid lines the reduced PDF act independently as expected and as confirmed by our supercell calculations described in the 0 025 0 020 0 015 0 010 0 005 Mean squared displacements A 0 000 0 0 0 2 0 4 0 6 0 8 1 0 Composition x FIG 4 Square of the PDF peak widths for far neighbors top panel and nearest neighbors lower panel separated by sublattice type Symbols values from the data In the lower panel the open symbols are for the Ga As
103. ures somewhat distorted Ti Og octahedra one Ti O distance of approximately 1 9 A four Ti O distances of 2 0 A and one Ti O distance of 2 15 A The first peak in the PDF for nanocrystalline BaTiO however is very well reproduced by the tetragonal based model indicating that a model featuring slightly distorted Ti Og octahedra is compatible with the experi mental diffraction data Interestingly the tetragonal based model is superior over the cubic one mostly at low r distances 0 15 A as the data in Figure 10 a show It also agrees reasonably well with the PDF data at higher r distances and yields a better overall reliability factor Ry see Figure 7 A closer look at the behavior of the model data at higher r distances however shows see Figure 10b that the cubic based model somewhat outperforms the tetragonal based one as the corresponding reliability factors Rw this time calculated over a range of real space distances from 15 to 28 shows For longer range distances the tetragonal structure based model does not reproduce the intensities of the experimental PDF peaks as good as the cubic type models do and furthermore seems to produce a PDF that is not perfectly lined up with the experimental data for distances longer than 24 A The fact that the tetragonal type model is superior over the cubic type one mostly at distances shorter than 10 15 A shows that the fine tetragonal distortions in nanocrystalline BaTiO ar

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